Subgroups — C₂³×C₁₁×C₄⋊Q₈ #14639858

Order 2816.

12190 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₁₁	C₂	1	2	yes	maximal
2	C₂²×C₁₁×C₄⋊Q₈	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₁₁×C₄⋊C₄	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₁	—	2	32	no	—
6	C₂³×Q₈×C₁₁	C₂²	1	56	yes	—
7	C₂⁴×C₄×C₁₁	C₂²	1	4	yes	—
8	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
10	C₂×C₁₁×C₄⋊Q₈	C₂²	1	448	yes	direct product factor
11	C₂²×C₁₁×C₄⋊C₄	C₂²	1	112	yes	—
12	C₂²×C₄²×C₁₁	C₂²	1	28	yes	—
13	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
14	C₂²×Q₈×C₁₁	—	2	448	no	—
15	C₂²×Q₈×C₁₁	C₂³	1	224	yes	—
16	C₂³×C₄×C₁₁	—	2	32	no	—
17	C₂³×C₄×C₁₁	D₄	1	16	yes	—
18	C₂³×C₄×C₁₁	Q₈	1	32	yes	complemented
19	C₂³×C₄×C₁₁	C₂³	1	56	yes	—
20	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
21	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
22	C₁₁×C₄⋊Q₈	C₂³	1	512	yes	direct product factor
23	C₂×C₁₁×C₄⋊C₄	C₂³	1	448	yes	—
24	C₂×C₄²×C₁₁	C₂³	1	112	yes	—
25	C₂³×C₄⋊Q₈	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
27	C₂⁴×C₁₁	C₂×Q₈	1	16	yes	—
28	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
29	C₂×Q₈×C₁₁	—	2	896	no	—
30	C₂×Q₈×C₁₁	C₂⁴	1	128	yes	—
31	C₂²×C₄×C₁₁	—	2	224	no	—
32	C₂²×C₄×C₁₁	C₂×D₄	1	112	yes	—
33	C₂²×C₄×C₁₁	C₂×Q₈	1	224	yes	complemented
34	C₂²×C₄×C₁₁	C₂⁴	1	112	yes	—
35	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
36	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
37	C₁₁×C₄⋊C₄	C₂⁴	1	256	yes	—
38	C₄²×C₁₁	C₂⁴	1	64	yes	—
39	C₂⁴×Q₈	C₂×C₁₁	1	2	yes	—
40	C₂²×C₄⋊Q₈	C₂×C₁₁	1	56	yes	direct product factor
41	C₂³×C₄⋊C₄	C₂×C₁₁	1	4	yes	—
42	C₂³×C₄²	C₂×C₁₁	1	1	yes	characteristic
43	C₂³×C₁₁	C₄⋊Q₈	1	64	yes	direct product factor
44	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
45	C₂³×C₁₁	C₂²×Q₈	1	56	yes	—
46	C₂³×C₁₁	C₂⁵	1	7	yes	—
47	Q₈×C₁₁	—	2	256	no	—
48	C₂×C₄×C₁₁	—	2	224	no	—
49	C₂×C₄×C₁₁	C₂²×D₄	1	112	yes	—
50	C₂×C₄×C₁₁	C₂²×Q₈	1	224	yes	complemented
51	C₂×C₄×C₁₁	C₂⁵	1	32	yes	—
52	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
53	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
54	C₂³×Q₈	—	2	32	no	—
55	C₂³×Q₈	C₂²×C₁₁	1	56	yes	—
56	C₂⁴×C₄	C₂²×C₁₁	1	4	yes	—
57	C₂⁴×C₄	C₂²×C₁₁	1	1	yes	characteristic
58	C₂⁴×C₄	C₂²×C₁₁	1	2	yes	—
59	C₂×C₄⋊Q₈	C₂²×C₁₁	1	448	yes	direct product factor
60	C₂²×C₄⋊C₄	C₂²×C₁₁	1	112	yes	—
61	C₂²×C₄²	C₂²×C₁₁	1	28	yes	—
62	C₂²×C₁₁	C₂×C₄⋊Q₈	1	112	yes	direct product factor
63	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
64	C₂²×C₁₁	C₂³×Q₈	1	28	yes	—
65	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
66	C₄×C₁₁	—	2	32	no	—
67	C₄×C₁₁	C₂³×D₄	1	16	yes	—
68	C₄×C₁₁	C₂³×Q₈	1	32	yes	complemented
69	C₂⁵	C₂³×C₁₁	1	1	yes	characteristic
70	C₂²×Q₈	—	2	448	no	—
