Group #67856 — subgroups

Subgroups — C₃×#1296 #67856

Order 384.

188 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₃×#384	C₂	1	1	yes	maximal, characteristic
3	#1296	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂²×C₃×C₈	—	2	2	no	—
5	C₂×C₃×C₄⋊C₄	—	2	2	no	—
6	C₂×C₃×C₄⋊C₄	C₄	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₃×C₄⋊C₈	—	2	2	no	—
9	C₃×C₄⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₈	C₂×C₃	1	2	yes	—
11	#384	C₂×C₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	—	2	2	no	—
13	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₃×C₄	D₄	1	2	yes	—
15	C₂×C₃×C₈	—	2	4	no	—
16	C₂×C₃×C₈	—	4	2	no	—
17	C₂×C₃×C₈	—	4	2	no	—
18	C₃×C₄⋊C₄	—	4	2	no	—
19	C₃×C₄⋊C₄	—	4	2	no	—
20	C₃×C₄⋊C₄	—	4	2	no	—
21	C₃×C₄²	C₂×C₄	1	2	yes	—
22	C₃×C₄²	D₄	1	1	yes	characteristic
23	C₃×C₄²	Q₈	1	1	yes	characteristic
24	C₂²×C₈	—	2	2	no	—
25	C₂×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄⋊C₄	C₃×C₄	1	2	yes	—
27	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	C₃×C₄	1	4	yes	complemented
30	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	—	2	1	no	—
33	C₂×C₃×C₄	—	4	2	no	—
34	C₂×C₃×C₄	—	4	4	no	—
35	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
37	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
39	C₂×C₃×C₄	D₈	1	2	yes	—
40	C₂×C₃×C₄	SD₁₆	1	2	yes	—
41	C₂×C₃×C₄	SD₁₆	1	2	yes	—
42	C₂×C₃×C₄	Q₁₆	1	2	yes	—
43	C₃×C₈	—	4	4	no	—
44	C₂²×C₄	—	2	2	no	—
45	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
46	C₂²×C₄	C₃×D₄	1	2	yes	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄⋊C₄	—	4	2	no	—
53	C₄²	C₂×C₃×C₄	1	2	yes	—
54	C₄²	C₃×D₄	1	1	yes	characteristic
55	C₄²	C₃×Q₈	1	1	yes	characteristic
56	C₂²×C₃	#95	1	1	yes	characteristic
57	C₂²×C₃	#100	1	1	yes	characteristic
58	C₂²×C₃	#101	1	1	yes	characteristic
59	C₂²×C₃	#102	1	2	yes	—
60	C₂²×C₃	#103	1	2	yes	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	8	2	no	—
64	C₃×C₄	#102	1	2	yes	—
65	C₃×C₄	#103	1	2	yes	—
66	C₃×C₄	#106	1	2	yes	—
67	C₃×C₄	#107	1	2	yes	—
68	C₂³	C₃×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	4	4	no	—
73	C₂×C₄	C₃×C₄²	1	1	yes	characteristic
74	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
75	C₂×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
77	C₂×C₄	C₃×D₈	1	2	yes	—
78	C₂×C₄	C₃×SD₁₆	1	2	yes	—
79	C₂×C₄	C₃×SD₁₆	1	2	yes	—
80	C₂×C₄	C₃×Q₁₆	1	2	yes	—
81	C₈	—	4	4	no	—
82	C₂×C₃	#331	1	1	yes	characteristic
83	C₂×C₃	#332	1	2	yes	—
84	C₂×C₃	#333	1	1	yes	characteristic
85	C₂×C₃	#340	1	2	yes	—
86	C₂×C₃	#343	1	1	yes	characteristic
87	C₂²	C₃×#95	1	1	yes	characteristic
88	C₂²	C₃×#100	1	1	yes	characteristic
89	C₂²	C₃×#101	1	1	yes	characteristic
90	C₂²	C₃×#102	1	2	yes	—
91	C₂²	C₃×#103	1	2	yes	—
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	8	2	no	—
95	C₄	C₃×#102	1	2	yes	—
96	C₄	C₃×#103	1	2	yes	—
97	C₄	C₃×#106	1	2	yes	—
98	C₄	C₃×#107	1	2	yes	—
99	C₃	#1296	1	1	yes	characteristic, direct product factor, Sylow
100	C₂	C₃×#331	1	1	yes	characteristic
101	C₂	C₃×#332	1	2	yes	—
102	C₂	C₃×#333	1	1	yes	characteristic
103	C₂	C₃×#340	1	2	yes	—
104	C₂	C₃×#343	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₈

C₂

yes

maximal

C₃×#384

C₂

yes

maximal, characteristic

#1296

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×C₈

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₈

C₂×C₃

yes

—

#384

C₂×C₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

Q₈

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₃×C₄

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₃×C₄

yes

complemented

C₂³×C₃

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

D₈

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

Q₁₆

yes

—

C₃×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

C₃×D₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₃×C₄

yes

—

C₄²

C₃×D₄

yes

characteristic

C₄²

C₃×Q₈

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

#103

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#102

yes

—

C₃×C₄

#103

yes

—

C₃×C₄

#106

yes

—

C₃×C₄

#107

yes

—

C₂³

C₃×C₂²⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃×C₄²

yes

characteristic

C₂×C₄

C₃×C₂²⋊C₄

yes

—

C₂×C₄

C₃×C₄⋊C₄

yes

characteristic

C₂×C₄

C₃×C₄⋊C₄

yes

—

C₂×C₄

C₃×D₈

yes

—

C₂×C₄

C₃×SD₁₆

yes

—

C₂×C₄

C₃×SD₁₆

yes

—

C₂×C₄

C₃×Q₁₆

yes

—

C₈

—

C₂×C₃

#331

yes

characteristic

C₂×C₃

#332

yes

—

C₂×C₃

#333

yes

characteristic

C₂×C₃

#340

yes

—

C₂×C₃

#343

yes

characteristic

C₂²

C₃×#95

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₃×#103

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₃×#102

yes

—

C₄

C₃×#103

yes

—

C₄

C₃×#106

yes

—

C₄

C₃×#107

yes

—

C₃

#1296

yes

characteristic, direct product factor, Sylow

100

C₂

C₃×#331

yes

characteristic

101

C₂

C₃×#332

yes

—

102

C₂

C₃×#333

yes

characteristic

103

C₂

C₃×#340

yes

—

104

C₂

C₃×#343

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³×C₃

C₂²⋊C₄

commutator

C₂×C₄

C₃×C₄²

Frattini, cosocle

C₂×C₄²

C₂²×C₃

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

N	H ≅ G/N	type
3. #1296	C₃	direct (N × H)
9. C₃×C₄⋊C₈	C₄	semidirect
29. C₄⋊C₈	C₃×C₄	semidirect
99. C₃	#1296	direct (N × H)

H ≅ G/N

type

3. #1296

C₃

direct (N × H)

9. C₃×C₄⋊C₈

C₄

semidirect

29. C₄⋊C₈

C₃×C₄

semidirect

99. C₃

#1296

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
C₃×#384	2
#1296	3

subgroup

index

C₂×C₃×C₄⋊C₈

C₃×#384

#1296

Subgroups — C₃×#1296 #67856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1296 #67856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1296 #67856