Group #13212 — subgroups

Subgroups — #13212

Order 256.

235 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#362	C₂	1	4	yes	maximal
3	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆	—	2	1	no	—
6	C₂²×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈	—	2	1	no	—
8	C₂×M₃₂	C₂²	1	2	yes	—
9	#494	—	2	1	no	—
10	#494	—	2	1	no	—
11	#494	C₄	1	2	yes	—
12	#494	C₄	1	2	yes	—
13	C₂×#108	—	2	1	no	—
14	C₂×#108	C₂²	1	2	yes	—
15	#423	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#362	—	2	8	no	—
18	#346	—	2	1	no	—
19	C₄×C₁₆	—	2	1	no	—
20	C₂²×Q₈	—	2	2	no	—
21	C₂×Q₁₆	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	4	no	—
24	C₂×Q₁₆	—	2	2	no	—
25	C₂×Q₁₆	—	2	4	no	—
26	C₂×Q₁₆	—	4	2	no	—
27	C₂×Q₁₆	—	4	2	no	—
28	C₂×Q₁₆	C₂×C₄	1	4	yes	—
29	C₂×M₁₆	—	2	2	no	—
30	C₂²×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈	—	2	2	no	—
33	C₄⋊Q₈	—	2	2	no	—
34	C₄⋊Q₈	—	4	2	no	—
35	C₂×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	M₃₂	D₄	1	4	yes	—
39	C₂×C₁₆	—	2	2	no	—
40	#108	—	2	4	no	—
41	#108	—	4	2	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈	D₄	1	1	yes	characteristic
46	C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×Q₈	—	2	4	no	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	SD₁₆	1	2	yes	—
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	Q₁₆	—	2	8	no	—
57	Q₁₆	—	4	4	no	—
58	Q₁₆	—	4	4	no	—
59	Q₁₆	—	4	4	no	—
60	M₁₆	—	4	2	no	—
61	M₁₆	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈	D₈	1	2	yes	—
66	C₂×C₈	SD₁₆	1	2	yes	—
67	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈	#41	1	2	yes	—
71	C₄⋊C₄	—	4	4	no	—
72	C₄⋊C₄	—	4	4	no	—
73	C₄²	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²	D₈	1	2	yes	—
76	C₁₆	—	2	4	no	—
77	C₂³	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈	—	4	4	no	—
79	Q₈	—	4	4	no	—
80	Q₈	—	8	2	no	—
81	Q₈	—	8	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	8	2	no	—
85	C₂×C₄	—	8	2	no	—
86	C₂×C₄	#102	1	2	yes	—
87	C₂×C₄	#102	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄	#124	1	1	yes	characteristic
90	C₂×C₄	#127	1	1	yes	characteristic
91	C₂×C₄	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
94	C₈	—	8	2	no	—
95	C₈	#102	1	2	yes	—
96	C₈	#102	1	2	yes	—
97	C₈	C₄×D₄	1	2	yes	—
98	C₈	#121	1	2	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#102	1	1	yes	characteristic
101	C₂²	#486	1	1	yes	characteristic
102	C₂²	#492	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	8	4	no	—
105	C₄	#362	1	4	yes	complemented
106	C₄	#390	1	1	yes	characteristic
107	C₄	C₂×#102	1	1	yes	characteristic
108	C₄	#486	1	1	yes	characteristic
109	C₄	#493	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	#1957	1	1	yes	characteristic
113	C₂	C₂×#362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal

C₄×M₃₂

C₂

yes

maximal, characteristic

#1848

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂²

yes

—

#494

—

#494

—

#494

C₄

yes

—

#494

C₄

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

—

#423

—

C₂×C₄×C₈

C₂²

yes

characteristic

#362

—

#346

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

SD₁₆

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

D₈

yes

—

C₁₆

—

C₂³

C₂×SD₁₆

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#102

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₈

—

C₈

#102

yes

—

C₈

#102

yes

—

C₈

C₄×D₄

yes

—

C₈

#121

yes

—

C₂²

—

100

C₂²

C₂×#102

yes

characteristic

101

C₂²

#486

yes

characteristic

102

C₂²

#492

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#362

yes

complemented

106

C₄

#390

yes

characteristic

107

C₄

C₂×#102

yes

characteristic

108

C₄

#486

yes

characteristic

109

C₄

#493

yes

characteristic

110

C₂

—

111

C₂

—

112

C₂

#1957

yes

characteristic

113

C₂

C₂×#362

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#102
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#102

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

C₂×SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
105. C₄	#362	semidirect

H ≅ G/N

type

105. C₄

#362

semidirect

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

subgroup	index
C₂×#494	2
C₂×#362	2
C₄×M₃₂	2
#1848	2

subgroup

index

C₂×#494

C₂×#362

C₄×M₃₂

#1848

Subgroups — #13212

Conjugacy classes of subgroups

Complements

Maximal subgroups