Subgroups — C₃²×#5769 #13179146

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#1009	C₂	1	2	yes	maximal, complemented
2	C₃³×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#949	C₂	1	2	yes	maximal, complemented
4	C₃²×#919	C₂	1	2	yes	maximal, complemented
5	C₃²×#914	C₂	1	4	yes	maximal, complemented
6	C₃²×D₃×D₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₃²×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×#576	—	3	1	no	maximal
10	C₃×#5769	C₃	1	4	yes	maximal, direct product factor
11	C₃³×#41	C₂²	1	2	yes	complemented
12	C₃²×#239	C₂²	1	4	yes	complemented
13	C₃²×D₃×D₄	—	2	4	no	—
14	C₃²×D₃×D₄	C₂²	1	4	yes	complemented
15	C₃²×#235	—	2	2	no	—
16	C₃²×#235	C₂²	1	1	yes	characteristic
17	C₂×C₃²×D₁₂	—	2	2	no	—
18	C₂×C₃²×D₁₂	C₂²	1	2	yes	complemented
19	C₃³×Q₁₆	C₂²	1	1	yes	characteristic, complemented
20	C₃³×SD₁₆	C₂²	1	2	yes	complemented
21	C₃³×D₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic, complemented
23	C₃²×#215	C₂²	1	4	yes	complemented
24	C₃²×#213	C₂²	1	4	yes	complemented
25	C₃²×#208	C₂²	1	1	yes	characteristic, complemented
26	C₃²×D₂₄	C₂²	1	2	yes	complemented
27	C₃²×D₂₄	C₂²	1	2	yes	complemented
28	C₃²×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
29	C₃²×D₃×C₈	C₂²	1	2	yes	complemented
30	C₃²×H(5,𝔽₂)	—	3	2	no	—
31	C₃×#1009	C₂×C₃	1	8	yes	complemented
32	C₃²×AGL(1,ℤ/8)	—	3	2	no	—
33	C₃²×AGL(1,ℤ/8)	—	3	4	no	—
34	C₃²×#135	—	2	4	no	—
35	C₃²×#135	—	3	2	no	—
36	C₃²×#135	D₃	1	1	yes	characteristic, complemented
37	C₃²×#135	C₂×C₃	1	4	yes	complemented
38	C₂×C₃²×D₈	—	3	2	no	—
39	C₂×C₃²×D₈	—	3	1	no	—
40	C₃²×#131	—	3	1	no	—
41	C₃×#949	C₂×C₃	1	8	yes	complemented
42	C₃×#919	C₂×C₃	1	8	yes	complemented
43	C₃×#914	C₂×C₃	1	16	yes	complemented
44	C₃×D₃×D₈	C₂×C₃	1	8	yes	complemented
45	C₂×C₃×D₂₄	C₂×C₃	1	4	yes	complemented
46	C₃×#901	C₂×C₃	1	4	yes	complemented
47	C₂²×C₃²×D₃	—	2	2	no	—
48	C₂²×C₃²×D₃	—	2	4	no	—
49	C₃³×Q₈	C₂³	1	2	yes	—
50	C₃³×D₄	—	2	2	no	—
51	C₃³×D₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	—	2	2	no	—
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₃²×#67	—	2	4	no	—
55	C₃²×#67	D₄	1	2	yes	—
56	C₃²×D₁₂	—	2	4	no	—
57	C₃²×D₁₂	—	2	2	no	—
58	C₃²×D₁₂	D₄	1	1	yes	characteristic
59	C₃²×D₁₂	C₂³	1	4	yes	—
60	C₃²×C₄×D₃	—	2	4	no	—
61	C₃²×C₄×D₃	C₂³	1	2	yes	—
62	C₃²×Dic₆	D₄	1	1	yes	characteristic
63	C₃³×C₈	C₂³	1	1	yes	characteristic
64	C₃³×C₈	C₂³	1	1	yes	characteristic
