Group #11768198 — subgroups

Subgroups — C₂×Q₈²×C₉ #11768198

Order 1152.

1668 conjugacy classes of subgroups; only the 1452 normal subgroups are shown.

normal subgroups only

This group has 1668 conjugacy classes of subgroups — too many to draw. Only its 1452 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₉	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₉×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₂×C₄×Q₈×C₉	C₂	1	6	yes	maximal
4	C₂×C₃×Q₈²	C₃	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₉	C₂²	1	6	yes	—
6	C₂²×Q₈×C₉	C₂²	1	2	yes	—
7	C₉×C₄⋊Q₈	C₂²	1	72	yes	—
8	C₄×Q₈×C₉	C₂²	1	48	yes	—
9	C₂×C₉×C₄⋊C₄	C₂²	1	18	yes	—
10	C₂×C₄²×C₉	C₂²	1	9	yes	—
11	C₃×Q₈²	C₂×C₃	1	16	yes	—
12	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	9	yes	—
13	C₂×C₃×C₄×Q₈	C₂×C₃	1	6	yes	—
14	C₂×Q₈×C₉	—	4	12	no	—
15	C₂×Q₈×C₉	Q₈	1	16	yes	direct product factor
16	C₂×Q₈×C₉	C₂³	1	24	yes	—
17	C₂×Q₈×C₉	C₂³	1	8	yes	—
18	C₂²×C₄×C₉	C₂³	1	9	yes	—
19	C₂²×C₄×C₉	C₂³	1	6	yes	—
20	C₉×C₄⋊C₄	C₂³	1	72	yes	—
21	C₄²×C₉	C₂³	1	36	yes	—
22	C₂×Q₈²	C₉	1	1	yes	characteristic, direct product factor, Sylow
23	C₂²×C₃×Q₈	C₂²×C₃	1	6	yes	—
24	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
25	C₃×C₄⋊Q₈	C₂²×C₃	1	72	yes	—
26	C₃×C₄×Q₈	C₂²×C₃	1	48	yes	—
27	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	18	yes	—
28	C₂×C₃×C₄²	C₂²×C₃	1	9	yes	—
29	C₂³×C₉	C₂⁴	1	1	yes	characteristic
30	Q₈×C₉	—	4	24	no	—
31	Q₈×C₉	C₂×Q₈	1	32	yes	direct product factor
32	C₂×C₄×C₉	—	2	18	no	—
33	C₂×C₄×C₉	C₂×Q₈	1	24	yes	—
34	C₂×C₄×C₉	C₂⁴	1	18	yes	—
35	C₂×C₄×C₉	C₂⁴	1	12	yes	—
36	Q₈²	C₂×C₉	1	16	yes	direct product factor
37	C₂×C₄⋊Q₈	C₂×C₉	1	9	yes	—
38	C₂×C₄×Q₈	C₂×C₉	1	6	yes	—
39	C₂×C₃×Q₈	—	4	12	no	—
40	C₂×C₃×Q₈	C₃×Q₈	1	16	yes	—
41	C₂×C₃×Q₈	C₂³×C₃	1	24	yes	—
42	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
43	C₂²×C₃×C₄	C₂³×C₃	1	9	yes	—
44	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
45	C₃×C₄⋊C₄	C₂³×C₃	1	72	yes	—
46	C₃×C₄²	C₂³×C₃	1	36	yes	—
47	C₂²×C₉	C₂²×Q₈	1	4	yes	—
48	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
49	C₂²×C₉	C₂⁵	1	1	yes	characteristic
50	C₄×C₉	—	2	18	no	—
51	C₄×C₉	C₂²×Q₈	1	24	yes	—
52	C₂²×Q₈	C₂²×C₉	1	6	yes	—
53	C₂²×Q₈	C₂²×C₉	1	2	yes	—
54	C₄⋊Q₈	C₂²×C₉	1	72	yes	—
55	C₄×Q₈	C₂²×C₉	1	48	yes	—
56	C₂×C₄⋊C₄	C₂²×C₉	1	18	yes	—
57	C₂×C₄²	C₂²×C₉	1	9	yes	—
58	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
59	C₃×Q₈	—	4	24	no	—
60	C₃×Q₈	C₂×C₃×Q₈	1	32	yes	—
61	C₂×C₃×C₄	—	2	18	no	—
62	C₂×C₃×C₄	C₂×C₃×Q₈	1	24	yes	—
63	C₂×C₃×C₄	C₂⁴×C₃	1	18	yes	—
64	C₂×C₃×C₄	C₂⁴×C₃	1	12	yes	—
65	C₂×C₉	Q₈²	1	4	yes	direct product factor
66	C₂×C₉	C₂³×Q₈	1	2	yes	—
67	C₂×C₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
68	C₂×Q₈	—	4	12	no	—
69	C₂×Q₈	Q₈×C₉	1	16	yes	direct product factor
70	C₂×Q₈	C₂³×C₉	1	24	yes	—
71	C₂×Q₈	C₂³×C₉	1	8	yes	—
72	C₂²×C₄	C₂³×C₉	1	9	yes	—
73	C₂²×C₄	C₂³×C₉	1	6	yes	—
74	C₄⋊C₄	C₂³×C₉	1	72	yes	—
75	C₄²	C₂³×C₉	1	36	yes	—
76	C₂²×C₃	C₂²×C₃×Q₈	1	4	yes	—
77	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
79	C₃×C₄	—	2	18	no	—
80	C₃×C₄	C₂²×C₃×Q₈	1	24	yes	—
81	C₉	C₂×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
82	C₂³	C₂⁴×C₉	1	1	yes	characteristic
83	Q₈	—	4	24	no	—
84	Q₈	C₂×Q₈×C₉	1	32	yes	direct product factor
85	C₂×C₄	—	2	18	no	—
86	C₂×C₄	C₂×Q₈×C₉	1	24	yes	—
87	C₂×C₄	C₂⁴×C₉	1	18	yes	—
88	C₂×C₄	C₂⁴×C₉	1	12	yes	—
89	C₂×C₃	C₃×Q₈²	1	4	yes	—
90	C₂×C₃	C₂³×C₃×Q₈	1	2	yes	—
91	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
92	C₂²	C₂²×Q₈×C₉	1	4	yes	—
93	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
94	C₂²	C₂⁵×C₉	1	1	yes	characteristic
95	C₄	—	2	18	no	—
96	C₄	C₂²×Q₈×C₉	1	24	yes	—
97	C₃	C₂×C₃×Q₈²	1	1	yes	characteristic
98	C₂	Q₈²×C₉	1	4	yes	direct product factor
99	C₂	C₂³×Q₈×C₉	1	2	yes	—
100	C₂	C₂×C₉×H(5,𝔽₂)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₉

