Subgroups — #11797328

Order 1152.

337 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#624	C₂	1	1	yes	maximal, characteristic
2	#10589373	C₂	1	1	yes	maximal, characteristic
3	C₃×#4502	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589108	C₂	1	1	yes	maximal, characteristic
7	#10589103	C₂	1	1	yes	maximal, characteristic, complemented
8	#69275	—	3	1	no	maximal
9	#69073	—	3	1	no	maximal
10	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₄×#624	C₄	1	1	yes	characteristic
12	C₂×#3710	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
20	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
21	C₄×D₃×C₈	—	3	1	no	—
22	C₄×D₃×C₈	—	3	1	no	—
23	C₃×C₄⋊C₁₆	—	3	1	no	—
24	C₃×#346	—	3	1	no	—
25	(C₂×D₃)⋊C₁₆	—	3	1	no	—
26	(C₂×D₃)⋊C₁₆	—	3	1	no	—
27	(C₂×D₃)⋊C₁₆	—	3	1	no	—
28	(C₂×D₃)⋊C₁₆	—	3	1	no	—
29	(C₂×D₃)⋊C₁₆	—	6	1	no	—
30	#4502	—	3	1	no	—
31	#4502	D₃	1	1	yes	characteristic, complemented
32	Dic₃⋊C₁₆	—	3	1	no	—
33	Dic₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
34	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
35	#4461	—	3	1	no	—
36	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#624	C₈	1	1	yes	characteristic
38	C₂×#624	C₈	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₈×C₃⋊D₃	—	2	1	no	—
42	C₈×C₃⋊D₃	—	2	1	no	—
43	C₃×C₄×Dic₃	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	Dic₃²	C₈	1	1	yes	characteristic
48	Dic₃²	C₈	1	1	yes	characteristic
49	#3710	—	2	2	no	—
50	C₃²×C₁₆	—	2	1	no	—
51	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
52	C₃×C₃⋊C₁₆	—	2	1	no	—
53	C₃×C₃⋊C₁₆	—	2	1	no	—
54	#2163	—	9	1	no	Sylow
55	C₂×D₃×C₈	—	3	1	no	—
56	C₂×D₃×C₈	—	3	1	no	—
57	C₂×D₃×C₈	—	3	1	no	—
58	C₂×D₃×C₈	—	3	1	no	—
59	C₂×D₃×C₈	—	6	1	no	—
60	C₄²×D₃	—	3	1	no	—
61	C₄²×D₃	—	3	1	no	—
62	C₂×C₃×C₁₆	—	2	1	no	—
63	C₂×C₃×C₁₆	—	3	1	no	—
64	C₂×C₃×C₁₆	—	3	1	no	—
65	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
67	C₃×C₄×C₈	—	3	1	no	—
68	C₃×C₄×C₈	—	3	1	no	—
69	C₈×Dic₃	—	3	1	no	—
70	C₈×Dic₃	—	3	1	no	—
71	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
72	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃⋊C₁₆	—	3	1	no	—
74	C₂×C₃⋊C₁₆	—	3	1	no	—
75	C₂×C₃⋊C₁₆	—	6	1	no	—
76	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
77	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
78	C₄×C₃⋊C₈	—	3	1	no	—
79	C₄×C₃⋊C₈	—	3	1	no	—
80	C₂²×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
83	C₄×C₃⋊D₃	—	2	1	no	—
84	C₄×C₃⋊D₃	—	2	1	no	—
85	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
89	#624	—	2	2	no	—
90	#624	—	2	2	no	—
91	C₃²×C₈	#41	1	1	yes	characteristic
92	C₃²×C₈	#41	1	1	yes	characteristic
93	C₃⋊(C₃⋊C₈)	—	2	1	no	—
94	C₃×C₃⋊C₈	—	2	1	no	—
95	C₃×C₃⋊C₈	M₁₆	1	2	yes	—
96	C₂×C₄×C₈	—	9	1	no	—
97	C₄⋊C₁₆	—	3	1	no	—
98	C₄⋊C₁₆	—	9	1	no	—
99	C₂²⋊C₁₆	—	9	1	no	—
100	C₂²⋊C₁₆	—	9	1	no	—
101	#346	—	3	1	no	—
102	#346	—	9	1	no	—
103	C₂×C₄×D₃	—	3	1	no	—
104	C₂×C₄×D₃	—	3	1	no	—
105	C₂×C₄×D₃	—	3	1	no	—
106	C₂×C₄×D₃	—	3	1	no	—
107	C₂×C₄×D₃	—	3	1	no	—
108	C₂×C₄×D₃	—	3	1	no	—
109	C₂×C₄×D₃	—	6	1	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	3	1	no	—
