Group #33908 — subgroups

Subgroups — #33908

Order 256.

246 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3341	C₂	1	2	yes	maximal, complemented
2	#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	#2234	C₂	1	4	yes	maximal, complemented
4	#2226	C₂	1	2	yes	maximal, complemented
5	#2215	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#2158	C₂	1	2	yes	maximal, complemented
9	#555	—	2	2	no	—
10	C₂×D₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	2	yes	—
12	#500	C₂²	1	2	yes	—
13	#482	—	2	4	no	—
14	#476	—	2	2	no	—
15	#466	C₂²	1	4	yes	—
16	C₄×D₈	C₂²	1	2	yes	—
17	C₈×D₄	C₂²	1	1	yes	characteristic
18	#427	C₂²	1	4	yes	—
19	#367	C₂²	1	4	yes	—
20	#366	C₂²	1	1	yes	characteristic
21	#366	C₂²	1	2	yes	—
22	#366	C₂²	1	2	yes	—
23	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	#357	C₂²	1	2	yes	—
25	#357	C₂²	1	4	yes	—
26	C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×D₈	C₂³	1	2	yes	—
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₄⋊Q₈	—	2	2	no	—
31	#124	—	4	4	no	—
32	#121	—	2	4	no	—
33	#121	—	4	2	no	—
34	C₄×D₄	—	2	2	no	—
35	C₄×D₄	D₄	1	1	yes	characteristic
36	#117	—	2	4	no	—
37	#117	—	4	2	no	—
38	D₁₆	—	2	2	no	—
39	C₂×C₁₆	—	2	4	no	—
40	C₂×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₆	C₂³	1	1	yes	characteristic
43	#107	—	2	2	no	—
44	#107	C₂³	1	4	yes	—
45	#107	C₂³	1	2	yes	—
46	#106	—	2	4	no	—
47	C₄⋊C₈	D₄	1	1	yes	characteristic
48	#102	—	2	2	no	—
49	#102	—	2	4	no	—
50	C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×Q₈	—	4	2	no	—
53	C₂×D₄	—	2	2	no	—
54	C₂×D₄	—	4	4	no	—
55	C₂×D₄	D₈	1	1	yes	characteristic
56	C₂²×C₄	—	4	2	no	—
57	C₂²×C₄	C₂×D₄	1	2	yes	—
58	D₈	—	2	2	no	—
59	D₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈	C₂×D₄	1	2	yes	—
64	C₂×C₈	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	4	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	D₈	1	1	yes	characteristic
69	C₂²⋊C₄	—	4	4	no	—
70	C₂²⋊C₄	—	4	4	no	—
71	C₂²⋊C₄	—	4	2	no	—
72	C₂²⋊C₄	D₈	1	2	yes	—
73	C₄²	—	4	4	no	—
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₁₆	—	2	2	no	—
76	C₁₆	—	2	1	no	—
77	C₁₆	#41	1	2	yes	—
78	C₂³	—	4	2	no	—
79	C₂³	C₂×D₈	1	2	yes	—
80	Q₈	—	16	2	no	—
81	D₄	—	2	2	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	16	4	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	4	no	—
89	C₂×C₄	—	8	4	no	—
90	C₂×C₄	C₂×D₈	1	2	yes	—
91	C₂×C₄	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	2	1	no	—
96	C₈	C₂×#41	1	1	yes	characteristic
97	C₈	2₋¹⁺⁴	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	2	no	—
101	C₂²	—	8	2	no	—
102	C₂²	—	8	2	no	—
103	C₂²	—	8	2	no	—
104	C₂²	C₂²×D₈	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	1	no	—
107	C₄	—	8	2	no	—
108	C₄	—	8	4	no	—
109	C₄	#508	1	2	yes	—
110	C₄	#547	1	1	yes	characteristic
111	C₄	#578	1	1	yes	characteristic
112	C₂	—	4	2	no	—
113	C₂	—	16	2	no	—
114	C₂	#3334	1	1	yes	characteristic
115	C₂	C₂×#508	1	1	yes	characteristic
116	C₂	#3415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3341

C₂

yes

maximal, complemented

#2252

C₂

yes

maximal, characteristic, complemented

#2234

C₂

yes

maximal, complemented

#2226

C₂

yes

maximal, complemented

#2215

C₂

yes

maximal, complemented

C₄×D₁₆

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆

C₂

yes

maximal, characteristic, complemented

#2158

C₂

yes

maximal, complemented

#555

—

C₂×D₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

#500

C₂²

yes

—

#482

—

#476

—

#466

C₂²

yes

—

C₄×D₈

C₂²

yes

—

C₈×D₄

C₂²

yes

characteristic

#427

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

#366

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

characteristic

#357

C₂²

yes

—

#357

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×D₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

D₄

yes

characteristic

#117

—

#117

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#106

—

C₄⋊C₈

D₄

yes

characteristic

#102

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#41

yes

—

C₂³

—

C₂³

C₂×D₈

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

—

C₈

—

C₈

C₂×#41

yes

characteristic

C₈

2₋¹⁺⁴

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂²×D₈

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#508

yes

—

110

C₄

#547

yes

characteristic

111

C₄

#578

yes

characteristic

112

C₂

—

113

C₂

—

114

C₂

#3334

yes

characteristic

115

C₂

C₂×#508

yes

characteristic

116

C₂

#3415

yes

characteristic

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

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #3341	C₂	semidirect
2. #2252	C₂	semidirect
3. #2234	C₂	semidirect
4. #2226	C₂	semidirect
5. #2215	C₂	semidirect
6. C₄×D₁₆	C₂	semidirect
7. D₄×C₁₆	C₂	semidirect
8. #2158	C₂	semidirect

H ≅ G/N

type

1. #3341

C₂

semidirect

2. #2252

C₂

semidirect

3. #2234

C₂

semidirect

4. #2226

C₂

semidirect

5. #2215

C₂

semidirect

6. C₄×D₁₆

C₂

semidirect

7. D₄×C₁₆

C₂

semidirect

8. #2158

C₂

semidirect

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

subgroup	index
#3341	2
#2252	2
#2234	2
#2226	2
#2215	2
C₄×D₁₆	2
D₄×C₁₆	2
#2158	2

subgroup

index

C₄×D₁₆

D₄×C₁₆

#2158

Subgroups — #33908

Conjugacy classes of subgroups

Complements

Maximal subgroups