C₂

yes

maximal, direct product factor

C₂×C₉×C₄⋊Q₈

C₂

yes

maximal

C₂×C₄×Q₈×C₉

C₂

yes

maximal

C₂×C₃×Q₈²

C₃

yes

maximal, characteristic

C₂²×Q₈×C₉

C₂²

yes

—

C₂²×Q₈×C₉

C₂²

yes

—

C₉×C₄⋊Q₈

C₂²

yes

—

C₄×Q₈×C₉

C₂²

yes

—

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

—

C₃×Q₈²

C₂×C₃

yes

—

C₂×C₃×C₄⋊Q₈

C₂×C₃

yes

—

C₂×C₃×C₄×Q₈

C₂×C₃

yes

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

Q₈

yes

direct product factor

C₂×Q₈×C₉

C₂³

yes

—

C₂×Q₈×C₉

C₂³

yes

—

C₂²×C₄×C₉

C₂³

yes

—

C₂²×C₄×C₉

C₂³

yes

—

C₉×C₄⋊C₄

C₂³

yes

—

C₄²×C₉

C₂³

yes

—

C₂×Q₈²

C₉

yes

characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²×C₃

yes

—

C₂²×C₃×Q₈

C₂²×C₃

yes

—

C₃×C₄⋊Q₈

C₂²×C₃

yes

—

C₃×C₄×Q₈

C₂²×C₃

yes

—

C₂×C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₃×C₄²

C₂²×C₃

yes

—

C₂³×C₉

C₂⁴

yes

characteristic

Q₈×C₉

—

Q₈×C₉

C₂×Q₈

yes

direct product factor

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×Q₈

yes

—

C₂×C₄×C₉

C₂⁴

yes

—

C₂×C₄×C₉

C₂⁴

yes

—

Q₈²

C₂×C₉

yes

direct product factor

C₂×C₄⋊Q₈

C₂×C₉

yes

—

C₂×C₄×Q₈

C₂×C₉

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×Q₈

yes

—

C₂×C₃×Q₈

C₂³×C₃

yes

—

C₂×C₃×Q₈

C₂³×C₃

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

—

C₃×C₄⋊C₄

C₂³×C₃

yes

—

C₃×C₄²

C₂³×C₃

yes

—

C₂²×C₉

C₂²×Q₈

yes

—

C₂²×C₉

H(5,𝔽₂)