71	C₂²×Q₈	C₂³×C₁₁	1	224	yes	—
72	C₂³×C₄	—	2	32	no	—
73	C₂³×C₄	D₄×C₁₁	1	16	yes	—
74	C₂³×C₄	Q₈×C₁₁	1	32	yes	complemented
75	C₂³×C₄	C₂³×C₁₁	1	56	yes	—
76	C₂³×C₄	C₂³×C₁₁	1	14	yes	—
77	C₂³×C₄	C₂³×C₁₁	1	28	yes	—
78	C₄⋊Q₈	C₂³×C₁₁	1	512	yes	direct product factor
79	C₂×C₄⋊C₄	C₂³×C₁₁	1	448	yes	—
80	C₂×C₄²	C₂³×C₁₁	1	112	yes	—
81	C₂×C₁₁	C₂²×C₄⋊Q₈	1	28	yes	direct product factor
82	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
83	C₂×C₁₁	C₂⁴×Q₈	1	2	yes	—
84	C₂⁴	C₂×D₄×C₁₁	1	8	yes	—
85	C₂⁴	C₂×Q₈×C₁₁	1	16	yes	—
86	C₂⁴	C₂⁴×C₁₁	1	7	yes	—
87	C₂×Q₈	—	2	896	no	—
88	C₂×Q₈	C₂⁴×C₁₁	1	128	yes	—
89	C₂²×C₄	—	2	224	no	—
90	C₂²×C₄	C₂×D₄×C₁₁	1	112	yes	—
91	C₂²×C₄	C₂×Q₈×C₁₁	1	224	yes	complemented
92	C₂²×C₄	C₂⁴×C₁₁	1	112	yes	—
93	C₂²×C₄	C₂⁴×C₁₁	1	28	yes	—
94	C₂²×C₄	C₂⁴×C₁₁	1	56	yes	—
95	C₄⋊C₄	C₂⁴×C₁₁	1	256	yes	—
96	C₄²	C₂⁴×C₁₁	1	64	yes	—
97	C₁₁	C₂³×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
98	C₂³	C₁₁×C₄⋊Q₈	1	64	yes	direct product factor
99	C₂³	C₂²×D₄×C₁₁	1	28	yes	—
100	C₂³	C₂²×Q₈×C₁₁	1	56	yes	—
101	C₂³	C₂⁵×C₁₁	1	7	yes	—
102	Q₈	—	2	256	no	—
103	C₂×C₄	—	2	224	no	—
104	C₂×C₄	C₂²×D₄×C₁₁	1	112	yes	—
105	C₂×C₄	C₂²×Q₈×C₁₁	1	224	yes	complemented
106	C₂×C₄	C₂⁵×C₁₁	1	32	yes	—
107	C₂×C₄	C₂⁵×C₁₁	1	8	yes	—
108	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
109	C₂²	C₂×C₁₁×C₄⋊Q₈	1	112	yes	direct product factor
110	C₂²	C₂³×D₄×C₁₁	1	14	yes	—
111	C₂²	C₂³×Q₈×C₁₁	1	28	yes	—
112	C₂²	C₂⁶×C₁₁	1	1	yes	characteristic
113	C₄	—	2	32	no	—
114	C₄	C₂³×D₄×C₁₁	1	16	yes	—
115	C₄	C₂³×Q₈×C₁₁	1	32	yes	complemented
116	C₂	C₂²×C₁₁×C₄⋊Q₈	1	28	yes	direct product factor
117	C₂	C₂⁴×D₄×C₁₁	1	1	yes	characteristic
118	C₂	C₂⁴×Q₈×C₁₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₁₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₁×C₄⋊Q₈	C₂	direct (N × H)
10. C₂×C₁₁×C₄⋊Q₈	C₂²	direct (N × H)
18. C₂³×C₄×C₁₁	Q₈	semidirect
22. C₁₁×C₄⋊Q₈	C₂³	direct (N × H)
25. C₂³×C₄⋊Q₈	C₁₁	direct (N × H)
33. C₂²×C₄×C₁₁	C₂×Q₈	semidirect
40. C₂²×C₄⋊Q₈	C₂×C₁₁	direct (N × H)
43. C₂³×C₁₁	C₄⋊Q₈	direct (N × H)
50. C₂×C₄×C₁₁	C₂²×Q₈	semidirect
59. C₂×C₄⋊Q₈	C₂²×C₁₁	direct (N × H)
62. C₂²×C₁₁	C₂×C₄⋊Q₈	direct (N × H)
68. C₄×C₁₁	C₂³×Q₈	semidirect
74. C₂³×C₄	Q₈×C₁₁	semidirect
78. C₄⋊Q₈	C₂³×C₁₁	direct (N × H)
81. C₂×C₁₁	C₂²×C₄⋊Q₈	direct (N × H)
91. C₂²×C₄	C₂×Q₈×C₁₁	semidirect
97. C₁₁	C₂³×C₄⋊Q₈	direct (N × H)
98. C₂³	C₁₁×C₄⋊Q₈	direct (N × H)
105. C₂×C₄	C₂²×Q₈×C₁₁	semidirect
109. C₂²	C₂×C₁₁×C₄⋊Q₈	direct (N × H)
115. C₄	C₂³×Q₈×C₁₁	semidirect
116. C₂	C₂²×C₁₁×C₄⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₁₁	2
C₂²×C₁₁×C₄⋊Q₈	2
C₂³×C₁₁×C₄⋊C₄	2
C₂³×C₄²×C₁₁	2
C₂³×C₄⋊Q₈	11

Subgroups — C23×C11×C4⋊Q8 #14639858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₁×C₄⋊Q₈ #14639858