Group #33658 — subgroups

Subgroups — #33658

Order 256.

267 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3059	C₂	1	1	yes	maximal, characteristic
2	#2916	C₂	1	1	yes	maximal, characteristic
3	#2239	C₂	1	8	yes	maximal, complemented
4	#2155	C₂	1	2	yes	maximal, complemented
5	C₂×#362	C₂	1	2	yes	maximal, complemented
6	#2115	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
8	#568	—	2	1	no	—
9	C₂×#131	C₂²	1	1	yes	characteristic
10	#536	—	2	1	no	—
11	C₂×M₃₂	C₂²	1	2	yes	—
12	#470	C₂²	1	4	yes	—
13	#467	—	2	2	no	—
14	#430	—	2	1	no	—
15	#430	C₂²	1	4	yes	—
16	C₂×#108	—	2	2	no	—
17	C₂×#108	C₂²	1	2	yes	—
18	#427	C₂²	1	1	yes	characteristic
19	#419	—	2	1	no	—
20	#365	C₂²	1	8	yes	—
21	#362	C₂²	1	8	yes	—
22	#349	C₂²	1	4	yes	—
23	C₂×#41	D₄	1	1	yes	characteristic
24	C₂²×Q₈	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×Q₁₆	—	2	2	no	—
27	C₂×Q₁₆	—	2	2	no	—
28	C₂×Q₁₆	C₂³	1	2	yes	—
29	#131	—	2	2	no	—
30	#131	—	2	2	no	—
31	#131	—	4	4	no	—
32	C₂×M₁₆	—	2	4	no	—
33	C₂×M₁₆	—	2	2	no	—
34	C₂×M₁₆	D₄	1	1	yes	characteristic
35	C₂×M₁₆	C₂³	1	2	yes	—
36	C₂²×C₈	D₄	1	2	yes	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈	—	4	1	no	—
39	#124	—	4	1	no	—
40	#123	—	4	2	no	—
41	C₂²⋊Q₈	—	2	4	no	—
42	#117	—	2	1	no	—
43	M₃₂	—	2	2	no	—
44	M₃₂	C₂³	1	4	yes	—
45	C₂×C₁₆	—	2	2	no	—
46	#108	—	2	2	no	—
47	#108	—	2	4	no	—
48	#108	C₂³	1	4	yes	—
49	#107	—	2	1	no	—
50	#106	C₂³	1	2	yes	—
51	#103	—	2	4	no	—
52	#41	—	2	2	no	—
53	#41	—	2	2	no	—
54	C₂×Q₈	—	2	2	no	—
55	C₂×Q₈	—	4	1	no	—
56	C₂×Q₈	—	4	2	no	—
57	C₂×Q₈	—	4	2	no	—
58	C₂×Q₈	SD₁₆	1	1	yes	characteristic
59	C₂×D₄	SD₁₆	1	2	yes	—
60	C₂×D₄	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄	—	4	1	no	—
62	C₂²×C₄	C₂×D₄	1	2	yes	—
63	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
64	Q₁₆	—	2	4	no	—
65	Q₁₆	—	4	1	no	—
66	Q₁₆	—	4	1	no	—
67	M₁₆	—	2	2	no	—
68	M₁₆	—	2	4	no	—
69	M₁₆	—	2	4	no	—
70	M₁₆	—	4	4	no	—
71	M₁₆	—	4	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	4	4	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈	#41	1	2	yes	—
79	C₂×C₈	#41	1	2	yes	—
80	C₂×C₈	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	4	4	no	—
83	C₂²⋊C₄	—	4	1	no	—
84	C₂²⋊C₄	—	4	4	no	—
85	C₄²	—	4	1	no	—
86	C₁₆	—	2	4	no	—
87	C₂³	C₂×SD₁₆	1	2	yes	—
88	C₂³	C₂×SD₁₆	1	1	yes	characteristic
89	Q₈	—	2	2	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	8	2	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	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	#123	1	2	yes	—
101	C₂×C₄	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	4	4	no	—
107	C₈	#123	1	1	yes	characteristic
108	C₈	#123	1	1	yes	characteristic
109	C₈	C₂×#41	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	#481	1	2	yes	—
114	C₂²	C₂²×SD₁₆	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	8	2	no	—
117	C₄	—	8	2	no	—
118	C₄	#481	1	2	yes	—
119	C₄	C₂×#123	1	1	yes	characteristic
120	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	2	no	—
123	C₂	—	4	2	no	—
124	C₂	C₂×#481	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3059

C₂

yes

maximal, characteristic

#2916

C₂

yes

maximal, characteristic

#2239

C₂

yes

maximal, complemented

#2155

C₂

yes

maximal, complemented

C₂×#362

C₂

yes

maximal, complemented

#2115

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

#568

—

C₂×#131

C₂²

yes

characteristic

#536

—

C₂×M₃₂

C₂²

yes

—

#470

C₂²

yes

—

#467

—

#430

—

#430

C₂²

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

—

#427

C₂²

yes

characteristic

#419

—

#365

C₂²

yes

—

#362

C₂²

yes

—

#349

C₂²

yes

—

C₂×#41

D₄

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂³

yes

—

#131

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

M₃₂

—

M₃₂

C₂³

yes

—

C₂×C₁₆

—

#108

—

#108

—

#108

C₂³

yes

—

#107

—

#106

C₂³

yes

—

#103

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

SD₁₆

yes

characteristic

C₂×D₄

SD₁₆

yes

—

C₂×D₄

SD₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂³

C₂×SD₁₆

yes

—

C₂³

C₂×SD₁₆

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#123

yes

—

101

C₂×C₄

C₂×SD₁₆

yes

—

102

C₂×C₄

C₂×SD₁₆

yes

characteristic

103

C₂×C₄

AGL(1,ℤ/8)

yes

—

104

C₂×C₄

C₂²×D₄

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

#123

yes

characteristic

108

C₈

#123

yes

characteristic

109

C₈

C₂×#41

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

#481

yes

—

114

C₂²

C₂²×SD₁₆

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#481

yes

—

119

C₄

C₂×#123

yes

characteristic

120

C₄

C₂×AGL(1,ℤ/8)

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

C₂×#481

yes

characteristic

	subgroup	G/H
center, socle	C₂	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

C₂×#481

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

C₂×#41

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #2239	C₂	semidirect
4. #2155	C₂	semidirect
5. C₂×#362	C₂	semidirect

H ≅ G/N

type

3. #2239

C₂

semidirect

4. #2155

C₂

semidirect

5. C₂×#362

C₂

semidirect

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

subgroup	index
#3059	2
#2916	2
#2239	2
#2155	2
C₂×#362	2
#2115	2

subgroup

index

Subgroups — #33658

Conjugacy classes of subgroups

Complements

Maximal subgroups