Group #2451 — subgroups

Subgroups — #2451

Order 128.

242 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 120 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	#398	C₂	1	2	yes	maximal
4	#395	C₂	1	2	yes	maximal
5	#391	C₂	1	2	yes	maximal
6	#391	C₂	1	1	yes	maximal, characteristic
7	#391	C₂	1	1	yes	maximal, characteristic
8	#389	C₂	1	1	yes	maximal, characteristic
9	#389	C₂	1	1	yes	maximal, characteristic
10	#387	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈	C₂²	1	2	yes	—
13	C₂²×Q₈	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈	—	2	4	no	—
15	C₄×Q₈	—	2	4	no	—
16	C₂×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²	C₂²	1	1	yes	characteristic
26	C₂×C₄²	C₂²	1	2	yes	—
27	C₂×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₄²	C₂²	1	1	yes	characteristic
29	#95	C₂²	1	1	yes	characteristic
30	#95	C₂²	1	1	yes	characteristic
31	#95	C₂²	1	2	yes	—
32	#95	C₂²	1	2	yes	—
33	#95	C₂²	1	2	yes	—
34	#95	C₂²	1	2	yes	—
35	#95	C₂²	1	2	yes	—
36	#95	C₂²	1	2	yes	—
37	C₂×Q₈	—	2	2	no	—
38	C₂×Q₈	—	2	4	no	—
39	C₂×Q₈	—	2	2	no	—
40	C₂×Q₈	—	2	4	no	—
41	C₂×Q₈	—	4	2	no	—
42	C₂×Q₈	Q₈	1	4	yes	complemented
43	C₂²×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₄	C₂³	1	2	yes	—
45	C₂²×C₄	C₂³	1	2	yes	—
46	C₂²×C₄	C₂³	1	1	yes	characteristic
47	C₂²×C₄	C₂³	1	2	yes	—
48	C₂²×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₄	C₂³	1	2	yes	—
50	C₂²×C₄	C₂³	1	1	yes	characteristic
51	C₂²×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₄	C₂³	1	1	yes	characteristic
54	C₄⋊C₄	—	2	4	no	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	D₄	1	4	yes	—
62	C₄⋊C₄	Q₈	1	4	yes	complemented
63	C₄²	—	2	4	no	—
64	C₄²	—	2	2	no	—
65	C₄²	—	2	2	no	—
66	C₄²	—	2	2	no	—
67	C₂³	C₂⁴	1	1	yes	characteristic
68	Q₈	—	2	8	no	—
69	Q₈	—	4	8	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	C₂×D₄	1	4	yes	—
95	C₂×C₄	C₂×D₄	1	2	yes	—
96	C₂×C₄	C₂×Q₈	1	2	yes	—
97	C₂×C₄	C₂×Q₈	1	2	yes	—
98	C₂×C₄	C₂×Q₈	1	2	yes	—
99	C₂×C₄	C₂×Q₈	1	4	yes	—
100	C₂×C₄	C₂×Q₈	1	2	yes	—
101	C₂×C₄	#41	1	2	yes	—
102	C₂×C₄	#41	1	2	yes	—
103	C₂²	C₂²×D₄	1	1	yes	characteristic
104	C₂²	C₂²×Q₈	1	1	yes	characteristic
105	C₂²	C₂²×Q₈	1	1	yes	characteristic
106	C₂²	C₂×#41	1	1	yes	characteristic
107	C₂²	C₂×#41	1	1	yes	characteristic
108	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
110	C₄	—	2	4	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	4	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	2	no	—
115	C₄	—	2	2	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	2	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	2	no	—
120	C₄	C₂²⋊Q₈	1	4	yes	—
121	C₂	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂	#533	1	1	yes	characteristic
123	C₂	#542	1	1	yes	characteristic
124	C₂	#546	1	1	yes	characteristic
125	C₂	D₄×Q₈	1	1	yes	characteristic
126	C₂	#557	1	1	yes	characteristic
127	C₂	Q₈²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂

yes

maximal, characteristic

#398

C₂

yes

maximal

#395

C₂

yes

maximal

#391

C₂

yes

maximal

#391

C₂

yes

maximal, characteristic

#391

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#387

C₂

yes

maximal

C₄×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂²×Q₈

C₂²

yes

—

C₂²×Q₈

C₂²

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

#95

C₂²

yes

characteristic

#95

C₂²

yes

characteristic

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈

yes

complemented

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

Q₈

yes

complemented

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂⁴

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

100

C₂×C₄

C₂×Q₈

yes

—

101

C₂×C₄

#41

yes

—

102

C₂×C₄

#41

yes

—

103

C₂²

C₂²×D₄

yes

characteristic

104

C₂²

C₂²×Q₈

yes

characteristic

105

C₂²

C₂²×Q₈

yes

characteristic

106

C₂²

C₂×#41

yes

characteristic

107

C₂²

C₂×#41

yes

characteristic

108

C₂²

H(5,𝔽₂)

yes

characteristic

109

C₂²

2₋¹⁺⁴

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂²⋊Q₈

yes

—

121

C₂

C₂×C₂²⋊Q₈

yes

characteristic

122

C₂

#533

yes

characteristic

123

C₂

#542

yes

characteristic

124

C₂

#546

yes

characteristic

125

C₂

D₄×Q₈

yes

characteristic

126

C₂

#557

yes

characteristic

127

C₂

Q₈²

yes

characteristic

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

subgroup

G/H

center, commutator, Frattini, socle, prime radical, cosocle

C₂³

C₂⁴

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
42. C₂×Q₈	Q₈	semidirect
62. C₄⋊C₄	Q₈	semidirect

H ≅ G/N

type

42. C₂×Q₈

Q₈

semidirect

62. C₄⋊C₄

Q₈

semidirect

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

subgroup	index
C₂×C₄⋊Q₈	2
C₂×C₄×Q₈	2
#398	2
#395	2
#391	2
#389	2
#387	2
C₄×C₄⋊C₄	2

subgroup

index

C₂×C₄⋊Q₈

C₂×C₄×Q₈

C₄×C₄⋊C₄

Subgroups — #2451

Conjugacy classes of subgroups

Complements

Maximal subgroups