Subgroups — #11773552

Order 1152.

507 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592384	C₂	1	1	yes	maximal, characteristic
2	C₄×#3712	C₂	1	1	yes	maximal, characteristic
3	C₄×#3710	C₂	1	1	yes	maximal, characteristic
4	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589078	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈²	C₂	1	1	yes	maximal, characteristic, complemented
8	#68067	—	3	1	no	maximal
9	C₈×ΓL(1,𝔽₂₅)	—	3	1	no	maximal
10	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂×#3712	C₄	1	2	yes	—
12	C₂×#3710	C₄	1	2	yes	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#63647	C₄	1	2	yes	—
15	#63641	C₄	1	2	yes	—
16	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	#63555	C₄	1	2	yes	—
22	#63553	C₄	1	2	yes	—
23	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
24	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
25	#4685	—	3	1	no	—
26	#4685	—	3	1	no	—
27	#4685	—	6	1	no	—
28	C₄×D₃×C₈	—	3	1	no	—
29	C₄×D₃×C₈	—	3	1	no	—
30	C₃×#322	—	3	1	no	—
31	C₃×C₈²	—	3	1	no	—
32	#4455	—	3	1	no	—
33	#4454	—	3	1	no	—
34	#4454	D₃	1	1	yes	characteristic, complemented
35	C₈×C₃⋊C₈	—	3	1	no	—
36	C₈×C₃⋊C₈	—	3	1	no	—
37	C₈×C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
38	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₃²×C₄²	C₂³	1	1	yes	characteristic
43	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	#3712	C₈	1	8	yes	complemented
52	#3710	—	2	4	no	—
53	C₈×M₁₆	—	9	1	no	Sylow
54	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
55	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
56	C₂×D₃×C₈	—	3	2	no	—
57	C₂×D₃×C₈	—	3	2	no	—
58	C₄²×D₃	—	3	1	no	—
59	C₄²×D₃	—	3	1	no	—
60	C₄²×D₃	—	6	1	no	—
61	C₃×C₄×C₈	—	2	1	no	—
62	C₃×C₄×C₈	—	3	1	no	—
63	C₃×C₄×C₈	—	3	1	no	—
64	C₃×C₄×C₈	—	3	1	no	—
65	C₃×C₄×C₈	—	3	1	no	—
66	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
68	(C₂×D₃)⋊C₈	—	3	2	no	—
69	(C₂×D₃)⋊C₈	—	3	2	no	—
70	(C₂×D₃)⋊C₈	—	6	2	no	—
71	#815	—	3	2	no	—
72	#815	—	3	2	no	—
73	Dic₃⋊C₈	—	3	2	no	—
74	Dic₃⋊C₈	—	3	2	no	—
75	Dic₃⋊C₈	—	6	2	no	—
76	C₈×Dic₃	—	3	2	no	—
77	C₈×Dic₃	—	3	2	no	—
78	C₄×C₃⋊C₈	—	3	1	no	—
79	C₄×C₃⋊C₈	—	3	1	no	—
80	C₄×C₃⋊C₈	—	6	1	no	—
81	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
82	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
83	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
84	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
85	C₂²×C₃⋊D₃	C₄²	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃⋊Dic₃	C₄²	1	1	yes	characteristic
90	C₂×C₃⋊Dic₃	C₄²	1	1	yes	characteristic
91	C₂×C₃⋊Dic₃	C₄²	1	1	yes	characteristic
92	C₄×C₃⋊D₃	—	2	2	no	—
93	C₄×C₃⋊D₃	—	2	2	no	—
94	C₄×C₃⋊D₃	C₂×C₈	1	4	yes	—
95	C₃²×C₈	—	2	2	no	—
96	C₃⋊(C₃⋊C₈)	—	2	2	no	—
97	C₃×C₃⋊C₈	—	2	2	no	—
98	C₃×C₃⋊C₈	C₂×C₈	1	4	yes	complemented
99	C₃×C₃⋊C₈	C₂×C₈	1	4	yes	complemented
100	C₃×C₃⋊C₈	M₁₆	1	4	yes	complemented
101	#431	—	9	1	no	—
102	C₄×M₁₆	—	9	1	no	—
103	C₂×C₄×C₈	—	9	1	no	—
104	#322	—	3	1	no	—
105	#322	—	9	1	no	—
