Group #33957 — subgroups

Subgroups — #33957

Order 256.

222 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 107 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	#3383	C₂	1	2	yes	maximal
2	#2255	C₂	1	2	yes	maximal
3	#2253	C₂	1	1	yes	maximal, characteristic
4	#2229	C₂	1	2	yes	maximal
5	#2229	C₂	1	2	yes	maximal
6	#2227	C₂	1	2	yes	maximal
7	#2183	C₂	1	1	yes	maximal, characteristic
8	#2176	C₂	1	2	yes	maximal
9	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
10	#557	—	2	2	no	—
11	C₂×Q₃₂	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	2	yes	—
13	#499	C₂²	1	2	yes	—
14	#499	C₂²	1	2	yes	—
15	#479	—	2	2	no	—
16	#479	—	2	2	no	—
17	#475	—	2	2	no	—
18	C₈×Q₈	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆	C₂²	1	2	yes	—
20	C₄×Q₁₆	C₂²	1	2	yes	—
21	C₄×Q₁₆	C₂²	1	2	yes	—
22	#367	C₂²	1	2	yes	—
23	#367	C₂²	1	2	yes	—
24	#366	C₂²	1	1	yes	characteristic
25	#366	C₂²	1	1	yes	characteristic
26	#366	C₂²	1	2	yes	—
27	#366	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	#358	C₂²	1	2	yes	—
31	#358	C₂²	1	2	yes	—
32	#358	C₂²	1	2	yes	—
33	#346	C₂²	1	2	yes	—
34	C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆	C₂³	1	2	yes	—
36	C₄⋊Q₈	—	2	2	no	—
37	C₄⋊Q₈	—	4	2	no	—
38	#125	—	2	2	no	—
39	#125	—	2	2	no	—
40	#125	—	4	2	no	—
41	#125	—	4	2	no	—
42	C₄×Q₈	—	2	2	no	—
43	C₄×Q₈	—	2	2	no	—
44	C₄×Q₈	—	2	2	no	—
45	C₄×Q₈	—	4	2	no	—
46	C₄×Q₈	D₄	1	1	yes	characteristic
47	Q₃₂	Q₈	1	4	yes	complemented
48	C₂×C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₆	C₂³	1	1	yes	characteristic
51	#107	—	2	2	no	—
52	#107	C₂³	1	2	yes	—
53	#107	C₂³	1	2	yes	—
54	#107	C₂³	1	2	yes	—
55	#106	—	2	2	no	—
56	#106	—	2	2	no	—
57	C₄⋊C₈	D₄	1	2	yes	—
58	C₄⋊C₈	D₄	1	1	yes	characteristic
59	#103	—	2	2	no	—
60	#103	—	2	2	no	—
61	#103	—	2	2	no	—
62	C₄×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₈	C₂³	1	2	yes	—
64	C₂×Q₈	—	2	2	no	—
65	C₂×Q₈	—	4	2	no	—
66	C₂×Q₈	D₈	1	1	yes	characteristic
67	Q₁₆	—	2	2	no	—
68	Q₁₆	C₂×Q₈	1	4	yes	—
69	C₂×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₈	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	D₈	1	2	yes	—
84	C₄⋊C₄	D₈	1	1	yes	characteristic
85	C₄²	—	4	2	no	—
86	C₄²	—	4	2	no	—
87	C₄²	—	4	2	no	—
88	C₄²	C₂×D₄	1	1	yes	characteristic
89	C₄²	C₂×D₄	1	2	yes	—
90	C₁₆	—	2	1	no	—
91	C₁₆	—	2	2	no	—
92	C₁₆	C₂×Q₈	1	2	yes	—
93	Q₈	—	2	4	no	—
94	Q₈	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	8	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄	C₂×D₈	1	2	yes	—
103	C₂×C₄	C₂×D₈	1	2	yes	—
104	C₂×C₄	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	2	2	no	—
108	C₈	C₂²×Q₈	1	1	yes	characteristic
109	C₈	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂²	C₂²×D₈	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	8	2	no	—
117	C₄	—	8	2	no	—
118	C₄	—	8	2	no	—
119	C₄	D₄×Q₈	1	1	yes	characteristic
120	C₄	#578	1	1	yes	characteristic
121	C₂	Q₈×D₈	1	1	yes	characteristic
122	C₂	#3414	1	1	yes	characteristic
123	C₂	#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3383

C₂

yes

maximal

#2255

C₂

yes

maximal

#2253

C₂

yes

maximal, characteristic

#2229

C₂

yes

maximal

#2229

C₂

yes

maximal

#2227

C₂

yes

maximal

#2183

C₂

yes

maximal, characteristic

#2176

C₂

yes

maximal

C₄×Q₃₂

C₂

yes

maximal, characteristic

#557

—

C₂×Q₃₂

C₂²

yes

characteristic

#500

C₂²

yes

—

#499

C₂²

yes

—

#499

C₂²

yes

—

#479

—

#479

—

#475

—

C₈×Q₈

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

#367

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

#366

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

characteristic

#358

C₂²

yes

—

#358

C₂²

yes

—

#358

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂³

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

characteristic

Q₃₂

Q₈

yes

complemented

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#106

—

#106

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

—

—

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

characteristic

Q₁₆

—

Q₁₆

C₂×Q₈

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₄⋊C₄

D₈

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₂×Q₈

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×D₈

yes

characteristic

102

C₂×C₄

C₂×D₈

yes

—

103

C₂×C₄

C₂×D₈

yes

—

104

C₂×C₄

C₂×D₈

yes

characteristic

105

C₂×C₄

C₂²×D₄

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

C₂²×Q₈

yes

characteristic

109

C₈

2₋¹⁺⁴

yes

characteristic

110

C₂²

C₂²×D₈

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

D₄×Q₈

yes

characteristic

120

C₄

#578

yes

characteristic

121

C₂

Q₈×D₈

yes

characteristic

122

C₂

#3414

yes

characteristic

123

C₂

#3417

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
47. Q₃₂	Q₈	semidirect

H ≅ G/N

type

47. Q₃₂

Q₈

semidirect

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

subgroup	index
#3383	2
#2255	2
#2253	2
#2229	2
#2227	2
#2183	2
#2176	2
C₄×Q₃₂	2

subgroup

index

C₄×Q₃₂

Subgroups — #33957

Conjugacy classes of subgroups

Complements

Maximal subgroups