Subgroups — #11786196

Order 1152.

581 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×#4932	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊(C₃⋊C₈)	C₂	1	4	yes	maximal
5	#70446	—	3	1	no	maximal
6	#69439	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
12	C₃×C₃⋊#95	C₄	1	2	yes	—
13	C₃×C₃⋊#95	C₄	1	2	yes	—
14	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	Dic₃⋊(C₃⋊C₈)	—	2	8	no	—
16	C₂×C₃×C₄⋊C₈	—	3	1	no	—
17	C₃×#376	—	3	1	no	—
18	C₂×Dic₃⋊C₈	—	3	4	no	—
19	#4932	D₃	1	1	yes	characteristic
20	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
29	C₃²×C₄²	D₄	1	2	yes	—
30	C₃²×C₄²	Q₈	1	2	yes	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
33	C₃×C₄×Dic₃	D₄	1	4	yes	—
34	C₃×C₄×Dic₃	Q₈	1	4	yes	—
35	C₂×C₃×C₃⋊C₈	—	2	4	no	—
36	C₂×C₃×C₃⋊C₈	—	2	2	no	—
37	#1843	—	9	1	no	Sylow
38	C₂²×C₃×C₈	—	3	2	no	—
39	C₂×C₃×C₄²	—	2	1	no	—
40	C₂×C₃×C₄²	—	3	2	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄×Dic₃	C₂×D₃	1	2	yes	—
44	C₂²×C₃⋊C₈	—	3	2	no	—
45	C₂²×C₃⋊C₈	—	3	2	no	—
46	C₂²×C₃⋊C₈	—	6	2	no	—
47	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
48	C₃×C₄⋊C₈	—	6	2	no	—
49	C₃×#95	—	3	2	no	—
50	C₃×#95	—	3	2	no	—
51	C₃⋊#95	Dic₃	1	2	yes	—
52	C₃⋊#95	Dic₃	1	2	yes	—
53	Dic₃⋊C₈	—	6	8	no	—
54	C₄⋊(C₃⋊C₈)	—	2	2	no	—
55	C₄⋊(C₃⋊C₈)	—	6	8	no	—
56	C₄⋊(C₃⋊C₈)	—	6	4	no	—
57	C₄⋊(C₃⋊C₈)	—	6	2	no	—
58	C₄⋊(C₃⋊C₈)	—	6	2	no	—
59	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	—	2	2	no	—
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
63	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
64	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
68	C₂×C₃×Dic₃	—	2	4	no	—
69	C₂×C₃×Dic₃	—	2	4	no	—
70	C₂×C₃×Dic₃	C₄⋊C₄	1	8	yes	—
71	C₃⋊(C₃⋊C₈)	—	4	2	no	—
72	C₃×C₃⋊C₈	—	4	2	no	—
73	C₂×C₄⋊C₈	—	3	1	no	—
74	C₂×C₄⋊C₈	—	9	4	no	—
75	C₂×C₄⋊C₈	—	9	1	no	—
76	#376	—	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	—	3	4	no	—
81	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂²×Dic₃	C₂×Dic₃	1	4	yes	—
88	C₂×C₃×C₈	—	6	4	no	—
89	C₂×C₃×C₈	—	6	2	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	—	2	2	no	—
92	C₃×C₄²	—	3	4	no	—
93	C₃×C₄²	—	3	4	no	—
94	C₃×C₄²	Dic₆	1	2	yes	—
95	C₃×C₄²	Dic₆	1	2	yes	—
96	C₃×C₄²	D₁₂	1	2	yes	—
97	C₃×C₄²	#67	1	2	yes	—
98	C₄×Dic₃	Dic₆	1	4	yes	—
99	C₄×Dic₃	D₁₂	1	4	yes	—
100	C₂×C₃⋊C₈	—	2	4	no	—
101	C₂×C₃⋊C₈	—	2	2	no	—
102	C₂×C₃⋊C₈	—	6	4	no	—
103	C₂×C₃⋊C₈	—	6	2	no	—
104	C₂×C₃⋊C₈	—	6	4	no	—
105	C₂×C₃⋊C₈	—	6	4	no	—
106	C₂×C₃⋊C₈	—	6	2	no	—
107	C₂×C₃⋊C₈	—	12	4	no	—
108	C₂²×C₃²	C₂×C₄⋊C₄	1	2	yes	—
109	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃²	#131	1	2	yes	—
111	C₂²×C₃²	#131	1	2	yes	—
112	C₃²×C₄	—	2	2	no	—
113	C₃²×C₄	—	2	2	no	—
114	C₃²×C₄	#127	1	1	yes	characteristic
115	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
116	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
117	C₃×Dic₃	—	2	8	no	—
118	C₂²×C₈	—	3	2	no	—
119	C₂²×C₈	—	9	2	no	—
120	C₂×C₄²	—	3	2	no	—
121	C₂×C₄²	D₃²	1	1	yes	characteristic
122	C₄⋊C₈	—	6	2	no	—
123	C₄⋊C₈	—	18	8	no	—
124	C₄⋊C₈	—	18	2	no	—
125	#95	—	3	2	no	—
126	#95	—	3	2	no	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
129	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	3	8	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	—	6	4	no	—
144	C₂×C₃×C₄	—	6	4	no	—
145	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
146	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
147	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
