Group #84191 — subgroups

Subgroups — C₃×#2327 #84191

Order 384.

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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×Q₈	C₂	1	2	yes	maximal
3	C₃×#391	C₂	1	2	yes	maximal
4	C₃×#387	C₂	1	8	yes	maximal
5	C₃×#382	C₂	1	1	yes	maximal, characteristic
6	C₃×#376	C₂	1	1	yes	maximal, characteristic
7	#2327	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×Q₈	C₂²	1	2	yes	—
9	C₃×C₄⋊Q₈	C₄	1	8	yes	complemented
10	C₃×C₄×Q₈	—	2	8	no	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²	C₂²	1	4	yes	—
16	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	C₃×#95	C₂²	1	4	yes	—
19	C₃×#95	C₂²	1	4	yes	—
20	C₃×#95	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄×Q₈	C₂×C₃	1	2	yes	—
23	#391	C₂×C₃	1	2	yes	—
24	#387	C₂×C₃	1	8	yes	—
25	#382	C₂×C₃	1	1	yes	characteristic
26	#376	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×Q₈	—	2	4	no	—
28	C₂×C₃×Q₈	—	2	4	no	—
29	C₂×C₃×Q₈	C₂×C₄	1	8	yes	—
30	C₂²×C₃×C₄	C₂³	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	4	yes	—
32	C₂²×C₃×C₄	C₂³	1	2	yes	—
33	C₂²×C₃×C₄	C₂³	1	4	yes	—
34	C₂²×C₃×C₄	C₂³	1	2	yes	—
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₃×C₄⋊C₄	—	2	8	no	—
37	C₃×C₄⋊C₄	—	2	4	no	—
38	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
39	C₃×C₄²	—	2	8	no	—
40	C₃×C₄²	—	2	2	no	—
41	C₃×C₄²	C₂×C₄	1	4	yes	—
42	C₃×C₄²	D₄	1	4	yes	—
43	C₂²×Q₈	C₂²×C₃	1	2	yes	—
44	C₄⋊Q₈	C₃×C₄	1	8	yes	complemented
45	C₄×Q₈	—	2	8	no	—
46	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
48	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₂×C₄²	C₂²×C₃	1	4	yes	—
50	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
53	#95	C₂²×C₃	1	4	yes	—
54	#95	C₂²×C₃	1	4	yes	—
55	#95	C₂²×C₃	1	8	yes	—
56	C₂³×C₃	C₂⁴	1	1	yes	characteristic
57	C₃×Q₈	—	2	16	no	—
58	C₂×C₃×C₄	—	2	4	no	—
59	C₂×C₃×C₄	—	2	4	no	—
60	C₂×C₃×C₄	—	2	4	no	—
61	C₂×C₃×C₄	—	2	4	no	—
62	C₂×C₃×C₄	—	2	4	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	8	no	—
65	C₂×C₃×C₄	—	2	4	no	—
66	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
67	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
68	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
69	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
70	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄	#41	1	2	yes	—
72	C₂×Q₈	—	2	4	no	—
73	C₂×Q₈	—	2	4	no	—
74	C₂×Q₈	C₂×C₃×C₄	1	8	yes	—
75	C₂²×C₄	C₂³×C₃	1	2	yes	—
76	C₂²×C₄	C₂³×C₃	1	4	yes	—
77	C₂²×C₄	C₂³×C₃	1	2	yes	—
78	C₂²×C₄	C₂³×C₃	1	4	yes	—
79	C₂²×C₄	C₂³×C₃	1	2	yes	—
80	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
81	C₄⋊C₄	—	2	8	no	—
82	C₄⋊C₄	—	2	4	no	—
83	C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	C₂×C₃×C₄	1	4	yes	—
87	C₄²	C₃×D₄	1	4	yes	—
88	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
89	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₃	C₂×#41	1	1	yes	characteristic
91	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
92	C₃×C₄	—	2	8	no	—
93	C₃×C₄	—	2	4	no	—
94	C₃×C₄	—	2	4	no	—
95	C₃×C₄	—	4	4	no	—
96	C₃×C₄	—	4	2	no	—
97	C₃×C₄	C₄×D₄	1	4	yes	—
98	C₂³	C₂⁴×C₃	1	1	yes	characteristic
99	Q₈	—	2	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	8	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
109	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
110	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
111	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
112	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
113	C₂×C₄	C₃×#41	1	2	yes	—
114	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂×C₃	#519	1	2	yes	—
116	C₂×C₃	#536	1	2	yes	—
117	C₂×C₃	#541	1	2	yes	—
118	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
119	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
120	C₂²	C₂×C₃×#41	1	1	yes	characteristic
121	C₂²	C₃×2₋¹⁺⁴	1	4	yes	—
122	C₄	—	2	8	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	—	4	4	no	—
126	C₄	—	4	2	no	—
127	C₄	C₃×C₄×D₄	1	4	yes	—
128	C₃	#2327	1	1	yes	characteristic, direct product factor, Sylow
129	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
130	C₂	C₃×#519	1	2	yes	—
131	C₂	C₃×#536	1	2	yes	—
132	C₂	C₃×#541	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₃×C₄×Q₈

