Group #11903713 — subgroups

Subgroups — Q₈²×C₃⋊D₃ #11903713

Order 1152.

3336 conjugacy classes of subgroups; only the 939 normal subgroups are shown.

normal subgroups only

This group has 3336 conjugacy classes of subgroups — too many to draw. Only its 939 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	C₃²×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊D₃×C₄⋊Q₈	C₂	1	9	yes	maximal
3	#10596193	C₂	1	9	yes	maximal, complemented
4	C₄×Q₈×C₃⋊D₃	C₂	1	6	yes	maximal
5	Q₈×#634	C₂	1	6	yes	maximal, complemented
6	D₃×Q₈²	—	3	4	no	maximal
7	C₂×Q₈×C₃⋊D₃	C₂²	1	6	yes	—
8	C₂×Q₈×C₃⋊D₃	C₂²	1	2	yes	—
9	C₃²×C₄⋊Q₈	C₂²	1	9	yes	—
10	C₃²×C₄×Q₈	C₂²	1	6	yes	—
11	Q₈×C₃⋊Dic₃	C₂²	1	6	yes	—
12	#64153	C₂²	1	18	yes	—
13	C₄⋊C₄×C₃⋊D₃	C₂²	1	18	yes	—
14	#64097	C₂²	1	36	yes	—
15	#64096	C₂²	1	18	yes	—
16	C₄²×C₃⋊D₃	C₂²	1	9	yes	—
17	#64078	C₂²	1	9	yes	—
18	C₄×#634	C₂²	1	18	yes	—
19	C₃×Q₈²	D₃	1	4	yes	complemented
20	D₃×C₄⋊Q₈	—	3	36	no	—
21	#5722	—	3	36	no	—
22	C₄×D₃×Q₈	—	3	24	no	—
23	Q₈×Dic₆	—	3	24	no	—
24	C₂×C₃²×Q₈	C₂³	1	6	yes	—
25	C₂×C₃²×Q₈	C₂³	1	2	yes	—
26	Q₈×C₃⋊D₃	—	4	12	no	—
27	Q₈×C₃⋊D₃	Q₈	1	16	yes	direct product factor
28	C₂×C₄×C₃⋊D₃	C₂³	1	9	yes	—
29	C₂×C₄×C₃⋊D₃	C₂³	1	6	yes	—
30	C₂×#634	C₂³	1	18	yes	—
31	C₂×#634	C₂³	1	6	yes	—
32	C₃²×C₄⋊C₄	C₂³	1	18	yes	—
33	C₃²×C₄²	C₂³	1	9	yes	—
34	C₄⋊(C₃⋊Dic₃)	C₂³	1	18	yes	—
35	#3750	C₂³	1	36	yes	—
36	C₄×C₃⋊Dic₃	C₂³	1	18	yes	—
37	C₄×C₃⋊Dic₃	C₂³	1	9	yes	—
38	C₂×Q₈²	—	9	1	no	Sylow
39	C₂×D₃×Q₈	—	3	8	no	—
40	C₂×D₃×Q₈	—	3	24	no	—
41	C₃×C₄⋊Q₈	C₂×D₃	1	36	yes	—
42	C₃×C₄×Q₈	C₂×D₃	1	24	yes	—
43	Q₈×Dic₃	—	3	24	no	—
44	#944	—	3	72	no	—
45	D₃×C₄⋊C₄	—	3	72	no	—
46	#888	—	3	144	no	—
47	#887	—	3	72	no	—
48	C₄²×D₃	—	3	36	no	—
49	#869	—	3	36	no	—
50	C₄×Dic₆	—	3	72	no	—
51	C₂²×C₃⋊D₃	C₂⁴	1	1	yes	characteristic
52	C₃²×Q₈	—	4	6	no	—
53	C₃²×Q₈	C₂×Q₈	1	8	yes	complemented
54	C₂×C₃²×C₄	C₂⁴	1	9	yes	—
55	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
56	C₂×C₃⋊Dic₃	C₂⁴	1	9	yes	—
57	C₂×C₃⋊Dic₃	C₂⁴	1	6	yes	—
58	C₄×C₃⋊D₃	—	2	18	no	—
59	C₄×C₃⋊D₃	C₂×Q₈	1	24	yes	—
60	#634	—	4	18	no	—
61	#634	C₂×Q₈	1	24	yes	complemented
62	Q₈²	—	9	9	no	—
63	Q₈²	—	9	6	no	—
64	Q₈²	C₃⋊D₃	1	1	yes	characteristic, direct product factor
65	C₂×C₄⋊Q₈	—	9	9	no	—
66	C₂×C₄×Q₈	—	9	6	no	—
67	C₂×C₃×Q₈	C₂²×D₃	1	24	yes	—
68	C₂×C₃×Q₈	C₂²×D₃	1	8	yes	—
69	D₃×Q₈	—	3	64	no	—
70	D₃×Q₈	—	12	48	no	—
71	C₂×C₄×D₃	—	3	24	no	—
72	C₂×C₄×D₃	—	3	36	no	—
73	C₂×Dic₆	—	3	24	no	—
