Subgroups — C₃²×#5770 #13179147

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#1010	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#1009	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#951	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#949	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#918	C₂	1	2	yes	maximal, complemented
7	C₃²×#916	C₂	1	2	yes	maximal, complemented
8	C₃²×D₃×SD₁₆	C₂	1	2	yes	maximal, complemented
9	C₃²×#911	C₂	1	2	yes	maximal, complemented
10	C₂×C₃²×#204	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃²×#901	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃²×#577	—	3	1	no	maximal
13	C₃×#5770	C₃	1	4	yes	maximal, direct product factor
14	C₃³×#41	C₂²	1	1	yes	characteristic
15	C₃³×#41	C₂²	1	1	yes	characteristic, complemented
16	C₃²×#239	C₂²	1	2	yes	—
17	C₃²×D₃×Q₈	C₂²	1	2	yes	complemented
18	C₃²×#237	—	2	2	no	—
19	C₃²×#237	C₂²	1	2	yes	complemented
20	C₃²×D₃×D₄	—	2	2	no	—
21	C₃²×D₃×D₄	C₂²	1	2	yes	—
22	C₃²×#235	—	2	1	no	—
23	C₃²×#235	—	2	1	no	—
24	C₃²×#235	C₂²	1	1	yes	characteristic
25	C₂×C₃²×D₁₂	—	2	1	no	—
26	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
27	C₂×C₃²×Dic₆	—	2	1	no	—
28	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic, complemented
29	C₃³×Q₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₃³×SD₁₆	C₂²	1	1	yes	characteristic, complemented
31	C₃³×SD₁₆	C₂²	1	1	yes	characteristic, complemented
32	C₃³×D₈	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic, complemented
34	C₃²×#216	C₂²	1	2	yes	complemented
35	C₃²×#215	C₂²	1	2	yes	complemented
36	C₃²×#214	C₂²	1	2	yes	complemented
37	C₃²×#213	C₂²	1	2	yes	complemented
38	C₃²×#208	C₂²	1	1	yes	characteristic, complemented
39	C₃²×#204	C₂²	1	2	yes	complemented
40	C₃²×#204	C₂²	1	2	yes	complemented
41	C₃²×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
42	C₃²×D₃×C₈	C₂²	1	2	yes	complemented
43	C₃²×2₋¹⁺⁴	—	3	1	no	—
44	C₃²×H(5,𝔽₂)	—	3	1	no	—
45	C₃×#1010	C₂×C₃	1	4	yes	complemented
46	C₃×#1009	C₂×C₃	1	4	yes	complemented
47	C₃²×#137	—	3	1	no	—
48	C₃²×#137	—	3	2	no	—
49	C₃²×AGL(1,ℤ/8)	—	3	1	no	—
50	C₃²×AGL(1,ℤ/8)	—	3	2	no	—
51	C₃²×#135	—	2	4	no	—
52	C₃²×#135	—	3	2	no	—
53	C₃²×#135	D₃	1	1	yes	characteristic, complemented
54	C₃²×#135	C₂×C₃	1	4	yes	complemented
55	C₂×C₃²×SD₁₆	—	3	2	no	—
56	C₂×C₃²×SD₁₆	—	3	1	no	—
57	C₃²×#131	—	3	1	no	—
58	C₃×#951	C₂×C₃	1	4	yes	complemented
59	C₃×#949	C₂×C₃	1	4	yes	complemented
60	C₃×#918	C₂×C₃	1	8	yes	complemented
61	C₃×#916	C₂×C₃	1	8	yes	complemented
62	C₃×D₃×SD₁₆	C₂×C₃	1	8	yes	complemented
63	C₃×#911	C₂×C₃	1	8	yes	complemented
64	C₂×C₃×#204	C₂×C₃	1	4	yes	complemented
65	C₃×#901	C₂×C₃	1	4	yes	complemented
66	C₂²×C₃²×D₃	—	2	1	no	—
67	C₂²×C₃²×D₃	—	2	2	no	—
68	C₃³×Q₈	C₂³	1	1	yes	characteristic
69	C₃³×Q₈	C₂³	1	1	yes	characteristic
70	C₃³×D₄	—	2	1	no	—
71	C₃³×D₄	—	2	1	no	—
72	C₃³×D₄	C₂³	1	1	yes	characteristic
73	C₃³×D₄	C₂³	1	1	yes	characteristic
