Group #11799512 — subgroups

Subgroups — #11799512

Order 1152.

180 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 39 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	D₈×#157	C₂	1	1	yes	maximal, characteristic
2	#10593536	C₂	1	1	yes	maximal, characteristic, complemented
3	#10590193	C₂	1	2	yes	maximal
4	#10590085	C₂	1	2	yes	maximal, complemented
5	#10590082	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#157	—	2	1	no	—
7	C₄⋊PSU(3,𝔽₂)	—	2	1	no	—
8	D₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	#63782	—	2	1	no	—
10	#63769	C₂²	1	2	yes	—
11	#63769	C₂²	1	1	yes	characteristic
12	C₈×#157	C₂²	1	1	yes	characteristic
13	#63764	C₂²	1	2	yes	—
14	#63746	—	2	1	no	—
15	C₂²×#157	—	4	1	no	—
16	C₂×PSU(3,𝔽₂)	—	4	1	no	—
17	D₄×C₃⋊D₃	—	2	1	no	—
18	C₂²⋊#157	—	4	1	no	—
19	C₄⋊#157	—	2	2	no	—
20	C₄⋊#157	—	2	1	no	—
21	C₄×#157	D₄	1	1	yes	characteristic
22	#3787	D₄	1	1	yes	characteristic
23	#3777	—	4	1	no	—
24	C₃²×D₈	Q₈	1	1	yes	characteristic, complemented
25	#3753	—	2	1	no	—
26	#3745	Q₈	1	1	yes	characteristic, complemented
27	C₈×C₃⋊D₃	C₂³	1	1	yes	characteristic
28	#3708	—	2	2	no	—
29	#2224	—	9	1	no	maximal, Sylow
30	D₃×D₈	—	12	1	no	—
31	C₂²×C₃⋊D₃	—	4	1	no	—
32	C₂×#157	—	4	2	no	—
33	C₂×#157	—	4	1	no	—
34	C₂×#157	—	8	2	no	—
35	C₂×#157	D₈	1	1	yes	characteristic
36	C₂×#157	D₈	1	1	yes	characteristic
37	PSU(3,𝔽₂)	—	8	2	no	—
38	C₃²×D₄	—	2	1	no	—
39	#638	—	4	1	no	—
40	C₄⋊(C₃⋊D₃)	—	2	1	no	—
41	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
42	#622	—	2	1	no	—
43	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₃⋊(C₃⋊C₈)	#41	1	1	yes	characteristic
45	#500	—	9	1	no	—
46	C₄×D₈	—	9	1	no	—
47	#366	—	9	2	no	—
48	C₄⋊C₁₆	—	9	1	no	—
49	#357	—	9	2	no	—
50	D₃×D₄	—	12	2	no	—
51	C₃×D₈	—	4	1	no	—
52	#213	—	12	2	no	—
53	D₂₄	—	12	1	no	—
54	D₃×C₈	—	12	1	no	—
55	C₂²×C₃²	—	4	1	no	—
56	C₂×C₃⋊D₃	—	4	1	no	—
57	C₂×C₃⋊D₃	—	8	1	no	—
58	C₂×C₃⋊D₃	—	8	1	no	—
59	C₂×C₃⋊D₃	C₂×D₈	1	1	yes	characteristic
60	#157	—	2	1	no	—
61	#157	—	8	2	no	—
62	#157	—	8	1	no	—
