Group #14517839 — subgroups

Subgroups — Q₈×#6894 #14517839

Order 1944.

372 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 234 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×#6894	C₂	1	3	yes	maximal
2	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₃²×Q₈×C₉	C₃	1	1	yes	maximal, characteristic
4	Q₈×C₉⋊C₉	C₃	1	6	yes	maximal, complemented
5	Q₈×C₉⋊C₉	C₃	1	3	yes	maximal, complemented
6	Q₈×C₃²⋊C₉	C₃	1	2	yes	maximal
7	C₂×#6894	C₂²	1	1	yes	characteristic
8	C₃×C₄×3₋¹⁺²	C₂×C₃	1	3	yes	—
9	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
10	C₄×C₉⋊C₉	C₂×C₃	1	18	yes	—
11	C₄×C₉⋊C₉	C₂×C₃	1	9	yes	—
12	C₄×C₃²⋊C₉	C₂×C₃	1	6	yes	—
13	#6894	Q₈	1	1	yes	characteristic, direct product factor, Sylow
14	C₃³×Q₈	C₃²	1	1	yes	characteristic
15	Q₈×3₋¹⁺²	—	3	3	no	—
16	C₃×Q₈×C₉	—	3	3	no	—
17	C₃×Q₈×C₉	C₃²	1	6	yes	—
18	C₃×Q₈×C₉	C₃²	1	2	yes	—
19	C₃×Q₈×C₉	C₃²	1	3	yes	—
20	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
21	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	1	yes	characteristic
22	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₉⋊C₉	C₂²×C₃	1	6	yes	—
24	C₂×C₉⋊C₉	C₂²×C₃	1	3	yes	—
25	C₂×C₃²⋊C₉	C₂²×C₃	1	2	yes	—
26	C₃³×C₄	C₂×C₃²	1	3	yes	—
27	C₄×3₋¹⁺²	—	3	9	no	—
28	C₃×C₄×C₉	—	3	9	no	—
29	C₃×C₄×C₉	C₂×C₃²	1	18	yes	—
30	C₃×C₄×C₉	C₂×C₃²	1	6	yes	—
31	C₃×C₄×C₉	C₂×C₃²	1	9	yes	—
32	C₃×C₄×C₉	C₂×C₃²	1	3	yes	—
33	C₃×3₋¹⁺²	C₃×Q₈	1	1	yes	characteristic
34	C₃²×C₉	C₃×Q₈	1	1	yes	characteristic
35	C₉⋊C₉	C₃×Q₈	1	6	yes	complemented
36	C₉⋊C₉	C₃×Q₈	1	3	yes	complemented
37	C₃²⋊C₉	C₃×Q₈	1	2	yes	—
38	C₃²×Q₈	—	3	1	no	—
39	C₃²×Q₈	—	3	2	no	—
40	C₃²×Q₈	3₋¹⁺²	1	3	yes	—
41	C₃²×Q₈	C₃³	1	1	yes	characteristic
42	Q₈×C₉	—	3	2	no	—
43	Q₈×C₉	—	3	3	no	—
44	Q₈×C₉	—	3	6	no	—
45	Q₈×C₉	3₋¹⁺²	1	3	yes	complemented
46	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