65	C₃²×C₃⋊C₈	C₂³	1	2	yes	—
66	C₃×#576	—	3	4	no	—
67	#5769	C₃²	1	1	yes	characteristic, direct product factor
68	C₃²×#41	—	2	8	no	—
69	C₃²×#41	—	3	1	no	—
70	C₃²×#41	—	3	4	no	—
71	C₃²×#41	—	6	2	no	—
72	C₃²×#41	C₂×D₃	1	2	yes	complemented
73	C₃²×#41	C₂²×C₃	1	8	yes	complemented
74	C₂×C₃²×D₄	—	3	4	no	—
75	C₂×C₃²×D₄	—	3	2	no	—
76	C₂×C₃²×D₄	—	6	2	no	—
77	C₂×C₃²×D₄	—	6	4	no	—
78	C₃×#239	C₂²×C₃	1	16	yes	complemented
79	C₃×D₃×D₄	—	2	16	no	—
80	C₃×D₃×D₄	C₂²×C₃	1	16	yes	complemented
81	C₃×#235	—	2	8	no	—
82	C₃×#235	C₂²×C₃	1	4	yes	—
83	C₂×C₃×D₁₂	—	2	8	no	—
84	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	complemented
85	C₃²×Q₁₆	—	2	4	no	—
86	C₃²×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
87	C₃²×Q₁₆	C₂²×C₃	1	4	yes	complemented
88	C₃²×SD₁₆	—	2	8	no	—
89	C₃²×SD₁₆	—	3	4	no	—
90	C₃²×SD₁₆	C₂×D₃	1	2	yes	complemented
91	C₃²×SD₁₆	C₂²×C₃	1	8	yes	complemented
92	C₃²×D₈	—	2	4	no	—
93	C₃²×D₈	—	3	4	no	—
94	C₃²×D₈	—	3	2	no	—
95	C₃²×D₈	—	3	2	no	—
96	C₃²×D₈	C₂×D₃	1	1	yes	characteristic, complemented
97	C₃²×D₈	C₂²×C₃	1	4	yes	complemented
98	C₃²×M₁₆	—	3	1	no	—
99	C₃²×M₁₆	—	3	2	no	—
100	C₂×C₃²×C₈	—	2	4	no	—
101	C₂×C₃²×C₈	—	3	2	no	—
102	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
103	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	complemented
104	C₃×#215	C₂²×C₃	1	16	yes	complemented
105	C₃×#213	C₂²×C₃	1	16	yes	complemented
106	C₃×#208	C₂²×C₃	1	4	yes	complemented
107	C₃×D₂₄	C₂²×C₃	1	8	yes	complemented
108	C₃×D₂₄	C₂²×C₃	1	8	yes	complemented
109	C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	1	8	yes	complemented
110	C₃×D₃×C₈	C₂²×C₃	1	8	yes	complemented
111	C₂²×C₃³	—	2	2	no	—
112	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃²×D₃	—	2	4	no	—
114	C₂×C₃²×D₃	—	8	4	no	—
115	C₂×C₃²×D₃	—	8	2	no	—
116	C₂×C₃²×D₃	C₂×D₄	1	2	yes	—
117	C₃³×C₄	—	2	2	no	—
118	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
119	C₃³×C₄	C₂⁴	1	1	yes	characteristic
120	C₃²×Dic₃	C₂×D₄	1	2	yes	—
121	C₃×H(5,𝔽₂)	—	3	8	no	—
122	#1009	C₂×C₃²	1	2	yes	complemented
123	C₃×AGL(1,ℤ/8)	—	3	8	no	—
124	C₃×AGL(1,ℤ/8)	—	3	16	no	—
125	C₃×#135	—	2	4	no	—
126	C₃×#135	—	3	8	no	—
127	C₃×#135	C₃×D₃	1	4	yes	complemented
128	C₃×#135	C₂×C₃²	1	1	yes	characteristic, complemented
129	C₂×C₃×D₈	—	3	8	no	—
130	C₂×C₃×D₈	—	3	4	no	—
131	C₃×#131	—	3	4	no	—
132	#949	C₂×C₃²	1	2	yes	complemented
133	#919	C₂×C₃²	1	2	yes	complemented
134	#914	C₂×C₃²	1	4	yes	complemented
