Group #11919725 — subgroups

Subgroups — C₂×Q₈×Q₈⋊C₉ #11919725

Order 1152.

597 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 213 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	Q₈×Q₈⋊C₉	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₄×Q₈⋊C₉	C₂	1	3	yes	maximal
3	C₂×C₃×Q₈²	C₃	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₉	—	4	1	no	maximal
5	C₂²×Q₈⋊C₉	C₂²	1	1	yes	characteristic
6	C₄×Q₈⋊C₉	C₂²	1	6	yes	—
7	C₃×Q₈²	—	3	4	no	—
8	C₃×Q₈²	C₂×C₃	1	4	yes	—
9	C₂×C₃×C₄⋊Q₈	—	3	3	no	—
10	C₂×C₃×C₄×Q₈	—	3	1	no	—
11	C₂×C₃×C₄×Q₈	C₂×C₃	1	3	yes	—
12	C₂×Q₈×C₉	—	4	8	no	—
13	C₂×Q₈×C₉	—	4	4	no	—
14	C₂²×C₄×C₉	—	4	3	no	—
15	C₂×Q₈⋊C₉	Q₈	1	2	yes	direct product factor
16	C₂×Q₈⋊C₉	C₂³	1	1	yes	characteristic
17	C₂×Q₈²	C₉	1	1	yes	characteristic, complemented, Sylow
18	C₂²×C₃×Q₈	—	3	2	no	—
19	C₂²×C₃×Q₈	A₄	1	1	yes	characteristic
20	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
21	C₃×C₄⋊Q₈	—	3	12	no	—
22	C₃×C₄⋊Q₈	—	3	12	no	—
23	C₃×C₄×Q₈	—	3	4	no	—
24	C₃×C₄×Q₈	—	3	4	no	—
25	C₃×C₄×Q₈	—	3	6	no	—
26	C₃×C₄×Q₈	C₂²×C₃	1	6	yes	—
27	C₂×C₃×C₄⋊C₄	—	3	3	no	—
28	C₂×C₃×C₄⋊C₄	—	3	3	no	—
29	C₂×C₃×C₄²	—	3	3	no	—
30	C₂³×C₉	—	4	1	no	—
31	Q₈×C₉	—	4	16	no	—
32	C₂×C₄×C₉	—	4	12	no	—
33	C₂×C₄×C₉	—	4	6	no	—
34	Q₈⋊C₉	C₂×Q₈	1	1	yes	characteristic, direct product factor
35	Q₈²	—	3	4	no	—
36	Q₈²	C₂×C₉	1	4	yes	complemented
37	C₂×C₄⋊Q₈	—	3	3	no	—
38	C₂×C₄×Q₈	—	3	1	no	—
39	C₂×C₄×Q₈	C₂×C₉	1	3	yes	—
40	C₂×C₃×Q₈	—	3	8	no	—
41	C₂×C₃×Q₈	—	3	1	no	—
42	C₂×C₃×Q₈	—	3	2	no	—
43	C₂×C₃×Q₈	—	12	4	no	—
44	C₂×C₃×Q₈	SL(2,𝔽₃)	1	8	yes	—
45	C₂×C₃×Q₈	C₃×Q₈	1	2	yes	—
46	C₂×C₃×Q₈	C₂×A₄	1	4	yes	—
47	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
48	C₂²×C₃×C₄	—	3	1	no	—
49	C₂²×C₃×C₄	—	3	3	no	—
50	C₂²×C₃×C₄	C₂×A₄	1	3	yes	—
51	C₃×C₄⋊C₄	—	3	6	no	—
52	C₃×C₄⋊C₄	—	3	6	no	—
53	C₃×C₄⋊C₄	—	3	12	no	—
54	C₃×C₄²	—	3	6	no	—
55	C₃×C₄²	—	3	6	no	—
56	C₂²×C₉	—	4	2	no	—
57	C₂²×C₉	—	4	2	no	—
58	C₂²×C₉	—	4	2	no	—
59	C₂²×C₉	—	4	1	no	—
60	C₄×C₉	—	4	12	no	—
61	C₂²×Q₈	—	3	2	no	—
62	C₂²×Q₈	C₂²⋊C₉	1	1	yes	characteristic
63	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
64	C₄⋊Q₈	—	3	12	no	—
65	C₄⋊Q₈	—	3	12	no	—
66	C₄×Q₈	—	3	4	no	—
67	C₄×Q₈	—	3	4	no	—
68	C₄×Q₈	—	3	6	no	—
69	C₄×Q₈	C₂²×C₉	1	6	yes	—
70	C₂×C₄⋊C₄	—	3	3	no	—
71	C₂×C₄⋊C₄	—	3	3	no	—