47	C₂×3₋¹⁺²	—	3	3	no	—
48	C₂×C₃×C₉	—	3	3	no	—
49	C₂×C₃×C₉	C₂²×C₃²	1	6	yes	—
50	C₂×C₃×C₉	C₂²×C₃²	1	2	yes	—
51	C₂×C₃×C₉	C₂²×C₃²	1	3	yes	—
52	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
53	C₃²×C₄	—	3	3	no	—
54	C₃²×C₄	—	3	6	no	—
55	C₃²×C₄	C₂×3₋¹⁺²	1	9	yes	—
56	C₃²×C₄	C₂×C₃³	1	3	yes	—
57	C₄×C₉	—	3	6	no	—
58	C₄×C₉	—	3	9	no	—
59	C₄×C₉	—	3	18	no	—
60	C₄×C₉	C₂×3₋¹⁺²	1	9	yes	—
61	C₃³	C₃²×Q₈	1	1	yes	characteristic
62	3₋¹⁺²	—	3	3	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	C₃²×Q₈	1	6	yes	—
65	C₃×C₉	C₃²×Q₈	1	2	yes	—
66	C₃×C₉	C₃²×Q₈	1	3	yes	—
67	C₃×C₉	C₃²×Q₈	1	1	yes	characteristic
68	C₃×Q₈	—	3	3	no	—
69	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
70	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
71	C₃×Q₈	#737	1	2	yes	—
72	C₂×C₃²	—	3	1	no	—
73	C₂×C₃²	—	3	2	no	—
74	C₂×C₃²	C₂²×3₋¹⁺²	1	3	yes	—
75	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
76	C₂×C₉	—	3	2	no	—
77	C₂×C₉	—	3	3	no	—
78	C₂×C₉	—	3	6	no	—
79	C₂×C₉	C₂²×3₋¹⁺²	1	3	yes	—
80	C₃×C₄	—	3	9	no	—
81	C₃×C₄	C₂×C₃×3₋¹⁺²	1	3	yes	—
82	C₃×C₄	C₂×C₃×3₋¹⁺²	1	3	yes	—
83	C₃×C₄	C₂×#737	1	6	yes	—
84	C₃²	—	3	1	no	—
85	C₃²	—	3	2	no	—
86	C₃²	Q₈×3₋¹⁺²	1	3	yes	—
87	C₃²	C₃³×Q₈	1	1	yes	characteristic
88	C₉	—	3	2	no	—
89	C₉	—	3	3	no	—
90	C₉	—	3	6	no	—
91	C₉	Q₈×3₋¹⁺²	1	3	yes	complemented
92	Q₈	#6894	1	1	yes	characteristic, direct product factor, Sylow
93	C₂×C₃	—	3	3	no	—
94	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	1	yes	characteristic
95	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	1	yes	characteristic
96	C₂×C₃	C₂²×#737	1	2	yes	—
97	C₄	C₂×#6894	1	3	yes	—
98	C₃	—	3	3	no	—
99	C₃	C₃×Q₈×3₋¹⁺²	1	1	yes	characteristic
100	C₃	C₃×Q₈×3₋¹⁺²	1	1	yes	characteristic
101	C₃	Q₈×#737	1	2	yes	—
102	C₂	C₂²×#6894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#6894