C₂

yes

maximal

C₃×#391

C₂

yes

maximal

C₃×#387

C₂

yes

maximal

C₃×#382

C₂

yes

maximal, characteristic

C₃×#376

C₂

yes

maximal, characteristic

#2327

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₃×C₄⋊Q₈

C₄

yes

complemented

C₃×C₄×Q₈

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₂×C₄⋊Q₈

C₂×C₃

yes

characteristic

C₂×C₄×Q₈

C₂×C₃

yes

—

#391

C₂×C₃

yes

—

#387

C₂×C₃

yes

—

#382

C₂×C₃

yes

characteristic

#376

C₂×C₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₂²×Q₈

C₂²×C₃

yes

—

C₄⋊Q₈

C₃×C₄

yes

complemented

C₄×Q₈

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

C₂×C₄²

C₂²×C₃

yes

characteristic

C₂×C₄²

C₂²×C₃

yes

characteristic

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

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

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₃×C₄

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃×C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₃×C₄

yes

—

C₄²

C₃×D₄

yes

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂×#41

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×D₄

yes

—

C₂³

C₂⁴×C₃

yes

characteristic

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×C₃×C₄

yes

—

109

C₂×C₄

C₂²×C₃×C₄

yes

—

110

C₂×C₄

C₂²×C₃×C₄

yes

—

111

C₂×C₄

C₂×C₃×D₄

yes

—

112

C₂×C₄

C₂×C₃×D₄

yes

—

113

C₂×C₄

C₃×#41

yes

—

114

C₂×C₃

C₂×C₄×D₄

yes

characteristic

115

C₂×C₃

#519

yes

—

116

C₂×C₃

#536

yes

—

117

C₂×C₃

#541

yes

—

118

C₂²

C₂³×C₃×C₄

yes

characteristic

119

C₂²

C₂²×C₃×D₄

yes

characteristic

120

C₂²

C₂×C₃×#41

yes

characteristic

121

C₂²

C₃×2₋¹⁺⁴

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₃×C₄×D₄

yes

—

128

C₃

#2327

yes

characteristic, direct product factor, Sylow

129

C₂

C₂×C₃×C₄×D₄

yes

characteristic

130

C₂

C₃×#519

yes

—

131

C₂

C₃×#536

yes

—

132

C₂

C₃×#541

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃

C₂⁴

commutator

C₂²

C₂³×C₃×C₄

Frattini, cosocle

C₂³

C₂⁴×C₃

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ G

N	H ≅ G/N	type
7. #2327	C₃	direct (N × H)
9. C₃×C₄⋊Q₈	C₄	semidirect
44. C₄⋊Q₈	C₃×C₄	semidirect
128. C₃	#2327	direct (N × H)

H ≅ G/N

type

7. #2327

C₃

direct (N × H)

9. C₃×C₄⋊Q₈

C₄

semidirect

44. C₄⋊Q₈

C₃×C₄

semidirect

128. C₃

#2327

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×Q₈	2
C₃×#391	2
C₃×#387	2
C₃×#382	2
C₃×#376	2
#2327	3

subgroup

index

C₂×C₃×C₄⋊Q₈

C₂×C₃×C₄×Q₈

Subgroups — C₃×#2327 #84191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2327 #84191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2327 #84191