Subgroups — C₂²×#10622612 #15617389

Order 2816.

3006 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#4318	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622612	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₁×#117	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×#117	C₄	1	12	yes	—
8	C₂×C₁₁×#117	C₂²	1	12	yes	—
9	C₂²×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
10	C₂²×C₁₁×C₂²⋊C₄	C₄	1	2	yes	—
11	C₂²×C₄²×C₁₁	C₂²	1	2	yes	—
12	C₂²×#4318	—	2	8	no	—
13	C₂²×#4318	C₂²	1	12	yes	—
14	C₂³×C₁₁⋊C₈	—	2	8	no	—
15	C₂³×C₁₁⋊C₈	C₂²	1	2	yes	—
16	C₂³×C₁₁⋊C₈	C₂²	1	2	yes	—
17	C₂³×C₁₁⋊C₈	C₂²	1	12	yes	—
18	#10622612	C₂²	1	64	yes	direct product factor
19	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	48	yes	complemented
20	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₁	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	C₂×C₄	1	6	yes	—
24	C₂³×C₄×C₁₁	D₄	1	8	yes	—
25	C₂³×C₄×C₁₁	Q₈	1	8	yes	—
26	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
27	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
28	C₁₁×#117	C₂×C₄	1	48	yes	—
29	C₁₁×#117	C₂³	1	16	yes	—
30	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	24	yes	—
31	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	24	yes	—
32	C₂×C₄²×C₁₁	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₁₁	C₂³	1	12	yes	—
34	C₂×#4318	—	2	48	no	—
35	C₂×#4318	C₂³	1	16	yes	—
36	C₂²×C₁₁⋊C₈	—	2	8	no	—
37	C₂²×C₁₁⋊C₈	—	2	48	no	—
38	C₂²×C₁₁⋊C₈	—	2	8	no	—
39	C₂²×C₁₁⋊C₈	C₂³	1	12	yes	—
40	C₂²×C₁₁⋊C₈	C₂³	1	12	yes	—
41	C₂²×C₁₁⋊C₈	C₂³	1	16	yes	—
42	C₄⋊(C₁₁⋊C₈)	C₂³	1	64	yes	complemented
43	C₂²×#424	—	11	1	no	maximal, Sylow
44	C₂⁴×C₁₁	—	2	8	no	—
45	C₂⁴×C₁₁	C₄⋊C₄	1	8	yes	—
46	C₂⁴×C₁₁	C₂²×C₄	1	6	yes	—
47	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	—	2	16	no	—
49	C₂²×C₄×C₁₁	C₄⋊C₄	1	48	yes	—
50	C₂²×C₄×C₁₁	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₁₁	C₂²×C₄	1	24	yes	—
52	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
54	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₁₁	C₂×D₄	1	24	yes	—
57	C₂²×C₄×C₁₁	C₂×Q₈	1	24	yes	—
58	C₂²×C₄×C₁₁	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
61	C₁₁×C₄⋊C₄	C₂²×C₄	1	32	yes	—
62	C₁₁×C₂²⋊C₄	C₂²×C₄	1	32	yes	—
63	C₄²×C₁₁	C₂²×C₄	1	24	yes	—
64	C₄²×C₁₁	C₂⁴	1	8	yes	—
65	#4318	—	2	32	no	—
66	C₂×C₁₁⋊C₈	—	2	24	no	—
67	C₂×C₁₁⋊C₈	—	2	32	no	—
68	C₂×C₁₁⋊C₈	—	2	24	no	—
69	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	—
70	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	—
71	C₂³×M₁₆	—	11	1	no	—
72	C₂⁴×C₈	—	11	1	no	—
73	C₂²×#117	D₁₁	1	1	yes	characteristic
74	C₂×#424	—	11	24	no	—
75	C₂²×C₄⋊C₈	—	11	4	no	—
76	C₂³×C₁₁	—	2	24	no	—
77	C₂³×C₁₁	—	2	32	no	—
78	C₂³×C₁₁	C₂×C₄⋊C₄	1	24	yes	—
