Subgroups — #11795942

Order 1152.

343 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#10590693	C₂	1	1	yes	maximal, characteristic
3	D₃×#812	C₂	1	1	yes	maximal, characteristic
4	#10589378	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4512	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589106	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589105	C₂	1	1	yes	maximal, characteristic, complemented
8	#71425	—	3	1	no	maximal
9	#69087	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	#63855	C₄	1	1	yes	characteristic
12	C₂×#3711	C₄	1	1	yes	characteristic
13	C₃²×M₃₂	C₂²	1	1	yes	characteristic
14	#63642	C₂²	1	1	yes	characteristic
15	#63639	C₂²	1	1	yes	characteristic
16	C₃×#815	C₂²	1	1	yes	characteristic
17	C₃×#812	C₂²	1	1	yes	characteristic
18	C₃×#812	C₂²	1	1	yes	characteristic
19	C₃×#798	C₄	1	2	yes	complemented
20	C₃×D₃×C₁₆	C₄	1	2	yes	complemented
21	#63553	C₄	1	1	yes	characteristic
22	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
23	#63543	C₄	1	2	yes	complemented
24	D₃×C₃⋊C₁₆	C₄	1	2	yes	complemented
25	C₂×C₃×M₃₂	—	3	1	no	—
26	#5101	—	3	1	no	—
27	C₂×#812	—	3	1	no	—
28	D₃×M₃₂	—	3	1	no	—
29	D₃×M₃₂	D₃	1	1	yes	characteristic
30	#4708	—	3	1	no	—
31	C₃×#347	—	3	1	no	—
32	#4512	—	3	1	no	—
33	#4512	—	3	1	no	—
34	#4512	—	3	1	no	—
35	#4512	—	3	1	no	—
36	#4512	—	3	1	no	—
37	#4512	—	6	1	no	—
38	#4512	D₃	1	1	yes	characteristic, complemented
39	#4466	—	3	1	no	—
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
42	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
50	C₃×D₃×C₈	C₂×C₄	1	1	yes	characteristic
51	Dic₃⋊Dic₃	—	2	1	no	—
52	(C₂×D₃)⋊Dic₃	—	2	1	no	—
53	#3711	—	2	1	no	—
54	#3711	—	2	1	no	—
55	C₃²×C₁₆	C₂×C₄	1	2	yes	—
56	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
57	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
58	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
59	#2110	—	9	1	no	Sylow
60	C₂²×C₃×C₈	—	3	1	no	—
61	#926	—	3	1	no	—
62	C₂×#208	—	3	1	no	—
63	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
64	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
65	#872	—	3	1	no	—
66	C₃×M₃₂	—	2	1	no	—
67	C₃×M₃₂	—	3	1	no	—
68	C₃×M₃₂	—	3	1	no	—
69	C₃×M₃₂	—	3	2	no	—
70	C₃×M₃₂	C₂×D₃	1	1	yes	characteristic
71	C₃×M₃₂	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃×C₁₆	—	3	2	no	—
73	C₃×#97	—	3	1	no	—
74	#815	—	3	1	no	—
75	#815	—	3	1	no	—
76	#815	—	6	1	no	—
77	#815	C₂×D₃	1	1	yes	characteristic
78	C₈×Dic₃	—	3	1	no	—
79	C₈×Dic₃	—	3	1	no	—
80	C₈×Dic₃	—	3	1	no	—
81	#812	—	3	1	no	—
82	#812	—	3	1	no	—
83	#812	—	3	2	no	—
84	#812	—	6	1	no	—
85	#812	C₂×D₃	1	1	yes	characteristic
86	#812	C₂×D₃	1	1	yes	characteristic
87	C₂×C₃⋊C₁₆	—	3	2	no	—
88	C₄×C₃⋊C₈	—	3	1	no	—
89	#798	—	3	2	no	—
90	#798	Dic₃	1	2	yes	complemented
91	D₃×C₁₆	—	3	2	no	—
92	D₃×C₁₆	Dic₃	1	2	yes	complemented
93	C₂²×C₃×D₃	M₁₆	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃⋊Dic₃	—	2	1	no	—
96	C₂×C₃×Dic₃	—	2	1	no	—
97	C₂×C₃×Dic₃	M₁₆	1	1	yes	characteristic
98	C₃×C₄×D₃	C₄²	1	1	yes	characteristic
99	C₃×C₄×D₃	C₄²	1	1	yes	characteristic
100	C₃×C₄×D₃	M₁₆	1	1	yes	characteristic
101	C₃×C₄×D₃	M₁₆	1	1	yes	characteristic
102	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₃⋊(C₃⋊C₈)	—	2	1	no	—
105	C₃×C₃⋊C₈	—	2	1	no	—
106	C₃×C₃⋊C₈	C₄²	1	1	yes	characteristic
