Subgroups — C₃²×#5768 #13179145

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₃×#41	C₂	1	2	yes	maximal, complemented
2	C₃³×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#950	C₂	1	2	yes	maximal, complemented
4	C₃²×#918	C₂	1	2	yes	maximal, complemented
5	C₃²×#915	C₂	1	2	yes	maximal, complemented
6	C₃²×#914	C₂	1	2	yes	maximal, complemented
7	C₃²×#911	C₂	1	2	yes	maximal, complemented
8	C₃²×#904	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃²×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃²×#575	—	3	1	no	maximal
11	C₃×#5768	C₃	1	4	yes	maximal, direct product factor
12	C₃³×#41	C₂²	1	2	yes	complemented
13	C₃²×#239	C₂²	1	2	yes	—
14	C₃²×D₃×Q₈	C₂²	1	2	yes	complemented
15	C₃²×#237	—	2	2	no	—
16	C₃²×#237	C₂²	1	2	yes	—
17	C₃²×D₃×D₄	—	2	2	no	—
18	C₃²×D₃×D₄	C₂²	1	2	yes	complemented
19	C₃²×#235	—	2	2	no	—
20	C₃²×#235	C₂²	1	2	yes	complemented
21	C₂×C₃²×C₄×D₃	—	2	2	no	—
22	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
23	C₃³×Q₁₆	C₂²	1	1	yes	characteristic, complemented
24	C₃³×SD₁₆	C₂²	1	2	yes	complemented
25	C₃³×D₈	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic, complemented
27	C₃²×#216	C₂²	1	2	yes	complemented
28	C₃²×#215	C₂²	1	2	yes	complemented
29	C₃²×#214	C₂²	1	2	yes	complemented
30	C₃²×#213	C₂²	1	2	yes	complemented
31	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₃²×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
33	C₃²×D₂₄	C₂²	1	1	yes	characteristic, complemented
34	C₃²×#204	C₂²	1	2	yes	complemented
35	C₃²×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
36	C₃²×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
37	C₂×C₃²×#41	—	3	2	no	—
38	C₃×D₃×#41	C₂×C₃	1	8	yes	complemented
39	C₃²×#137	—	3	2	no	—
40	C₃²×#137	—	3	2	no	—
41	C₃²×AGL(1,ℤ/8)	—	3	2	no	—
42	C₃²×AGL(1,ℤ/8)	—	3	2	no	—
43	C₃²×#135	—	2	4	no	—
44	C₃²×#135	—	3	2	no	—
45	C₃²×#135	—	3	1	no	—
46	C₃²×#135	D₃	1	1	yes	characteristic, complemented
47	C₃²×#135	C₂×C₃	1	4	yes	complemented
48	C₂×C₃²×M₁₆	—	3	1	no	—
49	C₃×#950	C₂×C₃	1	8	yes	complemented
50	C₃×#918	C₂×C₃	1	8	yes	complemented
51	C₃×#915	C₂×C₃	1	8	yes	complemented
52	C₃×#914	C₂×C₃	1	8	yes	complemented
53	C₃×#911	C₂×C₃	1	8	yes	complemented
54	C₃×#904	C₂×C₃	1	4	yes	complemented
55	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	4	yes	complemented
56	C₂²×C₃²×D₃	—	2	2	no	—
57	C₂²×C₃²×D₃	D₄	1	1	yes	characteristic
58	C₃³×Q₈	C₂³	1	2	yes	—
59	C₃³×D₄	—	2	2	no	—
60	C₃³×D₄	C₂³	1	2	yes	—
61	C₂×C₃³×C₄	—	2	2	no	—