63	#157	D₁₆	1	2	yes	—
64	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
65	C₃⋊Dic₃	#137	1	1	yes	characteristic
66	C₂×D₈	—	9	1	no	—
67	C₄⋊Q₈	—	18	1	no	—
68	C₄×D₄	—	18	1	no	—
69	C₂×C₁₆	—	9	2	no	—
70	#107	—	9	2	no	—
71	#107	—	9	1	no	—
72	#107	—	18	1	no	—
73	#102	—	18	1	no	—
74	C₄×C₈	—	9	1	no	—
75	C₂²×D₃	—	24	2	no	—
76	C₃×D₄	—	4	2	no	—
77	#67	—	24	2	no	—
78	D₁₂	—	12	2	no	—
79	C₄×D₃	—	12	1	no	—
80	C₃×C₈	—	4	1	no	—
81	C₃⋊C₈	—	12	1	no	—
82	C₂×C₃²	—	8	1	no	—
83	C₂×C₃²	#474	1	1	yes	characteristic
84	C₃⋊D₃	—	8	1	no	—
85	C₃⋊D₃	C₂×D₁₆	1	1	yes	characteristic
86	C₃⋊D₃	#510	1	1	yes	characteristic
87	C₂×Q₈	—	36	1	no	—
88	C₂×D₄	—	18	1	no	—
89	C₂²×C₄	—	36	1	no	—
90	D₈	—	9	1	no	—
91	D₈	—	18	1	no	—
92	D₈	PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
93	C₂×C₈	—	9	1	no	—
94	C₂×C₈	—	9	1	no	—
95	C₄⋊C₄	—	18	1	no	—
96	C₄⋊C₄	—	18	2	no	—
97	C₄⋊C₄	—	36	1	no	—
98	C₂²⋊C₄	—	36	1	no	—
99	C₄²	—	9	1	no	—
100	C₁₆	—	18	2	no	—
101	C₂²×C₃	—	8	2	no	—
102	C₂×D₃	—	12	1	no	—
103	C₂×D₃	—	24	2	no	—
104	C₂×D₃	—	48	2	no	—
105	C₂×D₃	—	48	2	no	—
106	C₃×C₄	—	4	1	no	—
107	Dic₃	—	12	1	no	—
108	C₃²	#2224	1	1	yes	characteristic, complemented, Sylow
109	C₂³	—	36	1	no	—
110	Q₈	—	72	2	no	—
111	D₄	—	2	1	no	—
112	D₄	—	18	1	no	—
113	D₄	—	36	1	no	—
114	C₂×C₄	—	9	1	no	—
115	C₂×C₄	—	9	1	no	—
116	C₂×C₄	—	9	1	no	—
117	C₂×C₄	—	36	1	no	—
118	C₂×C₄	—	36	2	no	—
119	C₂×C₄	—	72	2	no	—
120	C₈	—	9	1	no	—
121	C₈	—	18	1	no	—
122	C₈	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
123	C₂×C₃	—	4	1	no	—
124	C₂×C₃	—	16	2	no	—
125	D₃	—	12	1	no	—
126	D₃	—	12	1	no	—
127	D₃	—	48	2	no	—
128	C₂²	—	4	1	no	—
129	C₂²	—	9	1	no	—
130	C₂²	—	36	1	no	—
131	C₂²	—	72	1	no	—
132	C₂²	—	72	1	no	—
133	C₄	—	9	2	no	—
134	C₄	—	9	1	no	—
135	C₄	—	18	1	no	—
136	C₄	—	72	1	no	—
137	C₄	—	72	2	no	—
138	C₄	C₂²⋊PSU(3,𝔽₂)	1	1	yes	characteristic
139	C₃	—	4	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	9	1	no	—
142	C₂	—	9	1	no	—
143	C₂	—	72	1	no	—
144	C₂	#10593558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₈×#157

