Subgroups — #11772314

Order 1152.

325 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63639	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#815	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	#67812	—	3	1	no	maximal
5	#67664	—	3	1	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
9	#63639	—	2	1	no	—
10	#63639	C₄	1	2	yes	complemented
11	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
12	C₃×#815	—	2	1	no	—
13	C₃×#815	—	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	C₂×C₃×#97	—	3	1	no	—
19	C₂×#815	D₃	1	1	yes	characteristic
20	C₂×#812	—	3	1	no	—
21	C₂×#812	—	3	1	no	—
22	C₂×#812	—	6	1	no	—
23	C₂×#812	D₃	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×Dic₃	C₈	1	2	yes	—
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
32	C₃×C₄×Dic₃	—	2	1	no	—
33	C₃×C₄×Dic₃	—	2	1	no	—
34	C₂×C₃×C₃⋊C₈	—	2	1	no	—
35	C₂×C₃×C₃⋊C₈	—	2	1	no	—
36	C₂×C₃×C₃⋊C₈	C₈	1	2	yes	—
37	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
38	C₃×C₃⋊C₁₆	—	2	2	no	—
39	#1377	—	9	1	no	Sylow
40	C₂²×C₃×C₈	—	2	1	no	—
41	C₂²×C₃×C₈	—	3	1	no	—
42	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄²	—	3	1	no	—
45	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
46	C₂²×C₃⋊C₈	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×#97	—	6	1	no	—
52	#815	—	2	1	no	—
53	#815	—	2	1	no	—
54	#812	—	2	1	no	—
55	#812	—	3	2	no	—
56	#812	—	3	2	no	—
57	#812	—	6	2	no	—
58	#812	—	6	2	no	—
59	#812	—	6	1	no	—
60	#812	—	6	1	no	—
61	#812	Dic₃	1	2	yes	complemented
62	C₂×C₃⋊C₁₆	—	2	1	no	—
63	C₂×C₃⋊C₁₆	—	6	2	no	—
64	C₂×C₃⋊C₁₆	—	6	1	no	—
65	C₂×C₃⋊C₁₆	—	6	1	no	—
66	C₂³×C₃²	C₂×C₈	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	M₁₆	1	1	yes	characteristic
72	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
73	C₂×C₃×Dic₃	—	2	2	no	—
74	C₂×C₃×Dic₃	—	4	2	no	—
75	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₃×C₃⋊C₈	—	2	2	no	—
80	C₂×M₃₂	—	3	1	no	—
81	C₂×M₃₂	—	9	1	no	—
82	C₂×#97	—	3	1	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	—	3	2	no	—
85	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
87	C₂²×Dic₃	C₃⋊C₈	1	2	yes	—
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	—	2	1	no	—
94	C₂×C₃×C₈	—	3	2	no	—
95	C₂×C₃×C₈	—	6	1	no	—
96	C₂×C₃×C₈	—	6	1	no	—
97	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
98	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
99	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
101	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
103	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
105	C₂×C₃×C₈	#67	1	1	yes	characteristic
106	C₂×C₃×C₈	#67	1	1	yes	characteristic
107	C₂×C₃×C₈	#67	1	1	yes	characteristic
108	C₂×C₃×C₈	#67	1	1	yes	characteristic
109	C₃×C₄²	—	6	1	no	—
110	C₃×C₄²	—	6	1	no	—
111	C₄×Dic₃	—	2	1	no	—
112	C₄×Dic₃	—	2	1	no	—
113	C₂×C₃⋊C₈	—	2	1	no	—
114	C₂×C₃⋊C₈	—	2	1	no	—
115	C₂×C₃⋊C₈	C₃⋊C₈	1	2	yes	—
116	C₃×C₁₆	—	6	2	no	—
117	C₃⋊C₁₆	—	2	2	no	—
118	C₃⋊C₁₆	—	6	4	no	—
119	C₃⋊C₁₆	—	6	2	no	—
120	C₃⋊C₁₆	—	6	2	no	—
121	C₂²×C₃²	—	2	2	no	—
122	C₂²×C₃²	C₄×C₈	1	1	yes	characteristic
123	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
125	C₃²×C₄	#95	1	1	yes	characteristic
126	C₃²×C₄	#97	1	1	yes	characteristic
127	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
128	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
129	C₃×Dic₃	—	4	2	no	—
130	C₂²×C₈	—	3	1	no	—
131	C₂²×C₈	D₃²	1	1	yes	characteristic
132	C₂×C₄²	—	3	1	no	—
133	M₃₂	—	3	2	no	—
134	M₃₂	—	6	1	no	—
135	M₃₂	—	9	2	no	—
136	M₃₂	—	18	1	no	—
137	C₂×C₁₆	—	6	1	no	—
138	C₂×C₁₆	—	18	1	no	—
139	#97	—	6	1	no	—
140	#97	—	6	1	no	—
141	C₂³×C₃	—	2	1	no	—
142	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
143	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
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	—	2	1	no	—
150	C₂×C₃×C₄	—	6	2	no	—
151	C₂×C₃×C₄	—	12	2	no	—
152	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
153	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
154	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
156	C₂×C₃×C₄	#208	1	1	yes	characteristic
157	C₂×C₃×C₄	#208	1	1	yes	characteristic
