Subgroups — #11797604

Order 1152.

330 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592839	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#5144	C₂	1	1	yes	maximal, characteristic
3	#10590622	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4561	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589134	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589112	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589103	C₂	1	1	yes	maximal, characteristic, complemented
8	#70924	—	3	1	no	maximal
9	#69123	—	3	1	no	maximal
10	C₃²×#131	C₂²	1	1	yes	characteristic
11	C₃×#950	C₄	1	1	yes	characteristic
12	C₃×#901	C₂²	1	1	yes	characteristic
13	#63639	—	2	1	no	—
14	#63639	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
16	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
17	C₃×#837	C₄	1	1	yes	characteristic
18	C₃×#825	C₄	1	1	yes	characteristic
19	C₃×#822	C₄	1	1	yes	characteristic
20	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
21	C₃×#812	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	#63543	—	2	1	no	—
24	D₃×C₃⋊C₁₆	—	2	1	no	—
25	C₃×#443	—	3	1	no	—
26	#5144	D₃	1	1	yes	characteristic
27	#5140	—	3	1	no	—
28	#5140	—	3	1	no	—
29	#5140	—	3	1	no	—
30	#5140	—	6	1	no	—
31	#4907	—	3	1	no	—
32	C₃×#350	—	3	1	no	—
33	#4561	—	3	1	no	—
34	#4561	D₃	1	1	yes	characteristic, complemented
35	#4511	—	3	1	no	—
36	#4508	—	3	1	no	—
37	#4502	—	3	1	no	—
38	#4463	—	3	1	no	—
39	#4461	—	3	1	no	—
40	C₃²×#41	C₂×C₄	1	1	yes	characteristic
41	C₃×#235	C₂×C₄	1	1	yes	characteristic
42	C₃²×M₁₆	—	2	1	no	—
43	C₃²×M₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	—	2	1	no	—
45	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
46	C₃×#216	C₈	1	1	yes	characteristic
47	C₃×#215	C₈	1	1	yes	characteristic
48	C₃×#214	C₈	1	1	yes	characteristic
49	C₃×#213	C₈	1	1	yes	characteristic
50	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₃×#208	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
54	C₃×D₃×C₈	—	2	1	no	—
55	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
56	C₃⋊(C₃⋊C₁₆)	D₄	1	1	yes	characteristic
57	C₃⋊(C₃⋊C₁₆)	D₄	1	1	yes	characteristic
58	C₃×C₃⋊C₁₆	—	2	1	no	—
59	C₃×C₃⋊C₁₆	—	2	1	no	—
60	#2171	—	9	1	no	Sylow
61	C₃×#135	—	3	1	no	—
62	C₃×#131	—	2	1	no	—
63	C₃×#131	—	3	1	no	—
64	C₃×#131	C₂×D₃	1	1	yes	characteristic
65	C₃×#131	C₂×D₃	1	1	yes	characteristic
66	#950	Dic₃	1	1	yes	characteristic
67	#901	C₂×D₃	1	1	yes	characteristic
68	C₃×M₃₂	—	3	1	no	—
69	C₂×C₃×C₁₆	—	3	1	no	—
70	C₃×#108	—	3	1	no	—
71	C₃×#104	—	3	1	no	—
72	C₃×#104	—	3	1	no	—
73	C₃×C₄×C₈	—	3	1	no	—
74	#837	Dic₃	1	1	yes	characteristic
75	#825	Dic₃	1	1	yes	characteristic
76	#822	Dic₃	1	1	yes	characteristic
77	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
78	#812	—	3	1	no	—
79	#812	—	3	1	no	—
80	#812	—	6	1	no	—
81	#812	—	6	1	no	—
82	#812	—	6	1	no	—
83	#812	—	6	1	no	—
84	#812	—	6	2	no	—
85	#812	C₂×D₃	1	1	yes	characteristic
86	C₂×C₃⋊C₁₆	—	3	1	no	—
87	C₂×C₃⋊C₁₆	—	3	1	no	—
88	C₂×C₃⋊C₁₆	—	6	1	no	—
89	C₂×C₃⋊C₁₆	—	6	1	no	—
90	C₂×C₃⋊C₁₆	—	6	1	no	—
91	C₂×C₃⋊C₁₆	—	6	1	no	—
92	C₂×C₃⋊C₁₆	—	6	2	no	—
93	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
94	#798	—	6	1	no	—
95	D₃×C₁₆	—	6	1	no	—
96	C₃²×Q₈	C₂×C₈	1	1	yes	characteristic
97	C₃²×D₄	—	2	1	no	—
98	C₃²×D₄	C₂×C₈	1	1	yes	characteristic
99	C₂×C₃²×C₄	—	2	1	no	—