112	C₂×C₃×C₈	—	3	1	no	—
113	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₃×C₄²	—	3	1	no	—
116	C₃×C₄²	—	3	1	no	—
117	C₄×Dic₃	—	3	1	no	—
118	C₄×Dic₃	—	3	1	no	—
119	C₄×Dic₃	—	3	1	no	—
120	C₄×Dic₃	—	3	1	no	—
121	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
122	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
123	C₂×C₃⋊C₈	—	3	1	no	—
124	C₂×C₃⋊C₈	—	3	1	no	—
125	C₂×C₃⋊C₈	—	6	1	no	—
126	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
127	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
128	D₃×C₈	—	6	2	no	—
129	D₃×C₈	—	6	1	no	—
130	D₃×C₈	—	6	1	no	—
131	D₃×C₈	—	6	1	no	—
132	D₃×C₈	—	6	1	no	—
133	D₃×C₈	—	6	2	no	—
134	D₃×C₈	—	12	1	no	—
135	D₃×C₈	—	12	1	no	—
136	C₃×C₁₆	—	2	1	no	—
137	C₃×C₁₆	—	2	1	no	—
138	C₃×C₁₆	—	4	1	no	—
139	C₃×C₁₆	—	6	1	no	—
140	C₃×C₁₆	—	6	1	no	—
141	C₃⋊C₁₆	—	2	1	no	—
142	C₃⋊C₁₆	—	2	1	no	—
143	C₃⋊C₁₆	—	6	1	no	—
144	C₃⋊C₁₆	—	6	1	no	—
145	C₃⋊C₁₆	—	12	1	no	—
146	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₃⋊D₃	—	2	1	no	—
148	C₂×C₃⋊D₃	—	2	1	no	—
149	C₂×C₃⋊D₃	M₃₂	1	2	yes	—
150	C₃²×C₄	#117	1	1	yes	characteristic
151	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
152	C₃⋊Dic₃	—	2	1	no	—
153	C₃×Dic₃	—	2	1	no	—
154	C₃×Dic₃	—	2	1	no	—
155	C₃×Dic₃	—	2	1	no	—
156	C₃×Dic₃	M₃₂	1	2	yes	—
157	C₂²×C₈	—	9	1	no	—
158	C₂²×C₈	—	9	1	no	—
159	C₂×C₄²	—	9	1	no	—
160	C₂×C₁₆	—	3	1	no	—
161	C₂×C₁₆	—	3	1	no	—
162	C₂×C₁₆	—	9	1	no	—
163	C₂×C₁₆	D₃²	1	1	yes	characteristic
164	C₄×C₈	—	3	1	no	—
165	C₄×C₈	—	3	1	no	—
166	C₄×C₈	—	9	1	no	—
167	C₄×C₈	—	9	1	no	—
168	C₂²×D₃	—	3	1	no	—
169	C₂²×D₃	—	3	1	no	—
170	C₂²×D₃	—	6	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	3	1	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	3	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×Dic₃	—	3	1	no	—
179	C₂×Dic₃	—	3	1	no	—
180	C₂×Dic₃	—	6	1	no	—
181	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
182	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
183	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
184	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
185	C₄×D₃	—	6	2	no	—
186	C₄×D₃	—	6	2	no	—
187	C₄×D₃	—	6	2	no	—
188	C₄×D₃	—	6	1	no	—
189	C₄×D₃	—	6	1	no	—
190	C₄×D₃	—	6	1	no	—
191	C₄×D₃	—	6	1	no	—
192	C₄×D₃	—	6	2	no	—
193	C₄×D₃	—	12	1	no	—
194	C₄×D₃	—	12	1	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	1	no	—
197	C₃×C₈	—	3	2	no	—
198	C₃×C₈	—	6	1	no	—
199	C₃×C₈	#235	1	1	yes	characteristic
200	C₃×C₈	#235	1	1	yes	characteristic
201	C₃×C₈	#237	1	1	yes	characteristic
202	C₃×C₈	#239	1	1	yes	characteristic
203	C₃⋊C₈	—	2	1	no	—
204	C₃⋊C₈	—	6	1	no	—
205	C₃⋊C₈	—	6	1	no	—
206	C₃⋊C₈	—	12	1	no	—
207	C₃⋊C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
208	C₂×C₃²	#431	1	1	yes	characteristic
209	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
210	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
211	C₃⋊D₃	—	2	2	no	—
212	C₂²×C₄	—	9	1	no	—
213	C₂²×C₄	—	9	1	no	—
214	C₂²×C₄	—	9	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	9	1	no	—
218	C₂×C₈	—	18	2	no	—
219	C₂×C₈	—	18	1	no	—
220	C₂×C₈	—	18	1	no	—