135	D₃×D₈	C₂×C₃²	1	2	yes	complemented
136	C₂×D₂₄	C₂×C₃²	1	1	yes	characteristic, complemented
137	#901	C₂×C₃²	1	1	yes	characteristic, complemented
138	C₂³×C₃²	—	6	4	no	—
139	C₂³×C₃²	—	6	2	no	—
140	C₂²×C₃×D₃	—	2	8	no	—
141	C₂²×C₃×D₃	—	2	16	no	—
142	C₃²×Q₈	—	2	8	no	—
143	C₃²×Q₈	—	3	1	no	—
144	C₃²×Q₈	C₂²×D₃	1	2	yes	—
145	C₃²×Q₈	C₂³×C₃	1	8	yes	—
146	C₃²×D₄	—	2	2	no	—
147	C₃²×D₄	—	2	8	no	—
148	C₃²×D₄	—	2	8	no	—
149	C₃²×D₄	—	3	2	no	—
150	C₃²×D₄	—	3	1	no	—
151	C₃²×D₄	—	3	4	no	—
152	C₃²×D₄	—	4	8	no	—
153	C₃²×D₄	—	6	4	no	—
154	C₃²×D₄	—	6	4	no	—
155	C₃²×D₄	—	6	2	no	—
156	C₃²×D₄	C₂²×D₃	1	2	yes	—
157	C₃²×D₄	C₂³×C₃	1	8	yes	—
158	C₂×C₃²×C₄	—	2	2	no	—
159	C₂×C₃²×C₄	—	2	4	no	—
160	C₂×C₃²×C₄	—	2	8	no	—
161	C₂×C₃²×C₄	—	3	2	no	—
162	C₂×C₃²×C₄	—	4	8	no	—
163	C₂×C₃²×C₄	—	6	4	no	—
164	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
165	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
166	C₃×#67	—	2	16	no	—
167	C₃×#67	C₃×D₄	1	8	yes	—
168	C₃×D₁₂	—	2	16	no	—
169	C₃×D₁₂	—	2	8	no	—
170	C₃×D₁₂	C₃×D₄	1	4	yes	—
171	C₃×D₁₂	C₂³×C₃	1	16	yes	—
172	C₃×C₄×D₃	—	2	16	no	—
173	C₃×C₄×D₃	C₂³×C₃	1	8	yes	—
174	C₃×Dic₆	C₃×D₄	1	4	yes	—
175	C₃²×C₈	—	2	4	no	—
176	C₃²×C₈	—	2	4	no	—
177	C₃²×C₈	—	3	2	no	—
178	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
179	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
180	C₃²×C₈	C₂³×C₃	1	4	yes	—
181	C₃²×C₈	C₂³×C₃	1	4	yes	—
182	C₃×C₃⋊C₈	C₂³×C₃	1	8	yes	—
183	#576	—	3	1	no	Sylow
184	C₂×C₃³	—	2	1	no	—
185	C₂×C₃³	—	4	2	no	—
186	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
187	C₃²×D₃	—	2	2	no	—
188	C₃²×D₃	—	4	4	no	—
189	C₃×#41	—	2	8	no	—
190	C₃×#41	—	3	4	no	—
191	C₃×#41	—	3	16	no	—
192	C₃×#41	—	6	8	no	—
193	C₃×#41	C₂×C₃×D₃	1	8	yes	complemented
194	C₃×#41	C₂²×C₃²	1	2	yes	complemented
195	C₂×C₃×D₄	—	3	16	no	—
196	C₂×C₃×D₄	—	3	8	no	—
197	C₂×C₃×D₄	—	6	8	no	—
198	C₂×C₃×D₄	—	6	16	no	—
199	#239	C₂²×C₃²	1	4	yes	complemented
200	D₃×D₄	—	2	4	no	—
201	D₃×D₄	C₂²×C₃²	1	4	yes	complemented
202	#235	—	2	2	no	—
203	#235	C₂²×C₃²	1	1	yes	characteristic
204	C₂×D₁₂	—	2	2	no	—
205	C₂×D₁₂	C₂²×C₃²	1	2	yes	complemented
206	C₃×Q₁₆	—	2	4	no	—
207	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	complemented
208	C₃×Q₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
209	C₃×SD₁₆	—	2	8	no	—
210	C₃×SD₁₆	—	3	16	no	—
211	C₃×SD₁₆	C₂×C₃×D₃	1	8	yes	complemented
212	C₃×SD₁₆	C₂²×C₃²	1	2	yes	complemented