74	C₂×C₃³×C₄	—	2	1	no	—
75	C₂×C₃³×C₄	—	2	1	no	—
76	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
77	C₃²×#67	—	2	2	no	—
78	C₃²×#67	—	2	2	no	—
79	C₃²×#67	D₄	1	2	yes	—
80	C₂×C₃²×Dic₃	—	2	1	no	—
81	C₂×C₃²×Dic₃	—	2	2	no	—
82	C₃²×D₁₂	—	2	2	no	—
83	C₃²×D₁₂	—	2	1	no	—
84	C₃²×D₁₂	D₄	1	1	yes	characteristic
85	C₃²×D₁₂	C₂³	1	2	yes	—
86	C₃²×C₄×D₃	—	2	2	no	—
87	C₃²×C₄×D₃	—	2	2	no	—
88	C₃²×C₄×D₃	C₂³	1	2	yes	—
89	C₃²×Dic₆	—	2	1	no	—
90	C₃²×Dic₆	—	2	2	no	—
91	C₃²×Dic₆	D₄	1	1	yes	characteristic
92	C₃²×Dic₆	C₂³	1	2	yes	—
93	C₃³×C₈	C₂³	1	1	yes	characteristic
94	C₃³×C₈	C₂³	1	1	yes	characteristic
95	C₃²×C₃⋊C₈	C₂³	1	2	yes	—
96	C₃×#577	—	3	4	no	—
97	#5770	C₃²	1	1	yes	characteristic, direct product factor
98	C₃²×#41	—	2	4	no	—
99	C₃²×#41	—	2	4	no	—
100	C₃²×#41	—	3	1	no	—
101	C₃²×#41	—	3	2	no	—
102	C₃²×#41	—	3	2	no	—
103	C₃²×#41	—	6	1	no	—
104	C₃²×#41	—	6	1	no	—
105	C₃²×#41	—	6	2	no	—
106	C₃²×#41	C₂×D₃	1	1	yes	characteristic
107	C₃²×#41	C₂×D₃	1	1	yes	characteristic, complemented
108	C₃²×#41	C₂²×C₃	1	4	yes	—
109	C₃²×#41	C₂²×C₃	1	4	yes	complemented
110	C₂×C₃²×Q₈	—	3	2	no	—
111	C₂×C₃²×Q₈	—	3	1	no	—
112	C₂×C₃²×Q₈	—	6	1	no	—
113	C₂×C₃²×D₄	—	3	2	no	—
114	C₂×C₃²×D₄	—	3	1	no	—
115	C₂×C₃²×D₄	—	6	1	no	—
116	C₂×C₃²×D₄	—	6	2	no	—
117	C₃×#239	C₂²×C₃	1	8	yes	—
118	C₃×D₃×Q₈	C₂²×C₃	1	8	yes	complemented
119	C₃×#237	—	2	8	no	—
120	C₃×#237	C₂²×C₃	1	8	yes	complemented
121	C₃×D₃×D₄	—	2	8	no	—
122	C₃×D₃×D₄	C₂²×C₃	1	8	yes	—
123	C₃×#235	—	2	4	no	—
124	C₃×#235	—	2	4	no	—
125	C₃×#235	C₂²×C₃	1	4	yes	—
126	C₂×C₃×D₁₂	—	2	4	no	—
127	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
128	C₂×C₃×Dic₆	—	2	4	no	—
129	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	complemented
130	C₃²×Q₁₆	—	2	4	no	—
131	C₃²×Q₁₆	—	3	2	no	—
132	C₃²×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
133	C₃²×Q₁₆	C₂²×C₃	1	4	yes	complemented
134	C₃²×SD₁₆	—	2	4	no	—
135	C₃²×SD₁₆	—	2	4	no	—
136	C₃²×SD₁₆	—	3	2	no	—
137	C₃²×SD₁₆	—	3	2	no	—
138	C₃²×SD₁₆	—	3	2	no	—
139	C₃²×SD₁₆	—	3	2	no	—
140	C₃²×SD₁₆	C₂×D₃	1	1	yes	characteristic, complemented
141	C₃²×SD₁₆	C₂×D₃	1	1	yes	characteristic, complemented
142	C₃²×SD₁₆	C₂²×C₃	1	4	yes	complemented
143	C₃²×SD₁₆	C₂²×C₃	1	4	yes	complemented
144	C₃²×D₈	—	2	4	no	—
145	C₃²×D₈	—	3	2	no	—
146	C₃²×D₈	C₂×D₃	1	1	yes	characteristic, complemented
147	C₃²×D₈	C₂²×C₃	1	4	yes	complemented
148	C₃²×M₁₆	—	3	1	no	—
149	C₃²×M₁₆	—	3	2	no	—
150	C₂×C₃²×C₈	—	2	4	no	—
151	C₂×C₃²×C₈	—	3	2	no	—
152	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
153	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	complemented
154	C₃×#216	C₂²×C₃	1	8	yes	complemented
155	C₃×#215	C₂²×C₃	1	8	yes	complemented
156	C₃×#214	C₂²×C₃	1	8	yes	complemented
157	C₃×#213	C₂²×C₃	1	8	yes	complemented
158	C₃×#208	C₂²×C₃	1	4	yes	complemented
159	C₃×#204	C₂²×C₃	1	8	yes	complemented
160	C₃×#204	C₂²×C₃	1	8	yes	complemented