C₂

yes

maximal, characteristic

#10593536

C₂

yes

maximal, characteristic, complemented

#10590193

C₂

yes

maximal

#10590085

C₂

yes

maximal, complemented

#10590082

C₂

yes

maximal, characteristic, complemented

D₄×#157

—

C₄⋊PSU(3,𝔽₂)

—

D₈×C₃⋊D₃

C₂²

yes

characteristic

#63782

—

#63769

C₂²

yes

—

#63769

C₂²

yes

characteristic

C₈×#157

C₂²

yes

characteristic

#63764

C₂²

yes

—

#63746

—

C₂²×#157

—

C₂×PSU(3,𝔽₂)

—

D₄×C₃⋊D₃

—

C₂²⋊#157

—

C₄⋊#157

—

C₄⋊#157

—

C₄×#157

D₄

yes

characteristic

#3787

D₄

yes

characteristic

#3777

—

C₃²×D₈

Q₈

yes

characteristic, complemented

#3753

—

#3745

Q₈

yes

characteristic, complemented

C₈×C₃⋊D₃

C₂³

yes

—

—

D₃×D₈

—

C₂²×C₃⋊D₃

—

—

—

—

D₈

yes

characteristic

C₂×#157

D₈

yes

characteristic

PSU(3,𝔽₂)

—

C₃²×D₄

—

#638

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

#622

—

C₃²×C₈

C₂×Q₈

yes

C₃⋊(C₃⋊C₈)

yes

—

C₄×D₈

—

#366

—

C₄⋊C₁₆

—

#357

—

D₃×D₄

—

C₃×D₈

—

#213

—

D₂₄

—

D₃×C₈

—

C₂²×C₃²

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

C₂×D₈

yes

—

—

—

D₁₆

yes

—

C₃²×C₄

C₂²⋊Q₈

yes

characteristic

C₃⋊Dic₃

#137

yes

characteristic

C₂×D₈

—

C₄⋊Q₈

—

C₄×D₄

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

—

C₂²×D₃

—

C₃×D₄

—

#67

—

D₁₂

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#474

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

C₂×D₁₆

yes

characteristic

C₃⋊D₃

#510

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

D₈

PSU(3,𝔽₂)

yes

characteristic, complemented

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₁₆

—

101

C₂²×C₃

—

102

C₂×D₃

—

103

C₂×D₃

—

104

C₂×D₃

—

105

C₂×D₃

—

106

C₃×C₄

—

107

Dic₃

—

108

C₃²

#2224

yes

characteristic, complemented, Sylow

109

C₂³

—

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₈

—

121

C₈

—

122

C₈

C₂×PSU(3,𝔽₂)

yes

characteristic

123

C₂×C₃

—

124

C₂×C₃

—

125

D₃

—

126

D₃

—

127

D₃

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₂²⋊PSU(3,𝔽₂)

yes

characteristic

139

C₃

—

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#10593558

yes

characteristic

	subgroup	G/H
center	C₂	#10593558
commutator, cosocle	C₈×C₃⋊D₃	C₂³
Frattini	C₈	C₂×PSU(3,𝔽₂)
Fitting	C₃²×D₈	Q₈
socle	C₂×C₃²	#474
prime radical	D₈×C₃⋊D₃	C₂²
hypercenter	D₈	PSU(3,𝔽₂)
hypocenter	C₃²	#2224
LCS	G ▷ C₈×C₃⋊D₃ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

subgroup

G/H

center

C₂

#10593558

commutator, cosocle

C₈×C₃⋊D₃

C₂³

Frattini

C₈

C₂×PSU(3,𝔽₂)

Fitting

C₃²×D₈

Q₈

socle

C₂×C₃²

#474

prime radical

D₈×C₃⋊D₃

C₂²

D₈

C₃²

G ▷ C₈×C₃⋊D₃ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

N	H ≅ G/N	type
2. #10593536	C₂	semidirect
4. #10590085	C₂	semidirect
5. #10590082	C₂	semidirect
24. C₃²×D₈	Q₈	semidirect
26. #3745	Q₈	semidirect
92. D₈	PSU(3,𝔽₂)	semidirect
108. C₃²	#2224	semidirect

H ≅ G/N

type

2. #10593536

C₂

semidirect

4. #10590085

C₂

semidirect

5. #10590082

C₂

semidirect

24. C₃²×D₈

Q₈

semidirect

26. #3745

Q₈

semidirect

92. D₈

PSU(3,𝔽₂)

semidirect

108. C₃²

#2224

semidirect

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

subgroup	index
D₈×#157	2
#10593536	2
#10590193	2
#10590085	2
#10590082	2
#2224	9

subgroup

index

Subgroups — #11799512

Conjugacy classes of subgroups

Complements

Maximal subgroups