213	C₃×D₈	—	2	4	no	—
214	C₃×D₈	—	3	16	no	—
215	C₃×D₈	—	3	8	no	—
216	C₃×D₈	—	3	8	no	—
217	C₃×D₈	C₂×C₃×D₃	1	4	yes	complemented
218	C₃×D₈	C₂²×C₃²	1	1	yes	characteristic, complemented
219	C₃×M₁₆	—	3	4	no	—
220	C₃×M₁₆	—	3	8	no	—
221	C₂×C₃×C₈	—	2	4	no	—
222	C₂×C₃×C₈	—	3	8	no	—
223	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	complemented
224	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
225	#215	C₂²×C₃²	1	4	yes	complemented
226	#213	C₂²×C₃²	1	4	yes	complemented
227	#208	C₂²×C₃²	1	1	yes	characteristic, complemented
228	D₂₄	C₂²×C₃²	1	2	yes	complemented
229	D₂₄	C₂²×C₃²	1	2	yes	complemented
230	ΓL(1,𝔽₂₅)	C₂²×C₃²	1	2	yes	complemented
231	D₃×C₈	C₂²×C₃²	1	2	yes	complemented
232	C₂²×C₃²	—	2	2	no	—
233	C₂²×C₃²	—	2	4	no	—
234	C₂²×C₃²	—	2	8	no	—
235	C₂²×C₃²	—	3	2	no	—
236	C₂²×C₃²	—	4	8	no	—
237	C₂²×C₃²	—	6	4	no	—
238	C₂²×C₃²	—	24	2	no	—
239	C₂²×C₃²	—	24	4	no	—
240	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
241	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
242	C₂×C₃×D₃	—	2	16	no	—
243	C₂×C₃×D₃	—	8	16	no	—
244	C₂×C₃×D₃	—	8	8	no	—
245	C₂×C₃×D₃	C₂×C₃×D₄	1	8	yes	—
246	C₃²×C₄	—	2	2	no	—
247	C₃²×C₄	—	2	4	no	—
248	C₃²×C₄	—	2	4	no	—
249	C₃²×C₄	—	2	8	no	—
250	C₃²×C₄	—	3	2	no	—
251	C₃²×C₄	—	4	8	no	—
252	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
253	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
254	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
255	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
256	C₃×Dic₃	C₂×C₃×D₄	1	8	yes	—
257	H(5,𝔽₂)	—	3	2	no	—
258	AGL(1,ℤ/8)	—	3	2	no	—
259	AGL(1,ℤ/8)	—	3	4	no	—
260	#135	—	3	2	no	—
261	#135	C₃²×D₃	1	1	yes	characteristic, complemented
262	C₂×D₈	—	3	2	no	—
263	C₂×D₈	—	3	1	no	—
264	#131	—	3	1	no	—
265	C₃³	#576	1	1	yes	characteristic, complemented, Sylow
266	C₂³×C₃	—	6	16	no	—
267	C₂³×C₃	—	6	8	no	—
268	C₂²×D₃	—	2	2	no	—
269	C₂²×D₃	—	2	4	no	—
270	C₃×Q₈	—	2	8	no	—
271	C₃×Q₈	—	3	4	no	—
272	C₃×Q₈	C₂²×C₃×D₃	1	8	yes	—
273	C₃×Q₈	C₂³×C₃²	1	2	yes	—
274	C₃×D₄	—	2	8	no	—
275	C₃×D₄	—	2	2	no	—
276	C₃×D₄	—	2	8	no	—
277	C₃×D₄	—	3	8	no	—
278	C₃×D₄	—	3	16	no	—
279	C₃×D₄	—	3	4	no	—
280	C₃×D₄	—	4	8	no	—
281	C₃×D₄	—	6	16	no	—
282	C₃×D₄	—	6	16	no	—
283	C₃×D₄	—	6	8	no	—
284	C₃×D₄	C₂²×C₃×D₃	1	8	yes	—
285	C₃×D₄	C₂³×C₃²	1	2	yes	—
286	C₂×C₃×C₄	—	2	8	no	—
287	C₂×C₃×C₄	—	2	2	no	—
288	C₂×C₃×C₄	—	2	4	no	—
289	C₂×C₃×C₄	—	3	8	no	—
290	C₂×C₃×C₄	—	4	8	no	—
