Subgroups — #11795861

Order 1152.

437 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63625	C₂	1	1	yes	maximal, characteristic
2	C₈×#624	C₂	1	1	yes	maximal, characteristic
3	C₂×#63544	C₂	1	1	yes	maximal, characteristic
4	C₃×#4502	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589103	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#69071	—	3	1	no	maximal
9	C₄×#798	—	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	#63625	C₄	1	4	yes	complemented
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
21	Dic₃×C₃⋊C₈	C₄	1	1	yes	characteristic
22	#63544	C₄	1	4	yes	complemented
23	C₂×#798	—	3	1	no	—
24	C₂×#798	—	3	1	no	—
25	C₂×#798	—	3	1	no	—
26	C₂×#798	—	3	1	no	—
27	C₂×#798	—	6	1	no	—
28	C₄×D₃×C₈	—	3	1	no	—
29	C₄×D₃×C₈	—	3	1	no	—
30	C₃×#346	—	3	1	no	—
31	C₃×C₄×C₁₆	—	3	1	no	—
32	#4502	—	3	1	no	—
33	#4502	D₃	1	1	yes	characteristic, complemented
34	Dic₃×C₁₆	—	3	1	no	—
35	Dic₃×C₁₆	D₃	1	1	yes	characteristic, complemented
36	#4461	—	3	1	no	—
37	C₄×C₃⋊C₁₆	—	3	1	no	—
38	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×#624	C₈	1	2	yes	—
40	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
43	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
44	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	Dic₃²	C₈	1	2	yes	—
49	#3710	C₈	1	4	yes	—
50	C₃²×C₁₆	C₂×C₄	1	2	yes	complemented
51	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	complemented
52	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
53	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
54	C₄×M₃₂	—	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	#798	—	3	4	no	—
81	#798	—	3	4	no	—
82	#798	—	3	4	no	—
83	#798	—	3	4	no	—
84	#798	—	6	4	no	—
85	C₂²×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
88	C₄×C₃⋊D₃	C₄²	1	2	yes	—
89	C₄×C₃⋊D₃	C₂×C₈	1	2	yes	—
90	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
91	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
92	#624	—	2	4	no	—
93	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₃⋊(C₃⋊C₈)	C₄²	1	2	yes	—
96	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	—
97	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	—
98	C₂×M₃₂	—	9	1	no	—
99	C₂×M₃₂	—	9	1	no	—
100	C₂×C₄×C₈	—	9	1	no	—
101	#346	—	3	1	no	—
102	#346	—	9	1	no	—
103	C₄×C₁₆	—	3	1	no	—
104	C₄×C₁₆	—	9	1	no	—
105	C₂×C₄×D₃	—	3	1	no	—
106	C₂×C₄×D₃	—	3	1	no	—
107	C₂×C₄×D₃	—	3	2	no	—
108	C₂×C₄×D₃	—	3	2	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	2	no	—
118	C₄×Dic₃	—	3	2	no	—
119	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
120	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
121	C₂×C₃⋊C₈	—	3	1	no	—
122	C₂×C₃⋊C₈	—	3	1	no	—
123	C₂×C₃⋊C₈	—	6	1	no	—
124	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
125	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
126	D₃×C₈	—	3	2	no	—
127	D₃×C₈	—	3	2	no	—
128	D₃×C₈	—	3	2	no	—
129	D₃×C₈	—	3	2	no	—
130	D₃×C₈	—	3	4	no	—
131	D₃×C₈	—	3	4	no	—
132	D₃×C₈	—	6	2	no	—
133	D₃×C₈	—	6	2	no	—
134	C₃×C₁₆	—	2	2	no	—
135	C₃×C₁₆	—	3	2	no	—
136	C₃×C₁₆	—	3	2	no	—
137	C₃×C₁₆	C₄×D₃	1	2	yes	complemented
138	C₃×C₁₆	C₄×D₃	1	2	yes	complemented
139	C₃⋊C₁₆	—	3	2	no	—
140	C₃⋊C₁₆	—	3	2	no	—
141	C₃⋊C₁₆	—	6	2	no	—
142	C₃⋊C₁₆	C₄×D₃	1	2	yes	complemented
143	C₃⋊C₁₆	C₄×D₃	1	2	yes	complemented
144	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₃⋊D₃	—	2	2	no	—
146	C₂×C₃⋊D₃	C₄×C₈	1	2	yes	—
147	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
148	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
149	C₃⋊Dic₃	C₄×C₈	1	2	yes	—
150	C₃×Dic₃	—	2	2	no	—
151	C₃×Dic₃	M₃₂	1	4	yes	complemented
152	C₂²×C₈	—	9	1	no	—
153	C₂²×C₈	—	9	1	no	—
154	C₂×C₄²	—	9	1	no	—
155	M₃₂	—	9	4	no	—
156	M₃₂	—	9	4	no	—
157	C₂×C₁₆	—	3	1	no	—
158	C₂×C₁₆	—	3	1	no	—
159	C₂×C₁₆	—	9	1	no	—
160	C₂×C₁₆	D₃²	1	1	yes	characteristic
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	9	1	no	—
164	C₄×C₈	—	9	1	no	—
165	C₂²×D₃	—	3	1	no	—
166	C₂²×D₃	—	3	1	no	—
167	C₂²×D₃	—	6	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	3	2	no	—
170	C₂×C₃×C₄	—	3	2	no	—