100	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₃×#67	—	2	1	no	—
102	C₂×C₃×Dic₃	—	2	1	no	—
103	C₃×D₁₂	C₂×C₈	1	1	yes	characteristic
104	C₃×C₄×D₃	—	2	1	no	—
105	C₃×Dic₆	C₂×C₈	1	1	yes	characteristic
106	C₃²×C₈	—	2	1	no	—
107	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₃²×C₈	#41	1	1	yes	characteristic
109	C₃×C₃⋊C₈	—	2	1	no	—
110	C₃×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
111	C₃×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
112	#504	—	9	1	no	—
113	#504	—	9	1	no	—
114	#443	—	3	1	no	—
115	#364	—	9	1	no	—
116	#350	—	3	1	no	—
117	#350	—	9	1	no	—
118	#346	—	9	1	no	—
119	C₃×#41	—	2	1	no	—
120	C₃×#41	—	3	1	no	—
121	C₃×#41	C₄×D₃	1	1	yes	characteristic
122	C₃×#41	C₂×Dic₃	1	1	yes	characteristic
123	#235	C₂×Dic₃	1	1	yes	characteristic
124	C₃×Q₁₆	—	3	1	no	—
125	C₃×SD₁₆	—	3	1	no	—
126	C₃×SD₁₆	—	3	1	no	—
127	C₃×D₈	—	3	1	no	—
128	C₃×M₁₆	—	2	2	no	—
129	C₃×M₁₆	—	2	1	no	—
130	C₃×M₁₆	—	2	1	no	—
131	C₃×M₁₆	—	2	1	no	—
132	C₃×M₁₆	—	3	1	no	—
133	C₃×M₁₆	—	6	1	no	—
134	C₃×M₁₆	C₄×D₃	1	1	yes	characteristic
135	C₃×M₁₆	C₂×Dic₃	1	1	yes	characteristic
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	3	1	no	—
141	C₂×C₃×C₈	—	6	1	no	—
142	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₃×C₄²	—	3	1	no	—
145	#216	C₃⋊C₈	1	1	yes	characteristic
146	#215	C₃⋊C₈	1	1	yes	characteristic
147	#214	C₃⋊C₈	1	1	yes	characteristic
148	#213	C₃⋊C₈	1	1	yes	characteristic
149	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
150	#208	C₂×Dic₃	1	1	yes	characteristic
151	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
152	ΓL(1,𝔽₂₅)	—	2	1	no	—
153	D₃×C₈	—	2	1	no	—
154	C₃×C₁₆	—	6	1	no	—
155	C₃×C₁₆	—	6	1	no	—
156	C₃⋊C₁₆	—	2	1	no	—
157	C₃⋊C₁₆	—	2	1	no	—
158	C₃⋊C₁₆	—	3	1	no	—
159	C₃⋊C₁₆	—	3	1	no	—
160	C₃⋊C₁₆	—	3	1	no	—
161	C₃⋊C₁₆	—	3	1	no	—
162	C₃⋊C₁₆	—	6	1	no	—
163	C₃⋊C₁₆	—	6	1	no	—
164	C₃⋊C₁₆	—	6	2	no	—
165	C₃⋊C₁₆	—	6	1	no	—
166	C₃⋊C₁₆	—	6	1	no	—
167	C₂²×C₃²	—	2	1	no	—
168	C₂²×C₃²	#131	1	1	yes	characteristic
169	C₂×C₃×D₃	—	2	1	no	—
170	C₃²×C₄	—	2	1	no	—
171	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
172	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
173	C₃×Dic₃	—	2	1	no	—
174	C₃×Dic₃	—	4	1	no	—
175	#135	—	3	1	no	—
176	#131	—	3	1	no	—
177	#131	D₃²	1	1	yes	characteristic
178	M₃₂	—	3	1	no	—
179	M₃₂	—	9	1	no	—
180	M₃₂	—	18	1	no	—
181	M₃₂	—	18	1	no	—
182	C₂×C₁₆	—	3	1	no	—
183	C₂×C₁₆	—	9	1	no	—
184	C₂×C₁₆	—	18	1	no	—
185	C₂×C₁₆	—	18	1	no	—
186	#108	—	3	1	no	—
187	#104	—	3	1	no	—
188	#104	—	3	1	no	—
189	C₄×C₈	—	3	1	no	—
190	C₃×Q₈	—	2	1	no	—
191	C₃×Q₈	—	3	1	no	—
192	C₃×Q₈	D₃×C₈	1	1	yes	characteristic
193	C₃×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
194	C₃×D₄	—	2	2	no	—
195	C₃×D₄	—	2	1	no	—
196	C₃×D₄	—	2	1	no	—
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	—	3	1	no	—
199	C₃×D₄	—	6	1	no	—
200	C₃×D₄	D₃×C₈	1	1	yes	characteristic
201	C₃×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	6	1	no	—
207	C₂×C₃×C₄	—	6	1	no	—
208	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
210	#67	—	2	1	no	—
211	C₂×Dic₃	—	2	1	no	—
212	D₁₂	C₂×C₃⋊C₈	1	1	yes	characteristic
213	C₄×D₃	—	2	1	no	—
214	Dic₆	C₂×C₃⋊C₈	1	1	yes	characteristic
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	2	1	no	—
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	2	1	no	—
220	C₃×C₈	—	3	1	no	—
221	C₃×C₈	—	3	1	no	—
222	C₃×C₈	—	6	1	no	—
