Group #13136104 — subgroups

Subgroups — 3₋¹⁺²×#367 #13136104

Order 1728.

184 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×C₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×#107	C₂	1	2	yes	maximal
3	C₃²×#367	C₃	1	1	yes	maximal, characteristic
4	C₉×#367	C₃	1	3	yes	maximal, complemented
5	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
6	C₄⋊C₄×3₋¹⁺²	—	2	2	no	—
7	C₁₆×3₋¹⁺²	C₄	1	2	yes	complemented
8	C₂×C₃²×C₁₆	C₂×C₃	1	1	yes	characteristic
9	C₃²×#107	C₂×C₃	1	2	yes	—
10	C₂×C₉×C₁₆	C₂×C₃	1	3	yes	—
11	C₉×#107	C₂×C₃	1	6	yes	—
12	C₂×C₄×3₋¹⁺²	—	4	2	no	—
13	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
14	C₈×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
15	C₈×3₋¹⁺²	Q₈	1	1	yes	characteristic
16	C₃×#367	—	3	1	no	—
17	C₃×#367	C₃²	1	1	yes	characteristic
18	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
19	C₃²×C₄⋊C₄	—	2	2	no	—
20	C₃²×C₁₆	C₃×C₄	1	2	yes	—
21	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
22	C₉×C₄⋊C₄	—	2	6	no	—
23	C₉×C₁₆	C₃×C₄	1	6	yes	complemented
24	C₂²×3₋¹⁺²	D₈	1	1	yes	characteristic
25	C₄×3₋¹⁺²	—	8	2	no	—
26	C₄×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
27	C₄×3₋¹⁺²	Q₁₆	1	1	yes	characteristic
28	C₂×C₃×C₁₆	—	3	1	no	—
29	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
30	C₃×#107	—	3	2	no	—
31	C₃×#107	C₂×C₃²	1	2	yes	—
32	C₂×C₃²×C₄	—	4	2	no	—
33	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
34	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
35	C₃²×C₈	C₃×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₉	—	4	6	no	—
37	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
38	C₈×C₉	C₂×C₃×C₄	1	3	yes	—
39	C₈×C₉	C₃×Q₈	1	3	yes	—
40	#367	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×3₋¹⁺²	#107	1	1	yes	characteristic
42	C₂×3₋¹⁺²	SD₃₂	1	2	yes	—
43	C₂×C₃×C₈	—	3	1	no	—
44	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	2	no	—
46	C₃×C₄⋊C₄	—	6	2	no	—
47	C₃×C₁₆	—	3	2	no	—
48	C₃×C₁₆	C₃²×C₄	1	2	yes	—
49	C₂²×C₃²	C₃×D₈	1	1	yes	characteristic
50	C₃²×C₄	—	8	2	no	—
51	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
52	C₃²×C₄	C₃×Q₁₆	1	1	yes	characteristic
53	C₂²×C₉	C₃×D₈	1	3	yes	—
54	C₄×C₉	—	8	6	no	—
55	C₄×C₉	C₃×C₄⋊C₄	1	3	yes	—
56	C₄×C₉	C₃×Q₁₆	1	3	yes	—
57	C₂×C₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
58	#107	C₂×3₋¹⁺²	1	2	yes	—
59	3₋¹⁺²	#367	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₃×C₄	—	3	1	no	—
61	C₂×C₃×C₄	—	4	2	no	—
62	C₂×C₃×C₄	—	12	2	no	—
63	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
64	C₃×C₈	—	3	1	no	—
65	C₃×C₈	—	3	1	no	—
66	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
67	C₃×C₈	C₃²×Q₈	1	1	yes	characteristic
68	C₂×C₃²	C₃×#107	1	1	yes	characteristic
69	C₂×C₃²	C₃×SD₃₂	1	2	yes	—
70	C₂×C₉	C₃×#107	1	3	yes	—
71	C₂×C₉	C₃×SD₃₂	1	6	yes	—
72	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₁₆	C₄×3₋¹⁺²	1	2	yes	complemented
75	C₂²×C₃	—	3	1	no	—
76	C₂²×C₃	C₃²×D₈	1	1	yes	characteristic
77	C₃×C₄	—	3	1	no	—
78	C₃×C₄	—	3	1	no	—
79	C₃×C₄	—	8	2	no	—
80	C₃×C₄	—	24	2	no	—
81	C₃×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
82	C₃×C₄	C₃²×Q₁₆	1	1	yes	characteristic
83	C₃²	C₃×#367	1	1	yes	characteristic
84	C₉	C₃×#367	1	3	yes	complemented
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
87	C₈	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
88	C₈	Q₈×3₋¹⁺²	1	1	yes	characteristic
89	C₂×C₃	—	3	1	no	—
90	C₂×C₃	—	3	2	no	—
91	C₂×C₃	C₃²×#107	1	1	yes	characteristic
92	C₂×C₃	C₃²×SD₃₂	1	2	yes	—
93	C₂²	D₈×3₋¹⁺²	1	1	yes	characteristic
94	C₄	—	8	2	no	—
95	C₄	C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
96	C₄	Q₁₆×3₋¹⁺²	1	1	yes	characteristic
97	C₃	—	3	1	no	—
98	C₃	C₃²×#367	1	1	yes	characteristic
99	C₂	3₋¹⁺²×#107	1	1	yes	characteristic
100	C₂	3₋¹⁺²×SD₃₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic

