Subgroups — #11770825

Order 1152.

363 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592624	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
4	#67741	—	3	1	no	maximal
5	#67661	—	3	1	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₃⋊C₈	C₄	1	2	yes	—
8	#63657	C₄	1	2	yes	—
9	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
10	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
11	C₃×#830	C₄	1	2	yes	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×#803	—	2	1	no	—
14	C₃×#803	—	2	1	no	—
15	C₃×C₄×C₃⋊C₈	—	2	1	no	—
16	C₃×#431	—	3	1	no	—
17	C₂×C₃×#97	—	3	1	no	—
18	#4925	—	3	1	no	—
19	#4925	—	3	1	no	—
20	#4925	—	6	1	no	—
21	#4925	D₃	1	1	yes	characteristic
22	C₂×#803	D₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₄²	D₄	1	1	yes	characteristic
27	C₃²×C₄²	D₄	1	1	yes	characteristic
28	C₃²×C₄²	D₄	1	1	yes	characteristic
29	C₃²×C₄²	Q₈	1	1	yes	characteristic
30	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
31	C₂×C₃×C₃⋊C₈	—	2	2	no	—
32	C₂×C₃×C₃⋊C₈	—	2	2	no	—
33	C₂×C₃×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	C₈	1	4	yes	complemented
35	#1281	—	9	1	no	Sylow
36	C₂²×C₃×C₈	—	3	2	no	—
37	C₂×C₃×C₄²	—	2	1	no	—
38	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃⋊C₈	Dic₃	1	2	yes	—
41	C₃×C₄⋊C₈	—	6	1	no	—
42	C₃×C₂²⋊C₈	—	3	2	no	—
43	C₃×#97	—	6	1	no	—
44	C₃×#97	—	6	1	no	—
45	C₃×C₄×C₈	—	6	1	no	—
46	#830	—	3	2	no	—
47	#830	—	3	2	no	—
48	#830	—	6	2	no	—
49	#830	Dic₃	1	2	yes	—
50	C₄⋊(C₃⋊C₈)	—	2	1	no	—
51	C₄⋊(C₃⋊C₈)	—	6	2	no	—
52	C₄⋊(C₃⋊C₈)	—	6	1	no	—
53	C₄⋊(C₃⋊C₈)	—	6	1	no	—
54	#803	—	2	1	no	—
55	#803	—	2	1	no	—
56	C₄×C₃⋊C₈	—	2	1	no	—
57	C₄×C₃⋊C₈	—	6	2	no	—
58	C₄×C₃⋊C₈	—	6	1	no	—
59	C₄×C₃⋊C₈	—	6	1	no	—
60	C₂³×C₃²	C₄²	1	1	yes	characteristic
61	C₂×C₃²×C₄	—	4	1	no	—
62	C₂×C₃²×C₄	—	4	1	no	—
63	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
70	C₂×C₃²×C₄	M₁₆	1	2	yes	—
71	C₃⋊(C₃⋊C₈)	—	4	2	no	—
72	C₃×C₃⋊C₈	—	2	4	no	—
73	C₃×C₃⋊C₈	—	4	2	no	—
74	#431	—	3	1	no	—
75	#431	—	9	1	no	—
76	C₂×#97	—	3	1	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₃×C₈	—	3	4	no	—
87	C₂×C₃×C₈	—	6	2	no	—
88	C₂×C₃×C₈	—	6	2	no	—
89	C₂×C₃×C₈	—	6	2	no	—
90	C₃×C₄²	—	2	1	no	—
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	—	2	1	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	Dic₆	1	1	yes	characteristic
95	C₃×C₄²	Dic₆	1	1	yes	characteristic
96	C₃×C₄²	D₁₂	1	1	yes	characteristic
97	C₃×C₄²	D₁₂	1	1	yes	characteristic
98	C₃×C₄²	#67	1	1	yes	characteristic
99	C₃×C₄²	#67	1	1	yes	characteristic
100	C₃×C₄²	#67	1	1	yes	characteristic
101	C₃×C₄²	#67	1	1	yes	characteristic
102	C₂×C₃⋊C₈	—	2	2	no	—
103	C₂×C₃⋊C₈	—	2	2	no	—
104	C₂×C₃⋊C₈	—	2	2	no	—
105	C₂×C₃⋊C₈	—	6	4	no	—
106	C₂×C₃⋊C₈	—	6	2	no	—
107	C₂×C₃⋊C₈	—	6	2	no	—
108	C₂×C₃⋊C₈	C₃⋊C₈	1	4	yes	complemented
109	C₂²×C₃²	—	2	1	no	—
110	C₂²×C₃²	—	2	1	no	—
111	C₂²×C₃²	#95	1	1	yes	characteristic
112	C₂²×C₃²	C₄×C₈	1	1	yes	characteristic
113	C₂²×C₃²	#97	1	1	yes	characteristic
114	C₃²×C₄	—	2	1	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	—	2	1	no	—
117	C₃²×C₄	—	2	1	no	—
118	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
119	C₃²×C₄	C₄⋊C₈	1	2	yes	—
120	C₂²×C₈	—	3	2	no	—
121	C₂×C₄²	D₃²	1	1	yes	characteristic
122	C₄⋊C₈	—	6	1	no	—
123	C₄⋊C₈	—	18	1	no	—
124	C₂²⋊C₈	—	3	2	no	—
125	C₂²⋊C₈	—	9	2	no	—
126	#97	—	6	1	no	—
127	#97	—	6	1	no	—
128	C₄×C₈	—	6	1	no	—
129	C₄×C₈	—	18	1	no	—
130	C₂³×C₃	—	2	1	no	—
131	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
132	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	4	1	no	—
142	C₂×C₃×C₄	—	4	1	no	—
143	C₂×C₃×C₄	—	4	1	no	—
144	C₂×C₃×C₄	—	4	1	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
148	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