291	C₂×C₃×C₄	—	6	16	no	—
292	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
293	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
294	#67	—	2	4	no	—
295	#67	C₃²×D₄	1	2	yes	—
296	D₁₂	—	2	4	no	—
297	D₁₂	—	2	2	no	—
298	D₁₂	C₃²×D₄	1	1	yes	characteristic
299	D₁₂	C₂³×C₃²	1	4	yes	—
300	C₄×D₃	—	2	4	no	—
301	C₄×D₃	C₂³×C₃²	1	2	yes	—
302	Dic₆	C₃²×D₄	1	1	yes	characteristic
303	C₃×C₈	—	2	4	no	—
304	C₃×C₈	—	2	4	no	—
305	C₃×C₈	—	3	8	no	—
306	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
307	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
308	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
309	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
310	C₃⋊C₈	C₂³×C₃²	1	2	yes	—
311	C₂×C₃²	—	2	1	no	—
312	C₂×C₃²	—	2	4	no	—
313	C₂×C₃²	—	2	4	no	—
314	C₂×C₃²	—	4	4	no	—
315	C₂×C₃²	—	4	2	no	—
316	C₂×C₃²	—	4	8	no	—
317	C₂×C₃²	—	6	2	no	—
318	C₂×C₃²	—	8	8	no	—
319	C₂×C₃²	—	12	4	no	—
320	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
321	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
322	C₃×D₃	—	2	8	no	—
323	C₃×D₃	—	4	16	no	—
324	#41	—	3	1	no	—
325	#41	—	3	4	no	—
326	#41	—	6	2	no	—
327	#41	C₂×C₃²×D₃	1	2	yes	complemented
328	C₂×D₄	—	3	4	no	—
329	C₂×D₄	—	3	2	no	—
330	C₂×D₄	—	6	2	no	—
331	C₂×D₄	—	6	4	no	—
332	Q₁₆	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
333	SD₁₆	—	3	4	no	—
334	SD₁₆	C₂×C₃²×D₃	1	2	yes	complemented
335	D₈	—	3	4	no	—
336	D₈	—	3	2	no	—
337	D₈	—	3	2	no	—
338	D₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
339	M₁₆	—	3	1	no	—
340	M₁₆	—	3	2	no	—
341	C₂×C₈	—	3	2	no	—
342	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
343	C₂²×C₃	—	2	8	no	—
344	C₂²×C₃	—	2	2	no	—
345	C₂²×C₃	—	2	4	no	—
346	C₂²×C₃	—	3	8	no	—
347	C₂²×C₃	—	4	8	no	—
348	C₂²×C₃	—	6	16	no	—
349	C₂²×C₃	—	24	8	no	—
350	C₂²×C₃	—	24	16	no	—
351	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
352	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
353	C₂×D₃	—	2	4	no	—
354	C₂×D₃	—	8	4	no	—
355	C₂×D₃	—	8	2	no	—
356	C₂×D₃	C₂×C₃²×D₄	1	2	yes	—
357	C₃×C₄	—	2	8	no	—
358	C₃×C₄	—	2	2	no	—
359	C₃×C₄	—	2	4	no	—
360	C₃×C₄	—	2	4	no	—
361	C₃×C₄	—	3	8	no	—
362	C₃×C₄	—	4	8	no	—
363	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
364	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
365	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
366	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
