Group #22725 — subgroups

Subgroups — #22725

Order 256.

375 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 215 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#2355	C₂	1	2	yes	maximal
3	#1949	C₂	1	4	yes	maximal
4	#1947	C₂	1	2	yes	maximal
5	#1846	C₂	1	2	yes	maximal
6	#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×#106	C₂	1	2	yes	maximal
8	C₄×#106	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₈×Q₈	C₄	1	8	yes	complemented
11	C₂×#107	C₂²	1	4	yes	—
12	C₂×#106	C₂²	1	2	yes	—
13	C₂×#106	C₂²	1	1	yes	characteristic
14	C₂×#106	C₂²	1	2	yes	—
15	C₂×#106	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	#391	—	2	2	no	—
21	#387	—	2	4	no	—
22	#384	C₂²	1	2	yes	—
23	#382	C₂²	1	4	yes	—
24	C₄×C₄⋊C₄	—	2	2	no	—
25	C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄	C₂²	1	4	yes	—
27	#340	C₂²	1	4	yes	—
28	#340	C₂²	1	2	yes	—
29	C₂²×Q₈	D₄	1	1	yes	characteristic
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	2	yes	—
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	C₄×Q₈	C₂×C₄	1	4	yes	—
34	C₄×Q₈	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄	D₄	1	2	yes	—
39	C₂×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₄²	—	2	2	no	—
43	C₂×C₄²	—	2	4	no	—
44	C₂×C₄²	C₂³	1	2	yes	—
45	C₂×C₄²	C₂³	1	1	yes	characteristic
46	#107	—	4	4	no	—
47	#106	—	2	4	no	—
48	#106	—	2	2	no	—
49	#106	—	4	2	no	—
50	#106	Q₈	1	4	yes	complemented
51	C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₄×C₈	C₂×C₄	1	4	yes	—
54	C₄×C₈	C₂×C₄	1	8	yes	—
55	#95	—	2	4	no	—
56	#95	—	2	2	no	—
57	#95	—	2	4	no	—
58	#95	—	2	4	no	—
59	C₂×Q₈	—	2	4	no	—
60	C₂×Q₈	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	—	2	4	no	—
64	C₂²×C₄	C₂×D₄	1	2	yes	—
65	C₂²×C₄	C₂×D₄	1	2	yes	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₈	C₂²×C₄	1	4	yes	—
75	C₂×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₈	C₂×Q₈	1	2	yes	—
77	C₂×C₈	#41	1	2	yes	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	4	4	no	—
84	C₄⋊C₄	—	8	2	no	—
85	C₄⋊C₄	C₄⋊C₄	1	8	yes	—
86	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
87	C₄⋊C₄	C₂×Q₈	1	4	yes	—
88	C₄²	—	4	4	no	—
89	C₄²	—	8	4	no	—
90	C₄²	C₂²×C₄	1	4	yes	—
91	C₄²	C₂²×C₄	1	2	yes	—
92	C₄²	C₂×Q₈	1	4	yes	—
93	C₄²	C₂×Q₈	1	2	yes	—
94	C₂³	C₂²×D₄	1	1	yes	characteristic
95	Q₈	—	2	8	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	4	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄	#135	1	4	yes	—
113	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄	C₂×#41	1	1	yes	characteristic
117	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	2	4	no	—
121	C₈	C₄×Q₈	1	4	yes	—
122	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²	D₄×Q₈	1	1	yes	characteristic
124	C₂²	#550	1	1	yes	characteristic
125	C₂²	C₂×#135	1	2	yes	—
126	C₂²	#577	1	2	yes	—
127	C₄	—	2	4	no	—
128	C₄	—	2	4	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	4	no	—
131	C₄	—	8	2	no	—
132	C₄	—	8	4	no	—
133	C₄	#427	1	4	yes	—
134	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄	#520	1	1	yes	characteristic
136	C₄	#557	1	1	yes	characteristic
137	C₄	Q₈²	1	1	yes	characteristic
138	C₂	Q₈×C₄⋊C₄	1	1	yes	characteristic
139	C₂	C₂×#427	1	1	yes	characteristic
140	C₂	#2912	1	1	yes	characteristic
141	C₂	#3381	1	2	yes	—
142	C₂	#3394	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

#2355

C₂

yes

maximal

#1949

C₂

yes

maximal

#1947

C₂

yes

maximal

#1846

C₂

yes

maximal

#1844

C₂

yes

maximal, characteristic

C₄×#106

C₂

yes

maximal

C₄×#106

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

C₄

yes

complemented

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

C₂×#106

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

#391

—

#387

—

#384

C₂²

yes

—

#382

C₂²

yes

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

#340

C₂²

yes

—

#340

C₂²

yes

—

C₂²×Q₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

Q₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

—

—

—

Q₈

yes

complemented

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₄⋊C₄

yes

—

C₄⋊C₄

C₂×Q₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

C₂×Q₈

yes

—

C₂³

C₂²×D₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₄⋊C₄

yes

—

109

C₂×C₄

C₂×C₄⋊C₄

yes

—

110

C₂×C₄

C₂×C₄⋊C₄

yes

—

111

C₂×C₄

C₂×C₄⋊C₄

yes

—

112

C₂×C₄

#135

yes

—

113

C₂×C₄

C₂³×C₄

yes

characteristic

114

C₂×C₄

C₂²×Q₈

yes

characteristic

115

C₂×C₄

C₂²×Q₈

yes

characteristic

116

C₂×C₄

C₂×#41

yes

characteristic

117

C₂×C₄

H(5,𝔽₂)

yes

characteristic

118

C₂×C₄

2₋¹⁺⁴

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

C₄×Q₈

yes

—

122

C₂²

C₂²×C₄⋊C₄

yes

characteristic

123

C₂²

D₄×Q₈

yes

characteristic

124

C₂²

#550

yes

characteristic

125

C₂²

C₂×#135

yes

—

126

C₂²

#577

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#427

yes

—

134

C₄

C₂×C₄×Q₈

yes

characteristic

135

C₄

#520

yes

characteristic

136

C₄

#557

yes

characteristic

137

C₄

Q₈²

yes

characteristic

138

C₂

Q₈×C₄⋊C₄

yes

characteristic

139

C₂

C₂×#427

yes

characteristic

140

C₂

#2912

yes

characteristic

141

C₂

#3381

yes

—

142

C₂

#3394

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. C₈×Q₈	C₄	semidirect
50. #106	Q₈	semidirect

H ≅ G/N

type

10. C₈×Q₈

C₄

semidirect

50. #106

Q₈

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
#2355	2
#1949	2
#1947	2
#1846	2
#1844	2
C₄×#106	2

subgroup

index

C₂×C₈×Q₈

Subgroups — #22725

Conjugacy classes of subgroups

Complements

Maximal subgroups