62	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
63	C₃²×#67	—	2	2	no	—
64	C₃²×#67	—	2	2	no	—
65	C₃²×#67	—	2	2	no	—
66	C₂×C₃²×Dic₃	—	2	2	no	—
67	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
68	C₃²×D₁₂	—	2	2	no	—
69	C₃²×D₁₂	C₂³	1	2	yes	—
70	C₃²×C₄×D₃	—	2	2	no	—
71	C₃²×C₄×D₃	—	2	2	no	—
72	C₃²×C₄×D₃	—	2	2	no	—
73	C₃²×C₄×D₃	D₄	1	2	yes	—
74	C₃²×C₄×D₃	C₂³	1	2	yes	—
75	C₃²×Dic₆	—	2	2	no	—
76	C₃²×Dic₆	C₂³	1	2	yes	—
77	C₃³×C₈	C₂³	1	1	yes	characteristic
78	C₃³×C₈	C₂³	1	1	yes	characteristic
79	C₃²×C₃⋊C₈	C₂³	1	2	yes	—
80	C₃×#575	—	3	4	no	—
81	#5768	C₃²	1	1	yes	characteristic, direct product factor
82	C₃²×#41	—	2	8	no	—
83	C₃²×#41	—	3	2	no	—
84	C₃²×#41	—	3	2	no	—
85	C₃²×#41	—	3	2	no	—
86	C₃²×#41	—	6	2	no	—
87	C₃²×#41	—	6	2	no	—
88	C₃²×#41	C₂×D₃	1	2	yes	complemented
89	C₃²×#41	C₂²×C₃	1	8	yes	complemented
90	C₂×C₃²×Q₈	—	3	2	no	—
91	C₂×C₃²×D₄	—	3	2	no	—
92	C₂×C₃²×D₄	—	6	2	no	—
93	C₂²×C₃²×C₄	—	3	1	no	—
94	C₂²×C₃²×C₄	—	6	2	no	—
95	C₃×#239	C₂²×C₃	1	8	yes	—
96	C₃×D₃×Q₈	C₂²×C₃	1	8	yes	complemented
97	C₃×#237	—	2	8	no	—
98	C₃×#237	C₂²×C₃	1	8	yes	—
99	C₃×D₃×D₄	—	2	8	no	—
100	C₃×D₃×D₄	C₂²×C₃	1	8	yes	complemented
101	C₃×#235	—	2	8	no	—
102	C₃×#235	C₂²×C₃	1	8	yes	complemented
103	C₂×C₃×C₄×D₃	—	2	8	no	—
104	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
105	C₃²×Q₁₆	—	2	4	no	—
106	C₃²×Q₁₆	—	3	2	no	—
107	C₃²×Q₁₆	—	3	1	no	—
108	C₃²×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
109	C₃²×Q₁₆	C₂²×C₃	1	4	yes	complemented
110	C₃²×SD₁₆	—	2	8	no	—
111	C₃²×SD₁₆	—	3	2	no	—
112	C₃²×SD₁₆	—	3	2	no	—
113	C₃²×SD₁₆	—	3	2	no	—
114	C₃²×SD₁₆	C₂×D₃	1	2	yes	complemented
115	C₃²×SD₁₆	C₂²×C₃	1	8	yes	complemented
116	C₃²×D₈	—	2	4	no	—
117	C₃²×D₈	—	3	2	no	—
118	C₃²×D₈	—	3	1	no	—
119	C₃²×D₈	C₂×D₃	1	1	yes	characteristic, complemented
120	C₃²×D₈	C₂²×C₃	1	4	yes	complemented
121	C₃²×M₁₆	—	3	2	no	—
122	C₃²×M₁₆	—	3	2	no	—
123	C₂×C₃²×C₈	—	2	4	no	—
124	C₂×C₃²×C₈	—	3	1	no	—
125	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
126	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	complemented
127	C₃×#216	C₂²×C₃	1	8	yes	complemented
128	C₃×#215	C₂²×C₃	1	8	yes	complemented
129	C₃×#214	C₂²×C₃	1	8	yes	complemented
130	C₃×#213	C₂²×C₃	1	8	yes	complemented
131	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	complemented
132	C₃×Dic₁₂	C₂²×C₃	1	4	yes	complemented
133	C₃×D₂₄	C₂²×C₃	1	4	yes	complemented
134	C₃×#204	C₂²×C₃	1	8	yes	complemented
135	C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	1	8	yes	complemented
