Subgroups — #11776817

Order 1152.

441 conjugacy classes of subgroups.

The subgroup lattice has 241 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×#4925	C₂	1	2	yes	maximal
3	#10589100	C₂	1	2	yes	maximal, complemented
4	#10589098	C₂	1	2	yes	maximal, complemented
5	#70745	—	3	2	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
9	C₄×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
10	C₄×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
11	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃×#830	—	2	2	no	—
14	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
15	C₃×C₄×C₃⋊C₈	—	2	2	no	—
16	C₃×#431	—	3	2	no	—
17	#4925	D₃	1	2	yes	—
18	C₂×C₄×C₃⋊C₈	—	3	2	no	—
19	C₂×C₄×C₃⋊C₈	—	6	1	no	—
20	#4480	—	3	4	no	—
21	#4479	—	3	4	no	—
22	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
25	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₄²	D₄	1	1	yes	characteristic
27	C₃²×C₄²	Q₈	1	1	yes	characteristic
28	C₃²×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
30	C₂×C₃⋊(C₃⋊C₈)	C₈	1	4	yes	complemented
31	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₃⋊C₈	—	2	4	no	—
36	#1564	—	9	1	no	Sylow
37	C₂×C₃×C₄²	—	2	1	no	—
38	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
39	C₂²×C₃⋊C₈	—	3	2	no	—
40	C₂²×C₃⋊C₈	—	3	2	no	—
41	C₂²×C₃⋊C₈	—	6	1	no	—
42	C₂²×C₃⋊C₈	—	6	1	no	—
43	C₃×C₄⋊C₈	—	3	4	no	—
44	C₃×C₂²⋊C₈	—	6	2	no	—
45	C₃×C₄×C₈	—	6	2	no	—
46	#830	—	2	2	no	—
47	C₄⋊(C₃⋊C₈)	C₂×D₃	1	4	yes	—
48	C₄×C₃⋊C₈	—	2	2	no	—
49	C₄×C₃⋊C₈	—	3	2	no	—
50	C₄×C₃⋊C₈	—	3	2	no	—
51	C₄×C₃⋊C₈	—	3	2	no	—
52	C₄×C₃⋊C₈	—	3	2	no	—
53	C₄×C₃⋊C₈	—	6	1	no	—
54	C₄×C₃⋊C₈	—	6	1	no	—
55	C₄×C₃⋊C₈	—	6	1	no	—
56	C₄×C₃⋊C₈	—	6	1	no	—
57	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	—	2	2	no	—
59	C₂×C₃²×C₄	—	2	2	no	—
60	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
64	C₂×C₃²×C₄	M₁₆	1	2	yes	—
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
68	C₃⋊(C₃⋊C₈)	—	2	1	no	—
69	C₃⋊(C₃⋊C₈)	—	2	1	no	—
70	C₃⋊(C₃⋊C₈)	C₂×C₈	1	2	yes	—
71	C₃⋊(C₃⋊C₈)	C₂×C₈	1	2	yes	—
72	C₃×C₃⋊C₈	—	4	4	no	—
73	#431	—	3	2	no	—
74	C₂×C₄×C₈	—	9	1	no	—
75	#335	—	9	2	no	—
76	#334	—	9	2	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	Dic₆	1	2	yes	—
81	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
82	C₂²×C₃×C₄	#67	1	2	yes	—
83	C₂×C₃×C₈	—	6	4	no	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	—	2	1	no	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	Dic₆	1	2	yes	—
89	C₃×C₄²	C₄×D₃	1	2	yes	—
90	C₃×C₄²	#67	1	2	yes	—
91	C₃×C₄²	C₂²×D₃	1	2	yes	—
92	C₂×C₃⋊C₈	—	2	4	no	—
93	C₂×C₃⋊C₈	—	3	8	no	—
94	C₂×C₃⋊C₈	—	3	2	no	—
95	C₂×C₃⋊C₈	—	3	2	no	—
96	C₂×C₃⋊C₈	—	3	2	no	—
97	C₂×C₃⋊C₈	—	3	2	no	—
98	C₂×C₃⋊C₈	—	6	1	no	—
99	C₂×C₃⋊C₈	—	6	1	no	—
100	C₂×C₃⋊C₈	—	6	1	no	—
101	C₂×C₃⋊C₈	—	6	1	no	—
102	C₂×C₃⋊C₈	—	6	4	no	—
103	C₂×C₃⋊C₈	—	6	4	no	—
104	C₂×C₃⋊C₈	—	6	4	no	—
105	C₂²×C₃²	—	2	1	no	—
106	C₂²×C₃²	—	2	1	no	—
107	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
108	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₃²×C₄	—	2	1	no	—
110	C₃²×C₄	—	2	1	no	—
111	C₃²×C₄	C₄⋊C₈	1	2	yes	—
112	C₃²×C₄	#108	1	2	yes	—
113	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
114	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
115	C₃²×C₄	#135	1	2	yes	—
116	C₂²×C₈	—	9	1	no	—
117	C₂²×C₈	—	9	1	no	—
