Group #33458 — subgroups

Subgroups — #33458

Order 256.

323 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 159 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
2	#2950	C₂	1	2	yes	maximal
3	#2176	C₂	1	4	yes	maximal, complemented
4	C₄×Q₃₂	C₂	1	4	yes	maximal, complemented
5	#2158	C₂	1	1	yes	maximal, characteristic
6	#2141	C₂	1	2	yes	maximal
7	#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	C₂²	1	2	yes	—
9	#519	—	2	2	no	—
10	C₂×Q₃₂	C₄	1	8	yes	complemented
11	C₂×Q₃₂	C₂²	1	2	yes	—
12	C₂×Q₃₂	C₂²	1	2	yes	—
13	C₂×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆	C₂²	1	1	yes	characteristic
15	#441	—	2	4	no	—
16	C₄×Q₁₆	C₂²	1	8	yes	—
17	#427	C₂²	1	2	yes	—
18	#419	—	2	2	no	—
19	#405	C₂²	1	1	yes	characteristic
20	#367	C₂²	1	2	yes	—
21	#366	C₂²	1	2	yes	—
22	#358	C₂²	1	8	yes	—
23	#346	C₂²	1	2	yes	—
24	C₄×C₁₆	C₂²	1	2	yes	—
25	C₂²×Q₈	—	2	2	no	—
26	C₂×Q₁₆	—	2	2	no	—
27	C₂×Q₁₆	—	2	4	no	—
28	C₂×Q₁₆	C₂×C₄	1	8	yes	—
29	C₂×Q₁₆	C₂³	1	4	yes	—
30	C₂×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	C₄×Q₈	—	2	8	no	—
33	C₄×Q₈	—	4	4	no	—
34	#117	—	2	2	no	—
35	#117	—	4	4	no	—
36	#117	D₄	1	1	yes	characteristic
37	Q₃₂	C₂×C₄	1	8	yes	—
38	Q₃₂	C₂×C₄	1	8	yes	—
39	M₃₂	—	2	2	no	—
40	C₂×C₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₆	C₂³	1	1	yes	characteristic
44	#107	C₂³	1	4	yes	—
45	#106	—	2	2	no	—
46	#103	—	2	8	no	—
47	#97	D₄	1	2	yes	—
48	C₄×C₈	C₂³	1	2	yes	—
49	C₂×Q₈	—	2	8	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	4	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	Q₁₆	—	2	8	no	—
56	Q₁₆	—	2	4	no	—
57	Q₁₆	C₂²×C₄	1	8	yes	—
58	M₁₆	—	2	2	no	—
59	C₂×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₈	C₂×D₄	1	2	yes	—
61	C₂×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈	#41	1	2	yes	—
63	C₂×C₈	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	4	8	no	—
66	C₄⋊C₄	D₈	1	2	yes	—
67	C₂²⋊C₄	—	4	2	no	—
68	C₂²⋊C₄	D₈	1	2	yes	—
69	C₄²	—	4	2	no	—
70	C₄²	—	4	8	no	—
71	C₄²	C₂×D₄	1	2	yes	—
72	C₁₆	—	2	2	no	—
73	C₁₆	C₂²×C₄	1	2	yes	—
74	C₁₆	C₂²×C₄	1	2	yes	—
75	C₂³	C₂×D₈	1	1	yes	characteristic
76	Q₈	—	2	8	no	—
77	Q₈	—	4	4	no	—
78	Q₈	—	4	8	no	—
79	C₂×C₄	—	4	4	no	—
80	C₂×C₄	—	4	8	no	—
81	C₂×C₄	—	4	4	no	—
82	C₂×C₄	C₄×D₄	1	2	yes	—
83	C₂×C₄	C₂×D₈	1	4	yes	—
84	C₂×C₄	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄	#135	1	2	yes	—
86	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	C₄×D₄	1	2	yes	—
89	C₈	C₂³×C₄	1	1	yes	characteristic
90	C₈	C₂×#41	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	C₄×D₈	1	2	yes	—
93	C₂²	C₂²×D₈	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	4	8	no	—
96	C₄	—	8	4	no	—
97	C₄	C₄×D₈	1	2	yes	—
98	C₄	C₂×C₄×D₄	1	1	yes	characteristic
99	C₄	C₂×#135	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	C₂×C₄×D₈	1	1	yes	characteristic
102	C₂	#3417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂

C₂

yes

maximal, characteristic

#2950

C₂

yes

maximal

#2176

C₂

yes

maximal, complemented

C₄×Q₃₂

C₂

yes

maximal, complemented

#2158

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal

#2108

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

—

#519

—

C₂×Q₃₂

C₄

yes

complemented

C₂×Q₃₂

C₂²

yes

—

C₂×Q₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#441

—

C₄×Q₁₆

C₂²

yes

—

#427

C₂²

yes

—

#419

—

#405

C₂²

yes

characteristic

#367

C₂²

yes

—

#366

C₂²

yes

—

#358

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

C₂³

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

#117

D₄

yes

characteristic

Q₃₂

C₂×C₄

yes

—

Q₃₂

C₂×C₄

yes

—

M₃₂

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

#107

C₂³

yes

—

#106

—

#103

—

#97

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

C₂²×C₄

yes

—

M₁₆

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₁₆

—

C₁₆

C₂²×C₄

yes

—

C₁₆

C₂²×C₄

yes

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

#135

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

—

C₈

C₄×D₄

yes

—

C₈

C₂³×C₄

yes

characteristic

C₈

C₂×#41

yes

characteristic

C₂²

—

C₂²

C₄×D₈

yes

—

C₂²

C₂²×D₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₄×D₈

yes

—

C₄

C₂×C₄×D₄

yes

characteristic

C₄

C₂×#135

yes

characteristic

100

C₂

—

101

C₂

C₂×C₄×D₈

yes

characteristic

102

C₂

#3417

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂×D₈
LCS	G ▷ 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

C₈

C₂³×C₄

Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

C₂³

C₂×D₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #2176	C₂	semidirect
4. C₄×Q₃₂	C₂	semidirect
10. C₂×Q₃₂	C₄	semidirect

H ≅ G/N

type

3. #2176

C₂

semidirect

4. C₄×Q₃₂

C₂

semidirect

10. C₂×Q₃₂

C₄

semidirect

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

subgroup	index
C₂²×Q₃₂	2
#2950	2
#2176	2
C₄×Q₃₂	2
#2158	2
#2141	2
#2108	2

subgroup

index

C₂²×Q₃₂

C₄×Q₃₂

Subgroups — #33458

Conjugacy classes of subgroups

Complements

Maximal subgroups