Group #13512 — subgroups

Subgroups — #13512

Order 256.

116 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 51 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	#2161	C₂	1	1	yes	maximal, characteristic
2	#2155	C₂	1	1	yes	maximal, characteristic
3	#1951	C₂	1	1	yes	maximal, characteristic
4	#1833	C₂	1	1	yes	maximal, characteristic
5	#1386	C₂	1	1	yes	maximal, characteristic
6	#1383	C₂	1	1	yes	maximal, characteristic
7	#1378	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#108	C₂²	1	1	yes	characteristic
11	C₂×#108	C₂²	1	1	yes	characteristic
12	#428	—	2	1	no	—
13	#427	C₂²	1	1	yes	characteristic
14	C₂×#106	C₂²	1	1	yes	characteristic
15	#405	C₂²	1	1	yes	characteristic
16	#368	—	2	2	no	—
17	#365	—	2	2	no	—
18	#344	—	2	1	no	—
19	#343	—	2	1	no	—
20	H(3,ℤ/4)	—	2	1	no	—
21	C₂×M₁₆	—	2	1	no	—
22	C₂×M₁₆	—	2	1	no	—
23	C₂×M₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	#117	—	2	1	no	—
26	#117	D₄	1	1	yes	characteristic
27	C₂×C₄⋊C₄	—	2	1	no	—
28	M₃₂	—	2	1	no	—
29	M₃₂	—	2	1	no	—
30	M₃₂	—	2	1	no	—
31	M₃₂	Q₈	1	2	yes	—
32	C₂×C₁₆	—	2	1	no	—
33	C₂×C₁₆	Q₈	1	2	yes	—
34	#108	—	2	1	no	—
35	#108	—	2	1	no	—
36	#108	—	2	1	no	—
37	#108	D₄	1	2	yes	—
38	#107	—	2	1	no	—
39	#107	—	4	1	no	—
40	#106	—	2	1	no	—
41	#106	—	2	1	no	—
42	#106	—	2	1	no	—
43	#106	—	4	1	no	—
44	#97	—	2	1	no	—
45	C₄×C₈	—	2	1	no	—
46	C₂²×C₄	—	4	1	no	—
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	M₁₆	—	2	2	no	—
49	M₁₆	—	2	1	no	—
50	M₁₆	—	2	1	no	—
51	M₁₆	—	4	1	no	—
52	M₁₆	—	4	1	no	—
53	M₁₆	—	4	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	4	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈	#41	1	1	yes	characteristic
61	C₂×C₈	#41	1	1	yes	characteristic
62	C₂×C₈	#41	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₂²⋊C₄	—	2	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₄²	—	2	1	no	—
70	C₄²	—	4	1	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	2	2	no	—
73	C₂³	#135	1	1	yes	characteristic
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	8	1	no	—
77	C₂×C₄	—	16	1	no	—
78	C₂×C₄	#121	1	1	yes	characteristic
79	C₂×C₄	#123	1	1	yes	characteristic
80	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄	#135	1	1	yes	characteristic
82	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄	#137	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	—	4	1	no	—
86	C₈	—	8	1	no	—
87	C₈	C₂²⋊Q₈	1	1	yes	characteristic
88	C₈	C₂²⋊Q₈	1	1	yes	characteristic
89	C₈	#125	1	1	yes	characteristic
90	C₈	#125	1	1	yes	characteristic
91	C₂²	—	4	1	no	—
92	C₂²	#464	1	1	yes	characteristic
93	C₂²	#482	1	1	yes	characteristic
94	C₂²	#499	1	1	yes	characteristic
95	C₄	—	8	1	no	—
96	C₄	—	16	1	no	—
97	C₄	—	16	1	no	—
98	C₄	#398	1	1	yes	characteristic
99	C₄	#463	1	1	yes	characteristic
100	C₄	#485	1	1	yes	characteristic
101	C₄	#501	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#2085	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2161

C₂

yes

maximal, characteristic

#2155

C₂

yes

maximal, characteristic

#1951

C₂

yes

maximal, characteristic

#1833

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal, characteristic

#1383

C₂

yes

maximal, characteristic

#1378

C₂

yes

maximal, characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#428

—

#427

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

#405

C₂²

yes

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

D₄

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

Q₈

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

—

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

#135

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

C₂²⋊Q₈

yes

characteristic

C₈

C₂²⋊Q₈

yes

C₈

yes

C₈

yes

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#485

yes

characteristic

101

C₄

#501

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

—

105

C₂

#2085

yes

characteristic

	subgroup	G/H
center, socle	C₂	#2085
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#2085

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#2161	2
#2155	2
#1951	2
#1833	2
#1386	2
#1383	2
#1378	2

subgroup

index

Subgroups — #13512

Conjugacy classes of subgroups

Maximal subgroups