Group #67837 — subgroups

Subgroups — C₃×#1277 #67837

Order 384.

180 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 120 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×C₃×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#1277	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂²×C₃×C₈	—	2	2	no	—
5	C₂²×C₃×C₈	C₄	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₃×C₄⋊C₈	—	2	2	no	—
8	C₃×C₄⋊C₈	C₄	1	4	yes	—
9	C₃×C₄×C₈	—	2	2	no	—
10	C₂×C₄⋊C₈	C₂×C₃	1	2	yes	—
11	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
14	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
15	C₂×C₃×C₈	—	2	2	no	—
16	C₂×C₃×C₈	—	2	2	no	—
17	C₂×C₃×C₈	—	2	4	no	—
18	C₂×C₃×C₈	—	4	2	no	—
19	C₂×C₃×C₈	—	4	2	no	—
20	C₂×C₃×C₈	C₈	1	4	yes	complemented
21	C₃×C₄²	C₂×C₄	1	2	yes	—
22	C₃×C₄²	D₄	1	2	yes	—
23	C₂²×C₈	—	2	2	no	—
24	C₂²×C₈	C₃×C₄	1	2	yes	—
25	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
26	C₄⋊C₈	—	2	2	no	—
27	C₄⋊C₈	C₃×C₄	1	4	yes	—
28	C₄×C₈	—	2	2	no	—
29	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₃×C₄	—	2	2	no	—
31	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
32	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
34	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
35	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
36	C₂×C₃×C₄	M₁₆	1	2	yes	—
37	C₂×C₃×C₄	D₈	1	1	yes	characteristic
38	C₂×C₃×C₄	SD₁₆	1	2	yes	—
39	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
40	C₃×C₈	—	2	4	no	—
41	C₃×C₈	—	4	4	no	—
42	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
43	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
44	C₂²×C₄	C₃×Q₈	1	1	yes	characteristic
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	C₃×C₈	1	4	yes	complemented
51	C₄²	C₂×C₃×C₄	1	2	yes	—
52	C₄²	C₃×D₄	1	2	yes	—
53	C₂²×C₃	#95	1	1	yes	characteristic
54	C₂²×C₃	C₄⋊C₈	1	2	yes	—
55	C₂²×C₃	#106	1	1	yes	characteristic
56	C₂²×C₃	#107	1	1	yes	characteristic
57	C₂²×C₃	#108	1	2	yes	—
58	C₃×C₄	—	2	2	no	—
59	C₃×C₄	C₄×C₈	1	1	yes	characteristic
60	C₃×C₄	#97	1	1	yes	characteristic
61	C₃×C₄	C₂²⋊C₈	1	2	yes	—
62	C₃×C₄	#102	1	2	yes	—
63	C₃×C₄	#103	1	2	yes	—
64	C₂³	C₃×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	C₃×C₄²	1	1	yes	characteristic
67	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
69	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
70	C₂×C₄	C₂×C₃×C₈	1	2	yes	—
71	C₂×C₄	C₃×M₁₆	1	2	yes	—
72	C₂×C₄	C₃×D₈	1	1	yes	characteristic
73	C₂×C₄	C₃×SD₁₆	1	2	yes	—
74	C₂×C₄	C₃×Q₁₆	1	1	yes	characteristic
75	C₈	—	2	4	no	—
76	C₈	—	4	4	no	—
77	C₂×C₃	#334	1	2	yes	—
78	C₂×C₃	#335	1	2	yes	—
79	C₂×C₃	#336	1	1	yes	characteristic
80	C₂×C₃	#340	1	1	yes	characteristic
81	C₂×C₃	#341	1	1	yes	characteristic
82	C₂²	C₃×#95	1	1	yes	characteristic
83	C₂²	C₃×C₄⋊C₈	1	2	yes	—
84	C₂²	C₃×#106	1	1	yes	characteristic
85	C₂²	C₃×#107	1	1	yes	characteristic
86	C₂²	C₃×#108	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	C₃×C₄×C₈	1	1	yes	characteristic
89	C₄	C₃×#97	1	1	yes	characteristic
90	C₄	C₃×C₂²⋊C₈	1	2	yes	—
91	C₄	C₃×#102	1	2	yes	—
92	C₄	C₃×#103	1	2	yes	—
93	C₃	#1277	1	1	yes	characteristic, direct product factor, Sylow
94	C₂	C₃×#334	1	2	yes	—
95	C₂	C₃×#335	1	2	yes	—
96	C₂	C₃×#336	1	1	yes	characteristic
97	C₂	C₃×#340	1	1	yes	characteristic
98	C₂	C₃×#341	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₈