161	C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	1	8	yes	complemented
162	C₃×D₃×C₈	C₂²×C₃	1	8	yes	complemented
163	C₂²×C₃³	—	2	1	no	—
164	C₂²×C₃³	—	2	1	no	—
165	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
166	C₂×C₃²×D₃	—	2	2	no	—
167	C₂×C₃²×D₃	—	8	2	no	—
168	C₂×C₃²×D₃	—	8	1	no	—
169	C₂×C₃²×D₃	C₂×D₄	1	2	yes	—
170	C₃³×C₄	—	2	1	no	—
171	C₃³×C₄	—	2	1	no	—
172	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
173	C₃³×C₄	C₂⁴	1	1	yes	characteristic
174	C₃²×Dic₃	—	2	2	no	—
175	C₃²×Dic₃	C₂×D₄	1	2	yes	—
176	C₃×2₋¹⁺⁴	—	3	4	no	—
177	C₃×H(5,𝔽₂)	—	3	4	no	—
178	#1010	C₂×C₃²	1	1	yes	characteristic, complemented
179	#1009	C₂×C₃²	1	1	yes	characteristic, complemented
180	C₃×#137	—	3	4	no	—
181	C₃×#137	—	3	8	no	—
182	C₃×AGL(1,ℤ/8)	—	3	4	no	—
183	C₃×AGL(1,ℤ/8)	—	3	8	no	—
184	C₃×#135	—	2	4	no	—
185	C₃×#135	—	3	8	no	—
186	C₃×#135	C₃×D₃	1	4	yes	complemented
187	C₃×#135	C₂×C₃²	1	1	yes	characteristic, complemented
188	C₂×C₃×SD₁₆	—	3	8	no	—
189	C₂×C₃×SD₁₆	—	3	4	no	—
190	C₃×#131	—	3	4	no	—
191	#951	C₂×C₃²	1	1	yes	characteristic, complemented
192	#949	C₂×C₃²	1	1	yes	characteristic, complemented
193	#918	C₂×C₃²	1	2	yes	complemented
194	#916	C₂×C₃²	1	2	yes	complemented
195	D₃×SD₁₆	C₂×C₃²	1	2	yes	complemented
196	#911	C₂×C₃²	1	2	yes	complemented
197	C₂×#204	C₂×C₃²	1	1	yes	characteristic, complemented
198	#901	C₂×C₃²	1	1	yes	characteristic, complemented
199	C₂³×C₃²	—	6	2	no	—
200	C₂³×C₃²	—	6	1	no	—
201	C₂²×C₃×D₃	—	2	4	no	—
202	C₂²×C₃×D₃	—	2	8	no	—
203	C₃²×Q₈	—	2	4	no	—
204	C₃²×Q₈	—	2	4	no	—
205	C₃²×Q₈	—	3	1	no	—
206	C₃²×Q₈	—	3	2	no	—
207	C₃²×Q₈	—	6	2	no	—
208	C₃²×Q₈	—	6	1	no	—
209	C₃²×Q₈	C₂²×D₃	1	1	yes	characteristic
210	C₃²×Q₈	C₂²×D₃	1	1	yes	characteristic
211	C₃²×Q₈	C₂³×C₃	1	4	yes	—
212	C₃²×Q₈	C₂³×C₃	1	4	yes	—
213	C₃²×D₄	—	2	1	no	—
214	C₃²×D₄	—	2	1	no	—
215	C₃²×D₄	—	2	4	no	—
216	C₃²×D₄	—	2	4	no	—
217	C₃²×D₄	—	2	4	no	—
218	C₃²×D₄	—	2	4	no	—
219	C₃²×D₄	—	3	2	no	—
220	C₃²×D₄	—	3	1	no	—
221	C₃²×D₄	—	3	2	no	—
222	C₃²×D₄	—	4	4	no	—
223	C₃²×D₄	—	4	4	no	—
224	C₃²×D₄	—	6	2	no	—
225	C₃²×D₄	—	6	2	no	—
226	C₃²×D₄	—	6	2	no	—
227	C₃²×D₄	—	6	1	no	—
228	C₃²×D₄	C₂²×D₃	1	1	yes	characteristic
229	C₃²×D₄	C₂²×D₃	1	1	yes	characteristic
230	C₃²×D₄	C₂³×C₃	1	4	yes	—
231	C₃²×D₄	C₂³×C₃	1	4	yes	—
232	C₂×C₃²×C₄	—	2	1	no	—
233	C₂×C₃²×C₄	—	2	1	no	—
234	C₂×C₃²×C₄	—	2	4	no	—
235	C₂×C₃²×C₄	—	2	4	no	—
236	C₂×C₃²×C₄	—	2	4	no	—
237	C₂×C₃²×C₄	—	3	2	no	—
238	C₂×C₃²×C₄	—	4	4	no	—
239	C₂×C₃²×C₄	—	4	4	no	—
240	C₂×C₃²×C₄	—	6	2	no	—
241	C₂×C₃²×C₄	—	6	2	no	—
242	C₂×C₃²×C₄	—	6	2	no	—
243	C₂×C₃²×C₄	—	6	1	no	—
244	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
245	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
246	C₃×#67	—	2	8	no	—
247	C₃×#67	—	2	8	no	—
248	C₃×#67	C₃×D₄	1	8	yes	—
249	C₂×C₃×Dic₃	—	2	4	no	—
250	C₂×C₃×Dic₃	—	2	8	no	—
251	C₃×D₁₂	—	2	8	no	—
252	C₃×D₁₂	—	2	4	no	—
253	C₃×D₁₂	C₃×D₄	1	4	yes	—
254	C₃×D₁₂	C₂³×C₃	1	8	yes	—
255	C₃×C₄×D₃	—	2	8	no	—
