Group #13152259 — subgroups

Subgroups — C₃×Q₈⋊(Q₈⋊C₉) #13152259

Order 1728.

340 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 82 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	C₃	1	1	yes	maximal, characteristic
2	Q₈⋊(Q₈⋊C₉)	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×Q₈⋊C₉	—	4	3	no	maximal
4	C₂×C₃×Q₈⋊C₉	—	4	2	no	maximal
5	C₃²×C₄⋊Q₈	—	3	3	no	—
6	C₃²×C₄×Q₈	—	3	2	no	—
7	C₃×Q₈⋊C₉	—	4	3	no	—
8	C₃×Q₈⋊C₉	—	4	2	no	—
9	C₃×Q₈²	C₉	1	3	yes	complemented
10	C₃×Q₈²	C₃²	1	1	yes	characteristic
11	C₂×C₃²×Q₈	—	3	1	no	—
12	C₂×C₃²×Q₈	A₄	1	2	yes	—
13	C₂×C₃²×Q₈	A₄	1	3	yes	—
14	C₃²×C₄⋊C₄	—	3	6	no	—
15	C₃²×C₄²	—	3	3	no	—
16	C₂×Q₈⋊C₉	—	4	9	no	—
17	C₂×Q₈⋊C₉	—	4	6	no	—
18	C₂²×C₃×C₉	—	16	1	no	—
19	C₃×C₄⋊Q₈	—	3	9	no	—
20	C₃×C₄⋊Q₈	—	3	3	no	—
21	C₃×C₄×Q₈	—	3	6	no	—
22	C₃×C₄×Q₈	—	3	2	no	—
23	C₃²×Q₈	—	3	2	no	—
24	C₃²×Q₈	—	4	3	no	—
25	C₃²×Q₈	—	12	1	no	—
26	C₃²×Q₈	SL(2,𝔽₃)	1	2	yes	—
27	C₂×C₃²×C₄	—	3	2	no	—
28	C₂×C₃²×C₄	—	3	3	no	—
29	Q₈⋊C₉	—	4	9	no	—
30	Q₈⋊C₉	—	4	6	no	—
31	Q₈²	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃×C₉	—	16	2	no	—
33	C₂×C₃×C₉	—	16	1	no	—
34	C₂×C₃×Q₈	—	3	1	no	—
35	C₂×C₃×Q₈	—	3	3	no	—
36	C₂×C₃×Q₈	C₂²⋊C₉	1	6	yes	—
37	C₂×C₃×Q₈	C₂²⋊C₉	1	9	yes	—
38	C₂×C₃×Q₈	C₃×A₄	1	2	yes	—
39	C₂×C₃×Q₈	C₃×A₄	1	3	yes	—
40	C₃×C₄⋊C₄	—	3	6	no	—
41	C₃×C₄⋊C₄	—	3	18	no	—
42	C₃×C₄²	—	3	3	no	—
43	C₃×C₄²	—	3	9	no	—
44	C₂²×C₃²	C₂²⋊A₄	1	1	yes	characteristic
45	C₃²×C₄	—	3	2	no	—
46	C₃²×C₄	—	3	2	no	—
47	C₃²×C₄	—	6	3	no	—
48	C₂²×C₉	—	16	3	no	—
49	C₄⋊Q₈	—	3	3	no	—
50	C₄×Q₈	—	3	2	no	—
51	C₃×C₉	—	16	1	no	Sylow
52	C₃×Q₈	—	3	6	no	—
53	C₃×Q₈	—	3	2	no	—
54	C₃×Q₈	—	4	3	no	—
55	C₃×Q₈	—	4	9	no	—
56	C₃×Q₈	—	12	1	no	—
57	C₃×Q₈	—	12	3	no	—
58	C₃×Q₈	Q₈⋊C₉	1	6	yes	complemented
59	C₃×Q₈	C₃×SL(2,𝔽₃)	1	2	yes	—
60	C₂×C₃×C₄	—	3	2	no	—
61	C₂×C₃×C₄	—	3	6	no	—
62	C₂×C₃×C₄	—	3	3	no	—
63	C₂×C₃×C₄	—	3	9	no	—
64	C₂×C₃²	C₂²⋊SL(2,𝔽₃)	1	2	yes	—
65	C₂×C₃²	#997	1	1	yes	characteristic
66	C₂×C₉	—	16	6	no	—
67	C₂×C₉	—	16	3	no	—
68	C₂×Q₈	—	3	1	no	—
69	C₂×Q₈	C₃×C₂²⋊C₉	1	2	yes	—
70	C₂×Q₈	C₃×C₂²⋊C₉	1	3	yes	—
71	C₄⋊C₄	—	3	6	no	—
72	C₄²	—	3	3	no	—
73	C₂²×C₃	C₂²⋊(C₂²⋊C₉)	1	3	yes	—
74	C₂²×C₃	C₃×C₂²⋊A₄	1	1	yes	characteristic
75	C₃×C₄	—	3	2	no	—
76	C₃×C₄	—	3	2	no	—
77	C₃×C₄	—	3	6	no	—
78	C₃×C₄	—	3	6	no	—
79	C₃×C₄	—	6	3	no	—
80	C₃×C₄	—	6	9	no	—
81	C₃²	Q₈⋊SL(2,𝔽₃)	1	1	yes	characteristic
82	C₉	—	16	3	no	—
83	Q₈	—	3	2	no	—
84	Q₈	—	4	3	no	—
85	Q₈	—	12	1	no	—
86	Q₈	C₃×Q₈⋊C₉	1	2	yes	complemented
87	C₂×C₄	—	3	2	no	—
88	C₂×C₄	—	3	3	no	—
89	C₂×C₃	C₂²⋊(Q₈⋊C₉)	1	6	yes	—
90	C₂×C₃	#63703	1	3	yes	—
91	C₂×C₃	C₃×C₂²⋊SL(2,𝔽₃)	1	2	yes	—
92	C₂×C₃	C₃×#997	1	1	yes	characteristic
93	C₂²	C₃×C₂²⋊(C₂²⋊C₉)	1	1	yes	characteristic
94	C₄	—	3	2	no	—
95	C₄	—	3	2	no	—
96	C₄	—	6	3	no	—
97	C₃	Q₈⋊(Q₈⋊C₉)	1	3	yes	direct product factor
98	C₃	C₃×Q₈⋊SL(2,𝔽₃)	1	1	yes	characteristic
99	C₂	C₃×C₂²⋊(Q₈⋊C₉)	1	2	yes	—
100	C₂	C₃×#63703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×Q₈²