72	C₂×C₄²	—	3	3	no	—
73	C₂³×C₃	C₂²×A₄	1	1	yes	characteristic
74	C₃×Q₈	—	3	2	no	—
75	C₃×Q₈	—	3	1	no	—
76	C₃×Q₈	—	3	2	no	—
77	C₃×Q₈	—	12	8	no	—
78	C₃×Q₈	C₂×SL(2,𝔽₃)	1	16	yes	—
79	C₃×Q₈	C₂×C₃×Q₈	1	1	yes	characteristic
80	C₂×C₃×C₄	—	3	2	no	—
81	C₂×C₃×C₄	—	3	2	no	—
82	C₂×C₃×C₄	—	3	1	no	—
83	C₂×C₃×C₄	—	3	1	no	—
84	C₂×C₃×C₄	—	3	6	no	—
85	C₂×C₃×C₄	—	6	6	no	—
86	C₂×C₃×C₄	C₂×SL(2,𝔽₃)	1	12	yes	—
87	C₂×C₃×C₄	C₂²×A₄	1	6	yes	—
88	C₂×C₉	—	4	1	no	—
89	C₂×C₉	—	4	1	no	—
90	C₂×C₉	—	4	1	no	—
91	C₂×C₉	—	4	4	no	—
92	C₂×Q₈	—	3	8	no	—
93	C₂×Q₈	—	3	1	no	—
94	C₂×Q₈	—	3	2	no	—
95	C₂×Q₈	—	12	4	no	—
96	C₂×Q₈	Q₈⋊C₉	1	8	yes	direct product factor
97	C₂×Q₈	Q₈×C₉	1	2	yes	complemented
98	C₂×Q₈	C₂×C₂²⋊C₉	1	4	yes	—
99	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
100	C₂²×C₄	—	3	1	no	—
101	C₂²×C₄	—	3	3	no	—
102	C₂²×C₄	C₂×C₂²⋊C₉	1	3	yes	—
103	C₄⋊C₄	—	3	6	no	—
104	C₄⋊C₄	—	3	6	no	—
105	C₄⋊C₄	—	3	12	no	—
106	C₄²	—	3	6	no	—
107	C₄²	—	3	6	no	—
108	C₂²×C₃	C₂²×SL(2,𝔽₃)	1	2	yes	—
109	C₂²×C₃	Q₈×A₄	1	2	yes	—
110	C₂²×C₃	#994	1	2	yes	—
111	C₂²×C₃	C₂³×A₄	1	1	yes	characteristic
112	C₃×C₄	—	3	1	no	—
113	C₃×C₄	—	3	2	no	—
114	C₃×C₄	—	3	1	no	—
115	C₃×C₄	—	6	6	no	—
116	C₃×C₄	C₂²×SL(2,𝔽₃)	1	12	yes	—
117	C₉	—	4	1	no	Sylow
118	C₂³	C₂²×C₂²⋊C₉	1	1	yes	characteristic
119	Q₈	—	3	2	no	—
120	Q₈	—	3	1	no	—
121	Q₈	—	3	2	no	—
122	Q₈	—	12	8	no	—
123	Q₈	C₂×Q₈⋊C₉	1	16	yes	direct product factor
124	Q₈	C₂×Q₈×C₉	1	1	yes	characteristic, complemented
125	C₂×C₄	—	3	1	no	—
126	C₂×C₄	—	3	1	no	—
127	C₂×C₄	—	3	2	no	—
128	C₂×C₄	—	3	2	no	—
129	C₂×C₄	—	3	6	no	—
130	C₂×C₄	—	6	6	no	—
131	C₂×C₄	C₂×Q₈⋊C₉	1	12	yes	—
132	C₂×C₄	C₂²×C₂²⋊C₉	1	6	yes	—
133	C₂×C₃	Q₈×SL(2,𝔽₃)	1	4	yes	—
134	C₂×C₃	C₂³×SL(2,𝔽₃)	1	1	yes	characteristic
135	C₂×C₃	C₂×Q₈×A₄	1	1	yes	characteristic
136	C₂×C₃	C₂×#994	1	1	yes	characteristic
137	C₂²	C₂²×Q₈⋊C₉	1	2	yes	—
138	C₂²	Q₈×C₂²⋊C₉	1	2	yes	—
139	C₂²	#63700	1	2	yes	—
140	C₂²	C₂³×C₂²⋊C₉	1	1	yes	characteristic
141	C₄	—	3	1	no	—
142	C₄	—	3	1	no	—
143	C₄	—	3	2	no	—
144	C₄	—	6	6	no	—
145	C₄	C₂²×Q₈⋊C₉	1	12	yes	—
146	C₃	C₂×Q₈×SL(2,𝔽₃)	1	1	yes	characteristic
147	C₂	Q₈×Q₈⋊C₉	1	4	yes	direct product factor
148	C₂	C₂³×Q₈⋊C₉	1	1	yes	characteristic
149	C₂	C₂×Q₈×C₂²⋊C₉	1	1	yes	characteristic
150	C₂	C₂×#63700	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Q₈⋊C₉