118	C₂×C₄²	D₃²	1	1	yes	characteristic
119	C₄⋊C₈	—	3	4	no	—
120	C₂²⋊C₈	—	6	2	no	—
121	C₄×C₈	—	6	2	no	—
122	C₄×C₈	—	9	1	no	—
123	C₄×C₈	—	9	1	no	—
124	C₄×C₈	—	9	1	no	—
125	C₄×C₈	—	9	1	no	—
126	C₂³×C₃	—	2	1	no	—
127	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	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	D₃×C₈	1	4	yes	—
141	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
142	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
143	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
144	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
145	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
147	C₂×C₃×C₄	C₂×#67	1	2	yes	—
148	C₃×C₈	—	12	4	no	—
149	C₃⋊C₈	—	3	4	no	—
150	C₃⋊C₈	—	3	4	no	—
151	C₃⋊C₈	—	4	4	no	—
152	C₃⋊C₈	—	6	2	no	—
153	C₃⋊C₈	—	6	2	no	—
154	C₃⋊C₈	—	6	2	no	—
155	C₃⋊C₈	—	6	2	no	—
156	C₃⋊C₈	—	6	2	no	—
157	C₃⋊C₈	—	6	2	no	—
158	C₂×C₃²	—	2	2	no	—
159	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₂×C₃²	#427	1	1	yes	characteristic
161	C₂×C₃²	C₂×#108	1	1	yes	characteristic
162	C₂²×C₄	#624	1	1	yes	characteristic
163	C₂²×C₄	#625	1	1	yes	characteristic
164	C₂²×C₄	#627	1	1	yes	characteristic
165	C₂×C₈	—	6	4	no	—
166	C₂×C₈	—	9	4	no	—
167	C₂×C₈	—	9	1	no	—
168	C₂×C₈	—	9	1	no	—
169	C₂×C₈	—	9	1	no	—
170	C₂×C₈	—	9	1	no	—
171	C₂×C₈	—	18	2	no	—
172	C₄²	#624	1	1	yes	characteristic
173	C₄²	#625	1	1	yes	characteristic
174	C₄²	#627	1	1	yes	characteristic
175	C₄²	C₂×D₃²	1	1	yes	characteristic
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	Dic₃⋊C₈	1	4	yes	—
183	C₂²×C₃	C₂×Dic₃⋊C₄	1	2	yes	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	Dic₃⋊C₈	1	4	yes	—
194	C₃×C₄	#822	1	4	yes	—
195	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
196	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
197	C₃×C₄	#950	1	4	yes	—
198	C₃²	#1564	1	1	yes	characteristic, complemented, Sylow
199	C₂³	#3724	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	#3710	1	2	yes	—
203	C₂×C₄	#3712	1	2	yes	—
204	C₂×C₄	#3724	1	1	yes	characteristic
205	C₂×C₄	#3724	1	1	yes	characteristic
206	C₂×C₄	#3724	1	1	yes	characteristic
207	C₂×C₄	C₂×#624	1	1	yes	characteristic
208	C₂×C₄	C₂×#625	1	1	yes	characteristic
209	C₂×C₄	C₂×#627	1	1	yes	characteristic
210	C₈	—	9	2	no	—
211	C₈	—	9	2	no	—
212	C₈	—	12	4	no	—
213	C₈	—	18	1	no	—
214	C₈	—	18	1	no	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	#4998	1	2	yes	—
221	C₂×C₃	C₂×Dic₃⋊C₈	1	2	yes	—
222	C₂×C₃	C₂×#822	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	#63572	1	2	yes	—
226	C₂²	C₂×#3724	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	#63572	1	2	yes	—
230	C₄	#63578	1	2	yes	—
231	C₄	C₂×#3710	1	1	yes	characteristic
232	C₄	C₂×#3712	1	1	yes	characteristic
233	C₄	#63822	1	2	yes	—
234	C₃	—	2	1	no	—
235	C₃	#70745	1	2	yes	complemented
236	C₂	—	2	2	no	—
237	C₂	C₂×#63572	1	1	yes	characteristic
238	C₂	#10591138	1	1	yes	characteristic
239	C₂	C₂×#63578	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#625
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1564
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. #10589100	C₂	semidirect
4. #10589098	C₂	semidirect
30. C₂×C₃⋊(C₃⋊C₈)	C₈	semidirect
198. C₃²	#1564	semidirect
235. C₃	#70745	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×#4925	2
#10589100	2
#10589098	2
#70745	3