3₋¹⁺²×#107

C₂

yes

maximal

C₃²×#367

C₃

yes

maximal, characteristic

C₉×#367

C₃

yes

maximal, complemented

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₄⋊C₄×3₋¹⁺²

—

C₁₆×3₋¹⁺²

C₄

yes

complemented

C₂×C₃²×C₁₆

C₂×C₃

yes

characteristic

C₃²×#107

C₂×C₃

yes

—

C₂×C₉×C₁₆

C₂×C₃

yes

—

C₉×#107

C₂×C₃

yes

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₈×3₋¹⁺²

C₂×C₄

yes

characteristic

C₈×3₋¹⁺²

Q₈

yes

characteristic

C₃×#367

—

C₃×#367

C₃²

yes

characteristic

C₂×C₃²×C₈

C₂²×C₃

yes

characteristic

C₃²×C₄⋊C₄

—

C₃²×C₁₆

C₃×C₄

yes

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₉×C₄⋊C₄

—

C₉×C₁₆

C₃×C₄

yes

complemented

C₂²×3₋¹⁺²

D₈

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₄⋊C₄

yes

characteristic

C₄×3₋¹⁺²

Q₁₆

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₂×C₃²

yes

characteristic

C₃×#107

—

C₃×#107

C₂×C₃²

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃×D₄

yes

characteristic

C₃²×C₈

C₂×C₃×C₄

yes

characteristic

C₃²×C₈

C₃×Q₈

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₈×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

C₃×Q₈

yes

—

#367

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

#107

yes

characteristic

C₂×3₋¹⁺²

SD₃₂

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²×C₃²

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₁₆

—

C₃×C₁₆

C₃²×C₄

yes

—

C₂²×C₃²

C₃×D₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₃×C₄⋊C₄

yes

characteristic

C₃²×C₄

C₃×Q₁₆

yes

characteristic

C₂²×C₉

C₃×D₈

yes

—

C₄×C₉

—

C₄×C₉

C₃×C₄⋊C₄

yes

—

C₄×C₉

C₃×Q₁₆

yes

—

C₂×C₁₆

C₂×3₋¹⁺²

yes

characteristic

#107

C₂×3₋¹⁺²

yes

—

3₋¹⁺²

#367

yes

characteristic, direct product factor, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃²×D₄

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₃²×C₄

yes

characteristic

C₃×C₈

C₃²×Q₈

yes

characteristic

C₂×C₃²

C₃×#107

yes

characteristic

C₂×C₃²

C₃×SD₃₂

yes

—

C₂×C₉

C₃×#107

yes

—

C₂×C₉

C₃×SD₃₂

yes

—

C₂×C₈

C₂²×3₋¹⁺²

yes

characteristic

C₄⋊C₄

—

C₁₆

C₄×3₋¹⁺²

yes

complemented

C₂²×C₃

—

C₂²×C₃

C₃²×D₈

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃²×C₄⋊C₄

yes

characteristic

C₃×C₄

C₃²×Q₁₆

yes

C₃²

yes

C₉

yes

C₂×C₄

—

C₂×C₄

D₄×3₋¹⁺²

yes

characteristic

C₈

C₂×C₄×3₋¹⁺²

yes

characteristic

C₈

Q₈×3₋¹⁺²

yes

characteristic

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₃²×#107

yes

characteristic

C₂×C₃

C₃²×SD₃₂

yes

—

C₂²

D₈×3₋¹⁺²

yes

characteristic

C₄

—

C₄

C₄⋊C₄×3₋¹⁺²

yes

characteristic

C₄

Q₁₆×3₋¹⁺²

yes

characteristic

C₃

—

C₃

C₃²×#367

yes

characteristic

C₂

3₋¹⁺²×#107

yes

characteristic

100

C₂

3₋¹⁺²×SD₃₂

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×D₈
commutator	C₃×C₈	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₈	C₂²×C₃²
prime radical	C₂²×C₃²	C₃×D₈
LCS	G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ C₂×C₈×3₋¹⁺² ◁ G

subgroup

G/H

center, socle

C₂²×C₃

C₃²×D₈

commutator

C₃×C₈

C₂×C₃²×C₄

Frattini, cosocle

C₂×C₃×C₈

C₂²×C₃²

prime radical

C₂²×C₃²

C₃×D₈

LCS

G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ C₂×C₈×3₋¹⁺² ◁ G

N	H ≅ G/N	type
4. C₉×#367	C₃	semidirect
7. C₁₆×3₋¹⁺²	C₄	semidirect
23. C₉×C₁₆	C₃×C₄	semidirect
40. #367	3₋¹⁺²	direct (N × H)
59. 3₋¹⁺²	#367	direct (N × H)
74. C₁₆	C₄×3₋¹⁺²	semidirect
84. C₉	C₃×#367	semidirect

H ≅ G/N

type

4. C₉×#367

C₃

semidirect

7. C₁₆×3₋¹⁺²

C₄

semidirect

23. C₉×C₁₆

C₃×C₄

semidirect

40. #367

3₋¹⁺²

direct (N × H)

59. 3₋¹⁺²

#367

direct (N × H)

74. C₁₆

C₄×3₋¹⁺²

semidirect

84. C₉

C₃×#367

semidirect

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

subgroup	index
C₂×C₁₆×3₋¹⁺²	2
3₋¹⁺²×#107	2
C₃²×#367	3
C₉×#367	3

subgroup

index

C₂×C₁₆×3₋¹⁺²

3₋¹⁺²×#107

C₃²×#367

C₉×#367

Subgroups — 3₋¹⁺²×#367 #13136104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#367 #13136104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#367 #13136104