106	C₈²	—	3	1	no	—
107	C₈²	—	9	1	no	—
108	C₂×C₄×D₃	—	3	1	no	—
109	C₂×C₄×D₃	—	3	1	no	—
110	C₂×C₄×D₃	—	3	1	no	—
111	C₂×C₄×D₃	—	3	1	no	—
112	C₂×C₄×D₃	—	3	1	no	—
113	C₂×C₄×D₃	—	3	1	no	—
114	C₂×C₄×D₃	—	6	1	no	—
115	C₂×C₄×D₃	—	6	1	no	—
116	C₂×C₄×D₃	—	6	1	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₂×C₃×C₈	—	3	2	no	—
120	C₂×C₃×C₈	—	3	2	no	—
121	C₂×C₃×C₈	—	3	2	no	—
122	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
123	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
126	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
127	C₄×Dic₃	—	3	1	no	—
128	C₄×Dic₃	—	3	1	no	—
129	C₄×Dic₃	—	3	1	no	—
130	C₄×Dic₃	—	3	1	no	—
131	C₄×Dic₃	—	3	1	no	—
132	C₄×Dic₃	—	3	1	no	—
133	C₄×Dic₃	—	6	1	no	—
134	C₄×Dic₃	—	6	1	no	—
135	C₄×Dic₃	—	6	1	no	—
136	C₂×C₃⋊C₈	—	3	2	no	—
137	C₂×C₃⋊C₈	—	3	2	no	—
138	C₂×C₃⋊C₈	—	6	2	no	—
139	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
140	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
141	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
142	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
143	ΓL(1,𝔽₂₅)	—	3	8	no	—
144	ΓL(1,𝔽₂₅)	—	3	8	no	—
145	D₃×C₈	—	6	4	no	—
146	D₃×C₈	—	6	4	no	—
147	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₃⋊D₃	—	2	1	no	—
149	C₂×C₃⋊D₃	—	2	1	no	—
150	C₂×C₃⋊D₃	C₄×C₈	1	2	yes	—
151	C₃²×C₄	C₂²×C₈	1	2	yes	—
152	C₃²×C₄	C₂×M₁₆	1	2	yes	—
153	C₃²×C₄	#131	1	2	yes	—
154	C₃⋊Dic₃	—	2	1	no	—
155	C₃⋊Dic₃	—	2	1	no	—
156	C₃⋊Dic₃	C₄×C₈	1	2	yes	—
157	C₂×M₁₆	—	9	2	no	—
158	C₂²×C₈	—	9	2	no	—
159	C₂×C₄²	—	9	1	no	—
160	C₄⋊C₈	—	9	2	no	—
161	C₂²⋊C₈	—	9	2	no	—
162	#97	—	9	2	no	—
163	C₄×C₈	—	3	1	no	—
164	C₄×C₈	—	3	1	no	—
165	C₄×C₈	—	3	1	no	—
166	C₄×C₈	—	3	1	no	—
167	C₄×C₈	—	9	2	no	—
168	C₄×C₈	—	9	1	no	—
169	C₄×C₈	D₃²	1	1	yes	characteristic
170	C₂²×D₃	—	3	1	no	—
171	C₂²×D₃	—	3	1	no	—
172	C₂²×D₃	—	6	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×Dic₃	—	3	1	no	—
183	C₂×Dic₃	—	3	1	no	—
184	C₂×Dic₃	—	3	1	no	—
185	C₂×Dic₃	—	3	1	no	—
186	C₂×Dic₃	—	3	1	no	—
187	C₂×Dic₃	—	3	1	no	—
188	C₂×Dic₃	—	6	1	no	—
189	C₂×Dic₃	—	6	1	no	—
190	C₂×Dic₃	—	6	1	no	—
191	C₄×D₃	—	3	4	no	—
192	C₄×D₃	—	3	4	no	—
193	C₄×D₃	—	6	4	no	—
194	C₄×D₃	—	6	2	no	—
195	C₄×D₃	—	6	2	no	—
196	C₄×D₃	—	6	2	no	—
197	C₄×D₃	—	6	2	no	—
198	C₄×D₃	—	12	2	no	—
199	C₄×D₃	—	12	2	no	—
200	C₃×C₈	—	2	2	no	—
201	C₃×C₈	—	2	2	no	—
202	C₃×C₈	—	3	4	no	—
203	C₃×C₈	—	3	4	no	—
204	C₃×C₈	—	3	4	no	—
205	C₃×C₈	—	4	2	no	—
206	C₃×C₈	—	6	2	no	—
207	C₃⋊C₈	—	2	2	no	—
208	C₃⋊C₈	—	6	2	no	—
209	C₃⋊C₈	—	6	2	no	—
210	C₃⋊C₈	—	12	2	no	—
211	C₃⋊C₈	D₃×C₈	1	4	yes	complemented
212	C₃⋊C₈	D₃×C₈	1	4	yes	complemented
213	C₃⋊C₈	ΓL(1,𝔽₂₅)	1	4	yes	complemented
214	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
215	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
216	C₂×C₃²	#405	1	1	yes	characteristic
217	C₃⋊D₃	—	2	2	no	—
218	C₂²×C₄	—	9	1	no	—
219	C₂²×C₄	—	9	1	no	—
220	C₂²×C₄	—	9	1	no	—
221	M₁₆	—	9	8	no	—
222	C₂×C₈	—	3	2	no	—
223	C₂×C₈	—	3	2	no	—
224	C₂×C₈	—	3	2	no	—
