Group #33608 — subgroups

Subgroups — #33608

Order 256.

325 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504	C₂	1	1	yes	maximal, characteristic
3	#3059	C₂	1	2	yes	maximal
4	#2219	C₂	1	4	yes	maximal, complemented
5	#2216	C₂	1	4	yes	maximal, complemented
6	#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	#2141	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆	C₂²	1	2	yes	—
9	C₂×#131	C₂²	1	1	yes	characteristic
10	#536	—	2	2	no	—
11	C₂×Q₃₂	—	2	4	no	—
12	C₂×Q₃₂	C₂²	1	2	yes	—
13	C₂×Q₃₂	C₂²	1	2	yes	—
14	#504	—	2	4	no	—
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×M₃₂	C₂²	1	2	yes	—
17	C₂²×C₁₆	C₂²	1	2	yes	—
18	C₂²×C₁₆	C₂²	1	1	yes	characteristic
19	#470	—	2	4	no	—
20	#467	C₂²	1	8	yes	—
21	#427	C₂²	1	2	yes	—
22	#419	—	2	2	no	—
23	#367	C₂²	1	2	yes	—
24	#366	C₂²	1	2	yes	—
25	#358	C₂²	1	8	yes	—
26	C₂×#41	D₄	1	1	yes	characteristic
27	C₂²×Q₈	—	2	2	no	—
28	C₂×Q₁₆	—	2	4	no	—
29	C₂×Q₁₆	—	2	2	no	—
30	C₂×Q₁₆	—	2	4	no	—
31	C₂×Q₁₆	C₂³	1	4	yes	—
32	#131	—	2	2	no	—
33	#131	—	2	2	no	—
34	C₂×M₁₆	D₄	1	1	yes	characteristic
35	C₂×M₁₆	D₄	1	2	yes	—
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈	—	4	2	no	—
39	#124	—	4	2	no	—
40	#123	—	4	4	no	—
41	C₂²⋊Q₈	—	2	8	no	—
42	#117	—	2	2	no	—
43	Q₃₂	—	2	4	no	—
44	Q₃₂	—	2	4	no	—
45	M₃₂	—	2	2	no	—
46	M₃₂	—	2	4	no	—
47	C₂×C₁₆	—	2	4	no	—
48	C₂×C₁₆	D₄	1	4	yes	—
49	C₂×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₆	C₂³	1	1	yes	characteristic
52	#107	C₂³	1	4	yes	—
53	#106	—	2	2	no	—
54	#103	—	2	8	no	—
55	#41	—	2	2	no	—
56	#41	—	2	2	no	—
57	C₂×Q₈	—	2	4	no	—
58	C₂×Q₈	—	4	2	no	—
59	C₂×Q₈	—	4	4	no	—
60	C₂×Q₈	—	4	4	no	—
61	C₂×Q₈	D₈	1	1	yes	characteristic
62	C₂×D₄	D₈	1	2	yes	—
63	C₂×D₄	D₈	1	1	yes	characteristic
64	C₂²×C₄	—	4	2	no	—
65	C₂²×C₄	C₂×D₄	1	2	yes	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	Q₁₆	—	2	4	no	—
68	Q₁₆	—	2	4	no	—
69	Q₁₆	—	4	4	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	2	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₈	#41	1	2	yes	—
78	C₂×C₈	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	8	no	—
81	C₂²⋊C₄	—	4	2	no	—
82	C₂²⋊C₄	—	4	8	no	—
83	C₄²	—	4	2	no	—
84	C₁₆	—	2	2	no	—
85	C₁₆	C₂×D₄	1	2	yes	—
86	C₁₆	C₂×D₄	1	2	yes	—
87	C₂³	C₂×D₈	1	2	yes	—
88	C₂³	C₂×D₈	1	1	yes	characteristic
89	Q₈	—	2	2	no	—
90	Q₈	—	4	4	no	—
91	Q₈	—	4	4	no	—
92	Q₈	—	8	4	no	—
93	D₄	—	2	4	no	—
94	D₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	8	4	no	—
100	C₂×C₄	#121	1	2	yes	—
101	C₂×C₄	C₂×D₈	1	2	yes	—
102	C₂×C₄	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄	#135	1	2	yes	—
104	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	#121	1	2	yes	—
107	C₈	C₂²×D₄	1	1	yes	characteristic
108	C₈	C₂×#41	1	1	yes	characteristic
109	C₂²	—	2	2	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	#466	1	2	yes	—
113	C₂²	C₂²×D₈	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	8	4	no	—
116	C₄	—	8	4	no	—
117	C₄	#466	1	2	yes	—
118	C₄	C₂×#121	1	1	yes	characteristic
119	C₄	C₂×#135	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	C₂×#466	1	1	yes	characteristic
123	C₂	#3417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#504

C₂

yes

maximal, characteristic

#3059

C₂

yes

maximal

#2219

C₂

yes

maximal, complemented

#2216

C₂

yes

maximal, complemented

#2158

C₂

yes

maximal, characteristic, complemented

#2141

C₂

yes

maximal, complemented

C₂²×Q₁₆

C₂²

yes

—

C₂×#131

C₂²

yes

characteristic

#536

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂²

yes

—

C₂×Q₃₂

C₂²

yes

—

#504

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

#470

—

#467

C₂²

yes

—

#427

C₂²

yes

—

#419

—

#367

C₂²

yes

—

#366

C₂²

yes

—

#358

C₂²

yes

—

C₂×#41

D₄

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂³

yes

—

#131

—

#131

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

#107

C₂³

yes

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

characteristic

C₂×D₄

D₈

yes

—

C₂×D₄

D₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

C₂×D₄

yes

—

C₁₆

C₂×D₄

yes

—

C₂³

C₂×D₈

yes

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#121

yes

—

101

C₂×C₄

C₂×D₈

yes

—

102

C₂×C₄

C₂×D₈

yes

characteristic

103

C₂×C₄

#135

yes

—

104

C₂×C₄

C₂²×D₄

yes

characteristic

105

C₈

—

106

C₈

#121

yes

—

107

C₈

C₂²×D₄

yes

characteristic

108

C₈

C₂×#41

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

#466

yes

—

113

C₂²

C₂²×D₈

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#466

yes

—

118

C₄

C₂×#121

yes

characteristic

119

C₄

C₂×#135

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

C₂×#466

yes

characteristic

123

C₂

#3417

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

C₂×#41

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂²×Q₃₂	C₂	semidirect
4. #2219	C₂	semidirect
5. #2216	C₂	semidirect
6. #2158	C₂	semidirect
7. #2141	C₂	semidirect

H ≅ G/N

type

1. C₂²×Q₃₂

C₂

semidirect

4. #2219

C₂

semidirect

5. #2216

C₂

semidirect

6. #2158

C₂

semidirect

7. #2141

C₂

semidirect

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

subgroup	index
C₂²×Q₃₂	2
C₂×#504	2
#3059	2
#2219	2
#2216	2
#2158	2
#2141	2

subgroup

index

C₂²×Q₃₂

Subgroups — #33608

Conjugacy classes of subgroups

Complements

Maximal subgroups