Group #30601605 — subgroups

Subgroups — #30601605

Order 15616.

606 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 192 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	#21461751	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#92420	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92420	—	2	2	no	—
6	C₂×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×C₄×#92420	—	2	2	no	—
8	C₂×C₄×#92420	C₂²	1	2	yes	—
9	Q₈⋊#92420	—	2	8	no	—
10	C₂×Q₈×Dic₆₁	C₂²	1	2	yes	—
11	#16839669	C₄	1	2	yes	—
12	#16839661	—	2	2	no	—
13	#16839632	C₄	1	1	yes	characteristic
14	C₄⋊C₄×Dic₆₁	—	2	2	no	—
15	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
17	C₂²×#92420	—	2	2	no	—
18	C₂²×#92420	—	2	2	no	—
19	C₄⋊#92420	—	2	4	no	—
20	C₄⋊#92420	—	4	4	no	—
21	C₄×#92420	—	2	8	no	—
22	C₄×#92420	—	2	2	no	—
23	C₄×#92420	D₄	1	4	yes	—
24	Q₈×Dic₆₁	—	2	4	no	—
25	Q₈×Dic₆₁	—	2	4	no	—
26	C₂×C₄⋊Dic₆₁	—	2	2	no	—
27	C₂×C₄⋊Dic₆₁	—	2	2	no	—
28	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₆₁	—	2	2	no	—
30	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
32	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
33	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
34	#14519684	—	2	4	no	—
35	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	C₄⋊Q₈×C₆₁	C₈	1	4	yes	complemented
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×Q₈×C₆₁	—	2	2	no	—
41	C₂×Q₈×C₆₁	—	2	4	no	—
42	C₂×Q₈×C₆₁	—	2	4	no	—
43	C₂×Q₈×C₆₁	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₆₁	—	2	2	no	—
45	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₆₁	—	2	2	no	—
48	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
49	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×#92420	—	2	8	no	—
52	C₂×#92420	—	2	4	no	—
53	C₂×#92420	—	4	2	no	—
54	C₂×#92420	—	4	4	no	—
55	C₄⋊C₄×C₆₁	—	2	8	no	—
56	C₄²×C₆₁	C₂×C₈	1	2	yes	—
57	C₄²×C₆₁	M₁₆	1	2	yes	—
58	C₄⋊Dic₆₁	—	2	4	no	—
59	C₄⋊Dic₆₁	—	4	4	no	—
60	C₄⋊Dic₆₁	—	4	4	no	—
61	C₄×Dic₆₁	—	2	4	no	—
62	C₄×Dic₆₁	—	2	2	no	—
63	C₄×Dic₆₁	—	4	2	no	—
64	C₄×Dic₆₁	—	4	2	no	—
65	C₄×Dic₆₁	SD₁₆	1	4	yes	—
66	C₄×Dic₆₁	Q₁₆	1	4	yes	—
67	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
68	C₄×Dic₆₁	#41	1	2	yes	—
69	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	Q₈×C₆₁	—	2	8	no	—
71	Q₈×C₆₁	—	4	4	no	—
72	C₂×C₄×C₆₁	—	2	4	no	—
73	C₂×C₄×C₆₁	—	4	2	no	—
74	C₂×C₄×C₆₁	—	4	2	no	—
75	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
76	C₂×C₄×C₆₁	#103	1	4	yes	—
77	C₂×C₄×C₆₁	#104	1	4	yes	—
78	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#131	1	2	yes	—
82	C₂×Dic₆₁	—	2	2	no	—
83	C₂×Dic₆₁	—	2	2	no	—
84	C₂×Dic₆₁	—	2	2	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	C₂×Dic₆₁	—	4	4	no	—
87	C₂×Dic₆₁	#103	1	4	yes	—
88	C₂×Dic₆₁	#121	1	2	yes	—
89	C₂×Dic₆₁	#124	1	1	yes	characteristic
90	C₂×Dic₆₁	#127	1	1	yes	characteristic
91	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
92	C₂×Dic₆₁	C₂×Q₁₆	1	2	yes	—
93	#92420	—	2	8	no	—
94	#92420	—	8	2	no	—
95	#9637	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	#390	1	1	yes	characteristic
97	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₆₁	#408	1	1	yes	characteristic
99	C₂²×C₆₁	C₂×#103	1	2	yes	—
100	C₂²×C₆₁	C₂×#104	1	2	yes	—
101	C₄×C₆₁	—	4	4	no	—
102	C₄×C₆₁	Q₈⋊C₈	1	8	yes	complemented
103	C₄×C₆₁	C₈×D₄	1	2	yes	—
104	C₄×C₆₁	#436	1	2	yes	—
105	Dic₆₁	—	2	2	no	—
106	Dic₆₁	—	2	4	no	—
107	Dic₆₁	—	8	2	no	—
108	Dic₆₁	#487	1	2	yes	—
109	Dic₆₁	#492	1	1	yes	characteristic
