Group #84366 — subgroups

Subgroups — C₃×#2502 #84366

Order 384.

544 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 248 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	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#398	C₂	1	4	yes	maximal, complemented
4	C₃×#391	C₂	1	2	yes	maximal, complemented
5	C₃×#385	C₂	1	4	yes	maximal
6	C₃×#384	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
8	#2502	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₃×C₂²⋊Q₈	—	2	4	no	—
13	C₂×C₃×C₄⋊C₄	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₃×#95	C₂²	1	2	yes	—
23	C₃×#95	C₂²	1	8	yes	—
24	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
26	#398	C₂×C₃	1	4	yes	complemented
27	#391	C₂×C₃	1	2	yes	complemented
28	#385	C₂×C₃	1	4	yes	—
29	#384	C₂×C₃	1	2	yes	complemented
30	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
31	C₂⁴×C₃	C₂³	1	1	yes	characteristic
32	C₂×C₃×Q₈	—	2	2	no	—
33	C₂×C₃×Q₈	—	2	4	no	—
34	C₂²×C₃×C₄	—	2	4	no	—
35	C₂²×C₃×C₄	—	2	4	no	—
36	C₂²×C₃×C₄	—	4	4	no	—
37	C₂²×C₃×C₄	D₄	1	4	yes	—
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	4	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₃×C₄⋊C₄	—	2	4	no	—
45	C₃×C₄⋊C₄	—	2	4	no	—
46	C₃×C₄⋊C₄	—	2	2	no	—
47	C₃×C₄⋊C₄	—	2	4	no	—
48	C₃×C₄⋊C₄	—	4	8	no	—
49	C₃×C₂²⋊C₄	—	2	4	no	—
50	C₃×C₂²⋊C₄	Q₈	1	8	yes	complemented
51	C₃×C₄²	—	2	4	no	—
52	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
54	C₂³×C₄	C₂²×C₃	1	2	yes	—
55	C₂²⋊Q₈	—	2	4	no	—
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
62	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
63	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₄²	C₂²×C₃	1	2	yes	—
65	#95	C₂²×C₃	1	2	yes	—
66	#95	C₂²×C₃	1	8	yes	—
67	C₂³×C₃	—	2	2	no	—
68	C₂³×C₃	—	2	2	no	—
69	C₂³×C₃	C₂×D₄	1	2	yes	—
70	C₂³×C₃	C₂×Q₈	1	4	yes	—
71	C₂³×C₃	C₂⁴	1	1	yes	characteristic
72	C₃×Q₈	—	4	4	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	—	2	8	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	8	no	—
83	C₂×C₃×C₄	—	4	8	no	—
84	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₂×Q₈	1	8	yes	—
87	C₂×C₃×C₄	#41	1	4	yes	—
88	C₂⁴	C₂³×C₃	1	1	yes	characteristic
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	4	4	no	—
94	C₂²×C₄	C₃×D₄	1	4	yes	—
95	C₂²×C₄	C₂³×C₃	1	2	yes	—
96	C₂²×C₄	C₂³×C₃	1	4	yes	—
97	C₂²×C₄	C₂³×C₃	1	2	yes	—
98	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₃	1	4	yes	—
100	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	4	8	no	—
106	C₂²⋊C₄	—	2	4	no	—
107	C₂²⋊C₄	C₃×Q₈	1	8	yes	complemented
108	C₄²	—	2	4	no	—
109	C₂²×C₃	—	2	2	no	—
110	C₂²×C₃	—	2	2	no	—
111	C₂²×C₃	—	2	8	no	—
112	C₂²×C₃	C₄⋊Q₈	1	4	yes	—
113	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
114	C₂²×C₃	C₂²×Q₈	1	2	yes	—
115	C₂²×C₃	C₂×#41	1	2	yes	—
116	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
117	C₃×C₄	—	2	2	no	—
118	C₃×C₄	—	2	8	no	—
119	C₃×C₄	—	4	2	no	—
120	C₃×C₄	—	4	2	no	—
121	C₃×C₄	—	4	4	no	—
122	C₃×C₄	#123	1	4	yes	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	C₂×C₃×D₄	1	2	yes	—
126	C₂³	C₂×C₃×Q₈	1	4	yes	—
127	C₂³	C₂⁴×C₃	1	1	yes	characteristic
128	Q₈	—	4	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	4	8	no	—
140	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
141	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
142	C₂×C₄	C₂×C₃×Q₈	1	8	yes	—
143	C₂×C₄	C₃×#41	1	4	yes	—
144	C₂×C₃	—	2	4	no	—
145	C₂×C₃	C₂×#123	1	1	yes	characteristic
146	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂×C₃	#535	1	1	yes	characteristic
148	C₂×C₃	#554	1	4	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	8	no	—
152	C₂²	C₃×C₄⋊Q₈	1	4	yes	—
153	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
154	C₂²	C₂²×C₃×Q₈	1	2	yes	—
155	C₂²	C₂×C₃×#41	1	2	yes	—
156	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	8	no	—
159	C₄	—	4	2	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	2	no	—
162	C₄	C₃×#123	1	4	yes	—
163	C₃	#2502	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	—	2	4	no	—
165	C₂	C₂×C₃×#123	1	1	yes	characteristic
166	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
167	C₂	C₃×#535	1	1	yes	characteristic
168	C₂	C₃×#554	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂²×C₃×C₄⋊C₄