158	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	#217	1	1	yes	characteristic
164	C₂×Dic₃	—	2	2	no	—
165	C₂×Dic₃	—	4	2	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	6	2	no	—
171	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
172	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
173	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
174	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
175	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₃×C₈	#217	1	1	yes	characteristic
179	C₃⋊C₈	—	2	2	no	—
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	—	2	1	no	—
182	C₂×C₃²	#336	1	1	yes	characteristic
183	C₂×C₃²	#349	1	2	yes	—
184	C₂²×C₄	—	3	2	no	—
185	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
186	C₂×C₈	—	3	2	no	—
187	C₂×C₈	—	6	1	no	—
188	C₂×C₈	—	6	1	no	—
189	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
190	C₂×C₈	#624	1	1	yes	characteristic
191	C₂×C₈	#625	1	1	yes	characteristic
192	C₂×C₈	#626	1	1	yes	characteristic
193	C₂×C₈	#626	1	1	yes	characteristic
194	C₂×C₈	#627	1	1	yes	characteristic
195	C₄²	—	6	1	no	—
196	C₄²	—	6	1	no	—
197	C₁₆	—	6	2	no	—
198	C₁₆	—	18	2	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	C₄×C₃⋊C₈	1	1	yes	characteristic
206	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
207	C₂²×C₃	C₈×Dic₃	1	1	yes	characteristic
208	C₂²×C₃	Dic₃⋊C₈	1	1	yes	characteristic
209	C₂²×C₃	(C₂×D₃)⋊C₈	1	1	yes	characteristic
210	C₂²×C₃	#830	1	1	yes	characteristic
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	12	2	no	—
216	C₃×C₄	#803	1	1	yes	characteristic
217	C₃×C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
218	C₃×C₄	Dic₃⋊C₈	1	1	yes	characteristic
219	C₃×C₄	#815	1	1	yes	characteristic
220	C₃×C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
221	C₃×C₄	#830	1	1	yes	characteristic
222	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
223	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
224	Dic₃	—	4	2	no	—
225	C₃²	#1377	1	1	yes	characteristic, complemented, Sylow
226	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	—	12	2	no	—
229	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
230	C₂×C₄	#3711	1	1	yes	characteristic
231	C₂×C₄	#3711	1	1	yes	characteristic
232	C₂×C₄	Dic₃²	1	1	yes	characteristic
233	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
234	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
235	C₈	—	6	2	no	—
236	C₈	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
237	C₈	#3722	1	1	yes	characteristic
238	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
239	C₈	#3724	1	1	yes	characteristic
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	2	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	4	1	no	—
247	C₂×C₃	—	4	1	no	—
248	C₂×C₃	#4514	1	2	yes	—
249	C₂×C₃	#4524	1	1	yes	characteristic
250	C₂×C₃	#4550	1	1	yes	characteristic
251	C₂×C₃	#4553	1	2	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	Dic₃×C₃⋊C₈	1	1	yes	characteristic
254	C₂²	(C₂×D₃)⋊(C₃⋊C₈)	1	1	yes	characteristic
255	C₂²	Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic
256	C₄	—	12	2	no	—
257	C₄	#63554	1	1	yes	characteristic
258	C₄	(C₂×D₃)⋊(C₃⋊C₈)	1	1	yes	characteristic
259	C₄	Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic
260	C₄	#63579	1	1	yes	characteristic
261	C₃	—	2	1	no	—
262	C₃	#67664	1	1	yes	characteristic, complemented
263	C₃	#67812	1	1	yes	characteristic, complemented
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#10589109	1	2	yes	—
267	C₂	#10589155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊Dic₃
commutator	C₂²×C₃²	C₄×C₈
Frattini, hypercenter	C₂²×C₈	D₃²
Fitting, cosocle	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²⋊C₈
prime radical	C₂³×C₃²	C₂×C₈
hypocenter	C₃²	#1377
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
10. #63639	C₄	semidirect
15. C₃×#812	C₄	semidirect
61. #812	Dic₃	semidirect
225. C₃²	#1377	semidirect
262. C₃	#67664	semidirect
263. C₃	#67812	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63639	2
C₂×C₃×#815	2
C₂×C₃×#812	2
#67812	3
#67664	3