256	C₃×C₄×D₃	—	2	8	no	—
257	C₃×C₄×D₃	C₂³×C₃	1	8	yes	—
258	C₃×Dic₆	—	2	4	no	—
259	C₃×Dic₆	—	2	8	no	—
260	C₃×Dic₆	C₃×D₄	1	4	yes	—
261	C₃×Dic₆	C₂³×C₃	1	8	yes	—
262	C₃²×C₈	—	2	4	no	—
263	C₃²×C₈	—	2	4	no	—
264	C₃²×C₈	—	3	2	no	—
265	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
266	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
267	C₃²×C₈	C₂³×C₃	1	4	yes	—
268	C₃²×C₈	C₂³×C₃	1	4	yes	—
269	C₃×C₃⋊C₈	C₂³×C₃	1	8	yes	—
270	#577	—	3	1	no	Sylow
271	C₂×C₃³	—	2	1	no	—
272	C₂×C₃³	—	4	1	no	—
273	C₂×C₃³	—	4	1	no	—
274	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
275	C₃²×D₃	—	2	2	no	—
276	C₃²×D₃	—	4	2	no	—
277	C₃×#41	—	2	4	no	—
278	C₃×#41	—	2	4	no	—
279	C₃×#41	—	3	4	no	—
280	C₃×#41	—	3	8	no	—
281	C₃×#41	—	3	8	no	—
282	C₃×#41	—	6	4	no	—
283	C₃×#41	—	6	4	no	—
284	C₃×#41	—	6	8	no	—
285	C₃×#41	C₂×C₃×D₃	1	4	yes	—
286	C₃×#41	C₂×C₃×D₃	1	4	yes	complemented
287	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
288	C₃×#41	C₂²×C₃²	1	1	yes	characteristic, complemented
289	C₂×C₃×Q₈	—	3	8	no	—
290	C₂×C₃×Q₈	—	3	4	no	—
291	C₂×C₃×Q₈	—	6	4	no	—
292	C₂×C₃×D₄	—	3	8	no	—
293	C₂×C₃×D₄	—	3	4	no	—
294	C₂×C₃×D₄	—	6	4	no	—
295	C₂×C₃×D₄	—	6	8	no	—
296	#239	C₂²×C₃²	1	2	yes	—
297	D₃×Q₈	C₂²×C₃²	1	2	yes	complemented
298	#237	—	2	2	no	—
299	#237	C₂²×C₃²	1	2	yes	complemented
300	D₃×D₄	—	2	2	no	—
301	D₃×D₄	C₂²×C₃²	1	2	yes	—
302	#235	—	2	1	no	—
303	#235	—	2	1	no	—
304	#235	C₂²×C₃²	1	1	yes	characteristic
305	C₂×D₁₂	—	2	1	no	—
306	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
307	C₂×Dic₆	—	2	1	no	—
308	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic, complemented
309	C₃×Q₁₆	—	2	4	no	—
310	C₃×Q₁₆	—	3	8	no	—
311	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	complemented
312	C₃×Q₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
313	C₃×SD₁₆	—	2	4	no	—
314	C₃×SD₁₆	—	2	4	no	—
315	C₃×SD₁₆	—	3	8	no	—
316	C₃×SD₁₆	—	3	8	no	—
317	C₃×SD₁₆	—	3	8	no	—
318	C₃×SD₁₆	—	3	8	no	—
319	C₃×SD₁₆	C₂×C₃×D₃	1	4	yes	complemented
320	C₃×SD₁₆	C₂×C₃×D₃	1	4	yes	complemented
321	C₃×SD₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
322	C₃×SD₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
323	C₃×D₈	—	2	4	no	—
324	C₃×D₈	—	3	8	no	—
325	C₃×D₈	C₂×C₃×D₃	1	4	yes	complemented
326	C₃×D₈	C₂²×C₃²	1	1	yes	characteristic, complemented
327	C₃×M₁₆	—	3	4	no	—
328	C₃×M₁₆	—	3	8	no	—
329	C₂×C₃×C₈	—	2	4	no	—
330	C₂×C₃×C₈	—	3	8	no	—
331	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	complemented
332	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
333	#216	C₂²×C₃²	1	2	yes	complemented
334	#215	C₂²×C₃²	1	2	yes	complemented
335	#214	C₂²×C₃²	1	2	yes	complemented
336	#213	C₂²×C₃²	1	2	yes	complemented
337	#208	C₂²×C₃²	1	1	yes	characteristic, complemented
338	#204	C₂²×C₃²	1	2	yes	complemented
339	#204	C₂²×C₃²	1	2	yes	complemented
340	ΓL(1,𝔽₂₅)	C₂²×C₃²	1	2	yes	complemented
341	D₃×C₈	C₂²×C₃²	1	2	yes	complemented
342	C₂²×C₃²	—	2	1	no	—