79	C₂³×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
80	C₂³×C₁₁	#131	1	8	yes	—
81	C₂³×C₁₁	C₂³×C₄	1	4	yes	—
82	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₁₁	—	2	48	no	—
84	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	48	yes	—
85	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	12	yes	—
86	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	24	yes	—
87	C₂×C₄×C₁₁	C₂³×C₄	1	16	yes	—
88	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₁₁	C₂³×C₄	1	4	yes	—
90	C₂×C₄×C₁₁	C₂²×D₄	1	6	yes	—
91	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
92	C₂×C₄×C₁₁	C₂²×Q₈	1	8	yes	—
93	C₂×C₄×C₁₁	C₂²×Q₈	1	6	yes	—
94	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
95	C₁₁⋊C₈	—	2	8	no	—
96	C₁₁⋊C₈	—	2	8	no	—
97	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
98	C₂²×M₁₆	—	11	12	no	—
99	C₂²×M₁₆	—	22	8	no	—
100	C₂³×C₈	—	11	12	no	—
101	C₂³×C₈	—	11	2	no	—
102	C₂³×C₈	—	11	2	no	—
103	C₂³×C₈	—	22	8	no	—
104	C₂×#117	Dic₁₁	1	12	yes	—
105	C₂×#117	C₂×D₁₁	1	12	yes	—
106	C₂²×C₄⋊C₄	Dic₁₁	1	2	yes	—
107	C₂²×C₂²⋊C₄	Dic₁₁	1	2	yes	—
108	C₂²×C₄²	C₂×D₁₁	1	2	yes	—
109	#424	—	11	64	no	—
110	C₂×C₄⋊C₈	—	11	48	no	—
111	C₂²×C₁₁	—	2	48	no	—
112	C₂²×C₁₁	—	2	8	no	—
113	C₂²×C₁₁	#424	1	16	yes	complemented
114	C₂²×C₁₁	C₂²×C₄⋊C₄	1	8	yes	—
115	C₂²×C₁₁	C₂²×C₄⋊C₄	1	6	yes	—
116	C₂²×C₁₁	C₂×#131	1	12	yes	—
117	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
118	C₄×C₁₁	—	2	16	no	—
119	C₄×C₁₁	C₂²×C₄⋊C₄	1	6	yes	—
120	C₄×C₁₁	C₂²×C₄⋊C₄	1	8	yes	—
121	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
122	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
123	C₂⁵	C₂×Dic₁₁	1	1	yes	characteristic
124	C₂³×C₄	Dic₂₂	1	8	yes	—
125	C₂³×C₄	D₄₄	1	8	yes	—
126	C₂³×C₄	C₂×Dic₁₁	1	4	yes	—
127	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
128	C₂³×C₄	C₂×Dic₁₁	1	6	yes	—
129	C₂³×C₄	C₂²×D₁₁	1	6	yes	—
130	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
131	C₂×M₁₆	—	11	16	no	—
132	C₂×M₁₆	—	22	48	no	—
133	C₂²×C₈	—	11	12	no	—
134	C₂²×C₈	—	11	16	no	—
135	C₂²×C₈	—	11	12	no	—
136	C₂²×C₈	—	22	8	no	—
137	C₂²×C₈	—	22	48	no	—
138	C₂²×C₈	—	22	8	no	—
139	#117	C₂×Dic₁₁	1	48	yes	—
140	#117	C₂²×D₁₁	1	16	yes	—
141	C₂×C₄⋊C₄	C₂×Dic₁₁	1	24	yes	—
142	C₂×C₂²⋊C₄	C₂×Dic₁₁	1	24	yes	—
143	C₂×C₄²	C₂×Dic₁₁	1	12	yes	—
144	C₂×C₄²	C₂²×D₁₁	1	12	yes	—
145	C₄⋊C₈	—	11	64	no	—
146	C₂×C₁₁	—	2	8	no	—
147	C₂×C₁₁	C₂×#424	1	12	yes	complemented
148	C₂×C₁₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₁₁	C₂²×#131	1	2	yes	—
150	C₂⁴	—	2	8	no	—
151	C₂⁴	C₄⋊Dic₁₁	1	8	yes	—
152	C₂⁴	C₂²×Dic₁₁	1	6	yes	—
153	C₂⁴	C₂²×Dic₁₁	1	1	yes	characteristic
154	C₂²×C₄	—	2	16	no	—
155	C₂²×C₄	C₄⋊Dic₁₁	1	48	yes	—
156	C₂²×C₄	C₄⋊Dic₁₁	1	8	yes	—
157	C₂²×C₄	C₂×Dic₂₂	1	24	yes	—