150	C₂×C₃×C₄	#208	1	2	yes	—
151	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
157	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	(C₂×D₃)⋊C₄	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×C₃×C₄	#217	1	1	yes	characteristic
163	C₃×C₈	—	6	4	no	—
164	C₃×C₈	—	12	2	no	—
165	C₃⋊C₈	—	2	4	no	—
166	C₃⋊C₈	—	4	2	no	—
167	C₃⋊C₈	—	12	4	no	—
168	C₃⋊C₈	—	12	2	no	—
169	C₃⋊C₈	—	12	2	no	—
170	C₂×C₃²	—	2	1	no	—
171	C₂×C₃²	—	2	1	no	—
172	C₂×C₃²	#336	1	1	yes	characteristic
173	C₂×C₃²	#344	1	1	yes	characteristic
174	C₂×C₃²	#344	1	1	yes	characteristic
175	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
176	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
177	C₂²×C₄	#624	1	1	yes	characteristic
178	C₂×C₈	—	3	4	no	—
179	C₂×C₈	—	6	2	no	—
180	C₂×C₈	—	6	2	no	—
181	C₂×C₈	—	6	2	no	—
182	C₂×C₈	—	18	2	no	—
183	C₄²	#625	1	1	yes	characteristic
184	C₄²	#626	1	1	yes	characteristic
185	C₄²	#626	1	1	yes	characteristic
186	C₄²	#627	1	1	yes	characteristic
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	4	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	C₄×C₃⋊C₈	1	1	yes	characteristic
197	C₂²×C₃	#803	1	1	yes	characteristic
198	C₂²×C₃	C₈×Dic₃	1	1	yes	characteristic
199	C₂²×C₃	#815	1	1	yes	characteristic
200	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
201	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
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	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
217	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
218	C₃×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
219	C₃×C₄	#830	1	2	yes	—
220	C₃²	#1281	1	1	yes	characteristic, complemented, Sylow
221	C₂³	Dic₃²	1	1	yes	characteristic
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
225	C₂×C₄	#3711	1	2	yes	—
226	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
227	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
228	C₂×C₄	#3722	1	1	yes	characteristic
229	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
230	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
231	C₂×C₄	#3724	1	1	yes	characteristic
232	C₈	—	6	4	no	—
233	C₈	—	12	2	no	—
234	C₈	—	36	2	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	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	—	4	1	no	—
243	C₂×C₃	—	4	1	no	—
244	C₂×C₃	#4524	1	1	yes	characteristic
245	C₂×C₃	#4533	1	1	yes	characteristic
246	C₂×C₃	#4550	1	1	yes	characteristic
247	C₂×C₃	#4556	1	1	yes	characteristic
248	C₂×C₃	#4556	1	1	yes	characteristic
249	C₂×C₃	#4559	1	1	yes	characteristic
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	Dic₃×C₃⋊C₈	1	1	yes	characteristic
253	C₂²	#63554	1	1	yes	characteristic
254	C₂²	#63579	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	(C₂×D₃)⋊(C₃⋊C₈)	1	2	yes	—
260	C₄	Dic₃⋊(C₃⋊C₈)	1	2	yes	—
261	C₃	—	2	1	no	—
262	C₃	#67661	1	1	yes	characteristic, complemented
263	C₃	#67741	1	1	yes	characteristic, complemented
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#10589155	1	1	yes	characteristic
267	C₂	#10589169	1	1	yes	characteristic
268	C₂	#10589171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊Dic₃
commutator	C₂²×C₃²	C₄×C₈
Frattini, hypercenter	C₂×C₄²	D₃²
Fitting, cosocle	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄²
hypocenter	C₃²	#1281
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
34. C₂×C₃×C₃⋊C₈	C₈	semidirect
108. C₂×C₃⋊C₈	C₃⋊C₈	semidirect
220. C₃²	#1281	semidirect
262. C₃	#67661	semidirect
263. C₃	#67741	semidirect

Maximal subgroups

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

subgroup	index
#10592624	2
C₃×#4925	2
C₂×C₃×#803	2
#67741	3
#67661	3