367	Dic₃	C₂×C₃²×D₄	1	2	yes	—
368	C₃²	—	2	4	no	—
369	C₃²	#5769	1	1	yes	characteristic, direct product factor
370	C₃²	C₃×#576	1	4	yes	complemented
371	C₂³	—	6	4	no	—
372	C₂³	—	6	2	no	—
373	Q₈	—	3	1	no	—
374	Q₈	C₂²×C₃²×D₃	1	2	yes	—
375	D₄	—	2	2	no	—
376	D₄	—	3	2	no	—
377	D₄	—	3	4	no	—
378	D₄	—	3	1	no	—
379	D₄	—	6	4	no	—
380	D₄	—	6	4	no	—
381	D₄	—	6	2	no	—
382	D₄	C₂²×C₃²×D₃	1	2	yes	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	3	2	no	—
385	C₂×C₄	—	6	4	no	—
386	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
387	C₈	—	3	2	no	—
388	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
389	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
390	C₂×C₃	—	2	1	no	—
391	C₂×C₃	—	2	4	no	—
392	C₂×C₃	—	2	4	no	—
393	C₂×C₃	—	4	4	no	—
394	C₂×C₃	—	4	2	no	—
395	C₂×C₃	—	4	8	no	—
396	C₂×C₃	—	6	8	no	—
397	C₂×C₃	—	8	8	no	—
398	C₂×C₃	—	12	16	no	—
399	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
400	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
401	D₃	—	2	2	no	—
402	D₃	—	4	4	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	3	2	no	—
405	C₂²	—	6	4	no	—
406	C₂²	—	24	2	no	—
407	C₂²	—	24	4	no	—
408	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
409	C₄	—	2	2	no	—
410	C₄	—	3	2	no	—
411	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
412	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
413	C₃	—	2	4	no	—
414	C₃	C₃×#5769	1	4	yes	direct product factor
415	C₃	C₃²×#576	1	1	yes	characteristic, complemented
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	6	2	no	—
419	C₂	—	12	4	no	—
420	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×#576
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×#1009	C₂	semidirect
2. C₃³×#135	C₂	semidirect
3. C₃²×#949	C₂	semidirect
4. C₃²×#919	C₂	semidirect
5. C₃²×#914	C₂	semidirect
6. C₃²×D₃×D₈	C₂	semidirect
7. C₂×C₃²×D₂₄	C₂	semidirect
8. C₃²×#901	C₂	semidirect
10. C₃×#5769	C₃	direct (N × H)
11. C₃³×#41	C₂²	semidirect
12. C₃²×#239	C₂²	semidirect
14. C₃²×D₃×D₄	C₂²	semidirect
18. C₂×C₃²×D₁₂	C₂²	semidirect
19. C₃³×Q₁₆	C₂²	semidirect
20. C₃³×SD₁₆	C₂²	semidirect
21. C₃³×D₈	C₂²	semidirect
22. C₂×C₃³×C₈	C₂²	semidirect
23. C₃²×#215	C₂²	semidirect
24. C₃²×#213	C₂²	semidirect
25. C₃²×#208	C₂²	semidirect
26. C₃²×D₂₄	C₂²	semidirect
27. C₃²×D₂₄	C₂²	semidirect
28. C₃²×ΓL(1,𝔽₂₅)	C₂²	semidirect
29. C₃²×D₃×C₈	C₂²	semidirect
31. C₃×#1009	C₂×C₃	semidirect
36. C₃²×#135	D₃	semidirect