C₂

yes

maximal

C₂×C₃×C₄×C₈

C₂

yes

maximal, characteristic

#1277

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×C₈

—

C₂²×C₃×C₈

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

C₄

yes

—

C₃×C₄×C₈

—

C₂×C₄⋊C₈

C₂×C₃

yes

—

C₂×C₄×C₈

C₂×C₃

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

Q₈

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₈

yes

complemented

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₃×C₄

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₃×C₄

yes

—

C₄×C₈

—

C₂³×C₃

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

D₈

yes

characteristic

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

Q₁₆

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₂²×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

C₃×D₄

yes

characteristic

C₂²×C₄

C₃×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₃×C₈

yes

complemented

C₄²

C₂×C₃×C₄

yes

—

C₄²

C₃×D₄

yes

—

C₂²×C₃

#95

yes

characteristic

C₂²×C₃

C₄⋊C₈

yes

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₃×C₄

—

C₃×C₄

C₄×C₈

yes

characteristic

C₃×C₄

#97

yes

characteristic

C₃×C₄

C₂²⋊C₈

yes

—

C₃×C₄

#102

yes

—

C₃×C₄

#103

yes

—

C₂³

C₃×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

C₃×C₄²

yes

characteristic

C₂×C₄

C₃×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₃×C₂²⋊C₄

yes

—

C₂×C₄

C₃×C₄⋊C₄

yes

—

C₂×C₄

C₂×C₃×C₈

yes

—

C₂×C₄

C₃×M₁₆

yes

—

C₂×C₄

C₃×D₈

yes

characteristic

C₂×C₄

C₃×SD₁₆

yes

—

C₂×C₄

C₃×Q₁₆

yes

characteristic

C₈

—

C₈

—

C₂×C₃

#334

yes

—

C₂×C₃

#335

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂²

C₃×#95

yes

characteristic

C₂²

C₃×C₄⋊C₈

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

C₃×C₄×C₈

yes

characteristic

C₄

C₃×#97

yes

characteristic

C₄

C₃×C₂²⋊C₈

yes

—

C₄

C₃×#102

yes

—

C₄

C₃×#103

yes

—

C₃

#1277

yes

characteristic, direct product factor, Sylow

C₂

C₃×#334

yes

—

C₂

C₃×#335

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₃×C₄	D₄
commutator	C₄	C₃×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂²×C₃
socle, prime radical	C₂³×C₃	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

subgroup

G/H

center

C₂²×C₃×C₄

D₄

commutator

C₄

C₃×C₄×C₈

Frattini, cosocle

C₂×C₄²

C₂²×C₃

socle, prime radical

C₂³×C₃

C₄⋊C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

N	H ≅ G/N	type
3. #1277	C₃	direct (N × H)
20. C₂×C₃×C₈	C₈	semidirect
50. C₂×C₈	C₃×C₈	semidirect
93. C₃	#1277	direct (N × H)

H ≅ G/N

type

3. #1277

C₃

direct (N × H)

20. C₂×C₃×C₈

C₈

semidirect

50. C₂×C₈

C₃×C₈

semidirect

93. C₃

#1277

direct (N × H)

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

subgroup	index
C₂×C₃×C₄⋊C₈	2
C₂×C₃×C₄×C₈	2
#1277	3

subgroup

index

C₂×C₃×C₄⋊C₈

C₂×C₃×C₄×C₈

#1277

Subgroups — C₃×#1277 #67837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1277 #67837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1277 #67837