74	C₂×Dic₆	—	3	72	no	—
75	C₃×C₄⋊C₄	C₂²×D₃	1	72	yes	—
76	C₃×C₄²	C₂²×D₃	1	36	yes	—
77	C₄⋊Dic₃	—	3	72	no	—
78	Dic₃⋊C₄	—	3	144	no	—
79	C₄×Dic₃	—	3	72	no	—
80	C₄×Dic₃	—	3	36	no	—
81	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
82	C₂×C₃⋊D₃	C₂²×Q₈	1	4	yes	—
83	C₂×C₃⋊D₃	H(5,𝔽₂)	1	2	yes	—
84	C₃²×C₄	—	2	9	no	—
85	C₃²×C₄	C₂²×Q₈	1	12	yes	—
86	C₃⋊Dic₃	—	2	9	no	—
87	C₃⋊Dic₃	C₂²×Q₈	1	12	yes	—
88	C₂²×Q₈	—	9	2	no	—
89	C₂²×Q₈	—	9	6	no	—
90	C₄⋊Q₈	—	9	18	no	—
91	C₄⋊Q₈	—	9	36	no	—
92	C₄⋊Q₈	—	9	9	no	—
93	C₄⋊Q₈	C₂×C₃⋊D₃	1	9	yes	—
94	C₄×Q₈	—	9	6	no	—
95	C₄×Q₈	—	9	18	no	—
96	C₄×Q₈	—	9	18	no	—
97	C₄×Q₈	C₂×C₃⋊D₃	1	6	yes	—
98	C₂×C₄⋊C₄	—	9	18	no	—
99	C₂×C₄²	—	9	9	no	—
100	C₂²×D₃	—	3	4	no	—
101	C₃×Q₈	—	4	24	no	—
102	C₃×Q₈	D₃×Q₈	1	32	yes	complemented
103	C₂×C₃×C₄	C₂³×D₃	1	36	yes	—
104	C₂×C₃×C₄	C₂³×D₃	1	24	yes	—
105	C₂×Dic₃	—	3	24	no	—
106	C₂×Dic₃	—	3	36	no	—
107	C₄×D₃	—	3	96	no	—
108	C₄×D₃	—	6	72	no	—
109	Dic₆	—	3	96	no	—
110	Dic₆	—	12	72	no	—
111	C₂×C₃²	C₂³×Q₈	1	2	yes	—
112	C₂×C₃²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
113	C₃⋊D₃	Q₈²	1	4	yes	direct product factor
114	C₂×Q₈	—	9	16	no	—
115	C₂×Q₈	—	9	6	no	—
116	C₂×Q₈	—	9	18	no	—
117	C₂×Q₈	—	36	12	no	—
118	C₂×Q₈	C₂²×C₃⋊D₃	1	6	yes	—
119	C₂×Q₈	C₂²×C₃⋊D₃	1	2	yes	—
120	C₂²×C₄	—	9	6	no	—
121	C₂²×C₄	—	9	9	no	—
122	C₄⋊C₄	—	9	36	no	—
123	C₄⋊C₄	—	9	18	no	—
124	C₄⋊C₄	C₂²×C₃⋊D₃	1	18	yes	—
125	C₄²	—	9	9	no	—
126	C₄²	—	9	18	no	—
127	C₄²	C₂²×C₃⋊D₃	1	9	yes	—
128	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
129	C₂×D₃	—	3	16	no	—
130	C₂×D₃	—	3	8	no	—
131	C₃×C₄	—	2	36	no	—
132	C₃×C₄	C₂×D₃×Q₈	1	48	yes	—
133	Dic₃	—	3	48	no	—
134	Dic₃	—	6	36	no	—
135	C₃²	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
136	C₂³	—	9	1	no	—
137	Q₈	—	4	6	no	—
138	Q₈	—	9	24	no	—
139	Q₈	—	36	18	no	—
140	Q₈	Q₈×C₃⋊D₃	1	8	yes	direct product factor
141	C₂×C₄	—	9	6	no	—
142	C₂×C₄	—	9	24	no	—
143	C₂×C₄	—	9	9	no	—
144	C₂×C₄	—	18	18	no	—
145	C₂×C₄	C₂³×C₃⋊D₃	1	9	yes	—
146	C₂×C₄	C₂³×C₃⋊D₃	1	6	yes	—
147	C₂×C₃	C₂²×D₃×Q₈	1	8	yes	—
148	C₂×C₃	D₃×H(5,𝔽₂)	1	4	yes	—
149	D₃	—	3	16	no	—
150	C₂²	—	9	4	no	—
151	C₂²	—	9	2	no	—
152	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
153	C₄	—	2	9	no	—
154	C₄	—	9	12	no	—
155	C₄	—	18	9	no	—
156	C₄	C₂×Q₈×C₃⋊D₃	1	12	yes	—
157	C₃	D₃×Q₈²	1	4	yes	complemented
158	C₂	—	9	4	no	—
159	C₂	C₂²×Q₈×C₃⋊D₃	1	2	yes	—
160	C₂	C₃⋊D₃×H(5,𝔽₂)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×Q₈²

