Group #13152132 — subgroups

Subgroups — Q₈×C₉⋊SL(2,𝔽₃) #13152132

Order 1728.

281 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₄×C₉⋊SL(2,𝔽₃)	C₂	1	3	yes	maximal
2	C₃×Q₈×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	Q₈²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	Q₈×Q₈⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₂×Q₈×3₋¹⁺²	—	4	1	no	maximal
6	C₂×C₉⋊SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₃×C₄×SL(2,𝔽₃)	C₂×C₃	1	3	yes	—
8	C₉×C₄⋊Q₈	—	3	3	no	—
9	C₄×Q₈×C₉	—	3	1	no	—
10	C₄×Q₈×C₉	C₂×C₃	1	3	yes	—
11	C₄×Q₈⋊C₉	C₂×C₃	1	6	yes	—
12	Q₈×3₋¹⁺²	—	4	4	no	—
13	C₂×C₄×3₋¹⁺²	—	4	3	no	—
14	C₉⋊SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, direct product factor
15	C₃×Q₈²	C₃²	1	1	yes	characteristic
16	Q₈×SL(2,𝔽₃)	—	3	1	no	—
17	C₂×C₃²×Q₈	—	4	1	no	—
18	C₂×C₃×SL(2,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
19	C₂×Q₈×C₉	—	3	2	no	—
20	C₂×Q₈×C₉	—	4	2	no	—
21	C₂×Q₈×C₉	A₄	1	1	yes	characteristic
22	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
23	C₂×Q₈⋊C₉	C₂²×C₃	1	2	yes	—
24	C₉×C₄⋊C₄	—	3	3	no	—
25	C₉×C₄⋊C₄	—	3	3	no	—
26	C₄²×C₉	—	3	3	no	—
27	C₂²×3₋¹⁺²	—	4	1	no	—
28	C₄×3₋¹⁺²	—	4	6	no	—
29	C₃×C₄⋊Q₈	—	3	3	no	—
30	C₃×C₄×Q₈	—	3	1	no	—
31	C₃×C₄×Q₈	C₂×C₃²	1	3	yes	—
32	C₄×SL(2,𝔽₃)	—	3	3	no	—
33	C₃²×Q₈	—	4	4	no	—
34	C₂×C₃²×C₄	—	4	3	no	—
35	C₃×SL(2,𝔽₃)	C₃×Q₈	1	1	yes	characteristic
36	Q₈×C₉	—	3	1	no	—
37	Q₈×C₉	—	4	8	no	—
38	Q₈×C₉	—	12	2	no	—
39	Q₈×C₉	SL(2,𝔽₃)	1	4	yes	complemented
40	Q₈×C₉	C₃×Q₈	1	1	yes	characteristic, complemented
41	C₂×C₄×C₉	—	3	1	no	—
42	C₂×C₄×C₉	—	3	3	no	—
43	C₂×C₄×C₉	—	4	6	no	—
44	C₂×C₄×C₉	C₂×A₄	1	3	yes	—
45	Q₈⋊C₉	C₃×Q₈	1	2	yes	complemented
46	Q₈²	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
47	C₂×3₋¹⁺²	—	4	1	no	—
48	C₂×3₋¹⁺²	—	4	1	no	—
49	C₂×3₋¹⁺²	—	4	1	no	—
50	C₂×C₃×Q₈	—	3	2	no	—
51	C₂×C₃×Q₈	—	12	1	no	—
52	C₂×C₃×Q₈	C₃×A₄	1	1	yes	characteristic
53	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
54	C₂×SL(2,𝔽₃)	—	3	1	no	—
55	C₃×C₄⋊C₄	—	3	3	no	—
56	C₃×C₄⋊C₄	—	3	3	no	—
57	C₃×C₄²	—	3	3	no	—
58	C₂²×C₃²	—	4	1	no	—
59	C₃²×C₄	—	4	6	no	—
60	C₂²×C₉	—	4	2	no	—
61	C₂²×C₉	C₂²×A₄	1	1	yes	characteristic
62	C₄×C₉	—	3	1	no	—
63	C₄×C₉	—	3	1	no	—
64	C₄×C₉	—	4	12	no	—
65	C₄×C₉	—	6	3	no	—
66	C₄×C₉	C₂×SL(2,𝔽₃)	1	6	yes	—
67	C₄⋊Q₈	—	3	3	no	—
68	C₄×Q₈	—	3	1	no	—
69	C₄×Q₈	C₂×3₋¹⁺²	1	3	yes	—
70	3₋¹⁺²	—	4	1	no	Sylow
71	C₃×Q₈	—	3	1	no	—
72	C₃×Q₈	—	12	2	no	—
73	C₃×Q₈	—	12	4	no	—
74	C₃×Q₈	C₃×SL(2,𝔽₃)	1	4	yes	—
75	C₃×Q₈	C₃²×Q₈	1	1	yes	characteristic
76	C₂×C₃×C₄	—	3	1	no	—
77	C₂×C₃×C₄	—	3	3	no	—
78	C₂×C₃×C₄	—	12	3	no	—
79	C₂×C₃×C₄	C₂×C₃×A₄	1	3	yes	—
80	SL(2,𝔽₃)	—	3	1	no	—
81	C₂×C₃²	—	4	1	no	—
82	C₂×C₃²	—	4	1	no	—
83	C₂×C₃²	—	4	1	no	—
84	C₂×C₉	—	4	2	no	—
85	C₂×C₉	—	4	2	no	—
86	C₂×C₉	—	4	2	no	—
87	C₂×C₉	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic
88	C₂×C₉	Q₈×A₄	1	1	yes	characteristic
89	C₂×C₉	#994	1	1	yes	characteristic
90	C₂×Q₈	—	3	2	no	—
91	C₂×Q₈	C₉⋊A₄	1	1	yes	characteristic
92	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
93	C₄⋊C₄	—	3	3	no	—
94	C₄⋊C₄	—	3	3	no	—
95	C₄²	—	3	3	no	—
96	C₂²×C₃	—	12	1	no	—
97	C₂²×C₃	C₂²×C₃×A₄	1	1	yes	characteristic
98	C₃×C₄	—	3	1	no	—
99	C₃×C₄	—	3	1	no	—
100	C₃×C₄	—	6	3	no	—
101	C₃×C₄	—	12	6	no	—
102	C₃×C₄	C₂×C₃×SL(2,𝔽₃)	1	6	yes	—
103	C₃²	—	4	1	no	—
104	C₉	—	4	2	no	—
105	C₉	Q₈×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
106	Q₈	—	3	1	no	—
107	Q₈	—	12	2	no	—
108	Q₈	C₉⋊SL(2,𝔽₃)	1	4	yes	direct product factor
109	Q₈	Q₈×3₋¹⁺²	1	1	yes	characteristic, complemented
110	C₂×C₄	—	3	1	no	—
111	C₂×C₄	—	3	3	no	—
112	C₂×C₄	C₂×C₉⋊A₄	1	3	yes	—
113	C₂×C₃	—	12	1	no	—
114	C₂×C₃	—	12	1	no	—
115	C₂×C₃	—	12	1	no	—
116	C₂×C₃	C₂²×C₃×SL(2,𝔽₃)	1	1	yes	characteristic
117	C₂×C₃	C₃×Q₈×A₄	1	1	yes	characteristic
118	C₂×C₃	C₃×#994	1	1	yes	characteristic
119	C₂²	C₂²×C₉⋊A₄	1	1	yes	characteristic
120	C₄	—	3	1	no	—
121	C₄	—	3	1	no	—
122	C₄	—	6	3	no	—
123	C₄	C₂×C₉⋊SL(2,𝔽₃)	1	6	yes	—
124	C₃	—	12	1	no	—
125	C₃	C₃×Q₈×SL(2,𝔽₃)	1	1	yes	characteristic
126	C₂	C₂²×C₉⋊SL(2,𝔽₃)	1	1	yes	characteristic
127	C₂	Q₈×C₉⋊A₄	1	1	yes	characteristic
128	C₂	#11722557	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₉⋊SL(2,𝔽₃)