148	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
149	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
150	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
153	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
154	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
155	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
156	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
157	C₂×Dic₃	—	2	4	no	—
158	C₂×Dic₃	—	2	4	no	—
159	C₂×Dic₃	C₄⋊Dic₃	1	8	yes	—
160	C₃×C₈	—	12	2	no	—
161	C₃⋊C₈	—	4	2	no	—
162	C₃⋊C₈	—	12	4	no	—
163	C₃⋊C₈	—	12	2	no	—
164	C₃⋊C₈	—	12	2	no	—
165	C₂×C₃²	#384	1	1	yes	characteristic
166	C₂×C₃²	#424	1	4	yes	—
167	C₂×C₃²	#424	1	1	yes	characteristic
168	C₂×C₃²	#424	1	1	yes	characteristic
169	C₂²×C₄	—	3	4	no	—
170	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
171	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
173	C₂×C₈	—	6	4	no	—
174	C₂×C₈	—	6	2	no	—
175	C₂×C₈	—	18	4	no	—
176	C₂×C₈	—	18	2	no	—
177	C₄²	—	3	4	no	—
178	C₄²	—	3	4	no	—
179	C₄²	#626	1	2	yes	—
180	C₄²	#627	1	2	yes	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	D₃×C₄⋊C₄	1	2	yes	—
186	C₂²×C₃	#901	1	2	yes	—
187	C₂²×C₃	#907	1	2	yes	—
188	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
189	C₂²×C₃	C₂×C₄⋊Dic₃	1	2	yes	—
190	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
191	C₂²×C₃	#948	1	2	yes	—
192	C₂²×C₃	#948	1	2	yes	—
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	4	2	no	—
202	C₃×C₄	—	6	8	no	—
203	C₃×C₄	#869	1	1	yes	characteristic
204	C₃×C₄	#869	1	2	yes	—
205	C₃×C₄	#874	1	1	yes	characteristic
206	C₃×C₄	#888	1	2	yes	—
207	C₃×C₄	#940	1	1	yes	characteristic
208	C₃×C₄	#944	1	1	yes	characteristic
209	Dic₃	—	2	8	no	—
210	C₃²	#1843	1	1	yes	characteristic, complemented, Sylow
211	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	3	8	no	—
215	C₂×C₄	—	6	4	no	—
216	C₂×C₄	—	6	4	no	—
217	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
218	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
219	C₂×C₄	D₃×Dic₆	1	2	yes	—
220	C₂×C₄	D₃×D₁₂	1	2	yes	—
221	C₂×C₄	C₂×#626	1	1	yes	characteristic
222	C₂×C₄	C₂×#627	1	1	yes	characteristic
223	C₈	—	12	2	no	—
224	C₈	—	36	2	no	—
225	C₂×C₃	—	2	4	no	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	#4839	1	4	yes	—
230	C₂×C₃	#4935	1	1	yes	characteristic
231	C₂×C₃	#4975	1	1	yes	characteristic
232	C₂×C₃	#4999	1	4	yes	—
233	C₂×C₃	#4999	1	1	yes	characteristic
234	C₂×C₃	#4999	1	1	yes	characteristic
235	C₂×C₃	#5102	1	1	yes	characteristic
236	C₂×C₃	#5119	1	1	yes	characteristic
237	C₂²	#63814	1	2	yes	—
238	C₂²	#63815	1	2	yes	—
239	C₂²	D₃×C₄⋊Dic₃	1	2	yes	—
240	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	6	8	no	—
244	C₄	#63866	1	1	yes	characteristic
245	C₄	#63911	1	1	yes	characteristic
246	C₄	#63918	1	2	yes	—
247	C₃	—	2	1	no	—
248	C₃	#69439	1	1	yes	characteristic, complemented
249	C₃	#70446	1	1	yes	characteristic, complemented
250	C₂	#10590324	1	4	yes	—
251	C₂	#10591116	1	1	yes	characteristic
252	C₂	#10591117	1	1	yes	characteristic
253	C₂	#10591349	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄²	C₂²
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1843
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
210. C₃²	#1843	semidirect
248. C₃	#69439	semidirect
249. C₃	#70446	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×#4932	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₂×Dic₃⋊(C₃⋊C₈)	2
#70446	3
#69439	3