yes

—

C₂²×C₉

C₂⁵

yes

characteristic

C₄×C₉

—

C₄×C₉

C₂²×Q₈

yes

—

C₂²×Q₈

C₂²×C₉

yes

—

C₂²×Q₈

C₂²×C₉

yes

—

C₄⋊Q₈

C₂²×C₉

yes

—

C₄×Q₈

C₂²×C₉

yes

—

C₂×C₄⋊C₄

C₂²×C₉

yes

—

C₂×C₄²

C₂²×C₉

yes

—

C₂³×C₃

C₂⁴×C₃

yes

characteristic

C₃×Q₈

—

C₃×Q₈

C₂×C₃×Q₈

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×Q₈

yes

—

C₂×C₃×C₄

C₂⁴×C₃

yes

—

C₂×C₃×C₄

C₂⁴×C₃

yes

—

C₂×C₉

Q₈²

yes

direct product factor

C₂×C₉

C₂³×Q₈

yes

—

C₂×C₉

C₂×H(5,𝔽₂)

yes

characteristic

C₂×Q₈

—

C₂×Q₈

Q₈×C₉

yes

direct product factor

C₂×Q₈

C₂³×C₉

yes

—

C₂×Q₈

C₂³×C₉

yes

—

C₂²×C₄

C₂³×C₉

yes

—

C₂²×C₄

C₂³×C₉

yes

—

C₄⋊C₄

C₂³×C₉

yes

—

C₄²

C₂³×C₉

yes

—

C₂²×C₃

C₂²×C₃×Q₈

yes

—

C₂²×C₃

C₃×H(5,𝔽₂)

yes

—

C₂²×C₃

C₂⁵×C₃

yes

characteristic

C₃×C₄

—

C₃×C₄

C₂²×C₃×Q₈

yes

—

C₉

C₂×Q₈²

yes

characteristic, direct product factor, Sylow

C₂³

C₂⁴×C₉

yes

characteristic

Q₈

—

Q₈

C₂×Q₈×C₉

yes

direct product factor

C₂×C₄

—

C₂×C₄

C₂×Q₈×C₉

yes

—

C₂×C₄

C₂⁴×C₉

yes

—

C₂×C₄

C₂⁴×C₉

yes

—

C₂×C₃

C₃×Q₈²

yes

—

C₂×C₃

C₂³×C₃×Q₈

yes

—

C₂×C₃

C₂×C₃×H(5,𝔽₂)

yes

characteristic

C₂²

C₂²×Q₈×C₉

yes

—

C₂²

C₉×H(5,𝔽₂)

yes

—

C₂²

C₂⁵×C₉

yes

characteristic

C₄

—

C₄

C₂²×Q₈×C₉

yes

—

C₃

C₂×C₃×Q₈²

yes

characteristic

C₂

Q₈²×C₉

yes

direct product factor

C₂

C₂³×Q₈×C₉

yes

—

100

C₂

C₂×C₉×H(5,𝔽₂)

yes

characteristic

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle, prime radical	C₂³×C₃	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

subgroup

G/H

center

C₂³×C₉

C₂⁴

commutator

C₂²

C₂⁵×C₉

Frattini, cosocle

C₂²×C₃

C₂⁵×C₃

socle, prime radical

C₂³×C₃

C₂⁴×C₃

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₉ ◁ G

N	H ≅ G/N	type
1. Q₈²×C₉	C₂	direct (N × H)
15. C₂×Q₈×C₉	Q₈	direct (N × H)
22. C₂×Q₈²	C₉	direct (N × H)
31. Q₈×C₉	C₂×Q₈	direct (N × H)
36. Q₈²	C₂×C₉	direct (N × H)
65. C₂×C₉	Q₈²	direct (N × H)
69. C₂×Q₈	Q₈×C₉	direct (N × H)
81. C₉	C₂×Q₈²	direct (N × H)
84. Q₈	C₂×Q₈×C₉	direct (N × H)
98. C₂	Q₈²×C₉	direct (N × H)

H ≅ G/N

type

1. Q₈²×C₉

C₂

direct (N × H)

15. C₂×Q₈×C₉

Q₈

direct (N × H)

22. C₂×Q₈²

C₉

direct (N × H)

31. Q₈×C₉

C₂×Q₈

direct (N × H)

36. Q₈²

C₂×C₉

direct (N × H)

65. C₂×C₉

Q₈²

direct (N × H)

69. C₂×Q₈

Q₈×C₉

direct (N × H)

81. C₉

C₂×Q₈²

direct (N × H)

84. Q₈

C₂×Q₈×C₉

direct (N × H)

98. C₂

Q₈²×C₉

direct (N × H)

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

subgroup	index
Q₈²×C₉	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₄×Q₈×C₉	2
C₂×C₃×Q₈²	3

subgroup

index

Q₈²×C₉

C₂×C₉×C₄⋊Q₈

C₂×C₄×Q₈×C₉

C₂×C₃×Q₈²

Subgroups — C₂×Q₈²×C₉ #11768198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Q82×C9 #11768198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈²×C₉ #11768198