171	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×Dic₃	—	3	1	no	—
174	C₂×Dic₃	—	3	1	no	—
175	C₂×Dic₃	—	6	1	no	—
176	C₂×Dic₃	D₃×C₈	1	2	yes	—
177	C₂×Dic₃	D₃×C₈	1	2	yes	—
178	C₄×D₃	—	3	2	no	—
179	C₄×D₃	—	3	2	no	—
180	C₄×D₃	—	3	2	no	—
181	C₄×D₃	—	3	2	no	—
182	C₄×D₃	—	6	2	no	—
183	C₄×D₃	—	6	2	no	—
184	C₄×D₃	—	6	4	no	—
185	C₄×D₃	—	6	4	no	—
186	C₃×C₈	—	2	1	no	—
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	3	2	no	—
189	C₃×C₈	—	3	2	no	—
190	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
191	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
192	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
193	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
194	C₃⋊C₈	—	3	2	no	—
195	C₃⋊C₈	—	3	2	no	—
196	C₃⋊C₈	—	6	2	no	—
197	C₃⋊C₈	D₃×C₈	1	2	yes	—
198	C₃⋊C₈	D₃×C₈	1	2	yes	—
199	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
200	C₂×C₃²	C₂×M₃₂	1	2	yes	—
201	C₃⋊D₃	—	2	2	no	—
202	C₂²×C₄	—	9	2	no	—
203	C₂²×C₄	—	9	1	no	—
204	C₂×C₈	—	3	1	no	—
205	C₂×C₈	—	3	1	no	—
206	C₂×C₈	—	9	4	no	—
207	C₂×C₈	—	9	1	no	—
208	C₂×C₈	—	9	2	no	—
209	C₂×C₈	—	9	2	no	—
210	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
211	C₄²	—	3	1	no	—
212	C₄²	—	3	1	no	—
213	C₄²	—	9	2	no	—
214	C₁₆	—	3	2	no	—
215	C₁₆	—	3	2	no	—
216	C₁₆	—	9	2	no	—
217	C₁₆	#624	1	2	yes	complemented
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
220	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
221	C₂×D₃	—	3	2	no	—
222	C₂×D₃	—	3	2	no	—
223	C₂×D₃	—	6	2	no	—
224	C₂×D₃	—	6	2	no	—
225	C₂×D₃	—	6	2	no	—
226	C₂×D₃	—	12	2	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	3	4	no	—
230	C₃×C₄	—	6	2	no	—
231	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
232	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
233	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
234	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
235	Dic₃	—	2	2	no	—
236	Dic₃	—	3	2	no	—
237	Dic₃	—	3	2	no	—
238	Dic₃	—	6	2	no	—
239	Dic₃	#798	1	4	yes	complemented
240	C₃²	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	9	1	no	—
242	C₂×C₄	—	3	2	no	—
243	C₂×C₄	—	3	2	no	—
244	C₂×C₄	—	9	1	no	—
245	C₂×C₄	—	9	2	no	—
246	C₂×C₄	—	9	2	no	—
247	C₂×C₄	—	18	4	no	—
248	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
249	C₈	—	3	2	no	—
250	C₈	—	3	2	no	—
251	C₈	—	9	2	no	—
252	C₈	C₄×D₃²	1	1	yes	characteristic
253	C₈	C₂×#624	1	1	yes	characteristic
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
257	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
258	C₂×C₃	C₂×#798	1	2	yes	—
259	C₂×C₃	D₃×M₃₂	1	2	yes	—
260	D₃	—	6	2	no	—
261	D₃	—	6	2	no	—
262	D₃	—	12	2	no	—
263	C₂²	—	9	2	no	—
264	C₂²	—	18	2	no	—
265	C₂²	D₃²×C₈	1	1	yes	characteristic
266	C₄	—	3	4	no	—
267	C₄	—	6	2	no	—
268	C₄	—	9	2	no	—
269	C₄	D₃²×C₈	1	1	yes	characteristic
270	C₄	C₄×#624	1	1	yes	characteristic
271	C₃	—	2	1	no	—
272	C₃	C₄×#798	1	1	yes	characteristic, complemented
273	C₃	#69071	1	1	yes	characteristic, complemented
274	C₂	—	18	2	no	—
275	C₂	D₃×#798	1	2	yes	—
276	C₂	C₈×#624	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₂×C₃²×C₁₆	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂²×C₃⋊D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₁₆	D₃²
hypocenter	C₃²	C₄×M₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×#4502	C₂	semidirect
5. C₃×Dic₃×C₁₆	C₂	semidirect
6. #10589103	C₂	semidirect
7. Dic₃×C₃⋊C₁₆	C₂	semidirect
15. #63625	C₄	semidirect
22. #63544	C₄	semidirect
33. #4502	D₃	semidirect
35. Dic₃×C₁₆	D₃	semidirect
50. C₃²×C₁₆	C₂×C₄	semidirect
51. C₃⋊(C₃⋊C₁₆)	C₂×C₄	semidirect
52. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
53. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
137. C₃×C₁₆	C₄×D₃	semidirect
138. C₃×C₁₆	C₄×D₃	semidirect
142. C₃⋊C₁₆	C₄×D₃	semidirect
143. C₃⋊C₁₆	C₄×D₃	semidirect
151. C₃×Dic₃	M₃₂	semidirect
217. C₁₆	#624	semidirect
239. Dic₃	#798	semidirect
240. C₃²	C₄×M₃₂	semidirect
272. C₃	C₄×#798	semidirect
273. C₃	#69071	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63625	2
C₈×#624	2
C₂×#63544	2
C₃×#4502	2
C₃×Dic₃×C₁₆	2
#10589103	2
Dic₃×C₃⋊C₁₆	2
#69071	3
C₄×#798	3