136	C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	1	8	yes	complemented
137	C₂²×C₃³	—	2	2	no	—
138	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃²×D₃	—	2	2	no	—
140	C₂×C₃²×D₃	—	4	1	no	—
141	C₂×C₃²×D₃	—	4	4	no	—
142	C₂×C₃²×D₃	C₂×D₄	1	2	yes	—
143	C₃³×C₄	—	2	2	no	—
144	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
145	C₃³×C₄	C₂⁴	1	1	yes	characteristic
146	C₃²×Dic₃	—	2	2	no	—
147	C₃²×Dic₃	C₂×D₄	1	2	yes	—
148	C₂×C₃×#41	—	3	8	no	—
149	D₃×#41	C₂×C₃²	1	2	yes	complemented
150	C₃×#137	—	3	8	no	—
151	C₃×#137	—	3	8	no	—
152	C₃×AGL(1,ℤ/8)	—	3	8	no	—
153	C₃×AGL(1,ℤ/8)	—	3	8	no	—
154	C₃×#135	—	2	4	no	—
155	C₃×#135	—	3	8	no	—
156	C₃×#135	—	3	4	no	—
157	C₃×#135	C₃×D₃	1	4	yes	complemented
158	C₃×#135	C₂×C₃²	1	1	yes	characteristic, complemented
159	C₂×C₃×M₁₆	—	3	4	no	—
160	#950	C₂×C₃²	1	2	yes	complemented
161	#918	C₂×C₃²	1	2	yes	complemented
162	#915	C₂×C₃²	1	2	yes	complemented
163	#914	C₂×C₃²	1	2	yes	complemented
164	#911	C₂×C₃²	1	2	yes	complemented
165	#904	C₂×C₃²	1	1	yes	characteristic, complemented
166	C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	1	1	yes	characteristic, complemented
167	C₂³×C₃²	—	3	1	no	—
168	C₂³×C₃²	—	6	2	no	—
169	C₂²×C₃×D₃	—	2	8	no	—
170	C₂²×C₃×D₃	C₃×D₄	1	4	yes	—
171	C₃²×Q₈	—	2	8	no	—
172	C₃²×Q₈	—	3	2	no	—
173	C₃²×Q₈	—	6	2	no	—
174	C₃²×Q₈	C₂²×D₃	1	2	yes	—
175	C₃²×Q₈	C₂³×C₃	1	8	yes	—
176	C₃²×D₄	—	2	2	no	—
177	C₃²×D₄	—	2	8	no	—
178	C₃²×D₄	—	2	8	no	—
179	C₃²×D₄	—	3	2	no	—
180	C₃²×D₄	—	4	8	no	—
181	C₃²×D₄	—	6	2	no	—
182	C₃²×D₄	—	6	2	no	—
183	C₃²×D₄	—	6	2	no	—
184	C₃²×D₄	—	6	2	no	—
185	C₃²×D₄	C₂²×D₃	1	2	yes	—
186	C₃²×D₄	C₂³×C₃	1	8	yes	—
187	C₂×C₃²×C₄	—	2	2	no	—
188	C₂×C₃²×C₄	—	2	4	no	—
189	C₂×C₃²×C₄	—	2	8	no	—
190	C₂×C₃²×C₄	—	3	1	no	—
191	C₂×C₃²×C₄	—	3	2	no	—
192	C₂×C₃²×C₄	—	3	2	no	—
193	C₂×C₃²×C₄	—	4	8	no	—
194	C₂×C₃²×C₄	—	6	2	no	—
195	C₂×C₃²×C₄	—	6	2	no	—
196	C₂×C₃²×C₄	—	6	2	no	—
197	C₂×C₃²×C₄	—	6	2	no	—
198	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
199	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
200	C₃×#67	—	2	8	no	—
201	C₃×#67	—	2	8	no	—
202	C₃×#67	—	2	8	no	—
203	C₂×C₃×Dic₃	—	2	8	no	—
204	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
205	C₃×D₁₂	—	2	8	no	—
206	C₃×D₁₂	C₂³×C₃	1	8	yes	—
207	C₃×C₄×D₃	—	2	8	no	—
208	C₃×C₄×D₃	—	2	8	no	—
209	C₃×C₄×D₃	—	2	8	no	—
210	C₃×C₄×D₃	C₃×D₄	1	8	yes	—
211	C₃×C₄×D₃	C₂³×C₃	1	8	yes	—
212	C₃×Dic₆	—	2	8	no	—
213	C₃×Dic₆	C₂³×C₃	1	8	yes	—
214	C₃²×C₈	—	2	4	no	—
215	C₃²×C₈	—	2	4	no	—
216	C₃²×C₈	—	3	2	no	—
217	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