110	Dic₆₁	#494	1	1	yes	characteristic
111	C₄×C₄⋊Q₈	—	61	1	no	—
112	#1842	—	61	1	no	—
113	C₄²×C₈	—	61	1	no	—
114	C₂×Q₈⋊C₈	—	61	4	no	—
115	C₂×C₆₁	C₂×Q₈⋊C₈	1	2	yes	—
116	C₂×C₆₁	#1493	1	2	yes	—
117	C₂×C₆₁	#1954	1	1	yes	characteristic
118	C₂×C₆₁	#1956	1	1	yes	characteristic
119	C₂×C₆₁	#1962	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	#6923	1	1	yes	characteristic
121	C₂×C₄×Q₈	—	61	2	no	—
122	C₂×C₄⋊C₈	—	61	2	no	—
123	C₂×C₄⋊C₈	—	122	2	no	—
124	C₂×C₄×C₈	—	61	2	no	—
125	C₂×C₄×C₈	—	122	2	no	—
126	#391	—	61	2	no	—
127	#389	—	122	2	no	—
128	#384	—	61	1	no	—
129	C₄×C₄⋊C₄	—	122	2	no	—
130	C₄³	—	61	1	no	—
131	Q₈⋊C₈	—	122	8	no	—
132	C₆₁	#9637	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	C₂×#6923	1	2	yes	—
134	C₂²×C₈	—	122	2	no	—
135	C₂²×C₈	—	122	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	#92420	1	4	yes	complemented
138	C₄×Q₈	—	122	4	no	—
139	C₄×Q₈	—	122	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	61	2	no	—
142	C₂×C₄⋊C₄	—	122	2	no	—
143	C₂×C₄⋊C₄	—	122	2	no	—
144	C₂×C₄²	—	61	2	no	—
145	C₂×C₄²	—	61	1	no	—
146	C₂×C₄²	—	61	2	no	—
147	C₂×C₄²	—	61	1	no	—
148	C₂×C₄²	—	122	2	no	—
149	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
150	C₄⋊C₈	—	122	4	no	—
151	C₄⋊C₈	—	244	4	no	—
152	C₄×C₈	—	61	4	no	—
153	C₄×C₈	—	122	2	no	—
154	C₄×C₈	—	122	8	no	—
155	#95	—	122	4	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	C₂×#92420	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	61	1	no	—
162	C₂²×C₄	—	61	2	no	—
163	C₂²×C₄	—	61	1	no	—
164	C₂²×C₄	—	122	2	no	—
165	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
166	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
167	C₂×C₈	—	122	4	no	—
168	C₂×C₈	—	122	8	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	—	244	4	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	122	4	no	—
173	C₄⋊C₄	—	244	4	no	—
174	C₄⋊C₄	—	244	4	no	—
175	C₄²	—	61	4	no	—
176	C₄²	—	61	4	no	—
177	C₄²	—	61	2	no	—
178	C₄²	—	61	2	no	—
179	C₄²	—	122	4	no	—
180	C₄²	—	122	2	no	—
181	C₄²	—	244	2	no	—
182	C₄²	—	244	2	no	—
183	C₄²	C₂×#92420	1	2	yes	—
184	C₄²	#11758217	1	2	yes	—
185	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	Q₈	—	4	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	61	1	no	—
193	C₂×C₄	—	61	2	no	—
194	C₂×C₄	—	61	2	no	—
195	C₂×C₄	—	61	2	no	—
196	C₂×C₄	—	61	4	no	—
197	C₂×C₄	—	122	2	no	—
198	C₂×C₄	—	122	2	no	—
199	C₂×C₄	—	122	2	no	—
200	C₂×C₄	—	244	2	no	—
201	C₂×C₄	—	244	4	no	—
202	C₂×C₄	C₂²⋊#92420	1	4	yes	—
203	C₂×C₄	#14519840	1	4	yes	—
204	C₂×C₄	#14519842	1	4	yes	—
205	C₂×C₄	D₄×#6923	1	2	yes	—
206	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
207	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
208	C₂×C₄	#14519864	1	2	yes	—
209	C₈	—	122	8	no	—
210	C₈	—	488	2	no	—
211	C₂²	#16840714	1	1	yes	characteristic
212	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
213	C₂²	#16840736	1	1	yes	characteristic
214	C₂²	C₂×#14519840	1	2	yes	—
215	C₂²	C₂×#14519842	1	2	yes	—
216	C₄	—	4	4	no	—
217	C₄	—	61	2	no	—
218	C₄	—	61	1	no	—
219	C₄	—	61	1	no	—
220	C₄	—	122	4	no	—
221	C₄	—	122	2	no	—
222	C₄	—	488	2	no	—
223	C₄	Q₈⋊#92420	1	8	yes	complemented
224	C₄	D₄×#92420	1	2	yes	—
225	C₄	#16840797	1	2	yes	—
226	C₂	C₂×Q₈⋊#92420	1	2	yes	—
227	C₂	#21461168	1	2	yes	—
228	C₂	#21461940	1	1	yes	characteristic
229	C₂	#21461946	1	1	yes	characteristic
230	C₂	#21461952	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461751