C₂

yes

maximal, characteristic, complemented

C₃⋊D₃×C₄⋊Q₈

C₂

yes

maximal

#10596193

C₂

yes

maximal, complemented

C₄×Q₈×C₃⋊D₃

C₂

yes

maximal

Q₈×#634

C₂

yes

maximal, complemented

D₃×Q₈²

—

maximal

C₂×Q₈×C₃⋊D₃

C₂²

yes

—

C₂×Q₈×C₃⋊D₃

C₂²

yes

—

C₃²×C₄⋊Q₈

C₂²

yes

—

C₃²×C₄×Q₈

C₂²

yes

—

Q₈×C₃⋊Dic₃

C₂²

yes

—

#64153

C₂²

yes

—

C₄⋊C₄×C₃⋊D₃

C₂²

yes

—

#64097

C₂²

yes

—

#64096

C₂²

yes

—

C₄²×C₃⋊D₃

C₂²

yes

—

#64078

C₂²

yes

—

C₄×#634

C₂²

yes

—

C₃×Q₈²

D₃

yes

complemented

D₃×C₄⋊Q₈

—

#5722

—

C₄×D₃×Q₈

—

Q₈×Dic₆

—

C₂×C₃²×Q₈

C₂³

yes

—

C₂×C₃²×Q₈

C₂³

yes

—

Q₈×C₃⋊D₃

—

Q₈×C₃⋊D₃

Q₈

yes

direct product factor

C₂×C₄×C₃⋊D₃

C₂³

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

—

C₂×#634

C₂³

yes

—

C₂×#634

C₂³

yes

—

C₃²×C₄⋊C₄

C₂³

yes

—

C₃²×C₄²

C₂³

yes

—

C₄⋊(C₃⋊Dic₃)

C₂³

yes

—

#3750

C₂³

yes

—

C₄×C₃⋊Dic₃

C₂³

yes

—

C₄×C₃⋊Dic₃

C₂³

yes

—

C₂×Q₈²

—

Sylow

C₂×D₃×Q₈

—

C₂×D₃×Q₈

—

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₃×C₄×Q₈

C₂×D₃

yes

—

Q₈×Dic₃

—

#944

—

D₃×C₄⋊C₄

—

#888

—

144

—

#887

—

C₄²×D₃

—

#869

—

C₄×Dic₆

—

C₂²×C₃⋊D₃

C₂⁴

yes

characteristic

C₃²×Q₈

—

C₃²×Q₈

C₂×Q₈

yes

complemented

C₂×C₃²×C₄

C₂⁴

yes

—

C₂×C₃²×C₄

C₂⁴

yes

—

C₂×C₃⋊Dic₃

C₂⁴

yes

—

C₂×C₃⋊Dic₃

C₂⁴

yes

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

C₂×Q₈

yes

—

#634

—

#634

C₂×Q₈

yes

complemented

Q₈²

—

Q₈²

—

Q₈²

C₃⋊D₃

yes

characteristic, direct product factor

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×C₃×Q₈

C₂²×D₃

yes

—

C₂×C₃×Q₈

C₂²×D₃

yes

—

D₃×Q₈

—

D₃×Q₈

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂²×D₃

yes

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

144

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂⁵

yes

characteristic

C₂×C₃⋊D₃

C₂²×Q₈

yes

—

C₂×C₃⋊D₃

H(5,𝔽₂)

yes

—

C₃²×C₄

—

C₃²×C₄

C₂²×Q₈

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

C₂²×Q₈

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₃⋊D₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₃⋊D₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