C₂

yes

maximal, direct product factor

C₂×C₄×Q₈⋊C₉

C₂

yes

maximal

C₂×C₃×Q₈²

C₃

yes

maximal, characteristic

C₂²×Q₈×C₉

—

maximal

C₂²×Q₈⋊C₉

C₂²

yes

characteristic

C₄×Q₈⋊C₉

C₂²

yes

—

C₃×Q₈²

—

C₃×Q₈²

C₂×C₃

yes

—

C₂×C₃×C₄⋊Q₈

—

C₂×C₃×C₄×Q₈

—

C₂×C₃×C₄×Q₈

C₂×C₃

yes

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂²×C₄×C₉

—

C₂×Q₈⋊C₉

Q₈

yes

direct product factor

C₂×Q₈⋊C₉

C₂³

yes

characteristic

C₂×Q₈²

C₉

yes

characteristic, complemented, Sylow

C₂²×C₃×Q₈

—

C₂²×C₃×Q₈

A₄

yes

characteristic

C₂²×C₃×Q₈

C₂²×C₃

yes

characteristic

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

C₂²×C₃

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

—

C₂³×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

Q₈⋊C₉

C₂×Q₈

yes

characteristic, direct product factor

Q₈²

—

Q₈²

C₂×C₉

yes

complemented

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

C₂×C₉

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

SL(2,𝔽₃)

yes

—

C₂×C₃×Q₈

C₃×Q₈

yes

—

C₂×C₃×Q₈

C₂×A₄

yes

—

C₂×C₃×Q₈

C₂³×C₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×A₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₄×C₉

—

C₂²×Q₈

—

C₂²×Q₈

C₂²⋊C₉

yes

characteristic

C₂²×Q₈

C₂²×C₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂³×C₃

C₂²×A₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₂×SL(2,𝔽₃)

yes

—

C₃×Q₈

C₂×C₃×Q₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×SL(2,𝔽₃)

yes

—

C₂×C₃×C₄

C₂²×A₄

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈⋊C₉

yes

direct product factor

C₂×Q₈

Q₈×C₉

yes

complemented

C₂×Q₈

C₂×C₂²⋊C₉

yes

—

C₂×Q₈

C₂³×C₉

yes

characteristic

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂×C₂²⋊C₉

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₂²×C₃

C₂²×SL(2,𝔽₃)

yes

—

109

C₂²×C₃

Q₈×A₄

yes

—

110

C₂²×C₃

#994

yes

—

111

C₂²×C₃

C₂³×A₄

yes

characteristic

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

C₂²×SL(2,𝔽₃)