107	C₃×C₃⋊C₈	C₄²	1	1	yes	characteristic
108	C₂×M₃₂	—	3	1	no	—
109	C₂×M₃₂	—	9	1	no	—
110	#405	—	9	1	no	—
111	#347	—	3	1	no	—
112	#347	—	9	1	no	—
113	#347	—	9	1	no	—
114	#347	—	9	1	no	—
115	C₂²×C₃×C₄	—	3	1	no	—
116	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	3	1	no	—
119	C₂×C₃×C₈	—	3	1	no	—
120	C₂×C₃×C₈	—	3	1	no	—
121	C₂×C₃×C₈	—	3	1	no	—
122	C₂×C₃×C₈	—	3	1	no	—
123	C₂×C₃×C₈	—	3	1	no	—
124	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
126	C₃×C₄²	—	3	1	no	—
127	#217	—	6	1	no	—
128	(C₂×D₃)⋊C₄	—	6	1	no	—
129	C₄⋊Dic₃	—	6	1	no	—
130	Dic₃⋊C₄	—	6	1	no	—
131	C₄×Dic₃	—	3	1	no	—
132	C₄×Dic₃	—	3	1	no	—
133	C₄×Dic₃	—	6	1	no	—
134	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
135	#208	—	6	1	no	—
136	#208	—	6	1	no	—
137	C₂×C₃⋊C₈	—	3	1	no	—
138	C₂×C₃⋊C₈	—	3	1	no	—
139	C₂×C₃⋊C₈	—	6	1	no	—
140	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
141	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
142	ΓL(1,𝔽₂₅)	—	6	1	no	—
143	ΓL(1,𝔽₂₅)	—	6	1	no	—
144	D₃×C₈	C₄×D₃	1	1	yes	characteristic
145	D₃×C₈	C₄×D₃	1	1	yes	characteristic
146	D₃×C₈	C₂×Dic₃	1	1	yes	characteristic
147	D₃×C₈	C₂×Dic₃	1	1	yes	characteristic
148	C₃×C₁₆	—	2	2	no	—
149	C₃×C₁₆	—	3	2	no	—
150	C₃×C₁₆	—	3	2	no	—
151	C₃×C₁₆	C₄×D₃	1	2	yes	—
152	C₃×C₁₆	C₂×Dic₃	1	2	yes	—
153	C₃⋊C₁₆	—	3	2	no	—
154	C₃⋊C₁₆	—	3	2	no	—
155	C₃⋊C₁₆	—	6	2	no	—
156	C₃⋊C₁₆	C₄×D₃	1	2	yes	—
157	C₃⋊C₁₆	C₂×Dic₃	1	2	yes	—
158	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
159	C₂×C₃×D₃	—	4	1	no	—
160	C₂×C₃×D₃	#97	1	1	yes	characteristic
161	C₂×C₃×D₃	#97	1	1	yes	characteristic
162	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
163	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
164	C₃⋊Dic₃	—	4	1	no	—
165	C₃×Dic₃	—	4	1	no	—
166	C₃×Dic₃	#97	1	1	yes	characteristic
167	C₃×Dic₃	#97	1	1	yes	characteristic
168	C₂×M₁₆	—	9	1	no	—
169	C₂²×C₈	—	3	1	no	—
170	#117	—	9	1	no	—
171	M₃₂	—	3	1	no	—
172	M₃₂	—	3	1	no	—
173	M₃₂	—	3	2	no	—
174	M₃₂	—	9	2	no	—
175	M₃₂	—	9	1	no	—
176	M₃₂	D₃²	1	1	yes	characteristic
177	C₂×C₁₆	—	3	2	no	—
178	C₂×C₁₆	—	9	2	no	—
179	#97	—	3	1	no	—
180	#97	—	9	1	no	—
181	C₄×C₈	—	9	1	no	—
182	C₄×C₈	—	9	1	no	—
183	C₂³×C₃	—	3	1	no	—
184	C₂²×D₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	3	1	no	—
187	C₂×C₃×C₄	—	3	1	no	—
188	C₂×C₃×C₄	—	3	1	no	—
189	C₂×C₃×C₄	—	3	1	no	—
190	C₂×C₃×C₄	—	3	1	no	—
191	C₂×C₃×C₄	—	6	1	no	—
192	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
194	C₂×Dic₃	—	2	1	no	—
195	C₂×Dic₃	—	6	1	no	—
196	C₂×Dic₃	—	6	1	no	—
197	C₂×Dic₃	—	12	1	no	—
198	C₂×Dic₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
199	C₄×D₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
200	C₄×D₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
201	C₄×D₃	C₄×Dic₃	1	1	yes	characteristic
202	C₄×D₃	C₄×Dic₃	1	1	yes	characteristic
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	—	3	1	no	—
206	C₃×C₈	—	3	1	no	—
207	C₃×C₈	—	6	1	no	—
208	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
209	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
210	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
211	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
212	C₃⋊C₈	—	2	1	no	—
213	C₃⋊C₈	—	6	1	no	—
214	C₃⋊C₈	—	6	1	no	—
215	C₃⋊C₈	—	12	1	no	—