225	C₂×C₈	—	3	2	no	—
226	C₂×C₈	—	9	2	no	—
227	C₂×C₈	—	18	4	no	—
228	C₂×C₈	#624	1	2	yes	—
229	C₄²	—	9	1	no	—
230	C₄²	—	9	1	no	—
231	C₄²	—	9	1	no	—
232	C₄²	C₂×D₃²	1	1	yes	characteristic
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
235	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
236	C₂×D₃	—	3	2	no	—
237	C₂×D₃	—	3	2	no	—
238	C₂×D₃	—	6	2	no	—
239	C₂×D₃	—	6	1	no	—
240	C₂×D₃	—	6	1	no	—
241	C₂×D₃	—	6	1	no	—
242	C₂×D₃	—	6	1	no	—
243	C₂×D₃	—	12	1	no	—
244	C₂×D₃	—	12	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
249	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
250	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
251	C₃×C₄	#901	1	2	yes	—
252	C₃×C₄	D₃×M₁₆	1	2	yes	—
253	C₃×C₄	#907	1	2	yes	—
254	Dic₃	—	3	2	no	—
255	Dic₃	—	3	2	no	—
256	Dic₃	—	6	2	no	—
257	Dic₃	—	6	1	no	—
258	Dic₃	—	6	1	no	—
259	Dic₃	—	6	1	no	—
260	Dic₃	—	6	1	no	—
261	Dic₃	—	12	1	no	—
262	Dic₃	—	12	1	no	—
263	C₃²	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	9	1	no	—
265	C₂×C₄	—	9	1	no	—
266	C₂×C₄	—	9	4	no	—
267	C₂×C₄	—	9	1	no	—
268	C₂×C₄	—	9	1	no	—
269	C₂×C₄	—	18	2	no	—
270	C₂×C₄	—	18	2	no	—
271	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
272	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
273	C₂×C₄	C₂×#624	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	3	4	no	—
276	C₈	—	3	4	no	—
277	C₈	—	3	4	no	—
278	C₈	—	6	2	no	—
279	C₈	—	18	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
284	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
285	C₂×C₃	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
286	C₂×C₃	#4689	1	1	yes	characteristic
287	C₂×C₃	#4707	1	1	yes	characteristic
288	C₂×C₃	#4708	1	1	yes	characteristic
289	D₃	—	6	2	no	—
290	D₃	—	6	2	no	—
291	D₃	—	12	2	no	—
292	C₂²	—	9	2	no	—
293	C₂²	—	18	1	no	—
294	C₂²	—	18	1	no	—
295	C₂²	C₄×#624	1	1	yes	characteristic
296	C₄	—	9	2	no	—
297	C₄	—	18	1	no	—
298	C₄	—	18	1	no	—
299	C₄	D₃²×C₈	1	2	yes	—
300	C₄	D₃×ΓL(1,𝔽₂₅)	1	2	yes	—
301	C₄	#63817	1	2	yes	—
302	C₃	—	2	1	no	—
303	C₃	C₈×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
304	C₃	#68067	1	1	yes	characteristic, complemented
305	C₂	—	18	2	no	—
306	C₂	#10590321	1	1	yes	characteristic
307	C₂	C₈×#624	1	1	yes	characteristic
308	C₂	#10590664	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃×#4454	C₂	semidirect
5. C₃×C₈×C₃⋊C₈	C₂	semidirect
6. #10589078	C₂	semidirect
7. C₃⋊C₈²	C₂	semidirect
34. #4454	D₃	semidirect
37. C₈×C₃⋊C₈	D₃	semidirect
51. #3712	C₈	semidirect
98. C₃×C₃⋊C₈	C₂×C₈	semidirect
99. C₃×C₃⋊C₈	C₂×C₈	semidirect
100. C₃×C₃⋊C₈	M₁₆	semidirect
211. C₃⋊C₈	D₃×C₈	semidirect
212. C₃⋊C₈	D₃×C₈	semidirect
213. C₃⋊C₈	ΓL(1,𝔽₂₅)	semidirect
263. C₃²	C₈×M₁₆	semidirect
303. C₃	C₈×ΓL(1,𝔽₂₅)	semidirect
304. C₃	#68067	semidirect

Maximal subgroups

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

subgroup	index
#10592384	2
C₄×#3712	2
C₄×#3710	2
C₃×#4454	2
C₃×C₈×C₃⋊C₈	2
#10589078	2
C₃⋊C₈²	2
#68067	3
C₈×ΓL(1,𝔽₂₅)	3