Group #36593 — subgroups

Subgroups — #36593

Order 256.

1081 conjugacy classes of subgroups; only the 595 normal subgroups are shown.

normal subgroups only

This group has 1081 conjugacy classes of subgroups — too many to draw. Only its 595 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	C₂×#557	C₂	1	2	yes	maximal
2	#2868	C₂	1	3	yes	maximal
3	#2552	C₂	1	12	yes	maximal
4	#2546	C₂	1	6	yes	maximal
5	Q₈×C₄⋊C₄	C₂	1	4	yes	maximal
6	C₄×C₄⋊Q₈	C₂	1	3	yes	maximal
7	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
8	#557	—	2	16	no	—
9	C₂×C₄⋊Q₈	C₂²	1	3	yes	—
10	C₂×C₄⋊Q₈	C₂²	1	6	yes	—
11	C₂×C₄⋊Q₈	C₂²	1	6	yes	—
12	C₂×#125	C₂²	1	12	yes	—
13	C₂×C₄×Q₈	C₂²	1	6	yes	—
14	C₂×C₄×Q₈	C₂²	1	4	yes	—
15	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈	C₂²	1	2	yes	—
17	#398	C₂²	1	24	yes	—
18	#391	C₂²	1	6	yes	—
19	#389	C₂²	1	12	yes	—
20	#389	C₂²	1	24	yes	—
21	#384	C₂²	1	1	yes	characteristic
22	#384	C₂²	1	12	yes	—
23	#384	C₂²	1	12	yes	—
24	C₄×C₄⋊C₄	C₂²	1	12	yes	—
25	C₄×C₄⋊C₄	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄	C₂²	1	6	yes	—
27	C₄³	C₂²	1	3	yes	—
28	C₂²×Q₈	C₂³	1	6	yes	—
29	C₂²×Q₈	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈	—	2	24	no	—
31	C₄⋊Q₈	—	2	12	no	—
32	C₄⋊Q₈	—	4	12	no	—
33	#125	—	2	48	no	—
34	C₄×Q₈	—	2	16	no	—
35	C₄×Q₈	—	2	24	no	—
36	C₄×Q₈	D₄	1	8	yes	—
37	C₄×Q₈	Q₈	1	16	yes	complemented
38	C₂×C₄⋊C₄	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄	C₂³	1	12	yes	—
40	C₂×C₄⋊C₄	C₂³	1	12	yes	—
41	C₂×C₄⋊C₄	C₂³	1	12	yes	—
42	C₂×C₄⋊C₄	C₂³	1	12	yes	—
43	C₂×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄	C₂³	1	24	yes	—
45	C₂×C₄⋊C₄	C₂³	1	3	yes	—
46	C₂×C₄⋊C₄	C₂³	1	6	yes	—
47	C₂×C₄²	C₂³	1	12	yes	—
48	C₂×C₄²	C₂³	1	3	yes	—
49	C₂×C₄²	C₂³	1	6	yes	—
50	C₂×C₄²	C₂³	1	3	yes	—
51	C₂×C₄²	C₂³	1	6	yes	—
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	#95	C₂³	1	6	yes	—
54	#95	C₂³	1	24	yes	—
55	C₂×Q₈	—	2	12	no	—
56	C₂×Q₈	—	2	12	no	—
57	C₂×Q₈	—	2	12	no	—
58	C₂×Q₈	C₂×D₄	1	4	yes	—
59	C₂×Q₈	C₂×Q₈	1	8	yes	—
60	C₂²×C₄	C₂⁴	1	4	yes	—
61	C₂²×C₄	C₂⁴	1	12	yes	—
62	C₂²×C₄	C₂⁴	1	3	yes	—
63	C₂²×C₄	C₂⁴	1	6	yes	—
64	C₂²×C₄	C₂⁴	1	3	yes	—
65	C₂²×C₄	C₂⁴	1	1	yes	characteristic
66	C₂²×C₄	C₂⁴	1	2	yes	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	—	2	24	no	—
69	C₄⋊C₄	—	2	24	no	—
70	C₄⋊C₄	—	2	24	no	—
71	C₄⋊C₄	—	2	24	no	—
72	C₄⋊C₄	—	2	8	no	—
73	C₄⋊C₄	—	4	24	no	—
74	C₄⋊C₄	C₂×D₄	1	12	yes	—
75	C₄⋊C₄	C₂×Q₈	1	24	yes	—
76	C₄²	—	2	24	no	—
77	C₄²	—	2	6	no	—
78	C₄²	—	2	12	no	—
79	C₄²	C₂×D₄	1	12	yes	—
80	C₄²	C₂×Q₈	1	24	yes	—
81	C₄²	#41	1	4	yes	—
82	C₂³	C₂⁵	1	1	yes	characteristic
83	Q₈	—	4	24	no	—
84	Q₈	C₄⋊Q₈	1	16	yes	complemented
85	C₂×C₄	—	2	6	no	—
86	C₂×C₄	—	2	6	no	—
87	C₂×C₄	—	2	6	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	12	no	—
90	C₂×C₄	—	2	12	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	C₂²×D₄	1	6	yes	—
94	C₂×C₄	C₂²×D₄	1	6	yes	—
95	C₂×C₄	C₂²×D₄	1	2	yes	—
96	C₂×C₄	C₂²×Q₈	1	12	yes	—
97	C₂×C₄	C₂²×Q₈	1	12	yes	—
98	C₂×C₄	C₂²×Q₈	1	4	yes	—
99	C₂×C₄	C₂×#41	1	2	yes	—
100	C₂×C₄	C₂×#41	1	4	yes	—
101	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
102	C₂×C₄	2₋¹⁺⁴	1	4	yes	—
103	C₂²	C₂³×D₄	1	1	yes	characteristic
104	C₂²	C₂³×Q₈	1	2	yes	—
105	C₂²	C₂²×#41	1	1	yes	characteristic
106	C₂²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²	C₂×2₋¹⁺⁴	1	2	yes	—
108	C₄	—	2	6	no	—
109	C₄	—	2	12	no	—
110	C₄	—	4	12	no	—
111	C₄	—	4	4	no	—
112	C₄	C₂×C₄⋊Q₈	1	12	yes	—
113	C₄	#550	1	4	yes	—
114	C₄	#557	1	8	yes	complemented
115	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
116	C₂	C₂×#550	1	1	yes	characteristic
117	C₂	C₂×#557	1	2	yes	—
118	C₂	#3499	1	1	yes	characteristic
119	C₂	#3503	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#557

