Group #33950 — subgroups

Subgroups — #33950

Order 256.

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

normal subgroups only

This group has 224 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	#3383	C₂	1	2	yes	maximal
2	#2253	C₂	1	2	yes	maximal
3	#2252	C₂	1	1	yes	maximal, characteristic
4	#2229	C₂	1	4	yes	maximal
5	#2225	C₂	1	2	yes	maximal
6	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₃₂	C₂	1	2	yes	maximal
9	#557	—	2	2	no	—
10	C₂×Q₃₂	C₂²	1	1	yes	characteristic
11	#500	C₂²	1	2	yes	—
12	#499	C₂²	1	4	yes	—
13	#479	—	2	4	no	—
14	#475	—	2	2	no	—
15	C₈×Q₈	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆	C₂²	1	2	yes	—
17	C₄×Q₁₆	C₂²	1	4	yes	—
18	#367	C₂²	1	4	yes	—
19	#366	C₂²	1	2	yes	—
20	#366	C₂²	1	1	yes	characteristic
21	#366	C₂²	1	2	yes	—
22	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆	C₂²	1	2	yes	—
24	#358	C₂²	1	4	yes	—
25	#358	C₂²	1	2	yes	—
26	C₄×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₆	C₂²	1	2	yes	—
28	C₂×Q₁₆	C₂³	1	2	yes	—
29	C₄⋊Q₈	—	2	2	no	—
30	C₄⋊Q₈	—	4	2	no	—
31	#125	—	2	4	no	—
32	#125	—	4	4	no	—
33	C₄×Q₈	—	2	2	no	—
34	C₄×Q₈	—	2	4	no	—
35	C₄×Q₈	—	4	2	no	—
36	C₄×Q₈	D₄	1	1	yes	characteristic
37	Q₃₂	Q₈	1	4	yes	complemented
38	C₂×C₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₁₆	C₂³	1	2	yes	—
40	C₂×C₁₆	C₂³	1	1	yes	characteristic
41	#107	—	2	2	no	—
42	#107	C₂³	1	4	yes	—
43	#107	C₂³	1	2	yes	—
44	#106	—	2	4	no	—
45	C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₄⋊C₈	D₄	1	2	yes	—
47	#103	—	2	4	no	—
48	#103	—	2	2	no	—
49	C₄×C₈	C₂³	1	2	yes	—
50	C₄×C₈	C₂³	1	1	yes	characteristic
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂×Q₈	D₈	1	1	yes	characteristic
54	Q₁₆	—	2	2	no	—
55	Q₁₆	C₂×Q₈	1	4	yes	—
56	C₂×C₈	C₂×D₄	1	2	yes	—
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	2	4	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄⋊C₄	—	4	4	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	D₈	1	1	yes	characteristic
67	C₄⋊C₄	D₈	1	2	yes	—
68	C₄²	—	4	4	no	—
69	C₄²	—	4	2	no	—
70	C₄²	C₂×D₄	1	2	yes	—
71	C₄²	C₂×D₄	1	1	yes	characteristic
72	C₁₆	—	2	1	no	—
73	C₁₆	—	2	2	no	—
74	C₁₆	C₂×Q₈	1	2	yes	—
75	Q₈	—	2	2	no	—
76	Q₈	—	2	4	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	8	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	4	4	no	—
82	C₂×C₄	C₂×D₈	1	2	yes	—
83	C₂×C₄	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄	C₂×D₈	1	2	yes	—
85	C₂×C₄	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	2	1	no	—
89	C₈	C₂²×Q₈	1	1	yes	characteristic
90	C₈	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂²	C₂²×D₈	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	1	no	—
95	C₄	—	4	4	no	—
96	C₄	—	8	4	no	—
97	C₄	—	8	2	no	—
98	C₄	#508	1	2	yes	—
99	C₄	D₄×Q₈	1	1	yes	characteristic
100	C₄	#578	1	1	yes	characteristic
101	C₂	Q₈×D₈	1	1	yes	characteristic
102	C₂	C₂×#508	1	1	yes	characteristic
103	C₂	#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3383

C₂

yes

maximal

#2253

C₂

yes

maximal

#2252

C₂

yes

maximal, characteristic

#2229

C₂

yes

maximal

#2225

C₂

yes

maximal

Q₈×C₁₆

C₂

yes

maximal, characteristic

C₄×Q₃₂

C₂

yes

maximal, characteristic

C₄×Q₃₂

C₂

yes

maximal

#557

—

C₂×Q₃₂

C₂²

yes

characteristic

#500

C₂²

yes

—

#499

C₂²

yes

—

#479

—

#475

—

C₈×Q₈

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

#358

C₂²

yes

—

#358

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₄×C₁₆

C₂²

yes

—

C₂×Q₁₆

C₂³

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

characteristic

Q₃₂

Q₈

yes

complemented

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

C₄⋊C₈

D₄

yes

—

#103

—

#103

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

characteristic

Q₁₆

—

Q₁₆

C₂×Q₈

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

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

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₂×Q₈

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

—

C₈

—

C₈

C₂²×Q₈

yes

characteristic

C₈

2₋¹⁺⁴

yes

characteristic

C₂²

C₂²×D₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#508

yes

—

C₄

D₄×Q₈

yes

characteristic

100

C₄

#578

yes

characteristic

101

C₂

Q₈×D₈

yes

characteristic

102

C₂

C₂×#508

yes

characteristic

103

C₂

#3417

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
37. Q₃₂	Q₈	semidirect

H ≅ G/N

type

37. Q₃₂

Q₈

semidirect

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

subgroup	index
#3383	2
#2253	2
#2252	2
#2229	2
#2225	2
Q₈×C₁₆	2
C₄×Q₃₂	2

subgroup

index

Q₈×C₁₆

C₄×Q₃₂

Subgroups — #33950

Conjugacy classes of subgroups

Complements

Maximal subgroups