Subgroups — #11772356

Order 1152.

321 conjugacy classes of subgroups.

The subgroup lattice has 265 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₃×#5101	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	#67815	—	3	1	no	maximal
5	#67671	—	3	1	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₃×#926	C₄	1	1	yes	characteristic
8	C₂×C₃×#208	C₄	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
10	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
11	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
12	C₃×#815	—	2	1	no	—
13	C₃×C₈×Dic₃	—	2	1	no	—
14	C₃×#812	—	2	1	no	—
15	C₃×#812	C₄	1	2	yes	complemented
16	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
17	C₂×C₃×M₃₂	—	3	1	no	—
18	C₃×#405	—	3	1	no	—
19	#5101	D₃	1	1	yes	characteristic
20	C₂×#812	D₃	1	1	yes	characteristic
21	C₂²×C₃⋊C₁₆	—	3	1	no	—
22	C₂²×C₃⋊C₁₆	—	3	1	no	—
23	C₂²×C₃⋊C₁₆	—	6	1	no	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₃×#217	—	2	1	no	—
32	C₃×C₄⋊Dic₃	C₈	1	2	yes	—
33	C₃×C₄×Dic₃	—	2	1	no	—
34	C₃×#208	—	2	1	no	—
35	C₃×#208	C₈	1	2	yes	—
36	C₂×C₃×C₃⋊C₈	—	2	1	no	—
37	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
38	C₃×C₃⋊C₁₆	—	2	2	no	—
39	#1379	—	9	1	no	Sylow
40	C₂×C₃×M₁₆	—	3	1	no	—
41	C₂²×C₃×C₈	—	2	1	no	—
42	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₃×#117	—	3	1	no	—
45	#926	Dic₃	1	1	yes	characteristic
46	C₂×#208	Dic₃	1	1	yes	characteristic
47	C₃×M₃₂	—	3	2	no	—
48	C₃×M₃₂	—	6	1	no	—
49	C₂×C₃×C₁₆	—	6	1	no	—
50	C₃×#97	—	6	1	no	—
51	C₃×C₄×C₈	—	6	1	no	—
52	#815	—	2	1	no	—
53	C₈×Dic₃	—	2	1	no	—
54	#812	—	2	1	no	—
55	#812	Dic₃	1	2	yes	complemented
56	C₂×C₃⋊C₁₆	—	2	1	no	—
57	C₂×C₃⋊C₁₆	—	3	2	no	—
58	C₂×C₃⋊C₁₆	—	3	2	no	—
59	C₂×C₃⋊C₁₆	—	6	2	no	—
60	C₂×C₃⋊C₁₆	—	6	2	no	—
61	C₂×C₃⋊C₁₆	—	6	2	no	—
62	C₂×C₃⋊C₁₆	—	6	1	no	—
63	C₂×C₃⋊C₁₆	—	6	1	no	—
64	C₂×C₃⋊C₁₆	—	6	1	no	—
65	C₂×C₃⋊C₁₆	—	6	1	no	—
66	C₂³×C₃²	M₁₆	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
73	C₂×C₃×Dic₃	—	2	2	no	—
74	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₃×C₃⋊C₈	—	2	2	no	—
79	C₂×M₃₂	—	3	1	no	—
80	C₂²×C₁₆	—	9	1	no	—
81	#405	—	3	1	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₃×M₁₆	—	3	2	no	—
86	C₃×M₁₆	—	6	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	6	1	no	—
94	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
95	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
96	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
100	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₃×C₈	#67	1	1	yes	characteristic
103	C₂×C₃×C₈	#67	1	1	yes	characteristic
104	C₂×C₃×C₈	#67	1	1	yes	characteristic
105	C₂×C₃×C₈	#67	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	3	2	no	—
107	C₃×C₂²⋊C₄	—	6	1	no	—
108	C₃×C₄²	—	6	1	no	—
109	#217	—	2	1	no	—
110	C₄⋊Dic₃	C₃⋊C₈	1	2	yes	—
111	C₄×Dic₃	—	2	1	no	—
112	#208	—	2	1	no	—
113	#208	C₃⋊C₈	1	2	yes	—
114	C₂×C₃⋊C₈	—	2	1	no	—
115	C₃×C₁₆	—	6	2	no	—
116	C₃⋊C₁₆	—	2	2	no	—
117	C₃⋊C₁₆	—	6	4	no	—
118	C₃⋊C₁₆	—	6	2	no	—
119	C₃⋊C₁₆	—	6	2	no	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	#97	1	1	yes	characteristic
122	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
124	C₃²×C₄	#95	1	1	yes	characteristic
125	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
126	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
127	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
128	C₃×Dic₃	—	4	2	no	—
129	C₂×M₁₆	—	3	1	no	—
130	C₂²×C₈	D₃²	1	1	yes	characteristic
131	#117	—	3	1	no	—
132	M₃₂	—	3	2	no	—
133	M₃₂	—	6	1	no	—
134	C₂×C₁₆	—	6	1	no	—
135	C₂×C₁₆	—	9	2	no	—
136	C₂×C₁₆	—	18	1	no	—
137	C₂×C₁₆	—	18	1	no	—
138	#97	—	6	1	no	—
139	C₄×C₈	—	6	1	no	—
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
142	C₂³×C₃	#208	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	6	2	no	—
150	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
151	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
152	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
155	C₂×C₃×C₄	#208	1	1	yes	characteristic