37. C₃²×#135	C₂×C₃	semidirect
41. C₃×#949	C₂×C₃	semidirect
42. C₃×#919	C₂×C₃	semidirect
43. C₃×#914	C₂×C₃	semidirect
44. C₃×D₃×D₈	C₂×C₃	semidirect
45. C₂×C₃×D₂₄	C₂×C₃	semidirect
46. C₃×#901	C₂×C₃	semidirect
67. #5769	C₃²	direct (N × H)
72. C₃²×#41	C₂×D₃	semidirect
73. C₃²×#41	C₂²×C₃	semidirect
78. C₃×#239	C₂²×C₃	semidirect
80. C₃×D₃×D₄	C₂²×C₃	semidirect
84. C₂×C₃×D₁₂	C₂²×C₃	semidirect
86. C₃²×Q₁₆	C₂×D₃	semidirect
87. C₃²×Q₁₆	C₂²×C₃	semidirect
90. C₃²×SD₁₆	C₂×D₃	semidirect
91. C₃²×SD₁₆	C₂²×C₃	semidirect
96. C₃²×D₈	C₂×D₃	semidirect
97. C₃²×D₈	C₂²×C₃	semidirect
102. C₂×C₃²×C₈	C₂×D₃	semidirect
103. C₂×C₃²×C₈	C₂²×C₃	semidirect
104. C₃×#215	C₂²×C₃	semidirect
105. C₃×#213	C₂²×C₃	semidirect
106. C₃×#208	C₂²×C₃	semidirect
107. C₃×D₂₄	C₂²×C₃	semidirect
108. C₃×D₂₄	C₂²×C₃	semidirect
109. C₃×ΓL(1,𝔽₂₅)	C₂²×C₃	semidirect
110. C₃×D₃×C₈	C₂²×C₃	semidirect
122. #1009	C₂×C₃²	semidirect
127. C₃×#135	C₃×D₃	semidirect
128. C₃×#135	C₂×C₃²	semidirect
132. #949	C₂×C₃²	semidirect
133. #919	C₂×C₃²	semidirect
134. #914	C₂×C₃²	semidirect
135. D₃×D₈	C₂×C₃²	semidirect
136. C₂×D₂₄	C₂×C₃²	semidirect
137. #901	C₂×C₃²	semidirect
193. C₃×#41	C₂×C₃×D₃	semidirect
194. C₃×#41	C₂²×C₃²	semidirect
199. #239	C₂²×C₃²	semidirect
201. D₃×D₄	C₂²×C₃²	semidirect
205. C₂×D₁₂	C₂²×C₃²	semidirect
207. C₃×Q₁₆	C₂×C₃×D₃	semidirect
208. C₃×Q₁₆	C₂²×C₃²	semidirect
211. C₃×SD₁₆	C₂×C₃×D₃	semidirect
212. C₃×SD₁₆	C₂²×C₃²	semidirect
217. C₃×D₈	C₂×C₃×D₃	semidirect
218. C₃×D₈	C₂²×C₃²	semidirect
223. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
224. C₂×C₃×C₈	C₂²×C₃²	semidirect
225. #215	C₂²×C₃²	semidirect
226. #213	C₂²×C₃²	semidirect
227. #208	C₂²×C₃²	semidirect
228. D₂₄	C₂²×C₃²	semidirect
229. D₂₄	C₂²×C₃²	semidirect
230. ΓL(1,𝔽₂₅)	C₂²×C₃²	semidirect
231. D₃×C₈	C₂²×C₃²	semidirect
261. #135	C₃²×D₃	semidirect
265. C₃³	#576	semidirect
327. #41	C₂×C₃²×D₃	semidirect
332. Q₁₆	C₂×C₃²×D₃	semidirect
334. SD₁₆	C₂×C₃²×D₃	semidirect
338. D₈	C₂×C₃²×D₃	semidirect
342. C₂×C₈	C₂×C₃²×D₃	semidirect
369. C₃²	#5769	direct (N × H)
370. C₃²	C₃×#576	semidirect
414. C₃	C₃×#5769	direct (N × H)
415. C₃	C₃²×#576	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#1009	2
C₃³×#135	2
C₃²×#949	2
C₃²×#919	2
C₃²×#914	2
C₃²×D₃×D₈	2
C₂×C₃²×D₂₄	2
C₃²×#901	2
C₃²×#576	3
C₃×#5769	3

Subgroups — C32×#5769 #13179146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5769 #13179146