218	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
219	C₃²×C₈	C₂³×C₃	1	4	yes	—
220	C₃²×C₈	C₂³×C₃	1	4	yes	—
221	C₃×C₃⋊C₈	C₂³×C₃	1	8	yes	—
222	#575	—	3	1	no	Sylow
223	C₂×C₃³	—	2	1	no	—
224	C₂×C₃³	—	4	2	no	—
225	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
226	C₃²×D₃	—	2	2	no	—
227	C₃²×D₃	—	4	2	no	—
228	C₃×#41	—	2	8	no	—
229	C₃×#41	—	3	8	no	—
230	C₃×#41	—	3	8	no	—
231	C₃×#41	—	3	8	no	—
232	C₃×#41	—	6	8	no	—
233	C₃×#41	—	6	8	no	—
234	C₃×#41	C₂×C₃×D₃	1	8	yes	complemented
235	C₃×#41	C₂²×C₃²	1	2	yes	complemented
236	C₂×C₃×Q₈	—	3	8	no	—
237	C₂×C₃×D₄	—	3	8	no	—
238	C₂×C₃×D₄	—	6	8	no	—
239	C₂²×C₃×C₄	—	3	4	no	—
240	C₂²×C₃×C₄	—	6	8	no	—
241	#239	C₂²×C₃²	1	2	yes	—
242	D₃×Q₈	C₂²×C₃²	1	2	yes	complemented
243	#237	—	2	2	no	—
244	#237	C₂²×C₃²	1	2	yes	—
245	D₃×D₄	—	2	2	no	—
246	D₃×D₄	C₂²×C₃²	1	2	yes	complemented
247	#235	—	2	2	no	—
248	#235	C₂²×C₃²	1	2	yes	complemented
249	C₂×C₄×D₃	—	2	2	no	—
250	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
251	C₃×Q₁₆	—	2	4	no	—
252	C₃×Q₁₆	—	3	8	no	—
253	C₃×Q₁₆	—	3	4	no	—
254	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	complemented
255	C₃×Q₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
256	C₃×SD₁₆	—	2	8	no	—
257	C₃×SD₁₆	—	3	8	no	—
258	C₃×SD₁₆	—	3	8	no	—
259	C₃×SD₁₆	—	3	8	no	—
260	C₃×SD₁₆	C₂×C₃×D₃	1	8	yes	complemented
261	C₃×SD₁₆	C₂²×C₃²	1	2	yes	complemented
262	C₃×D₈	—	2	4	no	—
263	C₃×D₈	—	3	8	no	—
264	C₃×D₈	—	3	4	no	—
265	C₃×D₈	C₂×C₃×D₃	1	4	yes	complemented
266	C₃×D₈	C₂²×C₃²	1	1	yes	characteristic, complemented
267	C₃×M₁₆	—	3	8	no	—
268	C₃×M₁₆	—	3	8	no	—
269	C₂×C₃×C₈	—	2	4	no	—
270	C₂×C₃×C₈	—	3	4	no	—
271	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	complemented
272	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
273	#216	C₂²×C₃²	1	2	yes	complemented
274	#215	C₂²×C₃²	1	2	yes	complemented
275	#214	C₂²×C₃²	1	2	yes	complemented
276	#213	C₂²×C₃²	1	2	yes	complemented
277	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
278	Dic₁₂	C₂²×C₃²	1	1	yes	characteristic, complemented
279	D₂₄	C₂²×C₃²	1	1	yes	characteristic, complemented
280	#204	C₂²×C₃²	1	2	yes	complemented
281	ΓL(1,𝔽₂₅)	C₂²×C₃²	1	2	yes	complemented
282	ΓL(1,𝔽₂₅)	C₂²×C₃²	1	2	yes	complemented
283	C₂²×C₃²	—	2	2	no	—
284	C₂²×C₃²	—	2	4	no	—
285	C₂²×C₃²	—	2	8	no	—
286	C₂²×C₃²	—	3	2	no	—
287	C₂²×C₃²	—	4	8	no	—
288	C₂²×C₃²	—	6	2	no	—
289	C₂²×C₃²	—	12	1	no	—
290	C₂²×C₃²	—	12	4	no	—
291	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