156	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
158	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	#217	1	1	yes	characteristic
162	C₂×Dic₃	—	2	2	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	6	2	no	—
168	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
169	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
170	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
171	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
172	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₃×C₈	#217	1	1	yes	characteristic
176	C₃⋊C₈	—	2	2	no	—
177	C₂×C₃²	—	2	1	no	—
178	C₂×C₃²	—	2	1	no	—
179	C₂×C₃²	#336	1	1	yes	characteristic
180	C₂×C₃²	#350	1	2	yes	—
181	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
182	M₁₆	—	3	2	no	—
183	M₁₆	—	6	1	no	—
184	C₂×C₈	—	6	1	no	—
185	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
186	C₂×C₈	#624	1	1	yes	characteristic
187	C₂×C₈	#625	1	1	yes	characteristic
188	C₂×C₈	#626	1	1	yes	characteristic
189	C₂×C₈	#626	1	1	yes	characteristic
190	C₂×C₈	#627	1	1	yes	characteristic
191	C₄⋊C₄	—	3	2	no	—
192	C₂²⋊C₄	—	6	1	no	—
193	C₄²	—	6	1	no	—
194	C₁₆	—	6	2	no	—
195	C₁₆	—	18	2	no	—
196	C₂²×C₃	—	2	4	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	#803	1	1	yes	characteristic
203	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
204	C₂²×C₃	Dic₃⋊C₈	1	1	yes	characteristic
205	C₂²×C₃	#815	1	1	yes	characteristic
206	C₂²×C₃	(C₂×D₃)⋊C₈	1	1	yes	characteristic
207	C₂²×C₃	#830	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	12	2	no	—
213	C₃×C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
214	C₃×C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
215	C₃×C₄	C₈×Dic₃	1	1	yes	characteristic
216	C₃×C₄	Dic₃⋊C₈	1	1	yes	characteristic
217	C₃×C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
218	C₃×C₄	#830	1	1	yes	characteristic
219	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
220	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
221	Dic₃	—	4	2	no	—
222	C₃²	#1379	1	1	yes	characteristic, complemented, Sylow
223	C₂³	#3711	1	1	yes	characteristic
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
226	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
227	C₂×C₄	#3711	1	1	yes	characteristic
228	C₂×C₄	Dic₃²	1	1	yes	characteristic
229	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
230	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
231	C₈	—	6	2	no	—
232	C₈	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
233	C₈	#3722	1	1	yes	characteristic
234	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
235	C₈	#3724	1	1	yes	characteristic
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	2	2	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	#4515	1	2	yes	—
245	C₂×C₃	#4524	1	1	yes	characteristic
246	C₂×C₃	#4550	1	1	yes	characteristic
247	C₂×C₃	#4561	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	#63554	1	1	yes	characteristic
250	C₂²	(C₂×D₃)⋊(C₃⋊C₈)	1	1	yes	characteristic
251	C₂²	Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic
252	C₄	—	12	2	no	—
253	C₄	Dic₃×C₃⋊C₈	1	1	yes	characteristic
254	C₄	(C₂×D₃)⋊(C₃⋊C₈)	1	1	yes	characteristic
255	C₄	Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic
256	C₄	#63579	1	1	yes	characteristic
257	C₃	—	2	1	no	—
258	C₃	#67671	1	1	yes	characteristic, complemented
259	C₃	#67815	1	1	yes	characteristic, complemented
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#10589111	1	2	yes	—
263	C₂	#10589155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#625
commutator	C₃²×C₄	C₄×C₈
Frattini, hypercenter	C₂²×C₈	D₃²
Fitting, cosocle	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²⋊C₈
prime radical	C₂³×C₃²	M₁₆
hypocenter	C₃²	#1379
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×C₃⋊(C₃⋊C₁₆)	C₄	semidirect
15. C₃×#812	C₄	semidirect
55. #812	Dic₃	semidirect
222. C₃²	#1379	semidirect
258. C₃	#67671	semidirect
259. C₃	#67815	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₃×#5101	2
C₂×C₃×#812	2
#67815	3
#67671	3