Group #84681 — subgroups

Subgroups — C₃×#2817 #84681

Order 384.

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

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₃×#400	C₂	1	4	yes	maximal, complemented
3	C₃×#397	C₂	1	4	yes	maximal
4	C₃×#395	C₂	1	4	yes	maximal, complemented
5	C₃×#381	C₂	1	1	yes	maximal, characteristic
6	#2817	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₈	C₂²	1	2	yes	—
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₃×C₂²⋊Q₈	—	2	8	no	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₃×#95	C₂²	1	4	yes	—
16	C₃×#95	C₂²	1	8	yes	—
17	C₃×#95	C₂²	1	4	yes	—
18	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
19	#400	C₂×C₃	1	4	yes	complemented
20	#397	C₂×C₃	1	4	yes	—
21	#395	C₂×C₃	1	4	yes	complemented
22	#381	C₂×C₃	1	1	yes	characteristic
23	C₂⁴×C₃	C₂³	1	1	yes	characteristic
24	C₂×C₃×Q₈	—	2	4	no	—
25	C₂×C₃×Q₈	—	4	4	no	—
26	C₂²×C₃×C₄	—	4	2	no	—
27	C₂²×C₃×C₄	D₄	1	4	yes	—
28	C₂²×C₃×C₄	C₂³	1	4	yes	—
29	C₂²×C₃×C₄	C₂³	1	4	yes	—
30	C₂²×C₃×C₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₃×C₄⋊C₄	—	2	4	no	—
33	C₃×C₄⋊C₄	—	2	8	no	—
34	C₃×C₄⋊C₄	—	4	4	no	—
35	C₃×C₂²⋊C₄	—	2	8	no	—
36	C₃×C₂²⋊C₄	—	4	2	no	—
37	C₂²×Q₈	C₂²×C₃	1	2	yes	—
38	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
39	C₂²⋊Q₈	—	2	8	no	—
40	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
41	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
42	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
43	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
44	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
45	#95	C₂²×C₃	1	4	yes	—
46	#95	C₂²×C₃	1	8	yes	—
47	#95	C₂²×C₃	1	4	yes	—
48	C₂³×C₃	—	2	2	no	—
49	C₂³×C₃	—	2	4	no	—
50	C₂³×C₃	C₂×D₄	1	2	yes	—
51	C₂³×C₃	C₂⁴	1	1	yes	characteristic
52	C₃×Q₈	—	4	8	no	—
53	C₂×C₃×C₄	—	2	4	no	—
54	C₂×C₃×C₄	—	2	4	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	4	no	—
57	C₂×C₃×C₄	—	2	8	no	—
58	C₂×C₃×C₄	—	2	8	no	—
59	C₂×C₃×C₄	—	2	8	no	—
60	C₂×C₃×C₄	—	4	4	no	—
61	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
62	C₂⁴	C₂³×C₃	1	1	yes	characteristic
63	C₂×Q₈	—	2	4	no	—
64	C₂×Q₈	—	4	4	no	—
65	C₂²×C₄	—	4	2	no	—
66	C₂²×C₄	C₃×D₄	1	4	yes	—
67	C₂²×C₄	C₂³×C₃	1	4	yes	—
68	C₂²×C₄	C₂³×C₃	1	4	yes	—
69	C₂²×C₄	C₂³×C₃	1	4	yes	—
70	C₂²×C₄	C₂³×C₃	1	2	yes	—
71	C₄⋊C₄	—	2	4	no	—
72	C₄⋊C₄	—	2	8	no	—
73	C₄⋊C₄	—	4	4	no	—
74	C₂²⋊C₄	—	2	8	no	—
75	C₂²⋊C₄	—	4	2	no	—
76	C₂²×C₃	—	2	2	no	—
77	C₂²×C₃	—	2	4	no	—
78	C₂²×C₃	—	4	4	no	—
79	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
81	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
82	C₃×C₄	—	2	4	no	—
83	C₃×C₄	—	4	4	no	—
84	C₃×C₄	—	4	4	no	—
85	C₃×C₄	—	4	4	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	4	no	—
88	C₂³	C₂×C₃×D₄	1	2	yes	—
89	C₂³	C₂⁴×C₃	1	1	yes	characteristic
90	Q₈	—	4	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	8	no	—
96	C₂×C₄	—	2	8	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
100	C₂×C₃	—	4	2	no	—
101	C₂×C₃	#534	1	1	yes	characteristic
102	C₂×C₃	#536	1	2	yes	—
103	C₂×C₃	#563	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
108	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
109	C₂²	C₃×2₋¹⁺⁴	1	4	yes	—
110	C₄	—	2	4	no	—
111	C₄	—	4	4	no	—
112	C₄	—	4	4	no	—
113	C₄	—	4	4	no	—
114	C₃	#2817	1	1	yes	characteristic, direct product factor, Sylow
115	C₂	—	4	2	no	—
116	C₂	C₃×#534	1	1	yes	characteristic
117	C₂	C₃×#536	1	2	yes	—
118	C₂	C₃×#563	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₂²⋊Q₈

C₂

yes

maximal

C₃×#400

C₂

yes

maximal, complemented

C₃×#397

C₂

yes

maximal

C₃×#395

C₂

yes

maximal, complemented

C₃×#381

C₂

yes

maximal, characteristic

#2817

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₂³×C₃×C₄

C₂²

yes

characteristic

C₃×C₂²⋊Q₈

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₂×C₂²⋊Q₈

C₂×C₃

yes

—

#400

C₂×C₃

yes

complemented

#397

C₂×C₃

yes

—

#395

C₂×C₃

yes

complemented

#381

C₂×C₃

yes

characteristic

C₂⁴×C₃

C₂³

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₂²×Q₈

C₂²×C₃

yes

—

C₂³×C₄

C₂²×C₃

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×D₄

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂⁴

C₂³×C₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₃×D₄

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂³

—

C₂³

—

C₂³

C₂×C₃×D₄

yes

—

C₂³

C₂⁴×C₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₃×D₄

yes

—

100

C₂×C₃

—

101

C₂×C₃

#534

yes

characteristic

102

C₂×C₃

#536

yes

—

103

C₂×C₃

#563

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂²×C₃×D₄

yes

characteristic

108

C₂²

C₃×H(5,𝔽₂)

yes

—

109

C₂²

C₃×2₋¹⁺⁴

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₃

#2817

yes

characteristic, direct product factor, Sylow

115

C₂

—

116

C₂

C₃×#534

yes

characteristic

117

C₂

C₃×#536

yes

—

118

C₂

C₃×#563

yes

—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₃

prime radical

C₂⁴×C₃

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ G

N	H ≅ G/N	type
2. C₃×#400	C₂	semidirect
4. C₃×#395	C₂	semidirect
6. #2817	C₃	direct (N × H)
19. #400	C₂×C₃	semidirect
21. #395	C₂×C₃	semidirect
114. C₃	#2817	direct (N × H)

H ≅ G/N

type

2. C₃×#400

C₂

semidirect

4. C₃×#395

C₂

semidirect

6. #2817

C₃

direct (N × H)

19. #400

C₂×C₃

semidirect

21. #395

C₂×C₃

semidirect

114. C₃

#2817

direct (N × H)

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

subgroup	index
C₂×C₃×C₂²⋊Q₈	2
C₃×#400	2
C₃×#397	2
C₃×#395	2
C₃×#381	2
#2817	3

subgroup

index

C₂×C₃×C₂²⋊Q₈

Subgroups — C₃×#2817 #84681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2817 #84681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2817 #84681