Group #21712 — subgroups

Subgroups — #21712

Order 256.

411 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 223 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×#446	C₂	1	2	yes	maximal
2	C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	#1984	C₂	1	2	yes	maximal
4	#1939	C₂	1	4	yes	maximal
5	#1842	C₂	1	1	yes	maximal, characteristic
6	#1770	C₂	1	4	yes	maximal
7	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈	C₂²	1	2	yes	—
10	#446	—	2	8	no	—
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×#97	C₂²	1	4	yes	—
15	C₂×C₄×C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	#391	C₄	1	2	yes	—
20	#389	C₄	1	4	yes	—
21	#384	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄³	C₂²	1	1	yes	characteristic
24	#336	C₂²	1	8	yes	—
25	C₂²×Q₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈	C₂³	1	4	yes	—
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₈	C₂³	1	2	yes	—
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈	—	2	4	no	—
31	C₄×Q₈	—	2	4	no	—
32	C₄×Q₈	—	2	4	no	—
33	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₄²	C₂×C₄	1	2	yes	—
38	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂³	1	4	yes	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₂×C₄²	C₂³	1	2	yes	—
43	C₄⋊C₈	—	2	8	no	—
44	C₄⋊C₈	—	2	4	no	—
45	C₄⋊C₈	—	2	8	no	—
46	#97	—	2	8	no	—
47	C₄×C₈	—	2	2	no	—
48	C₄×C₈	—	2	4	no	—
49	C₄×C₈	D₄	1	4	yes	—
50	C₄×C₈	Q₈	1	8	yes	complemented
51	#95	C₂×C₄	1	8	yes	—
52	C₂×Q₈	—	2	4	no	—
53	C₂×Q₈	—	2	4	no	—
54	C₂×Q₈	—	2	4	no	—
55	C₂²×C₄	C₂²×C₄	1	4	yes	—
56	C₂²×C₄	C₂²×C₄	1	4	yes	—
57	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄	C₂²×C₄	1	2	yes	—
59	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄	C₂²×C₄	1	2	yes	—
61	C₂²×C₄	C₂⁴	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	2	8	no	—
67	C₂×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₈	C₂×Q₈	1	4	yes	—
70	C₂×C₈	C₂×Q₈	1	4	yes	—
71	C₄⋊C₄	—	2	4	no	—
72	C₄⋊C₄	—	2	8	no	—
73	C₄⋊C₄	—	2	8	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄²	—	2	2	no	—
76	C₄²	—	2	4	no	—
77	C₄²	—	2	4	no	—
78	C₄²	—	2	4	no	—
79	C₄²	M₁₆	1	4	yes	—
80	C₄²	C₂×D₄	1	2	yes	—
81	C₄²	C₂×Q₈	1	4	yes	—
82	C₄²	#41	1	2	yes	—
83	C₄²	#41	1	4	yes	—
84	C₂³	C₂³×C₄	1	1	yes	characteristic
85	Q₈	—	4	8	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×D₄	1	2	yes	—
95	C₂×C₄	C₄×D₄	1	2	yes	—
96	C₂×C₄	C₄×Q₈	1	4	yes	—
97	C₂×C₄	C₄×Q₈	1	4	yes	—
98	C₂×C₄	C₂×M₁₆	1	2	yes	—
99	C₂×C₄	C₂×M₁₆	1	4	yes	—
100	C₂×C₄	#131	1	2	yes	—
101	C₂×C₄	#131	1	4	yes	—
102	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄	C₂²×Q₈	1	2	yes	—
104	C₂×C₄	C₂×#41	1	2	yes	—
105	C₂×C₄	C₂×#41	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	2	4	no	—
108	C₈	—	4	4	no	—
109	C₈	C₄⋊Q₈	1	4	yes	complemented
110	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²	C₂×C₄×Q₈	1	2	yes	—
112	C₂²	C₂²×M₁₆	1	1	yes	characteristic
113	C₂²	C₂×#131	1	1	yes	characteristic
114	C₂²	C₂×#131	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	—	4	4	no	—
119	C₄	#436	1	4	yes	—
120	C₄	#446	1	8	yes	complemented
121	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄	#532	1	1	yes	characteristic
123	C₄	#533	1	2	yes	—
124	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
125	C₂	C₂×#436	1	1	yes	characteristic
126	C₂	#2931	1	1	yes	characteristic
127	C₂	C₂×#446	1	2	yes	—
128	C₂	#2960	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#446

C₂

yes

maximal

C₄×C₄⋊Q₈

C₂

yes

maximal, characteristic

#1984

C₂

yes

maximal

#1939

C₂

yes

maximal

#1842

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

#446

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

#391

C₄

yes

—

#389

C₄

yes

—

#384

C₄

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

C₄³

C₂²

yes

characteristic

#336

C₂²

yes

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

complemented

#95

C₂×C₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

M₁₆

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₄²

#41

yes

—

C₂³

C₂³×C₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂×M₁₆

yes

—

100

C₂×C₄

#131

yes

—

101

C₂×C₄

#131

yes

—

102

C₂×C₄

C₂²×D₄

yes

characteristic

103

C₂×C₄

C₂²×Q₈

yes

—

104

C₂×C₄

C₂×#41

yes

—

105

C₂×C₄

C₂×#41

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

C₄⋊Q₈

yes

complemented

110

C₂²

C₂×C₄×D₄

yes

characteristic

111

C₂²

C₂×C₄×Q₈

yes

—

112

C₂²

C₂²×M₁₆

yes

characteristic

113

C₂²

C₂×#131

yes

characteristic

114

C₂²

C₂×#131

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#436

yes

—

120

C₄

#446

yes

complemented

121

C₄

C₂×C₄⋊Q₈

yes

characteristic

122

C₄

#532

yes

characteristic

123

C₄

#533

yes

—

124

C₂

C₄×C₄⋊Q₈

yes

characteristic

125

C₂

C₂×#436

yes

characteristic

126

C₂

#2931

yes

characteristic

127

C₂

C₂×#446

yes

—

128

C₂

#2960

yes

—

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

subgroup

G/H

center, Frattini, cosocle

C₂²×C₄

C₂⁴

commutator, socle, prime radical

C₂³

C₂³×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
50. C₄×C₈	Q₈	semidirect
109. C₈	C₄⋊Q₈	semidirect
120. C₄	#446	semidirect

H ≅ G/N

type

50. C₄×C₈

Q₈

semidirect

109. C₈

C₄⋊Q₈

semidirect

120. C₄

#446

semidirect

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

subgroup	index
C₂×#446	2
C₄×C₄⋊Q₈	2
#1984	2
#1939	2
#1842	2
#1770	2
C₄²×C₈	2

subgroup

index

C₄×C₄⋊Q₈

C₄²×C₈

Subgroups — #21712

Conjugacy classes of subgroups

Complements

Maximal subgroups