Group #28829 — subgroups

Subgroups — #28829

Order 256.

339 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 159 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	#2552	C₂	1	2	yes	maximal
2	#2084	C₂	1	4	yes	maximal
3	#1948	C₂	1	2	yes	maximal
4	#1941	C₂	1	2	yes	maximal
5	#1844	C₂	1	2	yes	maximal
6	#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×#106	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×#125	—	2	2	no	—
10	C₂×#106	C₂²	1	2	yes	—
11	C₂×#106	C₂²	1	1	yes	characteristic
12	C₂×#106	C₂²	1	1	yes	characteristic
13	C₂×#106	C₂²	1	2	yes	—
14	C₂×#106	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#398	—	2	4	no	—
19	#389	—	2	2	no	—
20	#389	C₂²	1	4	yes	—
21	#384	C₂²	1	4	yes	—
22	#384	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	#340	C₂²	1	2	yes	—
26	#340	C₂²	1	4	yes	—
27	#336	C₂²	1	2	yes	—
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	#125	—	4	8	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	4	no	—
34	C₂×C₄⋊C₄	—	2	4	no	—
35	C₂×C₄⋊C₄	—	2	4	no	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄⋊C₄	D₄	1	2	yes	—
38	C₂×C₄⋊C₄	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₄²	—	2	2	no	—
42	C₂×C₄²	C₂³	1	2	yes	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	#106	—	2	2	no	—
45	#106	—	2	4	no	—
46	#106	—	2	2	no	—
47	#106	—	4	2	no	—
48	#106	—	4	4	no	—
49	C₄⋊C₈	Q₈	1	4	yes	—
50	C₄×C₈	—	2	4	no	—
51	C₄×C₈	Q₈	1	4	yes	—
52	#95	—	2	2	no	—
53	#95	—	2	4	no	—
54	#95	D₄	1	2	yes	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	4	yes	—
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄	C₂⁴	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₈	C₂×Q₈	1	4	yes	—
68	C₂×C₈	C₂×Q₈	1	2	yes	—
69	C₂×C₈	#41	1	2	yes	—
70	C₂×C₈	#41	1	2	yes	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	4	8	no	—
78	C₄⋊C₄	—	4	8	no	—
79	C₄⋊C₄	—	4	4	no	—
80	C₄⋊C₄	—	8	4	no	—
81	C₄⋊C₄	—	8	2	no	—
82	C₄⋊C₄	SD₁₆	1	8	yes	—
83	C₄²	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	4	4	no	—
86	C₄²	C₂×Q₈	1	2	yes	—
87	C₄²	C₂×Q₈	1	2	yes	—
88	C₂³	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₄⋊Q₈	1	2	yes	—
102	C₂×C₄	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄	C₂×SD₁₆	1	8	yes	—
104	C₂×C₄	C₂×SD₁₆	1	4	yes	—
105	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄	C₂×#41	1	1	yes	characteristic
108	C₂×C₄	C₂×#41	1	1	yes	characteristic
109	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	4	no	—
113	C₈	#125	1	4	yes	complemented
114	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂²	#547	1	1	yes	characteristic
116	C₂²	#550	1	1	yes	characteristic
117	C₂²	C₂²×SD₁₆	1	2	yes	—
118	C₂²	#577	1	2	yes	—
119	C₄	—	2	8	no	—
120	C₄	—	2	2	no	—
121	C₄	—	8	2	no	—
122	C₄	—	8	2	no	—
123	C₄	—	8	4	no	—
124	C₄	#498	1	4	yes	—
125	C₄	C₂×#125	1	1	yes	characteristic
126	C₄	#544	1	1	yes	characteristic
127	C₄	#554	1	1	yes	characteristic
128	C₄	#557	1	1	yes	characteristic
129	C₂	#2552	1	1	yes	characteristic
130	C₂	C₂×#498	1	1	yes	characteristic
131	C₂	#3163	1	1	yes	characteristic
132	C₂	#3335	1	2	yes	—
133	C₂	#3392	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2552

C₂

yes

maximal

#2084

C₂

yes

maximal

#1948

C₂

yes

maximal

#1941

C₂

yes

maximal

#1844

C₂

yes

maximal

#1844

C₂

yes

maximal, characteristic

C₄×#106

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#125

—

C₂×#106

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

C₂×#106

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

C₂²

yes

—

#384

C₂²

yes

—

#384

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

characteristic

#340

C₂²

yes

—

#340

C₂²

yes

—

#336

C₂²

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

C₄⋊C₈

Q₈

yes

—

C₄×C₈

—

C₄×C₈

Q₈

yes

—

#95

—

#95

—

#95

D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

SD₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Q₈

yes

—

C₄²

C₂×Q₈

yes

—

C₂³

C₂²×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Q₈

yes

—

102

C₂×C₄

C₄⋊Q₈

yes

—

103

C₂×C₄

C₂×SD₁₆

yes

—

104

C₂×C₄

C₂×SD₁₆

yes

—

105

C₂×C₄

C₂²×Q₈

yes

characteristic

106

C₂×C₄

C₂²×Q₈

yes

characteristic

107

C₂×C₄

C₂×#41

yes

characteristic

108

C₂×C₄

C₂×#41

yes

characteristic

109

C₂×C₄

H(5,𝔽₂)

yes

characteristic

110

C₂×C₄

2₋¹⁺⁴

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

#125

yes

complemented

114

C₂²

C₂×C₄⋊Q₈

yes

characteristic

115

C₂²

#547

yes

characteristic

116

C₂²

#550

yes

characteristic

117

C₂²

C₂²×SD₁₆

yes

—

118

C₂²

#577

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#498

yes

—

125

C₄

C₂×#125

yes

characteristic

126

C₄

#544

yes

characteristic

127

C₄

#554

yes

characteristic

128

C₄

#557

yes

characteristic

129

C₂

#2552

yes

characteristic

130

C₂

C₂×#498

yes

characteristic

131

C₂

#3163

yes

characteristic

132

C₂

#3335

yes

—

133

C₂

#3392

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
113. C₈	#125	semidirect

H ≅ G/N

type

113. C₈

#125

semidirect

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

subgroup	index
#2552	2
#2084	2
#1948	2
#1941	2
#1844	2
C₄×#106	2
C₈×C₄⋊C₄	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #28829

Conjugacy classes of subgroups

Complements

Maximal subgroups