Subgroups — #11773577

Order 1152.

457 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4685	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#10590261	C₂	1	1	yes	maximal, characteristic
4	#10589320	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589079	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈²	C₂	1	1	yes	maximal, characteristic, complemented
8	#70740	—	3	1	no	maximal
9	#68047	—	3	1	no	maximal
10	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
11	C₂×D₃×C₃⋊C₈	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₃×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
16	C₃×Dic₃⋊C₈	C₄	1	2	yes	—
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	Dic₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
21	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	2	yes	—
22	Dic₃×C₃⋊C₈	C₄	1	2	yes	—
23	C₃×#431	—	3	1	no	—
24	#4925	—	3	1	no	—
25	C₂×C₄×C₃⋊C₈	—	3	1	no	—
26	#4685	—	3	1	no	—
27	#4685	D₃	1	1	yes	characteristic
28	C₄×D₃×C₈	—	3	1	no	—
29	C₃×C₈²	—	3	1	no	—
30	#4454	—	3	1	no	—
31	#4454	—	3	1	no	—
32	#4454	—	3	1	no	—
33	#4454	—	3	1	no	—
34	#4454	—	6	1	no	—
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×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×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	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
44	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
45	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
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	D₃×C₃⋊C₈	C₈	1	8	yes	complemented
52	#1450	—	9	1	no	Sylow
53	C₂×C₃×C₄²	—	3	1	no	—
54	C₂²×C₃⋊C₈	—	3	2	no	—
55	C₂×D₃×C₈	—	3	2	no	—
56	C₄²×D₃	C₂×D₃	1	1	yes	characteristic
57	C₃×C₄⋊C₈	—	3	2	no	—
58	C₃×C₂²⋊C₈	—	3	2	no	—
59	C₃×C₄×C₈	—	2	1	no	—
60	C₃×C₄×C₈	—	3	1	no	—
61	C₃×C₄×C₈	—	3	1	no	—
62	C₃×C₄×C₈	—	3	1	no	—
63	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
64	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
65	#830	—	3	2	no	—
66	(C₂×D₃)⋊C₈	—	3	2	no	—
67	(C₂×D₃)⋊C₈	Dic₃	1	2	yes	—
68	Dic₃⋊C₈	—	3	2	no	—
69	Dic₃⋊C₈	Dic₃	1	2	yes	—
70	C₈×Dic₃	—	3	2	no	—
71	C₄⋊(C₃⋊C₈)	—	3	2	no	—
72	C₄×C₃⋊C₈	—	3	1	no	—
73	C₄×C₃⋊C₈	—	3	1	no	—
74	C₄×C₃⋊C₈	—	3	1	no	—
75	C₄×C₃⋊C₈	—	3	1	no	—
76	C₄×C₃⋊C₈	—	3	2	no	—
77	C₄×C₃⋊C₈	—	6	1	no	—
78	C₄×C₃⋊C₈	—	6	1	no	—
79	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
80	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
81	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₃×D₃	C₄²	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
89	C₃×C₄×D₃	—	2	2	no	—
90	C₃×C₄×D₃	—	2	2	no	—
91	C₃×C₄×D₃	C₂×C₈	1	4	yes	—
92	C₃²×C₈	—	2	2	no	—
93	C₃⋊(C₃⋊C₈)	—	2	2	no	—
94	C₃⋊(C₃⋊C₈)	C₂×C₈	1	4	yes	—
95	C₃×C₃⋊C₈	—	2	2	no	—
96	C₃×C₃⋊C₈	—	2	2	no	—
97	C₃×C₃⋊C₈	C₂×C₈	1	4	yes	—
98	#431	—	3	1	no	—
99	#431	—	9	1	no	—
100	C₂×C₄×C₈	—	9	1	no	—
101	#322	—	9	1	no	—
102	#322	—	9	1	no	—
103	C₈²	—	3	1	no	—
104	C₈²	—	9	1	no	—
105	C₂²×C₃×C₄	—	3	1	no	—
106	C₂²×C₃×C₄	—	3	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
109	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
110	C₂×C₄×D₃	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	—	3	2	no	—
114	C₂×C₃×C₈	—	3	2	no	—
115	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
116	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	3	1	no	—
119	C₃×C₄²	—	3	1	no	—
120	C₃×C₄²	—	3	1	no	—
121	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
122	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
123	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
124	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
125	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
126	C₂×C₃⋊C₈	—	3	2	no	—
127	C₂×C₃⋊C₈	—	3	2	no	—
128	C₂×C₃⋊C₈	—	3	8	no	—
129	C₂×C₃⋊C₈	—	3	2	no	—
130	C₂×C₃⋊C₈	—	3	2	no	—
131	C₂×C₃⋊C₈	—	6	2	no	—
132	C₂×C₃⋊C₈	—	6	2	no	—
133	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
134	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
135	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
136	D₃×C₈	—	3	8	no	—
137	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₃×D₃	—	2	1	no	—
139	C₂×C₃×D₃	—	2	1	no	—
140	C₂×C₃×D₃	C₄×C₈	1	2	yes	—
141	C₃²×C₄	C₂²×C₈	1	2	yes	—
142	C₃²×C₄	#131	1	2	yes	—
143	C₃²×C₄	#131	1	2	yes	—
144	C₃×Dic₃	—	2	1	no	—
145	C₃×Dic₃	—	2	1	no	—