292	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
293	C₂×C₃×D₃	—	2	8	no	—
294	C₂×C₃×D₃	—	4	4	no	—
295	C₂×C₃×D₃	—	4	16	no	—
296	C₂×C₃×D₃	C₂×C₃×D₄	1	8	yes	—
297	C₃²×C₄	—	2	2	no	—
298	C₃²×C₄	—	2	4	no	—
299	C₃²×C₄	—	2	4	no	—
300	C₃²×C₄	—	2	8	no	—
301	C₃²×C₄	—	3	2	no	—
302	C₃²×C₄	—	4	8	no	—
303	C₃²×C₄	—	6	2	no	—
304	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
305	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
306	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
307	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
308	C₃×Dic₃	—	2	8	no	—
309	C₃×Dic₃	C₂×C₃×D₄	1	8	yes	—
310	C₂×#41	—	3	2	no	—
311	#137	—	3	2	no	—
312	#137	—	3	2	no	—
313	AGL(1,ℤ/8)	—	3	2	no	—
314	AGL(1,ℤ/8)	—	3	2	no	—
315	#135	—	3	2	no	—
316	#135	—	3	1	no	—
317	#135	C₃²×D₃	1	1	yes	characteristic, complemented
318	C₂×M₁₆	—	3	1	no	—
319	C₃³	#575	1	1	yes	characteristic, complemented, Sylow
320	C₂³×C₃	—	3	4	no	—
321	C₂³×C₃	—	6	8	no	—
322	C₂²×D₃	—	2	2	no	—
323	C₂²×D₃	C₃²×D₄	1	1	yes	characteristic
324	C₃×Q₈	—	2	8	no	—
325	C₃×Q₈	—	3	8	no	—
326	C₃×Q₈	—	6	8	no	—
327	C₃×Q₈	C₂²×C₃×D₃	1	8	yes	—
328	C₃×Q₈	C₂³×C₃²	1	2	yes	—
329	C₃×D₄	—	2	8	no	—
330	C₃×D₄	—	2	2	no	—
331	C₃×D₄	—	2	8	no	—
332	C₃×D₄	—	3	8	no	—
333	C₃×D₄	—	4	8	no	—
334	C₃×D₄	—	6	8	no	—
335	C₃×D₄	—	6	8	no	—
336	C₃×D₄	—	6	8	no	—
337	C₃×D₄	—	6	8	no	—
338	C₃×D₄	C₂²×C₃×D₃	1	8	yes	—
339	C₃×D₄	C₂³×C₃²	1	2	yes	—
340	C₂×C₃×C₄	—	2	8	no	—
341	C₂×C₃×C₄	—	2	2	no	—
342	C₂×C₃×C₄	—	2	4	no	—
343	C₂×C₃×C₄	—	3	4	no	—
344	C₂×C₃×C₄	—	3	8	no	—
345	C₂×C₃×C₄	—	3	8	no	—
346	C₂×C₃×C₄	—	4	8	no	—
347	C₂×C₃×C₄	—	6	8	no	—
348	C₂×C₃×C₄	—	6	8	no	—
349	C₂×C₃×C₄	—	6	8	no	—
350	C₂×C₃×C₄	—	6	8	no	—
351	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
352	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
353	#67	—	2	2	no	—
354	#67	—	2	2	no	—
355	#67	—	2	2	no	—
356	C₂×Dic₃	—	2	2	no	—
357	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
358	D₁₂	—	2	2	no	—
359	D₁₂	C₂³×C₃²	1	2	yes	—
360	C₄×D₃	—	2	2	no	—
361	C₄×D₃	—	2	2	no	—
362	C₄×D₃	—	2	2	no	—
363	C₄×D₃	C₃²×D₄	1	2	yes	—
364	C₄×D₃	C₂³×C₃²	1	2	yes	—
365	Dic₆	—	2	2	no	—
366	Dic₆	C₂³×C₃²	1	2	yes	—
367	C₃×C₈	—	2	4	no	—
368	C₃×C₈	—	2	4	no	—
369	C₃×C₈	—	3	8	no	—
370	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
371	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
372	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
373	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
374	C₃⋊C₈	C₂³×C₃²	1	2	yes	—
375	C₂×C₃²	—	2	1	no	—
