Group #72700 — subgroups

Subgroups — C₃×#1985 #72700

Order 384.

280 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 136 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×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₃×#336	C₂	1	4	yes	maximal
5	#1985	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×Q₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₈	C₂²	1	2	yes	—
9	C₃×C₄×Q₈	—	2	2	no	—
10	C₃×C₄×Q₈	—	2	2	no	—
11	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊C₈	—	2	2	no	—
16	C₃×#97	—	2	2	no	—
17	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×#97	C₂×C₃	1	1	yes	characteristic
20	#336	C₂×C₃	1	4	yes	—
21	C₂×C₃×Q₈	—	2	2	no	—
22	C₂×C₃×Q₈	—	2	4	no	—
23	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₈	—	2	2	no	—
29	C₂×C₃×C₈	—	2	2	no	—
30	C₂×C₃×C₈	—	2	2	no	—
31	C₂×C₃×C₈	—	2	4	no	—
32	C₂×C₃×C₈	Q₈	1	4	yes	complemented
33	C₃×C₄⋊C₄	—	2	4	no	—
34	C₃×C₄⋊C₄	—	2	2	no	—
35	C₃×C₄²	—	2	2	no	—
36	C₃×C₄²	—	2	2	no	—
37	C₃×C₄²	D₄	1	2	yes	—
38	C₃×C₄²	D₄	1	2	yes	—
39	C₂²×Q₈	C₃×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂²×C₃	1	2	yes	—
41	C₂²×C₈	C₂²×C₃	1	2	yes	—
42	C₄×Q₈	—	2	2	no	—
43	C₄×Q₈	—	2	2	no	—
44	C₂×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄	C₃×C₄	1	2	yes	—
46	C₂×C₄²	C₂²×C₃	1	2	yes	—
47	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₄⋊C₈	—	2	2	no	—
49	#97	—	2	2	no	—
50	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
51	C₃×Q₈	—	2	8	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	—	2	4	no	—
58	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄	M₁₆	1	4	yes	—
61	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
64	C₂×C₃×C₄	#41	1	2	yes	—
65	C₃×C₈	—	2	4	no	—
66	C₃×C₈	—	4	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	—	2	4	no	—
69	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
70	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
73	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	C₃×Q₈	1	4	yes	complemented
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	—	2	2	no	—
83	C₄²	C₃×D₄	1	2	yes	—
84	C₄²	C₃×D₄	1	2	yes	—
85	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₃	C₄×Q₈	1	2	yes	—
87	C₂²×C₃	C₂×M₁₆	1	2	yes	—
88	C₂²×C₃	#131	1	2	yes	—
89	C₃×C₄	—	2	2	no	—
90	C₃×C₄	—	2	4	no	—
91	C₃×C₄	—	2	2	no	—
92	C₃×C₄	—	4	2	no	—
93	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
94	C₃×C₄	#124	1	1	yes	characteristic
95	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
96	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
97	Q₈	—	2	8	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄	C₃×M₁₆	1	4	yes	—
107	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
108	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
109	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
110	C₂×C₄	C₃×#41	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	—	4	2	no	—
113	C₂×C₃	#391	1	1	yes	characteristic
114	C₂×C₃	#407	1	1	yes	characteristic
115	C₂×C₃	#408	1	1	yes	characteristic
116	C₂×C₃	#446	1	4	yes	—
117	C₂²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²	C₃×C₄×Q₈	1	2	yes	—
119	C₂²	C₂×C₃×M₁₆	1	2	yes	—
120	C₂²	C₃×#131	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	2	4	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	2	no	—
125	C₄	C₃×C₂²⋊Q₈	1	2	yes	—
126	C₄	C₃×#124	1	1	yes	characteristic
127	C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
128	C₃	#1985	1	1	yes	characteristic, direct product factor, Sylow
129	C₂	C₃×#391	1	1	yes	characteristic
130	C₂	C₃×#407	1	1	yes	characteristic
131	C₂	C₃×#408	1	1	yes	characteristic
132	C₂	C₃×#446	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄×Q₈

