Group #30598046 — subgroups

Subgroups — #30598046

Order 15616.

414 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 140 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	C₄×#14519845	C₂	1	1	yes	maximal, characteristic
2	#21461616	C₂	1	2	yes	maximal
3	C₈×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#21460653	C₂	1	2	yes	maximal
5	#334×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519845	C₄	1	2	yes	—
7	C₂×#14519831	—	2	2	no	—
8	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊#6923	C₂²	1	2	yes	—
11	#16840636	C₄	1	2	yes	—
12	(C₂×#6923)⋊C₈	—	2	2	no	—
13	C₈×#92420	C₄	1	2	yes	complemented
14	C₈×#92420	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₆₁	C₂²	1	2	yes	—
16	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
17	#16839513	C₄	1	1	yes	characteristic, complemented
18	#16839512	C₄	1	2	yes	complemented
19	C₆₁×#334	C₄	1	1	yes	characteristic, complemented
20	C₂×D₆₁⋊C₈	—	2	1	no	—
21	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
22	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₄⋊#6923	C₂×C₄	1	4	yes	—
24	C₂×C₄×#6923	—	2	2	no	—
25	#14519845	—	2	4	no	—
26	#14519831	—	4	4	no	—
27	C₄×#92420	—	2	1	no	—
28	C₄×#92420	C₂×C₄	1	2	yes	—
29	C₄×#92420	C₂×C₄	1	2	yes	—
30	C₄×#92420	Q₈	1	2	yes	—
31	(C₂×#6923)⋊C₄	—	2	2	no	—
32	C₂×C₈×D₆₁	—	2	2	no	—
33	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊#92418	C₂×C₄	1	2	yes	—
37	#14519706	—	2	2	no	—
38	C₄×#92418	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×#6923	—	2	4	no	—
44	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
47	C₂×#92420	—	2	2	no	—
48	C₂×#92420	C₄⋊C₄	1	2	yes	—
49	C₂×#92420	C₄⋊C₄	1	2	yes	—
50	C₄⋊#6923	—	2	4	no	—
51	C₄⋊#6923	—	2	4	no	—
52	C₄×#6923	—	4	2	no	—
53	C₄×#6923	—	4	2	no	—
54	D₆₁⋊C₈	—	2	4	no	—
55	D₆₁⋊C₈	—	2	2	no	—
56	D₆₁⋊C₈	SD₁₆	1	4	yes	—
57	C₂×C₈×C₆₁	—	2	2	no	—
58	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
60	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
63	C₄×Dic₆₁	#41	1	1	yes	characteristic
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	C₄²	1	1	yes	characteristic
66	C₂×#92418	C₄²	1	1	yes	characteristic
67	C₈×D₆₁	—	2	2	no	—
68	C₈×D₆₁	—	4	2	no	—
69	C₈×D₆₁	—	4	2	no	—
70	C₈×D₆₁	M₁₆	1	4	yes	—
71	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
72	C₂×#6923	—	4	4	no	—
73	C₂×#6923	—	4	4	no	—
74	C₂×#6923	—	4	4	no	—
75	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
78	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×Dic₆₁	#117	1	1	yes	characteristic
80	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
81	C₄×D₆₁	—	2	2	no	—
82	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
83	C₄×D₆₁	#131	1	2	yes	—
84	C₄×D₆₁	C₂×SD₁₆	1	2	yes	—
85	C₄×D₆₁	#135	1	2	yes	—
86	#92420	—	2	4	no	—
87	#92420	—	2	2	no	—
88	#92420	#106	1	4	yes	complemented
89	C₈×C₆₁	—	2	1	no	—
90	C₈×C₆₁	—	4	2	no	—
91	C₈×C₆₁	#97	1	2	yes	complemented
92	#92418	—	2	1	no	—
93	#92418	—	4	2	no	—
94	#92418	#97	1	2	yes	complemented
95	#8283	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₂×D₆₁	#436	1	2	yes	—
98	C₂×D₆₁	C₄×SD₁₆	1	2	yes	—
99	C₂×D₆₁	#443	1	2	yes	—
100	#6923	—	8	4	no	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
103	C₄×C₆₁	#405	1	1	yes	characteristic