C₂

yes

maximal, characteristic

C₄²×#92420

C₂

yes

maximal, characteristic

C₂×Q₈⋊#92420

C₂

yes

maximal

C₄⋊Q₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊#92420

—

C₂×C₄⋊#92420

C₂²

yes

—

C₂×C₄×#92420

—

C₂×C₄×#92420

C₂²

yes

—

Q₈⋊#92420

—

C₂×Q₈×Dic₆₁

C₂²

yes

—

#16839669

C₄

yes

—

#16839661

—

#16839632

C₄

yes

characteristic

C₄⋊C₄×Dic₆₁

—

C₄²×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₂²×#92420

—

C₂²×#92420

—

—

—

—

—

D₄

yes

—

Q₈×Dic₆₁

—

Q₈×Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

C₂³

yes

characteristic

#14519684

—

C₂²×Q₈×C₆₁

C₂×C₄

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₈

yes

complemented

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

C₂²⋊C₄

yes

—

C₂²×Dic₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

C₂×D₄

yes

—

—

—

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

Q₁₆

yes

—

C₄×Dic₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

#41

yes

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

#104

yes

—

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#103

yes

—

C₂×Dic₆₁

#121

yes

—

C₂×Dic₆₁

yes

C₂×Dic₆₁

yes

C₂×Dic₆₁

C₂×SD₁₆

yes

—

C₂×Dic₆₁

C₂×Q₁₆

yes

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

100

C₂²×C₆₁

C₂×#104

yes

—

101

C₄×C₆₁

—

102

C₄×C₆₁

Q₈⋊C₈

yes

complemented

103

C₄×C₆₁

C₈×D₄

yes

—

104

C₄×C₆₁

#436

yes

—

105

Dic₆₁

—

106

Dic₆₁

—

107

Dic₆₁

—

108

Dic₆₁

#487

yes

—

109

Dic₆₁

#492

yes

characteristic

110

Dic₆₁

#494

yes

characteristic

111

C₄×C₄⋊Q₈

—

112

#1842

—

113

C₄²×C₈

—

114

C₂×Q₈⋊C₈

—

115

C₂×C₆₁

C₂×Q₈⋊C₈

yes

—

116

C₂×C₆₁

#1493

yes

—

117

C₂×C₆₁

#1954

yes

characteristic

118

C₂×C₆₁

#1956

yes

characteristic

119

C₂×C₆₁

#1962

yes

characteristic

120

C₂×C₄⋊Q₈

#6923

yes

characteristic

121

C₂×C₄×Q₈

—

122

C₂×C₄⋊C₈

—

123

C₂×C₄⋊C₈

—

122

—

124

C₂×C₄×C₈

—

125

C₂×C₄×C₈

—

122

—

126

#391

—

127

#389

—

122

—

128

#384

—

129

C₄×C₄⋊C₄

—

122

—

130

C₄³

—

131

Q₈⋊C₈

—

122

—

132

C₆₁

#9637

yes

characteristic, complemented, Sylow

133

C₂²×Q₈

C₂×#6923

yes

—

134

C₂²×C₈

—

122

—

135

C₂²×C₈

—

122

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

#92420

yes

complemented

138

C₄×Q₈

—

122

—

139

C₄×Q₈

—

122

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

—

122

—

143

C₂×C₄⋊C₄

—

122

—

144

C₂×C₄²

—

145

C₂×C₄²

—

146

C₂×C₄²

—

147

C₂×C₄²

—

148

C₂×C₄²

—

122

—

149

C₂×C₄²

C₂×#6923

yes

characteristic

150

C₄⋊C₈

—

122

—

151

C₄⋊C₈

—

244

—

152