376	C₂×C₃²	—	2	4	no	—
377	C₂×C₃²	—	2	4	no	—
378	C₂×C₃²	—	4	4	no	—
379	C₂×C₃²	—	4	2	no	—
380	C₂×C₃²	—	4	8	no	—
381	C₂×C₃²	—	6	2	no	—
382	C₂×C₃²	—	8	8	no	—
383	C₂×C₃²	—	12	2	no	—
384	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
385	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
386	C₃×D₃	—	2	8	no	—
387	C₃×D₃	—	4	8	no	—
388	#41	—	3	2	no	—
389	#41	—	3	2	no	—
390	#41	—	3	2	no	—
391	#41	—	6	2	no	—
392	#41	—	6	2	no	—
393	#41	C₂×C₃²×D₃	1	2	yes	complemented
394	C₂×Q₈	—	3	2	no	—
395	C₂×D₄	—	3	2	no	—
396	C₂×D₄	—	6	2	no	—
397	C₂²×C₄	—	3	1	no	—
398	C₂²×C₄	—	6	2	no	—
399	Q₁₆	—	3	2	no	—
400	Q₁₆	—	3	1	no	—
401	Q₁₆	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
402	SD₁₆	—	3	2	no	—
403	SD₁₆	—	3	2	no	—
404	SD₁₆	—	3	2	no	—
405	SD₁₆	C₂×C₃²×D₃	1	2	yes	complemented
406	D₈	—	3	2	no	—
407	D₈	—	3	1	no	—
408	D₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
409	M₁₆	—	3	2	no	—
410	M₁₆	—	3	2	no	—
411	C₂×C₈	—	3	1	no	—
412	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
413	C₂²×C₃	—	2	8	no	—
414	C₂²×C₃	—	2	2	no	—
415	C₂²×C₃	—	2	4	no	—
416	C₂²×C₃	—	3	8	no	—
417	C₂²×C₃	—	4	8	no	—
418	C₂²×C₃	—	6	8	no	—
419	C₂²×C₃	—	12	4	no	—
420	C₂²×C₃	—	12	16	no	—
421	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
422	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
423	C₂×D₃	—	2	2	no	—
424	C₂×D₃	—	4	1	no	—
425	C₂×D₃	—	4	4	no	—
426	C₂×D₃	C₂×C₃²×D₄	1	2	yes	—
427	C₃×C₄	—	2	8	no	—
428	C₃×C₄	—	2	2	no	—
429	C₃×C₄	—	2	4	no	—
430	C₃×C₄	—	2	4	no	—
431	C₃×C₄	—	3	8	no	—
432	C₃×C₄	—	4	8	no	—
433	C₃×C₄	—	6	8	no	—
434	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
435	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
436	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
437	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
438	Dic₃	—	2	2	no	—
439	Dic₃	C₂×C₃²×D₄	1	2	yes	—
440	C₃²	—	2	4	no	—
441	C₃²	#5768	1	1	yes	characteristic, direct product factor
442	C₃²	C₃×#575	1	4	yes	complemented
443	C₂³	—	3	1	no	—
444	C₂³	—	6	2	no	—
445	Q₈	—	3	2	no	—
446	Q₈	—	6	2	no	—
447	Q₈	C₂²×C₃²×D₃	1	2	yes	—
448	D₄	—	2	2	no	—
449	D₄	—	3	2	no	—
450	D₄	—	6	2	no	—
451	D₄	—	6	2	no	—
452	D₄	—	6	2	no	—
453	D₄	—	6	2	no	—
454	D₄	C₂²×C₃²×D₃	1	2	yes	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	3	1	no	—
457	C₂×C₄	—	3	2	no	—
458	C₂×C₄	—	3	2	no	—
459	C₂×C₄	—	6	2	no	—
460	C₂×C₄	—	6	2	no	—
461	C₂×C₄	—	6	2	no	—
462	C₂×C₄	—	6	2	no	—
463	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
464	C₈	—	3	2	no	—