C₂

yes

maximal

C₃×Q₈×SL(2,𝔽₃)

C₃

yes

maximal, characteristic

Q₈²×C₉

C₃

yes

maximal, characteristic, complemented

Q₈×Q₈⋊C₉

C₃

yes

maximal, complemented

C₂×Q₈×3₋¹⁺²

—

maximal

C₂×C₉⋊SL(2,𝔽₃)

C₂²

yes

characteristic

C₃×C₄×SL(2,𝔽₃)

C₂×C₃

yes

—

C₉×C₄⋊Q₈

—

C₄×Q₈×C₉

—

C₄×Q₈×C₉

C₂×C₃

yes

—

C₄×Q₈⋊C₉

C₂×C₃

yes

—

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₉⋊SL(2,𝔽₃)

Q₈

yes

characteristic, direct product factor

C₃×Q₈²

C₃²

yes

characteristic

Q₈×SL(2,𝔽₃)

—

C₂×C₃²×Q₈

—

C₂×C₃×SL(2,𝔽₃)

C₂²×C₃

yes

characteristic

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

A₄

yes

characteristic

C₂×Q₈×C₉

C₂²×C₃

yes

characteristic

C₂×Q₈⋊C₉

C₂²×C₃

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₂²×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₃×C₄⋊Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

C₂×C₃²

yes

—

C₄×SL(2,𝔽₃)

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₃×SL(2,𝔽₃)

C₃×Q₈

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

SL(2,𝔽₃)

yes

complemented

Q₈×C₉

C₃×Q₈

yes

characteristic, complemented

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×A₄

yes

—

Q₈⋊C₉

C₃×Q₈

yes

complemented

Q₈²

3₋¹⁺²

yes

characteristic, complemented, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×A₄

yes

characteristic

C₂×C₃×Q₈

C₂²×C₃²

yes

characteristic

C₂×SL(2,𝔽₃)

