Group #33729 — subgroups

Subgroups — #33729

Order 256.

237 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 135 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	#3164	C₂	1	2	yes	maximal
2	#2255	C₂	1	4	yes	maximal, complemented
3	#2253	C₂	1	2	yes	maximal, complemented
4	#2252	C₂	1	2	yes	maximal, complemented
5	#2159	C₂	1	2	yes	maximal
6	#2158	C₂	1	1	yes	maximal, characteristic
7	C₂×#366	C₂	1	1	yes	maximal, characteristic
8	#2108	C₂	1	1	yes	maximal, characteristic
9	#544	—	2	2	no	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#501	—	2	4	no	—
13	#500	C₂²	1	4	yes	—
14	#499	C₂²	1	4	yes	—
15	#428	—	2	2	no	—
16	#427	C₂²	1	2	yes	—
17	C₂×#107	C₂²	1	2	yes	—
18	#405	C₂²	1	1	yes	characteristic
19	#367	C₂²	1	2	yes	—
20	#367	C₂²	1	4	yes	—
21	#366	C₂²	1	2	yes	—
22	#366	C₂²	1	2	yes	—
23	#366	C₂²	1	1	yes	characteristic
24	#366	C₂²	1	1	yes	characteristic
25	#366	C₂²	1	4	yes	—
26	#346	C₂²	1	2	yes	—
27	C₄×C₁₆	C₂²	1	2	yes	—
28	C₂×M₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈	—	2	4	no	—
31	C₄⋊Q₈	—	4	2	no	—
32	#125	—	2	4	no	—
33	#125	—	4	2	no	—
34	C₂²⋊Q₈	—	4	4	no	—
35	#117	—	2	2	no	—
36	#117	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄	—	2	2	no	—
38	M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₁₆	Q₈	1	4	yes	complemented
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	—	2	4	no	—
45	#107	C₂³	1	4	yes	—
46	#107	C₂³	1	2	yes	—
47	#107	C₂³	1	2	yes	—
48	#106	—	2	2	no	—
49	#106	—	2	4	no	—
50	#97	D₄	1	2	yes	—
51	C₄×C₈	C₂³	1	2	yes	—
52	C₂×Q₈	—	4	4	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	M₁₆	Q₁₆	1	4	yes	—
56	C₂×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₈	C₂×Q₈	1	2	yes	—
59	C₂×C₈	C₂×Q₈	1	2	yes	—
60	C₂×C₈	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	2	4	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	4	no	—
66	C₄⋊C₄	—	4	4	no	—
67	C₄⋊C₄	—	4	4	no	—
68	C₄⋊C₄	D₈	1	1	yes	characteristic
69	C₄⋊C₄	D₈	1	1	yes	characteristic
70	C₂²⋊C₄	—	4	2	no	—
71	C₂²⋊C₄	D₈	1	2	yes	—
72	C₄²	—	4	2	no	—
73	C₄²	C₂×D₄	1	2	yes	—
74	C₁₆	C₂×Q₈	1	2	yes	—
75	C₁₆	C₂×Q₈	1	2	yes	—
76	C₁₆	C₂×Q₈	1	4	yes	—
77	C₂³	C₂×D₈	1	1	yes	characteristic
78	Q₈	—	16	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	4	4	no	—
82	C₂×C₄	—	16	2	no	—
83	C₂×C₄	C₄⋊Q₈	1	2	yes	—
84	C₂×C₄	C₂×D₈	1	2	yes	—
85	C₂×C₄	C₂×D₈	1	2	yes	—
86	C₂×C₄	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
89	C₈	C₄⋊Q₈	1	2	yes	—
90	C₈	C₂×Q₁₆	1	4	yes	—
91	C₈	C₂²×Q₈	1	1	yes	characteristic
92	C₈	C₂²×Q₈	1	1	yes	characteristic
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	#500	1	2	yes	—
96	C₂²	C₂²×D₈	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	8	4	no	—
100	C₄	—	8	2	no	—
101	C₄	—	8	2	no	—
102	C₄	#500	1	2	yes	—
103	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₄	C₂²×Q₁₆	1	1	yes	characteristic
105	C₂	—	2	2	no	—
106	C₂	C₂×#500	1	1	yes	characteristic
107	C₂	#3415	1	1	yes	characteristic
108	C₂	#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3164

C₂

yes

maximal

#2255

C₂

yes

maximal, complemented

#2253

C₂

yes

maximal, complemented

#2252

C₂

yes

maximal, complemented

#2159

C₂

yes

maximal

#2158

C₂

yes

maximal, characteristic

C₂×#366

C₂

yes

maximal, characteristic

#2108

C₂

yes

maximal, characteristic

#544

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#501

—

#500

C₂²

yes

—

#499

C₂²

yes

—

#428

—

#427

C₂²

yes

—

C₂×#107

C₂²

yes

—

#405

C₂²

yes

characteristic

#367

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

—

#366

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

D₄

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

Q₈

yes

complemented

C₂×C₁₆

Q₈

yes

complemented

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

—

C₂³

yes

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#106

—

#106

—

#97

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

Q₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

characteristic

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₁₆

C₂×Q₈

yes

—

C₁₆

C₂×Q₈

yes

—

C₁₆

C₂×Q₈

yes

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Q₈

yes

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

C₄⋊Q₈

yes

—

C₈

C₂×Q₁₆

yes

—

C₈

C₂²×Q₈

yes

characteristic

C₈

C₂²×Q₈

yes

characteristic

C₂²

—

C₂²

—

C₂²

#500

yes

—

C₂²

C₂²×D₈

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#500

yes

—

103

C₄

C₂×C₄⋊Q₈

yes

characteristic

104

C₄

C₂²×Q₁₆

yes

characteristic

105

C₂

—

106

C₂

C₂×#500

yes

characteristic

107

C₂

#3415

yes

characteristic

108

C₂

#3417

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

C₂³

C₂×D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2255	C₂	semidirect
3. #2253	C₂	semidirect
4. #2252	C₂	semidirect
38. M₃₂	Q₈	semidirect
39. C₂×C₁₆	Q₈	semidirect

H ≅ G/N

type

2. #2255

C₂

semidirect

3. #2253

C₂

semidirect

4. #2252

C₂

semidirect

38. M₃₂

Q₈

semidirect

39. C₂×C₁₆

Q₈

semidirect

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

subgroup	index
#3164	2
#2255	2
#2253	2
#2252	2
#2159	2
#2158	2
C₂×#366	2
#2108	2

subgroup

index

Subgroups — #33729

Conjugacy classes of subgroups

Complements

Maximal subgroups