343	C₂²×C₃²	—	2	1	no	—
344	C₂²×C₃²	—	2	4	no	—
345	C₂²×C₃²	—	2	4	no	—
346	C₂²×C₃²	—	2	4	no	—
347	C₂²×C₃²	—	3	2	no	—
348	C₂²×C₃²	—	4	4	no	—
349	C₂²×C₃²	—	4	4	no	—
350	C₂²×C₃²	—	6	2	no	—
351	C₂²×C₃²	—	24	1	no	—
352	C₂²×C₃²	—	24	2	no	—
353	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
354	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
355	C₂×C₃×D₃	—	2	8	no	—
356	C₂×C₃×D₃	—	8	8	no	—
357	C₂×C₃×D₃	—	8	4	no	—
358	C₂×C₃×D₃	C₂×C₃×D₄	1	8	yes	—
359	C₃²×C₄	—	2	1	no	—
360	C₃²×C₄	—	2	1	no	—
361	C₃²×C₄	—	2	4	no	—
362	C₃²×C₄	—	2	4	no	—
363	C₃²×C₄	—	2	4	no	—
364	C₃²×C₄	—	2	4	no	—
365	C₃²×C₄	—	3	2	no	—
366	C₃²×C₄	—	4	4	no	—
367	C₃²×C₄	—	4	4	no	—
368	C₃²×C₄	—	6	2	no	—
369	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
370	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
371	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
372	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
373	C₃×Dic₃	—	2	8	no	—
374	C₃×Dic₃	C₂×C₃×D₄	1	8	yes	—
375	2₋¹⁺⁴	—	3	1	no	—
376	H(5,𝔽₂)	—	3	1	no	—
377	#137	—	3	1	no	—
378	#137	—	3	2	no	—
379	AGL(1,ℤ/8)	—	3	1	no	—
380	AGL(1,ℤ/8)	—	3	2	no	—
381	#135	—	3	2	no	—
382	#135	C₃²×D₃	1	1	yes	characteristic, complemented
383	C₂×SD₁₆	—	3	2	no	—
384	C₂×SD₁₆	—	3	1	no	—
385	#131	—	3	1	no	—
386	C₃³	#577	1	1	yes	characteristic, complemented, Sylow
387	C₂³×C₃	—	6	8	no	—
388	C₂³×C₃	—	6	4	no	—
389	C₂²×D₃	—	2	1	no	—
390	C₂²×D₃	—	2	2	no	—
391	C₃×Q₈	—	2	4	no	—
392	C₃×Q₈	—	2	4	no	—
393	C₃×Q₈	—	3	4	no	—
394	C₃×Q₈	—	3	8	no	—
395	C₃×Q₈	—	6	8	no	—
396	C₃×Q₈	—	6	4	no	—
397	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
398	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
399	C₃×Q₈	C₂³×C₃²	1	1	yes	characteristic
400	C₃×Q₈	C₂³×C₃²	1	1	yes	characteristic
401	C₃×D₄	—	2	4	no	—
402	C₃×D₄	—	2	4	no	—
403	C₃×D₄	—	2	1	no	—
404	C₃×D₄	—	2	1	no	—
405	C₃×D₄	—	2	4	no	—
406	C₃×D₄	—	2	4	no	—
407	C₃×D₄	—	3	8	no	—
408	C₃×D₄	—	3	8	no	—
409	C₃×D₄	—	3	4	no	—
410	C₃×D₄	—	4	4	no	—
411	C₃×D₄	—	4	4	no	—
412	C₃×D₄	—	6	8	no	—
413	C₃×D₄	—	6	8	no	—
414	C₃×D₄	—	6	8	no	—
415	C₃×D₄	—	6	4	no	—
416	C₃×D₄	C₂²×C₃×D₃	1	4	yes	—
417	C₃×D₄	C₂²×C₃×D₃	1	4	yes	—
418	C₃×D₄	C₂³×C₃²	1	1	yes	characteristic
419	C₃×D₄	C₂³×C₃²	1	1	yes	characteristic
420	C₂×C₃×C₄	—	2	4	no	—
421	C₂×C₃×C₄	—	2	4	no	—
422	C₂×C₃×C₄	—	2	1	no	—
423	C₂×C₃×C₄	—	2	1	no	—
424	C₂×C₃×C₄	—	2	4	no	—
425	C₂×C₃×C₄	—	3	8	no	—
426	C₂×C₃×C₄	—	4	4	no	—
427	C₂×C₃×C₄	—	4	4	no	—
428	C₂×C₃×C₄	—	6	8	no	—
429	C₂×C₃×C₄	—	6	8	no	—
430	C₂×C₃×C₄	—	6	8	no	—
431	C₂×C₃×C₄	—	6	4	no	—
432	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
433	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
434	#67	—	2	2	no	—
435	#67	—	2	2	no	—
436	#67	C₃²×D₄	1	2	yes	—
437	C₂×Dic₃	—	2	1	no	—
438	C₂×Dic₃	—	2	2	no	—
439	D₁₂	—	2	2	no	—
440	D₁₂	—	2	1	no	—
441	D₁₂	C₃²×D₄	1	1	yes	characteristic
442	D₁₂	C₂³×C₃²	1	2	yes	—
443	C₄×D₃	—	2	2	no	—
444	C₄×D₃	—	2	2	no	—
445	C₄×D₃	C₂³×C₃²	1	2	yes	—