221	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
222	C₄²	—	3	1	no	—
223	C₄²	—	3	1	no	—
224	C₄²	—	9	1	no	—
225	C₄²	—	9	1	no	—
226	C₁₆	—	2	1	no	—
227	C₁₆	—	6	1	no	—
228	C₁₆	—	6	1	no	—
229	C₁₆	—	18	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
232	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
233	C₂×D₃	—	3	2	no	—
234	C₂×D₃	—	3	2	no	—
235	C₂×D₃	—	6	2	no	—
236	C₂×D₃	—	6	1	no	—
237	C₂×D₃	—	6	1	no	—
238	C₂×D₃	—	6	1	no	—
239	C₂×D₃	—	6	1	no	—
240	C₂×D₃	—	12	1	no	—
241	C₂×D₃	—	12	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	3	2	no	—
245	C₃×C₄	—	6	1	no	—
246	C₃×C₄	—	6	1	no	—
247	C₃×C₄	—	6	1	no	—
248	C₃×C₄	#872	1	1	yes	characteristic
249	C₃×C₄	#892	1	1	yes	characteristic
250	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
251	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
252	Dic₃	—	2	1	no	—
253	Dic₃	—	2	1	no	—
254	Dic₃	—	2	1	no	—
255	Dic₃	—	6	1	no	—
256	Dic₃	—	6	1	no	—
257	Dic₃	—	12	1	no	—
258	Dic₃	#798	1	2	yes	—
259	C₃²	#2163	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	9	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	1	no	—
263	C₂×C₄	—	3	1	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	9	1	no	—
266	C₂×C₄	—	18	2	no	—
267	C₂×C₄	—	18	2	no	—
268	C₂×C₄	—	18	1	no	—
269	C₂×C₄	—	18	1	no	—
270	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
271	C₈	—	3	2	no	—
272	C₈	—	6	1	no	—
273	C₈	—	18	1	no	—
274	C₈	#3799	1	1	yes	characteristic
275	C₈	#3804	1	1	yes	characteristic
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	#4685	1	1	yes	characteristic
280	C₂×C₃	#4833	1	1	yes	characteristic
281	C₂×C₃	C₂×#798	1	1	yes	characteristic
282	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
283	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
284	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
285	D₃	—	6	2	no	—
286	D₃	—	6	2	no	—
287	D₃	—	12	2	no	—
288	C₂²	—	9	2	no	—
289	C₂²	—	18	1	no	—
290	C₂²	—	18	1	no	—
291	C₂²	D₃²×C₈	1	1	yes	characteristic
292	C₄	—	3	2	no	—
293	C₄	—	6	1	no	—
294	C₄	—	6	1	no	—
295	C₄	—	6	1	no	—
296	C₄	—	18	1	no	—
297	C₄	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
298	C₄	#63836	1	1	yes	characteristic
299	C₃	—	2	1	no	—
300	C₃	#69073	1	1	yes	characteristic, complemented
301	C₃	#69275	1	1	yes	characteristic, complemented
302	C₂	—	18	2	no	—
303	C₂	D₃×#798	1	1	yes	characteristic
304	C₂	#10590226	1	1	yes	characteristic
305	C₂	#10590643	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₁₆	C₂²
prime radical	C₂²×C₃⋊D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₁₆	D₃²
hypocenter	C₃²	#2163
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×#4502	C₂	semidirect
4. C₃×Dic₃⋊C₁₆	C₂	semidirect
5. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
7. #10589103	C₂	semidirect
31. #4502	D₃	semidirect
33. Dic₃⋊C₁₆	D₃	semidirect
259. C₃²	#2163	semidirect
300. C₃	#69073	semidirect
301. C₃	#69275	semidirect

Maximal subgroups

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

subgroup	index
C₈×#624	2
#10589373	2
C₃×#4502	2
C₃×Dic₃⋊C₁₆	2
Dic₃⋊(C₃⋊C₁₆)	2
#10589108	2
#10589103	2
#69275	3
#69073	3