223	C₃×C₈	D₃×D₄	1	1	yes	characteristic
224	C₃×C₈	D₃×D₄	1	1	yes	characteristic
225	C₃×C₈	#237	1	1	yes	characteristic
226	C₃×C₈	#237	1	1	yes	characteristic
227	C₃⋊C₈	—	2	1	no	—
228	C₃⋊C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
229	C₃⋊C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
230	C₂×C₃²	—	2	1	no	—
231	C₂×C₃²	—	4	1	no	—
232	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
233	C₃×D₃	—	4	1	no	—
234	#41	—	3	1	no	—
235	#41	D₃×Dic₃	1	1	yes	characteristic
236	Q₁₆	—	3	1	no	—
237	SD₁₆	—	3	1	no	—
238	SD₁₆	—	3	1	no	—
239	D₈	—	3	1	no	—
240	M₁₆	—	2	1	no	—
241	M₁₆	—	3	1	no	—
242	M₁₆	—	6	1	no	—
243	M₁₆	D₃×Dic₃	1	1	yes	characteristic
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	3	1	no	—
246	C₂×C₈	—	6	1	no	—
247	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
248	C₄²	—	3	1	no	—
249	C₁₆	—	6	1	no	—
250	C₁₆	—	6	1	no	—
251	C₁₆	—	9	1	no	—
252	C₁₆	—	9	1	no	—
253	C₁₆	—	18	1	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	6	1	no	—
259	C₂²×C₃	#907	1	1	yes	characteristic
260	C₂²×C₃	#948	1	1	yes	characteristic
261	C₂×D₃	—	2	1	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	6	1	no	—
268	C₃×C₄	—	12	1	no	—
269	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
270	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
271	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
272	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
273	Dic₃	—	2	1	no	—
274	Dic₃	—	4	1	no	—
275	C₃²	#2171	1	1	yes	characteristic, complemented, Sylow
276	Q₈	—	3	1	no	—
277	Q₈	D₃×C₃⋊C₈	1	1	yes	characteristic
278	D₄	—	2	1	no	—
279	D₄	—	3	1	no	—
280	D₄	—	6	1	no	—
281	D₄	D₃×C₃⋊C₈	1	1	yes	characteristic
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	3	1	no	—
288	C₈	—	3	1	no	—
289	C₈	—	6	1	no	—
290	C₈	#3805	1	1	yes	characteristic
291	C₈	#3811	1	1	yes	characteristic
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	4	1	no	—
296	C₂×C₃	—	4	2	no	—
297	C₂×C₃	—	4	1	no	—
298	C₂×C₃	—	4	1	no	—
299	C₂×C₃	—	12	1	no	—
300	C₂×C₃	#67⋊C₈	1	1	yes	characteristic
301	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
302	D₃	—	4	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	6	1	no	—
305	C₂²	#63815	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	6	1	no	—
308	C₄	—	12	1	no	—
309	C₄	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
310	C₄	#67⋊Dic₃	1	1	yes	characteristic
311	C₃	—	2	1	no	—
312	C₃	#69123	1	1	yes	characteristic, complemented
313	C₃	#70924	1	1	yes	characteristic, complemented
314	C₂	—	2	1	no	—
315	C₂	—	4	1	no	—
316	C₂	—	12	1	no	—
317	C₂	#67⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#3811
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₃²×#131	C₂²
socle	C₂×C₃²	C₈×D₄
prime radical	C₃×#950	C₄
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	#131	D₃²
hypocenter	C₃²	#2171
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #10592839	C₂	semidirect
3. #10590622	C₂	semidirect
4. C₃×#4561	C₂	semidirect
5. #10589134	C₂	semidirect
6. #10589112	C₂	semidirect
7. #10589103	C₂	semidirect
34. #4561	D₃	semidirect
275. C₃²	#2171	semidirect
312. C₃	#69123	semidirect
313. C₃	#70924	semidirect

Maximal subgroups

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

subgroup	index
#10592839	2
C₃×#5144	2
#10590622	2
C₃×#4561	2
#10589134	2
#10589112	2
#10589103	2
#70924	3
#69123	3