158	C₂²×C₄	C₂×Dic₂₂	1	4	yes	—
159	C₂²×C₄	C₂×D₄₄	1	4	yes	—
160	C₂²×C₄	C₂×D₄₄	1	24	yes	—
161	C₂²×C₄	C₂²×Dic₁₁	1	24	yes	—
162	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
163	C₂²×C₄	C₂²×Dic₁₁	1	6	yes	—
164	C₂²×C₄	C₂²×Dic₁₁	1	12	yes	—
165	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
166	C₂²×C₄	C₂³×D₁₁	1	4	yes	—
167	M₁₆	—	22	32	no	—
168	C₂×C₈	—	11	8	no	—
169	C₂×C₈	—	11	8	no	—
170	C₂×C₈	—	22	24	no	—
171	C₂×C₈	—	22	32	no	—
172	C₂×C₈	—	22	24	no	—
173	C₄⋊C₄	C₂²×Dic₁₁	1	32	yes	—
174	C₂²⋊C₄	C₂²×Dic₁₁	1	32	yes	—
175	C₄²	C₂²×Dic₁₁	1	24	yes	—
176	C₄²	C₂³×D₁₁	1	8	yes	—
177	C₁₁	C₂²×#424	1	1	yes	characteristic, complemented, Sylow
178	C₂³	—	2	24	no	—
179	C₂³	—	2	32	no	—
180	C₂³	C₂×C₄⋊Dic₁₁	1	24	yes	—
181	C₂³	C₂×C₄⋊Dic₁₁	1	4	yes	—
182	C₂³	#67001	1	8	yes	—
183	C₂³	C₂³×Dic₁₁	1	4	yes	—
184	C₂³	C₂³×Dic₁₁	1	3	yes	—
185	C₂×C₄	—	2	48	no	—
186	C₂×C₄	C₂×C₄⋊Dic₁₁	1	48	yes	—
187	C₂×C₄	C₂×C₄⋊Dic₁₁	1	24	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₁₁	1	12	yes	—
189	C₂×C₄	C₂²×Dic₂₂	1	8	yes	—
190	C₂×C₄	C₂²×Dic₂₂	1	6	yes	—
191	C₂×C₄	C₂²×D₄₄	1	6	yes	—
192	C₂×C₄	C₂²×D₄₄	1	8	yes	—
193	C₂×C₄	C₂³×Dic₁₁	1	16	yes	—
194	C₂×C₄	C₂³×Dic₁₁	1	3	yes	—
195	C₂×C₄	C₂³×Dic₁₁	1	4	yes	—
196	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
197	C₈	—	22	8	no	—
198	C₈	—	22	8	no	—
199	C₂²	—	2	48	no	—
200	C₂²	—	2	8	no	—
201	C₂²	#10622612	1	16	yes	direct product factor
202	C₂²	C₂²×C₄⋊Dic₁₁	1	8	yes	—
203	C₂²	C₂²×C₄⋊Dic₁₁	1	6	yes	—
204	C₂²	C₂×#67001	1	12	yes	—
205	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
206	C₄	—	2	16	no	—
207	C₄	C₂²×C₄⋊Dic₁₁	1	6	yes	—
208	C₄	C₂²×C₄⋊Dic₁₁	1	8	yes	—
209	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
210	C₄	C₂³×D₄₄	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#10622612	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊Dic₁₁	1	1	yes	characteristic
214	C₂	C₂²×#67001	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂²×C₁₁×#117	C₂
socle	C₂⁴×C₁₁	C₂²×C₄
prime radical	C₂⁵×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂²×#117	D₁₁
hypocenter	C₁₁	C₂²×#424
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
3. C₂×#10622612	C₂	direct (N × H)
4. C₂²×C₄⋊(C₁₁⋊C₈)	C₂	semidirect
18. #10622612	C₂²	direct (N × H)
19. C₂×C₄⋊(C₁₁⋊C₈)	C₂²	semidirect
42. C₄⋊(C₁₁⋊C₈)	C₂³	semidirect
113. C₂²×C₁₁	#424	semidirect
147. C₂×C₁₁	C₂×#424	semidirect
177. C₁₁	C₂²×#424	semidirect
201. C₂²	#10622612	direct (N × H)
212. C₂	C₂×#10622612	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#4318	2
C₂⁴×C₁₁⋊C₈	2
C₂×#10622612	2
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂²×C₁₁×#117	2
C₂²×#424	11

Subgroups — C22×#10622612 #15617389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10622612 #15617389