100

C₂²×D₃

—

101

C₃×Q₈

—

102

C₃×Q₈

D₃×Q₈

yes

complemented

103

C₂×C₃×C₄

C₂³×D₃

yes

—

104

C₂×C₃×C₄

C₂³×D₃

yes

—

105

C₂×Dic₃

—

106

C₂×Dic₃

—

107

C₄×D₃

—

108

C₄×D₃

—

109

Dic₆

—

110

Dic₆

—

111

C₂×C₃²

C₂³×Q₈

yes

—

112

C₂×C₃²

C₂×H(5,𝔽₂)

yes

characteristic

113

C₃⋊D₃

Q₈²

yes

direct product factor

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

C₂²×C₃⋊D₃

yes

—

119

C₂×Q₈

C₂²×C₃⋊D₃

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×C₃⋊D₃

yes

—

125

C₄²

—

126

C₄²

—

127

C₄²

C₂²×C₃⋊D₃

yes

—

128

C₂²×C₃

C₂⁴×D₃

yes

—

129

C₂×D₃

—

130

C₂×D₃

—

131

C₃×C₄

—

132

C₃×C₄

C₂×D₃×Q₈

yes

—

133

Dic₃

—

134

Dic₃

—

135

C₃²

C₂×Q₈²

yes

characteristic, complemented, Sylow

136

C₂³

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

140

Q₈

Q₈×C₃⋊D₃

yes

direct product factor

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂³×C₃⋊D₃

yes

—

146

C₂×C₄

C₂³×C₃⋊D₃

yes

—

147

C₂×C₃

C₂²×D₃×Q₈

yes

—

148

C₂×C₃

D₃×H(5,𝔽₂)

yes

—

149

D₃

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂⁴×C₃⋊D₃

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂×Q₈×C₃⋊D₃

yes

—

157

C₃

D₃×Q₈²

yes

complemented

158

C₂

—

159

C₂

C₂²×Q₈×C₃⋊D₃

yes

—

160

C₂

C₃⋊D₃×H(5,𝔽₂)

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂⁵
Fitting	C₃²×Q₈²	C₂
prime radical	C₂²×C₃⋊D₃	C₂⁴
hypercenter	Q₈²	C₃⋊D₃
hypocenter	C₃²	C₂×Q₈²
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×C₃⋊D₃

commutator, socle, cosocle

C₂²×C₃²

C₂⁵

Fitting

C₃²×Q₈²

C₂

prime radical

C₂²×C₃⋊D₃

C₂⁴

hypercenter

Q₈²

C₃⋊D₃

hypocenter

C₃²

C₂×Q₈²

LCS

G ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. C₃²×Q₈²	C₂	semidirect
3. #10596193	C₂	semidirect
5. Q₈×#634	C₂	semidirect
19. C₃×Q₈²	D₃	semidirect
27. Q₈×C₃⋊D₃	Q₈	direct (N × H)
53. C₃²×Q₈	C₂×Q₈	semidirect
61. #634	C₂×Q₈	semidirect
64. Q₈²	C₃⋊D₃	direct (N × H)
102. C₃×Q₈	D₃×Q₈	semidirect
113. C₃⋊D₃	Q₈²	direct (N × H)
135. C₃²	C₂×Q₈²	semidirect
140. Q₈	Q₈×C₃⋊D₃	direct (N × H)
157. C₃	D₃×Q₈²	semidirect

H ≅ G/N

type

1. C₃²×Q₈²

C₂

semidirect

3. #10596193

C₂

semidirect

5. Q₈×#634

C₂

semidirect

19. C₃×Q₈²

D₃

semidirect

27. Q₈×C₃⋊D₃

Q₈

direct (N × H)

53. C₃²×Q₈

C₂×Q₈

semidirect

61. #634

C₂×Q₈

semidirect

64. Q₈²

C₃⋊D₃

direct (N × H)

102. C₃×Q₈

D₃×Q₈

semidirect

113. C₃⋊D₃

Q₈²

direct (N × H)

135. C₃²

C₂×Q₈²

semidirect

140. Q₈

Q₈×C₃⋊D₃

direct (N × H)

157. C₃

D₃×Q₈²

semidirect

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

subgroup	index
C₃²×Q₈²	2
C₃⋊D₃×C₄⋊Q₈	2
#10596193	2
C₄×Q₈×C₃⋊D₃	2
Q₈×#634	2
D₃×Q₈²	3

subgroup

index

C₃²×Q₈²

C₃⋊D₃×C₄⋊Q₈

#10596193

C₄×Q₈×C₃⋊D₃

Q₈×#634

D₃×Q₈²

Subgroups — Q₈²×C₃⋊D₃ #11903713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q82×C3⋊D3 #11903713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈²×C₃⋊D₃ #11903713