C₂

yes

maximal

#2868

C₂

yes

maximal

#2552

C₂

yes

maximal

#2546

C₂

yes

maximal

Q₈×C₄⋊C₄

C₂

yes

maximal

C₄×C₄⋊Q₈

C₂

yes

maximal

C₄²×Q₈

C₂

yes

maximal, characteristic

#557

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×#125

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

#398

C₂²

yes

—

#391

C₂²

yes

—

#389

C₂²

yes

—

#389

C₂²

yes

—

#384

C₂²

yes

characteristic

#384

C₂²

yes

—

#384

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄³

C₂²

yes

—

C₂²×Q₈

C₂³

yes

—

C₂²×Q₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

—

C₄×Q₈

Q₈

yes

complemented

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#95

C₂³

yes

—

#95

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×Q₈

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

C₂²×C₄

C₂⁴

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×Q₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

C₂⁵

yes

characteristic

Q₈

—

Q₈

C₄⋊Q₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₄

yes

—

C₂×C₄

C₂²×D₄

yes

—

C₂×C₄

C₂²×D₄

yes

—

C₂×C₄

C₂²×Q₈

yes

—

C₂×C₄

C₂²×Q₈

yes

—

C₂×C₄

C₂²×Q₈

yes

—

C₂×C₄

C₂×#41

yes

—

100

C₂×C₄

C₂×#41

yes

—

101

C₂×C₄

H(5,𝔽₂)

yes

—

102

C₂×C₄

2₋¹⁺⁴

yes

—

103

C₂²

C₂³×D₄

yes

characteristic

104

C₂²

C₂³×Q₈

yes

—

105

C₂²

C₂²×#41

yes

characteristic

106

C₂²

C₂×H(5,𝔽₂)

yes

characteristic

107

C₂²

C₂×2₋¹⁺⁴

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₂×C₄⋊Q₈

yes

—

113

C₄

#550

yes

—

114

C₄

#557

yes

complemented

115

C₂

C₂²×C₄⋊Q₈

yes

characteristic

116

C₂

C₂×#550

yes

characteristic

117

C₂

C₂×#557

yes

—

118

C₂

#3499

yes

characteristic

119

C₂

#3503

yes

—

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

subgroup

G/H

center, commutator, Frattini, socle, prime radical, cosocle

C₂³

C₂⁵

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
37. C₄×Q₈	Q₈	semidirect
84. Q₈	C₄⋊Q₈	semidirect
114. C₄	#557	semidirect

H ≅ G/N

type

37. C₄×Q₈

Q₈

semidirect

84. Q₈

C₄⋊Q₈

semidirect

114. C₄

#557

semidirect

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

subgroup	index
C₂×#557	2
#2868	2
#2552	2
#2546	2
Q₈×C₄⋊C₄	2
C₄×C₄⋊Q₈	2
C₄²×Q₈	2

subgroup

index

Q₈×C₄⋊C₄

C₄×C₄⋊Q₈

C₄²×Q₈

Subgroups — #36593

Conjugacy classes of subgroups

Complements

Maximal subgroups