146	C₃×Dic₃	C₄×C₈	1	2	yes	—
147	C₂²×C₈	—	9	2	no	—
148	C₂×C₄²	—	3	1	no	—
149	C₄⋊C₈	—	3	2	no	—
150	C₄⋊C₈	—	9	2	no	—
151	C₂²⋊C₈	—	3	2	no	—
152	C₂²⋊C₈	—	9	2	no	—
153	C₄×C₈	—	3	1	no	—
154	C₄×C₈	—	3	1	no	—
155	C₄×C₈	—	3	1	no	—
156	C₄×C₈	—	9	2	no	—
157	C₄×C₈	—	9	1	no	—
158	C₄×C₈	—	9	1	no	—
159	C₄×C₈	D₃²	1	1	yes	characteristic
160	C₂³×C₃	—	3	1	no	—
161	C₂²×D₃	C₄×Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	3	1	no	—
166	C₂×C₃×C₄	—	3	4	no	—
167	C₂×C₃×C₄	—	3	1	no	—
168	C₂×C₃×C₄	—	3	1	no	—
169	C₂×C₃×C₄	—	6	2	no	—
170	C₂×C₃×C₄	—	6	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×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
177	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
178	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
179	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
180	C₄×D₃	—	2	2	no	—
181	C₄×D₃	—	2	2	no	—
182	C₄×D₃	D₃×C₈	1	4	yes	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	—	3	4	no	—
186	C₃×C₈	—	4	2	no	—
187	C₃×C₈	—	6	2	no	—
188	C₃×C₈	—	6	2	no	—
189	C₃⋊C₈	—	2	2	no	—
190	C₃⋊C₈	—	2	2	no	—
191	C₃⋊C₈	—	3	4	no	—
192	C₃⋊C₈	—	3	4	no	—
193	C₃⋊C₈	—	6	4	no	—
194	C₃⋊C₈	—	6	2	no	—
195	C₃⋊C₈	—	6	2	no	—
196	C₃⋊C₈	—	12	2	no	—
197	C₃⋊C₈	D₃×C₈	1	4	yes	—
198	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
199	C₂×C₃²	#405	1	1	yes	characteristic
200	C₂×C₃²	#405	1	1	yes	characteristic
201	C₃×D₃	—	2	2	no	—
202	C₂²×C₄	—	3	1	no	—
203	C₂²×C₄	—	3	1	no	—
204	C₂²×C₄	—	3	1	no	—
205	C₂×C₈	—	3	2	no	—
206	C₂×C₈	—	3	2	no	—
207	C₂×C₈	—	3	2	no	—
208	C₂×C₈	—	9	8	no	—
209	C₂×C₈	—	9	2	no	—
210	C₂×C₈	—	9	2	no	—
211	C₂×C₈	D₃×Dic₃	1	2	yes	—
212	C₄²	—	3	1	no	—
213	C₄²	—	3	1	no	—
214	C₄²	—	3	1	no	—
215	C₄²	C₂×D₃²	1	1	yes	characteristic
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	3	2	no	—
218	C₂²×C₃	—	6	1	no	—
219	C₂²×C₃	—	6	1	no	—
220	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
221	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
222	C₂×D₃	—	2	1	no	—
223	C₂×D₃	—	2	1	no	—
224	C₂×D₃	C₈×Dic₃	1	2	yes	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	3	2	no	—
229	C₃×C₄	—	6	1	no	—
230	C₃×C₄	—	6	1	no	—
231	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
232	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
233	C₃×C₄	#901	1	2	yes	—
234	C₃×C₄	#901	1	2	yes	—
235	C₃×C₄	#907	1	2	yes	—
236	C₃×C₄	#948	1	2	yes	—
237	Dic₃	—	2	1	no	—
238	Dic₃	—	2	1	no	—
239	Dic₃	C₈×Dic₃	1	2	yes	—
240	C₃²	#1450	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	3	1	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	4	no	—
244	C₂×C₄	—	3	1	no	—
245	C₂×C₄	—	3	1	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	—	6	2	no	—
248	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
249	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
250	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	3	4	no	—
253	C₈	—	6	2	no	—
254	C₈	—	6	2	no	—
255	C₈	—	9	4	no	—
256	C₈	—	18	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	6	2	no	—
261	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
262	C₂×C₃	#4689	1	1	yes	characteristic
263	C₂×C₃	#4689	1	1	yes	characteristic
264	C₂×C₃	#4997	1	1	yes	characteristic
265	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
266	C₂×C₃	#5122	1	1	yes	characteristic
267	D₃	—	2	2	no	—
268	C₂²	—	3	2	no	—
269	C₂²	—	6	1	no	—
270	C₂²	—	6	1	no	—
271	C₂²	C₄×D₃×Dic₃	1	1	yes	characteristic
272	C₄	—	3	2	no	—
273	C₄	—	6	1	no	—
274	C₄	—	6	1	no	—
275	C₄	D₃²×C₈	1	2	yes	—
276	C₄	#63804	1	2	yes	—
277	C₄	#63814	1	2	yes	—
278	C₃	—	2	1	no	—
279	C₃	#68047	1	1	yes	characteristic, complemented
280	C₃	#70740	1	1	yes	characteristic, complemented
281	C₂	—	6	2	no	—
282	C₂	#10590318	1	1	yes	characteristic
283	C₂	D₃×C₈×Dic₃	1	1	yes	characteristic
284	C₂	#10590908	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	#1450
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. #10589320	C₂	semidirect
5. C₃×C₈×C₃⋊C₈	C₂	semidirect
6. #10589079	C₂	semidirect
7. C₃⋊C₈²	C₂	semidirect
37. C₈×C₃⋊C₈	D₃	semidirect
51. D₃×C₃⋊C₈	C₈	semidirect
240. C₃²	#1450	semidirect
279. C₃	#68047	semidirect
280. C₃	#70740	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4685	2
C₄×D₃×C₃⋊C₈	2
#10590261	2
#10589320	2
C₃×C₈×C₃⋊C₈	2
#10589079	2
C₃⋊C₈²	2
#70740	3
#68047	3