216	C₃⋊C₈	C₄×Dic₃	1	1	yes	characteristic
217	C₃⋊C₈	C₄×Dic₃	1	1	yes	characteristic
218	C₂×C₃²	—	2	1	no	—
219	C₂×C₃²	C₂×#97	1	1	yes	characteristic
220	C₃×D₃	—	2	1	no	—
221	C₃×D₃	—	2	1	no	—
222	C₂²×C₄	—	3	1	no	—
223	M₁₆	—	18	1	no	—
224	M₁₆	—	18	1	no	—
225	C₂×C₈	—	3	1	no	—
226	C₂×C₈	—	3	1	no	—
227	C₂×C₈	—	3	1	no	—
228	C₂×C₈	—	3	1	no	—
229	C₂×C₈	—	3	1	no	—
230	C₂×C₈	—	3	1	no	—
231	C₂×C₈	—	9	1	no	—
232	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
233	C₄⋊C₄	—	18	1	no	—
234	C₂²⋊C₄	—	18	1	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	9	1	no	—
237	C₁₆	—	3	2	no	—
238	C₁₆	—	3	2	no	—
239	C₁₆	—	9	2	no	—
240	C₁₆	D₃×Dic₃	1	2	yes	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	3	1	no	—
243	C₂²×C₃	—	3	1	no	—
244	C₂²×C₃	—	12	1	no	—
245	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
246	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
247	C₂×D₃	—	4	1	no	—
248	C₂×D₃	#815	1	1	yes	characteristic
249	C₂×D₃	#815	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	3	1	no	—
253	C₃×C₄	—	3	1	no	—
254	C₃×C₄	—	12	1	no	—
255	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
256	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
257	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
258	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
259	Dic₃	—	4	1	no	—
260	Dic₃	—	12	1	no	—
261	Dic₃	—	12	1	no	—
262	Dic₃	—	24	1	no	—
263	Dic₃	#815	1	1	yes	characteristic
264	Dic₃	#815	1	1	yes	characteristic
265	C₃²	#2110	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	6	1	no	—
273	C₂×C₄	—	18	1	no	—
274	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
275	C₈	—	3	1	no	—
276	C₈	—	3	1	no	—
277	C₈	—	6	1	no	—
278	C₈	—	18	1	no	—
279	C₈	C₄×D₃²	1	1	yes	characteristic
280	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	—	6	1	no	—
286	C₂×C₃	—	6	1	no	—
287	C₂×C₃	D₃×#97	1	1	yes	characteristic
288	C₂×C₃	C₂×#815	1	1	yes	characteristic
289	D₃	—	2	1	no	—
290	D₃	—	2	1	no	—
291	C₂²	—	3	1	no	—
292	C₂²	—	3	1	no	—
293	C₂²	—	12	1	no	—
294	C₂²	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
295	C₄	—	3	1	no	—
296	C₄	—	3	1	no	—
297	C₄	—	12	1	no	—
298	C₄	—	36	1	no	—
299	C₄	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
300	C₄	C₄×D₃×Dic₃	1	1	yes	characteristic
301	C₃	—	2	1	no	—
302	C₃	#69087	1	1	yes	characteristic, complemented
303	C₃	#71425	1	1	yes	characteristic, complemented
304	C₂	—	2	1	no	—
305	C₂	—	6	1	no	—
306	C₂	—	6	1	no	—
307	C₂	D₃×#815	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×D₃×Dic₃
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₃²×M₃₂	C₂²
socle	C₂×C₃²	C₂×#97
prime radical	C₂²×C₃×D₃	M₁₆
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	M₃₂	D₃²
hypocenter	C₃²	#2110
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
4. #10589378	C₂	semidirect
5. C₃×#4512	C₂	semidirect
6. #10589106	C₂	semidirect
7. #10589105	C₂	semidirect
19. C₃×#798	C₄	semidirect
20. C₃×D₃×C₁₆	C₄	semidirect
23. #63543	C₄	semidirect
24. D₃×C₃⋊C₁₆	C₄	semidirect
38. #4512	D₃	semidirect
90. #798	Dic₃	semidirect
92. D₃×C₁₆	Dic₃	semidirect
265. C₃²	#2110	semidirect
302. C₃	#69087	semidirect
303. C₃	#71425	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×M₃₂	2
#10590693	2
D₃×#812	2
#10589378	2
C₃×#4512	2
#10589106	2
#10589105	2
#71425	3
#69087	3