104	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
105	Dic₆₁	—	2	1	no	—
106	Dic₆₁	C₂×#106	1	1	yes	characteristic
107	Dic₆₁	#427	1	1	yes	characteristic
108	Dic₆₁	#446	1	2	yes	—
109	C₄×#106	—	61	1	no	—
110	#1770	—	61	2	no	—
111	C₂×#334	—	61	1	no	—
112	C₂×C₈²	—	61	1	no	—
113	#1279	—	61	2	no	—
114	C₂×C₆₁	#1770	1	1	yes	characteristic
115	C₂×C₆₁	#1772	1	1	yes	characteristic
116	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
117	D₆₁	#1590	1	4	yes	—
118	C₂×#106	—	61	2	no	—
119	C₂×C₄⋊C₈	—	61	2	no	—
120	C₂×#97	—	122	2	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₂×C₄×C₈	—	61	1	no	—
123	C₂×C₄×C₈	—	61	1	no	—
124	C₄×C₄⋊C₄	—	61	2	no	—
125	#340	—	61	2	no	—
126	#336	—	122	2	no	—
127	#334	—	61	2	no	—
128	#334	—	61	1	no	—
129	#334	#6923	1	1	yes	characteristic, complemented
130	C₈²	—	61	2	no	—
131	C₈²	—	61	2	no	—
132	C₆₁	#8283	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	61	1	no	—
136	C₂²×C₈	—	61	1	no	—
137	C₂²×C₈	—	122	2	no	—
138	C₂²×C₈	—	122	1	no	—
139	C₂×C₄⋊C₄	—	61	4	no	—
140	C₂×C₄²	—	61	1	no	—
141	C₂×C₄²	—	122	2	no	—
142	#106	—	122	4	no	—
143	C₄⋊C₈	—	61	2	no	—
144	C₄⋊C₈	—	122	2	no	—
145	C₄⋊C₈	C₂×#6923	1	2	yes	—
146	#97	—	244	4	no	—
147	C₄×C₈	—	61	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	61	1	no	—
150	C₄×C₈	—	61	1	no	—
151	C₄×C₈	—	61	2	no	—
152	C₄×C₈	—	61	1	no	—
153	C₄×C₈	—	122	1	no	—
154	C₄×C₈	C₂×#6923	1	1	yes	characteristic
155	#95	—	122	2	no	—
156	C₂²×C₄	—	61	1	no	—
157	C₂²×C₄	—	61	1	no	—
158	C₂²×C₄	—	61	1	no	—
159	C₂²×C₄	—	122	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	61	2	no	—
162	C₂×C₈	—	61	4	no	—
163	C₂×C₈	—	61	1	no	—
164	C₂×C₈	—	61	4	no	—
165	C₂×C₈	—	61	2	no	—
166	C₂×C₈	—	61	1	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₂×C₈	—	122	2	no	—
170	C₂×C₈	—	122	4	no	—
171	C₂×C₈	—	122	2	no	—
172	C₂×C₈	—	244	2	no	—
173	C₂×C₈	—	244	2	no	—
174	C₂×C₈	C₄×#6923	1	1	yes	characteristic
175	C₂×C₈	C₄×#6923	1	1	yes	characteristic
176	C₄⋊C₄	—	122	4	no	—
177	C₄⋊C₄	—	122	4	no	—
178	C₄²	—	61	1	no	—
179	C₄²	—	61	1	no	—
180	C₄²	—	61	1	no	—
181	C₄²	—	244	2	no	—
182	C₄²	—	244	2	no	—
183	C₄²	C₂²×#6923	1	1	yes	characteristic
184	C₂³	—	61	1	no	—
185	C₂×C₄	—	61	1	no	—
186	C₂×C₄	—	61	1	no	—
187	C₂×C₄	—	61	2	no	—
188	C₂×C₄	—	61	2	no	—
189	C₂×C₄	—	61	2	no	—
190	C₂×C₄	—	61	2	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	122	2	no	—
193	C₂×C₄	—	244	4	no	—
194	C₂×C₄	—	244	4	no	—
195	C₂×C₄	—	244	4	no	—
196	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
197	C₂×C₄	D₄×#6923	1	1	yes	characteristic
198	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	4	2	no	—
201	C₈	—	61	4	no	—
202	C₈	—	61	2	no	—
203	C₈	—	122	2	no	—
204	C₈	—	122	4	no	—
205	C₈	—	122	1	no	—
206	C₈	—	244	2	no	—
207	C₈	#14519831	1	2	yes	complemented
208	C₂²	—	61	2	no	—
209	C₂²	—	61	2	no	—
210	C₂²	—	61	2	no	—
211	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	61	2	no	—
214	C₄	—	61	1	no	—
215	C₄	—	61	1	no	—
216	C₄	—	122	1	no	—
217	C₄	—	488	4	no	—
218	C₄	C₂×#14519831	1	1	yes	characteristic
219	C₄	#16840728	1	1	yes	characteristic
220	C₄	SD₁₆×#6923	1	2	yes	—
221	C₂	—	61	4	no	—
222	C₂	#21461616	1	1	yes	characteristic
223	C₂	#21461624	1	1	yes	characteristic
224	C₂	#106×#6923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519845