C₂

yes

maximal, characteristic

C₃×#398

C₂

yes

maximal, complemented

C₃×#391

C₂

yes

maximal, complemented

C₃×#385

C₂

yes

maximal

C₃×#384

C₂

yes

maximal, complemented

C₃×C₄×C₂²⋊C₄

C₂

yes

maximal, characteristic

#2502

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂³×C₃×C₄

C₂²

yes

characteristic

C₂³×C₃×C₄

C₂²

yes

—

C₃×C₂²⋊Q₈

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

—

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

—

C₂×C₃×C₄²

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₂×C₂²⋊Q₈

C₂×C₃

yes

characteristic

C₂²×C₄⋊C₄

C₂×C₃

yes

characteristic

#398

C₂×C₃

yes

complemented

#391

C₂×C₃

yes

complemented

#385

C₂×C₃

yes

—

#384

C₂×C₃

yes

complemented

C₄×C₂²⋊C₄

C₂×C₃

yes

characteristic

C₂⁴×C₃

C₂³

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

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

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

Q₈

yes

complemented

C₃×C₄²

—

C₂²×Q₈

C₂²×C₃

yes

characteristic

C₂³×C₄

C₂²×C₃

yes

characteristic

C₂³×C₄

C₂²×C₃

yes

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

#95

C₂²×C₃

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×D₄

yes

—

C₂³×C₃

C₂×Q₈

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×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂⁴

C₂³×C₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

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

characteristic

C₂²×C₄

C₂³×C₃

yes

—

100

C₂²×C₄

C₂³×C₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

C₃×Q₈

yes

complemented

108

C₄²

—

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂²×C₃

C₄⋊Q₈

yes

—

113

C₂²×C₃

C₂²×D₄

yes

characteristic

114

C₂²×C₃

C₂²×Q₈

yes

—

115

C₂²×C₃

C₂×#41

yes

—

116

C₂²×C₃

H(5,𝔽₂)

yes

—

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

#123

yes

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₃×D₄

yes

—

126

C₂³

C₂×C₃×Q₈

yes

—

127

C₂³

C₂⁴×C₃

yes

characteristic

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×C₃×D₄

yes

—

141

C₂×C₄

C₂×C₃×D₄

yes

—

142

C₂×C₄

C₂×C₃×Q₈

yes

—

143

C₂×C₄

C₃×#41

yes

—

144

C₂×C₃

—

145

C₂×C₃

C₂×#123

yes

characteristic

146

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

147

C₂×C₃

#535

yes

characteristic

148

C₂×C₃

#554

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₃×C₄⋊Q₈

yes

—

153

C₂²

C₂²×C₃×D₄

yes

characteristic

154

C₂²

C₂²×C₃×Q₈

yes

—

155

C₂²

C₂×C₃×#41

yes

—

156

C₂²

C₃×H(5,𝔽₂)

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₃×#123

yes

—

163

C₃

#2502

yes

characteristic, direct product factor, Sylow

164

C₂

—

165

C₂

C₂×C₃×#123

yes

characteristic

166

C₂

C₂×C₃×C₄⋊Q₈

yes

characteristic

167

C₂

C₃×#535

yes

characteristic

168

C₂

C₃×#554

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
3. C₃×#398	C₂	semidirect
4. C₃×#391	C₂	semidirect
6. C₃×#384	C₂	semidirect
8. #2502	C₃	direct (N × H)
26. #398	C₂×C₃	semidirect
27. #391	C₂×C₃	semidirect
29. #384	C₂×C₃	semidirect
50. C₃×C₂²⋊C₄	Q₈	semidirect
107. C₂²⋊C₄	C₃×Q₈	semidirect
163. C₃	#2502	direct (N × H)

H ≅ G/N

type

3. C₃×#398

C₂

semidirect

4. C₃×#391

C₂

semidirect

6. C₃×#384

C₂

semidirect

8. #2502

C₃

direct (N × H)

26. #398

C₂×C₃

semidirect

27. #391

C₂×C₃

semidirect

29. #384

C₂×C₃

semidirect

50. C₃×C₂²⋊C₄

Q₈

semidirect

107. C₂²⋊C₄

C₃×Q₈

semidirect

163. C₃

#2502

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
C₂²×C₃×C₄⋊C₄	2
C₃×#398	2
C₃×#391	2
C₃×#385	2
C₃×#384	2
C₃×C₄×C₂²⋊C₄	2
#2502	3

subgroup

index

C₂×C₃×C₂²⋊Q₈

C₂²×C₃×C₄⋊C₄

C₃×C₄×C₂²⋊C₄

Subgroups — C₃×#2502 #84366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2502 #84366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2502 #84366