C₂

yes

maximal, characteristic

C₂×C₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×C₃×#97

C₂

yes

maximal, characteristic

C₃×#336

C₂

yes

maximal

#1985

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₄

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

C₂×C₃×C₄⋊C₄

C₄

yes

characteristic

C₂×C₃×C₄⋊C₄

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×#97

—

C₂×C₄×Q₈

C₂×C₃

yes

characteristic

C₂×C₄⋊C₈

C₂×C₃

yes

characteristic

C₂×#97

C₂×C₃

yes

characteristic

#336

C₂×C₃

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

Q₈

yes

complemented

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

D₄

yes

—

C₂²×Q₈

C₃×C₄

yes

characteristic

C₂²×C₈

C₂²×C₃

yes

—

C₂²×C₈

C₂²×C₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₃×C₄

yes

characteristic

C₂×C₄⋊C₄

C₃×C₄

yes

—

C₂×C₄²

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

C₄⋊C₈

—

#97

—

C₂³×C₃

C₂²×C₄

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

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₃×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×C₃×C₄

yes

—

C₂²×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

C₂×C₃×C₄

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₃×Q₈

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₃×D₄

yes

—

C₄²

C₃×D₄

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₄×Q₈

yes

—

C₂²×C₃

C₂×M₁₆

yes

—

C₂²×C₃

#131

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₃×C₄

#124

yes

characteristic

C₃×C₄

C₄⋊Q₈

yes

characteristic

C₂³

C₂²×C₃×C₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₃×C₂²⋊C₄

yes

—

105

C₂×C₄

C₃×C₂²⋊C₄

yes

—

106

C₂×C₄

C₃×M₁₆

yes

—

107

C₂×C₄

C₂×C₃×D₄

yes

characteristic

108

C₂×C₄

C₂×C₃×D₄

yes

characteristic

109

C₂×C₄

C₂×C₃×Q₈

yes

—

110

C₂×C₄

C₃×#41

yes

—

111

C₈

—

112

C₈

—

113

C₂×C₃

#391

yes

characteristic

114

C₂×C₃

#407

yes

characteristic

115

C₂×C₃

#408

yes

characteristic

116

C₂×C₃

#446

yes

—

117

C₂²

C₂×C₃×C₂²⋊C₄

yes

characteristic

118

C₂²

C₃×C₄×Q₈

yes

—

119

C₂²

C₂×C₃×M₁₆

yes

—

120

C₂²

C₃×#131

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₃×C₂²⋊Q₈

yes

—

126

C₄

C₃×#124

yes

characteristic

127

C₄

C₃×C₄⋊Q₈

yes

characteristic

128

C₃

#1985

yes

characteristic, direct product factor, Sylow

129

C₂

C₃×#391

yes

characteristic

130

C₂

C₃×#407

yes

characteristic

131

C₂

C₃×#408

yes

characteristic

132

C₂

C₃×#446

yes

—

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

subgroup

G/H

center

C₂²×C₃×C₄

C₂³

commutator

C₂³

C₂²×C₃×C₄

Frattini, cosocle

C₂²×C₄

C₂³×C₃

socle, prime radical

C₂³×C₃

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₃×C₄ ◁ G

N	H ≅ G/N	type
5. #1985	C₃	direct (N × H)
32. C₂×C₃×C₈	Q₈	semidirect
78. C₂×C₈	C₃×Q₈	semidirect
128. C₃	#1985	direct (N × H)

H ≅ G/N

type

5. #1985

C₃

direct (N × H)

32. C₂×C₃×C₈

Q₈

semidirect

78. C₂×C₈

C₃×Q₈

semidirect

128. C₃

#1985

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×Q₈	2
C₂×C₃×C₄⋊C₈	2
C₂×C₃×#97	2
C₃×#336	2
#1985	3

subgroup

index

C₂×C₃×C₄×Q₈

C₂×C₃×C₄⋊C₈

C₂×C₃×#97

C₃×#336

#1985

Subgroups — C₃×#1985 #72700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1985 #72700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1985 #72700