C₂

yes

maximal

C₃×Q₈×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃²×Q₈×C₉

C₃

yes

maximal, characteristic

Q₈×C₉⋊C₉

C₃

yes

maximal, complemented

Q₈×C₉⋊C₉

C₃

yes

maximal, complemented

Q₈×C₃²⋊C₉

C₃

yes

maximal

C₂×#6894

C₂²

yes

characteristic

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

—

C₄×C₉⋊C₉

C₂×C₃

yes

—

C₄×C₉⋊C₉

C₂×C₃

yes

—

C₄×C₃²⋊C₉

C₂×C₃

yes

—

#6894

Q₈

yes

characteristic, direct product factor, Sylow

C₃³×Q₈

C₃²

yes

characteristic

Q₈×3₋¹⁺²

—

C₃×Q₈×C₉

—

C₃×Q₈×C₉

C₃²

yes

—

C₃×Q₈×C₉

C₃²

yes

—

C₃×Q₈×C₉

C₃²

yes

—

C₃×Q₈×C₉

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

C₂²×C₃

yes

characteristic

C₂×C₃²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₉

C₂²×C₃

yes

—

C₂×C₉⋊C₉

C₂²×C₃

yes

—

C₂×C₃²⋊C₉

C₂²×C₃

yes

—

C₃³×C₄

C₂×C₃²

yes

—

C₄×3₋¹⁺²

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×3₋¹⁺²

C₃×Q₈

yes

characteristic

C₃²×C₉

C₃×Q₈

yes

characteristic

C₉⋊C₉

C₃×Q₈

yes

complemented

C₉⋊C₉

C₃×Q₈

yes

complemented

C₃²⋊C₉

C₃×Q₈

yes

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

3₋¹⁺²

yes

—

C₃²×Q₈

C₃³

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

3₋¹⁺²

yes

complemented

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×3₋¹⁺²

yes

—

C₃²×C₄

C₂×C₃³

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×3₋¹⁺²

yes

—

C₃³

C₃²×Q₈

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

C₃²×Q₈

yes

—

C₃×C₉

C₃²×Q₈

yes

—

C₃×C₉

C₃²×Q₈

yes

—

C₃×C₉

C₃²×Q₈

yes

characteristic

C₃×Q₈

—

C₃×Q₈

C₃×3₋¹⁺²

yes

characteristic

C₃×Q₈

C₃×3₋¹⁺²

yes

characteristic

C₃×Q₈

#737

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×3₋¹⁺²

yes

—

C₂×C₃²

C₂²×C₃³

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×3₋¹⁺²

yes

—

C₃×C₄

—

C₃×C₄

C₂×C₃×3₋¹⁺²

yes

—

C₃×C₄

C₂×C₃×3₋¹⁺²

yes

—

C₃×C₄

C₂×#737

yes

—

C₃²

—

C₃²

—

C₃²

Q₈×3₋¹⁺²

yes

—

C₃²

C₃³×Q₈

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

Q₈×3₋¹⁺²

yes

complemented

Q₈

#6894

yes

characteristic, direct product factor, Sylow

C₂×C₃

—

C₂×C₃

C₂²×C₃×3₋¹⁺²

yes

characteristic

C₂×C₃

C₂²×C₃×3₋¹⁺²

yes

characteristic

C₂×C₃

C₂²×#737

yes

—

C₄

C₂×#6894

yes

—

C₃

—

C₃

C₃×Q₈×3₋¹⁺²

yes

characteristic

100

C₃

C₃×Q₈×3₋¹⁺²

yes

characteristic

101

C₃

Q₈×#737

yes

—

102

C₂

C₂²×#6894

yes

characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃³	C₂²×C₃²
LCS	G ▷ C₂×C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂×C₃²

C₂²×C₃³

prime radical

C₂×C₃³

C₂²×C₃²

LCS

G ▷ C₂×C₃² ▷ C₁

UCS

C₁ ◁ C₂×C₃² ◁ G

N	H ≅ G/N	type
4. Q₈×C₉⋊C₉	C₃	semidirect
5. Q₈×C₉⋊C₉	C₃	semidirect
13. #6894	Q₈	direct (N × H)
35. C₉⋊C₉	C₃×Q₈	semidirect
36. C₉⋊C₉	C₃×Q₈	semidirect
45. Q₈×C₉	3₋¹⁺²	semidirect
91. C₉	Q₈×3₋¹⁺²	semidirect
92. Q₈	#6894	direct (N × H)

H ≅ G/N

type

4. Q₈×C₉⋊C₉

C₃

semidirect

5. Q₈×C₉⋊C₉

C₃

semidirect

13. #6894

Q₈

direct (N × H)

35. C₉⋊C₉

C₃×Q₈

semidirect

36. C₉⋊C₉

C₃×Q₈

semidirect

45. Q₈×C₉

3₋¹⁺²

semidirect

91. C₉

Q₈×3₋¹⁺²

semidirect

92. Q₈

#6894

direct (N × H)

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

subgroup	index
C₄×#6894	2
C₃×Q₈×3₋¹⁺²	3
C₃²×Q₈×C₉	3
Q₈×C₉⋊C₉	3
Q₈×C₃²⋊C₉	3

subgroup

index

C₄×#6894

C₃×Q₈×3₋¹⁺²

C₃²×Q₈×C₉

Q₈×C₉⋊C₉

Q₈×C₃²⋊C₉

Subgroups — Q₈×#6894 #14517839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×#6894 #14517839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6894 #14517839