446	Dic₆	—	2	1	no	—
447	Dic₆	—	2	2	no	—
448	Dic₆	C₃²×D₄	1	1	yes	characteristic
449	Dic₆	C₂³×C₃²	1	2	yes	—
450	C₃×C₈	—	2	4	no	—
451	C₃×C₈	—	2	4	no	—
452	C₃×C₈	—	3	8	no	—
453	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
454	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
455	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
456	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
457	C₃⋊C₈	C₂³×C₃²	1	2	yes	—
458	C₂×C₃²	—	2	1	no	—
459	C₂×C₃²	—	2	4	no	—
460	C₂×C₃²	—	2	4	no	—
461	C₂×C₃²	—	4	4	no	—
462	C₂×C₃²	—	4	1	no	—
463	C₂×C₃²	—	4	1	no	—
464	C₂×C₃²	—	4	4	no	—
465	C₂×C₃²	—	4	4	no	—
466	C₂×C₃²	—	6	2	no	—
467	C₂×C₃²	—	8	4	no	—
468	C₂×C₃²	—	8	4	no	—
469	C₂×C₃²	—	12	2	no	—
470	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
471	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
472	C₃×D₃	—	2	8	no	—
473	C₃×D₃	—	4	8	no	—
474	#41	—	3	1	no	—
475	#41	—	3	2	no	—
476	#41	—	3	2	no	—
477	#41	—	6	1	no	—
478	#41	—	6	1	no	—
479	#41	—	6	2	no	—
480	#41	C₂×C₃²×D₃	1	1	yes	characteristic
481	#41	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
482	C₂×Q₈	—	3	2	no	—
483	C₂×Q₈	—	3	1	no	—
484	C₂×Q₈	—	6	1	no	—
485	C₂×D₄	—	3	2	no	—
486	C₂×D₄	—	3	1	no	—
487	C₂×D₄	—	6	1	no	—
488	C₂×D₄	—	6	2	no	—
489	Q₁₆	—	3	2	no	—
490	Q₁₆	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
491	SD₁₆	—	3	2	no	—
492	SD₁₆	—	3	2	no	—
493	SD₁₆	—	3	2	no	—
494	SD₁₆	—	3	2	no	—
495	SD₁₆	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
496	SD₁₆	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
497	D₈	—	3	2	no	—
498	D₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
499	M₁₆	—	3	1	no	—
500	M₁₆	—	3	2	no	—
501	C₂×C₈	—	3	2	no	—
502	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
503	C₂²×C₃	—	2	4	no	—
504	C₂²×C₃	—	2	4	no	—
505	C₂²×C₃	—	2	1	no	—
506	C₂²×C₃	—	2	1	no	—
507	C₂²×C₃	—	2	4	no	—
508	C₂²×C₃	—	3	8	no	—
509	C₂²×C₃	—	4	4	no	—
510	C₂²×C₃	—	4	4	no	—
511	C₂²×C₃	—	6	8	no	—
512	C₂²×C₃	—	24	4	no	—
513	C₂²×C₃	—	24	8	no	—
514	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
515	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
516	C₂×D₃	—	2	2	no	—
517	C₂×D₃	—	8	2	no	—
518	C₂×D₃	—	8	1	no	—
519	C₂×D₃	C₂×C₃²×D₄	1	2	yes	—
520	C₃×C₄	—	2	4	no	—
521	C₃×C₄	—	2	4	no	—
522	C₃×C₄	—	2	1	no	—
523	C₃×C₄	—	2	1	no	—
524	C₃×C₄	—	2	4	no	—
525	C₃×C₄	—	2	4	no	—
526	C₃×C₄	—	3	8	no	—
527	C₃×C₄	—	4	4	no	—
528	C₃×C₄	—	4	4	no	—
529	C₃×C₄	—	6	8	no	—
530	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
531	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
532	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
533	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
534	Dic₃	—	2	2	no	—
535	Dic₃	C₂×C₃²×D₄	1	2	yes	—
536	C₃²	—	2	4	no	—
537	C₃²	#5770	1	1	yes	characteristic, direct product factor
538	C₃²	C₃×#577	1	4	yes	complemented
539	C₂³	—	6	2	no	—
540	C₂³	—	6	1	no	—
541	Q₈	—	3	1	no	—