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₂²×C₃²

—

C₃²×C₄

—

C₂²×C₉

—

C₂²×C₉

C₂²×A₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×SL(2,𝔽₃)

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×3₋¹⁺²

yes

—

3₋¹⁺²

—

Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃×SL(2,𝔽₃)

yes

—

C₃×Q₈

C₃²×Q₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×A₄

yes

—

SL(2,𝔽₃)

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×SL(2,𝔽₃)

yes

characteristic

C₂×C₉

Q₈×A₄

yes

characteristic

C₂×C₉

#994

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₉⋊A₄

yes

characteristic

C₂×Q₈

C₂²×3₋¹⁺²

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

C₂²×C₃×A₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

C₂×C₃×SL(2,𝔽₃)

yes

—

103

C₃²

—

104

C₉

—

105

C₉

Q₈×SL(2,𝔽₃)

yes

characteristic, complemented

106

Q₈

—

107

Q₈

—

108

Q₈

C₉⋊SL(2,𝔽₃)

yes

direct product factor

109

Q₈

Q₈×3₋¹⁺²

yes

characteristic, complemented

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×C₉⋊A₄

yes

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

—

116

C₂×C₃

C₂²×C₃×SL(2,𝔽₃)

yes

characteristic

117

C₂×C₃

C₃×Q₈×A₄

yes

characteristic

118

C₂×C₃

C₃×#994

yes

characteristic

119

C₂²

C₂²×C₉⋊A₄

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×C₉⋊SL(2,𝔽₃)

yes

—

124

C₃

—

125

C₃

C₃×Q₈×SL(2,𝔽₃)

yes

characteristic

126

C₂

C₂²×C₉⋊SL(2,𝔽₃)

yes

characteristic

127

C₂

Q₈×C₉⋊A₄

yes

characteristic

128

C₂

#11722557

yes

characteristic

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₂²×C₃×A₄
commutator, cosocle	C₂×C₃×Q₈	C₂²×C₃²
Fitting	Q₈²×C₉	C₃
prime radical	C₂×C₃×SL(2,𝔽₃)	C₂²×C₃
hypercenter	C₂×Q₈×C₉	A₄
hypocenter	Q₈	Q₈×3₋¹⁺²
LCS	G ▷ C₂×C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×Q₈×C₉

subgroup

G/H

center, Frattini, socle

C₂²×C₃

C₂²×C₃×A₄

commutator, cosocle

C₂×C₃×Q₈

C₂²×C₃²

Fitting

Q₈²×C₉

C₃

prime radical

C₂×C₃×SL(2,𝔽₃)

C₂²×C₃

hypercenter

C₂×Q₈×C₉

A₄

hypocenter

Q₈

Q₈×3₋¹⁺²

LCS

G ▷ C₂×C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂²×C₃ ◁ C₂×Q₈×C₉

N	H ≅ G/N	type
3. Q₈²×C₉	C₃	semidirect
4. Q₈×Q₈⋊C₉	C₃	semidirect
14. C₉⋊SL(2,𝔽₃)	Q₈	direct (N × H)
39. Q₈×C₉	SL(2,𝔽₃)	semidirect
40. Q₈×C₉	C₃×Q₈	semidirect
45. Q₈⋊C₉	C₃×Q₈	semidirect
46. Q₈²	3₋¹⁺²	semidirect
105. C₉	Q₈×SL(2,𝔽₃)	semidirect
108. Q₈	C₉⋊SL(2,𝔽₃)	direct (N × H)
109. Q₈	Q₈×3₋¹⁺²	semidirect

H ≅ G/N

type

3. Q₈²×C₉

C₃

semidirect

4. Q₈×Q₈⋊C₉

C₃

semidirect

14. C₉⋊SL(2,𝔽₃)

Q₈

direct (N × H)

39. Q₈×C₉

SL(2,𝔽₃)

semidirect

40. Q₈×C₉

C₃×Q₈

semidirect

45. Q₈⋊C₉

C₃×Q₈

semidirect

46. Q₈²

3₋¹⁺²

semidirect

105. C₉

Q₈×SL(2,𝔽₃)

semidirect

108. Q₈

C₉⋊SL(2,𝔽₃)

direct (N × H)

109. Q₈

Q₈×3₋¹⁺²

semidirect

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

subgroup	index
C₄×C₉⋊SL(2,𝔽₃)	2
C₃×Q₈×SL(2,𝔽₃)	3
Q₈²×C₉	3
Q₈×Q₈⋊C₉	3
C₂×Q₈×3₋¹⁺²	4

subgroup

index

C₄×C₉⋊SL(2,𝔽₃)

C₃×Q₈×SL(2,𝔽₃)

Q₈²×C₉

Q₈×Q₈⋊C₉

C₂×Q₈×3₋¹⁺²

Subgroups — Q₈×C₉⋊SL(2,𝔽₃) #13152132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C9⋊SL(2,𝔽3) #13152132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉⋊SL(2,𝔽₃) #13152132