465	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
466	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
467	C₂×C₃	—	2	1	no	—
468	C₂×C₃	—	2	4	no	—
469	C₂×C₃	—	2	4	no	—
470	C₂×C₃	—	4	4	no	—
471	C₂×C₃	—	4	2	no	—
472	C₂×C₃	—	4	8	no	—
473	C₂×C₃	—	6	8	no	—
474	C₂×C₃	—	8	8	no	—
475	C₂×C₃	—	12	8	no	—
476	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
477	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
478	D₃	—	2	2	no	—
479	D₃	—	4	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	3	2	no	—
482	C₂²	—	6	2	no	—
483	C₂²	—	12	1	no	—
484	C₂²	—	12	4	no	—
485	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
486	C₄	—	2	2	no	—
487	C₄	—	3	2	no	—
488	C₄	—	6	2	no	—
489	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
490	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
491	C₃	—	2	4	no	—
492	C₃	C₃×#5768	1	4	yes	direct product factor
493	C₃	C₃²×#575	1	1	yes	characteristic, complemented
494	C₂	—	2	1	no	—
495	C₂	—	4	2	no	—
496	C₂	—	6	2	no	—
497	C₂	—	12	2	no	—
498	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	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×#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₃²×#135

Complements

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

N	H ≅ G/N	type
1. C₃²×D₃×#41	C₂	semidirect
2. C₃³×#135	C₂	semidirect
3. C₃²×#950	C₂	semidirect
4. C₃²×#918	C₂	semidirect
5. C₃²×#915	C₂	semidirect
6. C₃²×#914	C₂	semidirect
7. C₃²×#911	C₂	semidirect
8. C₃²×#904	C₂	semidirect
9. C₂×C₃²×ΓL(1,𝔽₂₅)	C₂	semidirect
11. C₃×#5768	C₃	direct (N × H)
12. C₃³×#41	C₂²	semidirect
14. C₃²×D₃×Q₈	C₂²	semidirect
18. C₃²×D₃×D₄	C₂²	semidirect
20. C₃²×#235	C₂²	semidirect
23. C₃³×Q₁₆	C₂²	semidirect
24. C₃³×SD₁₆	C₂²	semidirect
25. C₃³×D₈	C₂²	semidirect
26. C₂×C₃³×C₈	C₂²	semidirect
27. C₃²×#216	C₂²	semidirect
28. C₃²×#215	C₂²	semidirect
29. C₃²×#214	C₂²	semidirect
30. C₃²×#213	C₂²	semidirect
31. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
32. C₃²×Dic₁₂	C₂²	semidirect
33. C₃²×D₂₄	C₂²	semidirect
34. C₃²×#204	C₂²	semidirect
35. C₃²×ΓL(1,𝔽₂₅)	C₂²	semidirect
36. C₃²×ΓL(1,𝔽₂₅)	C₂²	semidirect
38. C₃×D₃×#41	C₂×C₃	semidirect
46. C₃²×#135	D₃	semidirect
47. C₃²×#135	C₂×C₃	semidirect
49. C₃×#950	C₂×C₃	semidirect
50. C₃×#918	C₂×C₃	semidirect
51. C₃×#915	C₂×C₃	semidirect
52. C₃×#914	C₂×C₃	semidirect
53. C₃×#911	C₂×C₃	semidirect
54. C₃×#904	C₂×C₃	semidirect
55. C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	semidirect
81. #5768	C₃²	direct (N × H)
88. C₃²×#41	C₂×D₃	semidirect