yes

—

117

C₉

—

Sylow

118

C₂³

C₂²×C₂²⋊C₉

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

C₂×Q₈⋊C₉

yes

direct product factor

124

Q₈

C₂×Q₈×C₉

yes

characteristic, complemented

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂×Q₈⋊C₉

yes

—

132

C₂×C₄

C₂²×C₂²⋊C₉

yes

—

133

C₂×C₃

Q₈×SL(2,𝔽₃)

yes

—

134

C₂×C₃

C₂³×SL(2,𝔽₃)

yes

characteristic

135

C₂×C₃

C₂×Q₈×A₄

yes

characteristic

136

C₂×C₃

C₂×#994

yes

characteristic

137

C₂²

C₂²×Q₈⋊C₉

yes

—

138

C₂²

Q₈×C₂²⋊C₉

yes

—

139

C₂²

#63700

yes

—

140

C₂²

C₂³×C₂²⋊C₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

C₂²×Q₈⋊C₉

yes

—

146

C₃

C₂×Q₈×SL(2,𝔽₃)

yes

characteristic

147

C₂

Q₈×Q₈⋊C₉

yes

direct product factor

148

C₂

C₂³×Q₈⋊C₉

yes

characteristic

149

C₂

C₂×Q₈×C₂²⋊C₉

yes

characteristic

150

C₂

C₂×#63700

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×A₄
commutator	C₂×Q₈	C₂³×C₉
Frattini	C₂²×C₃	C₂³×A₄
Fitting	C₂×C₃×Q₈²	C₃
cosocle	C₂×C₃×Q₈	C₂³×C₃
hypercenter	C₂²×C₃×Q₈	A₄
hypocenter	Q₈	C₂×Q₈×C₉
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈

subgroup

G/H

center, socle, prime radical

C₂³×C₃

C₂²×A₄

commutator

C₂×Q₈

C₂³×C₉

Frattini

C₂²×C₃

C₂³×A₄

Fitting

C₂×C₃×Q₈²

C₃

cosocle

C₂×C₃×Q₈

C₂³×C₃

hypercenter

C₂²×C₃×Q₈

A₄

hypocenter

Q₈

C₂×Q₈×C₉

LCS

G ▷ C₂×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈

N	H ≅ G/N	type
1. Q₈×Q₈⋊C₉	C₂	direct (N × H)
15. C₂×Q₈⋊C₉	Q₈	direct (N × H)
17. C₂×Q₈²	C₉	semidirect
34. Q₈⋊C₉	C₂×Q₈	direct (N × H)
36. Q₈²	C₂×C₉	semidirect
96. C₂×Q₈	Q₈⋊C₉	direct (N × H)
97. C₂×Q₈	Q₈×C₉	semidirect
123. Q₈	C₂×Q₈⋊C₉	direct (N × H)
124. Q₈	C₂×Q₈×C₉	semidirect
147. C₂	Q₈×Q₈⋊C₉	direct (N × H)

H ≅ G/N

type

1. Q₈×Q₈⋊C₉

C₂

direct (N × H)

15. C₂×Q₈⋊C₉

Q₈

direct (N × H)

17. C₂×Q₈²

C₉

semidirect

34. Q₈⋊C₉

C₂×Q₈

direct (N × H)

36. Q₈²

C₂×C₉

semidirect

96. C₂×Q₈

Q₈⋊C₉

direct (N × H)

97. C₂×Q₈

Q₈×C₉

semidirect

123. Q₈

C₂×Q₈⋊C₉

direct (N × H)

124. Q₈

C₂×Q₈×C₉

semidirect

147. C₂

Q₈×Q₈⋊C₉

direct (N × H)

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

subgroup	index
Q₈×Q₈⋊C₉	2
C₂×C₄×Q₈⋊C₉	2
C₂×C₃×Q₈²	3
C₂²×Q₈×C₉	4

subgroup

index

Q₈×Q₈⋊C₉

C₂×C₄×Q₈⋊C₉

C₂×C₃×Q₈²

C₂²×Q₈×C₉

Subgroups — C₂×Q₈×Q₈⋊C₉ #11919725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Q8×Q8⋊C9 #11919725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Q₈⋊C₉ #11919725