C₃

yes

maximal, characteristic

Q₈⋊(Q₈⋊C₉)

C₃

yes

maximal, direct product factor

C₂×C₃×Q₈⋊C₉

—

maximal

C₂×C₃×Q₈⋊C₉

—

maximal

C₃²×C₄⋊Q₈

—

C₃²×C₄×Q₈

—

C₃×Q₈⋊C₉

—

C₃×Q₈⋊C₉

—

C₃×Q₈²

C₉

yes

complemented

C₃×Q₈²

C₃²

yes

characteristic

C₂×C₃²×Q₈

—

C₂×C₃²×Q₈

A₄

yes

—

C₂×C₃²×Q₈

A₄

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

—

C₂×Q₈⋊C₉

—

C₂×Q₈⋊C₉

—

C₂²×C₃×C₉

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

SL(2,𝔽₃)

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈²

C₃×C₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂²⋊C₉

yes

—

C₂×C₃×Q₈

C₂²⋊C₉

yes

—

C₂×C₃×Q₈

C₃×A₄

yes

—

C₂×C₃×Q₈

C₃×A₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₂²×C₃²

C₂²⋊A₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₂²×C₉

—

C₄⋊Q₈

—

C₄×Q₈

—

C₃×C₉

—

Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

Q₈⋊C₉

yes

complemented

C₃×Q₈

C₃×SL(2,𝔽₃)

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃²

C₂²⋊SL(2,𝔽₃)

yes

—

C₂×C₃²

#997

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×Q₈

—

C₂×Q₈

C₃×C₂²⋊C₉

yes

—

C₂×Q₈

C₃×C₂²⋊C₉

yes

—

C₄⋊C₄

—

C₄²

—

C₂²×C₃

C₂²⋊(C₂²⋊C₉)

yes

—

C₂²×C₃

C₃×C₂²⋊A₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

Q₈⋊SL(2,𝔽₃)

yes

characteristic

C₉

—

Q₈

—

Q₈

—

Q₈

—

Q₈

C₃×Q₈⋊C₉

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₃

C₂²⋊(Q₈⋊C₉)

yes

—

C₂×C₃

#63703

yes

—

C₂×C₃

C₃×C₂²⋊SL(2,𝔽₃)

yes

—

C₂×C₃

C₃×#997

yes

characteristic

C₂²

C₃×C₂²⋊(C₂²⋊C₉)

yes

characteristic

C₄

—

C₄

—

C₄

—

C₃

Q₈⋊(Q₈⋊C₉)

yes

direct product factor

C₃

C₃×Q₈⋊SL(2,𝔽₃)

yes

characteristic

C₂

C₃×C₂²⋊(Q₈⋊C₉)

yes

—

100

C₂

C₃×#63703

yes

characteristic

	subgroup	G/H
center, socle, prime radical, hypercenter	C₂²×C₃²	C₂²⋊A₄
commutator, hypocenter	Q₈²	C₃×C₉
Frattini	C₂²×C₃	C₃×C₂²⋊A₄
Fitting	C₃²×Q₈²	C₃
cosocle	C₃×Q₈²	C₃²
LCS	G ▷ Q₈²
UCS	C₁ ◁ C₂²×C₃²

subgroup

G/H

center, socle, prime radical, hypercenter

C₂²×C₃²

C₂²⋊A₄

commutator, hypocenter

Q₈²

C₃×C₉

Frattini

C₂²×C₃

C₃×C₂²⋊A₄

Fitting

C₃²×Q₈²

C₃

cosocle

C₃×Q₈²

C₃²

LCS

G ▷ Q₈²

UCS

C₁ ◁ C₂²×C₃²

N	H ≅ G/N	type
2. Q₈⋊(Q₈⋊C₉)	C₃	direct (N × H)
9. C₃×Q₈²	C₉	semidirect
31. Q₈²	C₃×C₉	semidirect
58. C₃×Q₈	Q₈⋊C₉	semidirect
86. Q₈	C₃×Q₈⋊C₉	semidirect
97. C₃	Q₈⋊(Q₈⋊C₉)	direct (N × H)

H ≅ G/N

type

2. Q₈⋊(Q₈⋊C₉)

C₃

direct (N × H)

9. C₃×Q₈²

C₉

semidirect

31. Q₈²

C₃×C₉

semidirect

58. C₃×Q₈

Q₈⋊C₉

semidirect

86. Q₈

C₃×Q₈⋊C₉

semidirect

97. C₃

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
C₃²×Q₈²	3
Q₈⋊(Q₈⋊C₉)	3
C₂×C₃×Q₈⋊C₉	4

subgroup

index

C₃²×Q₈²

Q₈⋊(Q₈⋊C₉)

C₂×C₃×Q₈⋊C₉

Subgroups — C₃×Q₈⋊(Q₈⋊C₉) #13152259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×Q8⋊(Q8⋊C9) #13152259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈⋊(Q₈⋊C₉) #13152259