89. C₃²×#41	C₂²×C₃	semidirect
96. C₃×D₃×Q₈	C₂²×C₃	semidirect
100. C₃×D₃×D₄	C₂²×C₃	semidirect
102. C₃×#235	C₂²×C₃	semidirect
108. C₃²×Q₁₆	C₂×D₃	semidirect
109. C₃²×Q₁₆	C₂²×C₃	semidirect
114. C₃²×SD₁₆	C₂×D₃	semidirect
115. C₃²×SD₁₆	C₂²×C₃	semidirect
119. C₃²×D₈	C₂×D₃	semidirect
120. C₃²×D₈	C₂²×C₃	semidirect
125. C₂×C₃²×C₈	C₂×D₃	semidirect
126. C₂×C₃²×C₈	C₂²×C₃	semidirect
127. C₃×#216	C₂²×C₃	semidirect
128. C₃×#215	C₂²×C₃	semidirect
129. C₃×#214	C₂²×C₃	semidirect
130. C₃×#213	C₂²×C₃	semidirect
131. C₂×C₃×C₃⋊C₈	C₂²×C₃	semidirect
132. C₃×Dic₁₂	C₂²×C₃	semidirect
133. C₃×D₂₄	C₂²×C₃	semidirect
134. C₃×#204	C₂²×C₃	semidirect
135. C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	semidirect
136. C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	semidirect
149. D₃×#41	C₂×C₃²	semidirect
157. C₃×#135	C₃×D₃	semidirect
158. C₃×#135	C₂×C₃²	semidirect
160. #950	C₂×C₃²	semidirect
161. #918	C₂×C₃²	semidirect
162. #915	C₂×C₃²	semidirect
163. #914	C₂×C₃²	semidirect
164. #911	C₂×C₃²	semidirect
165. #904	C₂×C₃²	semidirect
166. C₂×ΓL(1,𝔽₂₅)	C₂×C₃²	semidirect
234. C₃×#41	C₂×C₃×D₃	semidirect
235. C₃×#41	C₂²×C₃²	semidirect
242. D₃×Q₈	C₂²×C₃²	semidirect
246. D₃×D₄	C₂²×C₃²	semidirect
248. #235	C₂²×C₃²	semidirect
254. C₃×Q₁₆	C₂×C₃×D₃	semidirect
255. C₃×Q₁₆	C₂²×C₃²	semidirect
260. C₃×SD₁₆	C₂×C₃×D₃	semidirect
261. C₃×SD₁₆	C₂²×C₃²	semidirect
265. C₃×D₈	C₂×C₃×D₃	semidirect
266. C₃×D₈	C₂²×C₃²	semidirect
271. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
272. C₂×C₃×C₈	C₂²×C₃²	semidirect
273. #216	C₂²×C₃²	semidirect
274. #215	C₂²×C₃²	semidirect
275. #214	C₂²×C₃²	semidirect
276. #213	C₂²×C₃²	semidirect
277. C₂×C₃⋊C₈	C₂²×C₃²	semidirect
278. Dic₁₂	C₂²×C₃²	semidirect
279. D₂₄	C₂²×C₃²	semidirect
280. #204	C₂²×C₃²	semidirect
281. ΓL(1,𝔽₂₅)	C₂²×C₃²	semidirect
282. ΓL(1,𝔽₂₅)	C₂²×C₃²	semidirect
317. #135	C₃²×D₃	semidirect
319. C₃³	#575	semidirect
393. #41	C₂×C₃²×D₃	semidirect
401. Q₁₆	C₂×C₃²×D₃	semidirect
405. SD₁₆	C₂×C₃²×D₃	semidirect
408. D₈	C₂×C₃²×D₃	semidirect
412. C₂×C₈	C₂×C₃²×D₃	semidirect
441. C₃²	#5768	direct (N × H)
442. C₃²	C₃×#575	semidirect
492. C₃	C₃×#5768	direct (N × H)
493. C₃	C₃²×#575	semidirect

Maximal subgroups

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

subgroup	index
C₃²×D₃×#41	2
C₃³×#135	2
C₃²×#950	2
C₃²×#918	2
C₃²×#915	2
C₃²×#914	2
C₃²×#911	2
C₃²×#904	2
C₂×C₃²×ΓL(1,𝔽₂₅)	2
C₃²×#575	3
C₃×#5768	3

Subgroups — C32×#5768 #13179145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5768 #13179145