542	Q₈	—	3	2	no	—
543	Q₈	—	6	2	no	—
544	Q₈	—	6	1	no	—
545	Q₈	C₂²×C₃²×D₃	1	1	yes	characteristic
546	Q₈	C₂²×C₃²×D₃	1	1	yes	characteristic
547	D₄	—	2	1	no	—
548	D₄	—	2	1	no	—
549	D₄	—	3	2	no	—
550	D₄	—	3	2	no	—
551	D₄	—	3	1	no	—
552	D₄	—	6	2	no	—
553	D₄	—	6	2	no	—
554	D₄	—	6	2	no	—
555	D₄	—	6	1	no	—
556	D₄	C₂²×C₃²×D₃	1	1	yes	characteristic
557	D₄	C₂²×C₃²×D₃	1	1	yes	characteristic
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	3	2	no	—
561	C₂×C₄	—	6	2	no	—
562	C₂×C₄	—	6	2	no	—
563	C₂×C₄	—	6	2	no	—
564	C₂×C₄	—	6	1	no	—
565	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
566	C₈	—	3	2	no	—
567	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
568	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
569	C₂×C₃	—	2	1	no	—
570	C₂×C₃	—	2	4	no	—
571	C₂×C₃	—	2	4	no	—
572	C₂×C₃	—	4	4	no	—
573	C₂×C₃	—	4	1	no	—
574	C₂×C₃	—	4	1	no	—
575	C₂×C₃	—	4	4	no	—
576	C₂×C₃	—	4	4	no	—
577	C₂×C₃	—	6	8	no	—
578	C₂×C₃	—	8	4	no	—
579	C₂×C₃	—	8	4	no	—
580	C₂×C₃	—	12	8	no	—
581	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
582	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
583	D₃	—	2	2	no	—
584	D₃	—	4	2	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	3	2	no	—
588	C₂²	—	6	2	no	—
589	C₂²	—	24	1	no	—
590	C₂²	—	24	2	no	—
591	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	3	2	no	—
595	C₄	—	6	2	no	—
596	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
597	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
598	C₃	—	2	4	no	—
599	C₃	C₃×#5770	1	4	yes	direct product factor
600	C₃	C₃²×#577	1	1	yes	characteristic, complemented
601	C₂	—	2	1	no	—
602	C₂	—	4	1	no	—
603	C₂	—	4	1	no	—
604	C₂	—	6	2	no	—
605	C₂	—	12	2	no	—
606	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₃³×#135	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃²×#135	D₃
hypocenter	C₃	C₃²×#577
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#135

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃²×#1010	C₂	semidirect
2. C₃²×#1009	C₂	semidirect
3. C₃³×#135	C₂	semidirect
4. C₃²×#951	C₂	semidirect
5. C₃²×#949	C₂	semidirect
6. C₃²×#918	C₂	semidirect
7. C₃²×#916	C₂	semidirect
8. C₃²×D₃×SD₁₆	C₂	semidirect
9. C₃²×#911	C₂	semidirect
10. C₂×C₃²×#204	C₂	semidirect
11. C₃²×#901	C₂	semidirect
13. C₃×#5770	C₃	direct (N × H)
15. C₃³×#41	C₂²	semidirect
17. C₃²×D₃×Q₈	C₂²	semidirect
19. C₃²×#237	C₂²	semidirect
28. C₂×C₃²×Dic₆	C₂²	semidirect
29. C₃³×Q₁₆	C₂²	semidirect
30. C₃³×SD₁₆	C₂²	semidirect
31. C₃³×SD₁₆	C₂²	semidirect
32. C₃³×D₈	C₂²	semidirect
33. C₂×C₃³×C₈	C₂²	semidirect
34. C₃²×#216	C₂²	semidirect
35. C₃²×#215	C₂²	semidirect
36. C₃²×#214	C₂²	semidirect
37. C₃²×#213	C₂²	semidirect
38. C₃²×#208	C₂²	semidirect
39. C₃²×#204	C₂²	semidirect
40. C₃²×#204	C₂²	semidirect
41. C₃²×ΓL(1,𝔽₂₅)	C₂²	semidirect
42. C₃²×D₃×C₈	C₂²	semidirect
45. C₃×#1010	C₂×C₃	semidirect
46. C₃×#1009	C₂×C₃	semidirect
53. C₃²×#135	D₃	semidirect