C₄×C₈

—

153

C₄×C₈

—

122

—

154

C₄×C₈

—

122

—

155

#95

—

122

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

C₂×#92420

yes

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

122

—

165

C₂²×C₄

C₂²⋊#6923

yes

—

166

C₂²×C₄

C₂²×#6923

yes

characteristic

167

C₂×C₈

—

122

—

168

C₂×C₈

—

122

—

169

C₂×C₈

—

244

—

170

C₂×C₈

—

244

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

122

—

173

C₄⋊C₄

—

244

—

174

C₄⋊C₄

—

244

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

—

122

—

180

C₄²

—

122

—

181

C₄²

—

244

—

182

C₄²

—

244

—

183

C₄²

C₂×#92420

yes

—

184

C₄²

#11758217

yes

—

185

C₂³

C₂×C₂²⋊#6923

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

122

—

198

C₂×C₄

—

122

—

199

C₂×C₄

—

122

—

200

C₂×C₄

—

244

—

201

C₂×C₄

—

244

—

202

C₂×C₄

C₂²⋊#92420

yes

—

203

C₂×C₄

#14519840

yes

—

204

C₂×C₄

#14519842

yes

—

205

C₂×C₄

D₄×#6923

yes

—

206

C₂×C₄

C₂²×#92420

yes

characteristic

207

C₂×C₄

C₂×#11758217

yes

characteristic

208

C₂×C₄

#14519864

yes

—

209

C₈

—

122

—

210

C₈

—

488

—

211

C₂²

#16840714

yes

characteristic

212

C₂²

C₂×C₂²⋊#92420

yes

characteristic

213

C₂²

#16840736

yes

characteristic

214

C₂²

C₂×#14519840

yes

—

215

C₂²

C₂×#14519842

yes

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

122

—

221

C₄

—

122

—

222

C₄

—

488

—

223

C₄

Q₈⋊#92420

yes

complemented

224

C₄

D₄×#92420

yes

—

225

C₄

#16840797

yes

—

226

C₂

C₂×Q₈⋊#92420

yes

—

227

C₂

#21461168

yes

—

228

C₂

#21461940

yes

characteristic

229

C₂

#21461946

yes

characteristic

230

C₂

#21461952

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄⋊Q₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄⋊Q₈	#6923
hypocenter	C₆₁	#9637
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₂×C₄⋊Q₈×C₆₁

C₄

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄×Dic₆₁

C₂³

C₂×C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
37. C₄⋊Q₈×C₆₁	C₈	semidirect
102. C₄×C₆₁	Q₈⋊C₈	semidirect
132. C₆₁	#9637	semidirect
137. C₄⋊Q₈	#92420	semidirect
223. C₄	Q₈⋊#92420	semidirect

H ≅ G/N

type

37. C₄⋊Q₈×C₆₁

C₈

semidirect

102. C₄×C₆₁

Q₈⋊C₈

semidirect

132. C₆₁

#9637

semidirect

137. C₄⋊Q₈

#92420

semidirect

223. C₄

Q₈⋊#92420

semidirect

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

subgroup	index
#21461751	2
C₄²×#92420	2
C₂×Q₈⋊#92420	2
C₄⋊Q₈×Dic₆₁	2
#9637	61

subgroup

index

#21461751

C₄²×#92420

C₂×Q₈⋊#92420

C₄⋊Q₈×Dic₆₁

#9637

Subgroups — #30601605

Conjugacy classes of subgroups

Complements

Maximal subgroups