54. C₃²×#135	C₂×C₃	semidirect
58. C₃×#951	C₂×C₃	semidirect
59. C₃×#949	C₂×C₃	semidirect
60. C₃×#918	C₂×C₃	semidirect
61. C₃×#916	C₂×C₃	semidirect
62. C₃×D₃×SD₁₆	C₂×C₃	semidirect
63. C₃×#911	C₂×C₃	semidirect
64. C₂×C₃×#204	C₂×C₃	semidirect
65. C₃×#901	C₂×C₃	semidirect
97. #5770	C₃²	direct (N × H)
107. C₃²×#41	C₂×D₃	semidirect
109. C₃²×#41	C₂²×C₃	semidirect
118. C₃×D₃×Q₈	C₂²×C₃	semidirect
120. C₃×#237	C₂²×C₃	semidirect
129. C₂×C₃×Dic₆	C₂²×C₃	semidirect
132. C₃²×Q₁₆	C₂×D₃	semidirect
133. C₃²×Q₁₆	C₂²×C₃	semidirect
140. C₃²×SD₁₆	C₂×D₃	semidirect
141. C₃²×SD₁₆	C₂×D₃	semidirect
142. C₃²×SD₁₆	C₂²×C₃	semidirect
143. C₃²×SD₁₆	C₂²×C₃	semidirect
146. C₃²×D₈	C₂×D₃	semidirect
147. C₃²×D₈	C₂²×C₃	semidirect
152. C₂×C₃²×C₈	C₂×D₃	semidirect
153. C₂×C₃²×C₈	C₂²×C₃	semidirect
154. C₃×#216	C₂²×C₃	semidirect
155. C₃×#215	C₂²×C₃	semidirect
156. C₃×#214	C₂²×C₃	semidirect
157. C₃×#213	C₂²×C₃	semidirect
158. C₃×#208	C₂²×C₃	semidirect
159. C₃×#204	C₂²×C₃	semidirect
160. C₃×#204	C₂²×C₃	semidirect
161. C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	semidirect
162. C₃×D₃×C₈	C₂²×C₃	semidirect
178. #1010	C₂×C₃²	semidirect
179. #1009	C₂×C₃²	semidirect
186. C₃×#135	C₃×D₃	semidirect
187. C₃×#135	C₂×C₃²	semidirect
191. #951	C₂×C₃²	semidirect
192. #949	C₂×C₃²	semidirect
193. #918	C₂×C₃²	semidirect
194. #916	C₂×C₃²	semidirect
195. D₃×SD₁₆	C₂×C₃²	semidirect
196. #911	C₂×C₃²	semidirect
197. C₂×#204	C₂×C₃²	semidirect
198. #901	C₂×C₃²	semidirect
286. C₃×#41	C₂×C₃×D₃	semidirect
288. C₃×#41	C₂²×C₃²	semidirect
297. D₃×Q₈	C₂²×C₃²	semidirect
299. #237	C₂²×C₃²	semidirect
308. C₂×Dic₆	C₂²×C₃²	semidirect
311. C₃×Q₁₆	C₂×C₃×D₃	semidirect
312. C₃×Q₁₆	C₂²×C₃²	semidirect
319. C₃×SD₁₆	C₂×C₃×D₃	semidirect
320. C₃×SD₁₆	C₂×C₃×D₃	semidirect
321. C₃×SD₁₆	C₂²×C₃²	semidirect
322. C₃×SD₁₆	C₂²×C₃²	semidirect
325. C₃×D₈	C₂×C₃×D₃	semidirect
326. C₃×D₈	C₂²×C₃²	semidirect
331. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
332. C₂×C₃×C₈	C₂²×C₃²	semidirect
333. #216	C₂²×C₃²	semidirect
334. #215	C₂²×C₃²	semidirect
335. #214	C₂²×C₃²	semidirect
336. #213	C₂²×C₃²	semidirect
337. #208	C₂²×C₃²	semidirect
338. #204	C₂²×C₃²	semidirect
339. #204	C₂²×C₃²	semidirect
340. ΓL(1,𝔽₂₅)	C₂²×C₃²	semidirect
341. D₃×C₈	C₂²×C₃²	semidirect
382. #135	C₃²×D₃	semidirect
386. C₃³	#577	semidirect
481. #41	C₂×C₃²×D₃	semidirect
490. Q₁₆	C₂×C₃²×D₃	semidirect
495. SD₁₆	C₂×C₃²×D₃	semidirect
496. SD₁₆	C₂×C₃²×D₃	semidirect
498. D₈	C₂×C₃²×D₃	semidirect
502. C₂×C₈	C₂×C₃²×D₃	semidirect
537. C₃²	#5770	direct (N × H)
538. C₃²	C₃×#577	semidirect
599. C₃	C₃×#5770	direct (N × H)
600. C₃	C₃²×#577	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×#1010	2
C₃²×#1009	2
C₃³×#135	2
C₃²×#951	2
C₃²×#949	2
C₃²×#918	2
C₃²×#916	2
C₃²×D₃×SD₁₆	2
C₃²×#911	2
C₂×C₃²×#204	2
C₃²×#901	2
C₃²×#577	3
C₃×#5770	3

Subgroups — C32×#5770 #13179147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5770 #13179147