C₂

yes

maximal, characteristic

#21461616

C₂

yes

maximal

C₈×D₆₁⋊C₈

C₂

yes

maximal, characteristic

#21460653

C₂

yes

maximal

#334×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519845

C₄

yes

—

C₂×#14519831

—

C₄×D₆₁⋊C₈

C₂²

yes

characteristic

C₄×D₆₁⋊C₈

C₂²

yes

characteristic

C₄×C₄⋊#6923

C₂²

yes

—

#16840636

C₄

yes

—

(C₂×#6923)⋊C₈

—

C₈×#92420

C₄

yes

complemented

C₈×#92420

C₄

yes

complemented

C₄⋊C₈×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839513

C₄

yes

characteristic, complemented

#16839512

C₄

yes

complemented

C₆₁×#334

C₄

yes

characteristic, complemented

C₂×D₆₁⋊C₈

—

C₂×D₆₁⋊C₈

D₄

yes

characteristic

C₂×D₆₁⋊C₈

D₄

yes

characteristic

C₂×C₄⋊#6923

C₂×C₄

yes

—

C₂×C₄×#6923

—

—

—

—

C₂×C₄

yes

—

C₄×#92420

C₂×C₄

yes

—

C₄×#92420

Q₈

yes

—

(C₂×#6923)⋊C₄

—

C₂×C₈×D₆₁

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₄²×D₆₁

C₂³

yes

characteristic

C₄⋊#92418

C₂×C₄

yes

—

#14519706

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×#6923

—

C₂×C₄×D₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×D₄

yes

C₂×C₄×D₆₁

yes

—

C₄⋊C₄

yes

—

C₂×#92420

C₄⋊C₄

yes

—

—

—

—

—

D₆₁⋊C₈

—

D₆₁⋊C₈

—

D₆₁⋊C₈

SD₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×Q₈

yes

C₄×Dic₆₁

yes

—

C₄²

yes

characteristic

C₂×#92418

C₄²

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

M₁₆

yes

—

C₂²×D₆₁

C₄×D₄

yes

—

—

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₄×Q₈

yes

characteristic

C₂×Dic₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#117

yes

characteristic

C₂×Dic₆₁

C₄×Q₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₂×M₁₆

yes

—

C₄×D₆₁

#131

yes

—

C₄×D₆₁

C₂×SD₁₆

yes

—

C₄×D₆₁

#135

yes

—

—

—

yes

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

—

—

yes

—

C₂²×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₂×D₆₁

#436

yes

—

C₂×D₆₁

C₄×SD₁₆

yes

—

C₂×D₆₁

#443

yes

—

100

#6923

—

101

C₄×C₆₁

—

102

C₄×C₆₁

C₂×#97

yes

characteristic

103

C₄×C₆₁

#405

yes

characteristic

104

C₄×C₆₁

C₄×SD₁₆

yes

—

105

Dic₆₁

—

106

Dic₆₁

C₂×#106

yes

characteristic

107

Dic₆₁

#427

yes

characteristic

108

Dic₆₁

#446

yes

—

109

C₄×#106

—

110

#1770

—

111

C₂×#334

—

112

C₂×C₈²

—

113

#1279

—

114

C₂×C₆₁

#1770

yes

characteristic

115

C₂×C₆₁

#1772

yes

characteristic

116

C₂×C₆₁

C₄×#106

yes

characteristic

117

D₆₁

#1590

yes

—

118

C₂×#106

—

119

C₂×C₄⋊C₈

—

120

C₂×#97

—

122

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

—

123

C₂×C₄×C₈

—

124

C₄×C₄⋊C₄

—

125

#340

—

126

#336

—

122

—

127

#334

—

128

#334

—

129

#334

#6923

yes

characteristic, complemented

130

C₈²

—

131

C₈²

—

132

C₆₁

#8283

yes

characteristic, complemented, Sylow

133

C₂²×C₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

122

—

138

C₂²×C₈

—

122

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄²

—

141

C₂×C₄²

—

122

—

142

#106

—

122

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

122

—

145

C₄⋊C₈

C₂×#6923

yes

—

146

#97

—

244

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

122

—

154

C₄×C₈

C₂×#6923

yes

characteristic

155

#95

—

122

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

122

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

122

—

168

C₂×C₈

—

122

—

169

C₂×C₈

—

122

—

170

C₂×C₈

—

122

—

171

C₂×C₈

—

122

—

172

C₂×C₈

—

244

—

173

C₂×C₈

—

244

—

174

C₂×C₈

C₄×#6923

yes

characteristic

175

C₂×C₈

C₄×#6923

yes

characteristic

176

C₄⋊C₄

—

122

—

177

C₄⋊C₄

—

122

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

244

—

182

C₄²

—

244

—

183

C₄²

C₂²×#6923

yes

characteristic

184

C₂³

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

122

—

193

C₂×C₄

—

244

—

194

C₂×C₄

—

244

—

195

C₂×C₄

—

244

—

196

C₂×C₄

C₂×C₄×#6923

yes

characteristic

197

C₂×C₄

D₄×#6923

yes

characteristic

198

C₂×C₄

Q₈×#6923

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

122

—

204

C₈

—

122

—

205

C₈

—

122

—

206

C₈

—

244

—

207

C₈

#14519831

yes

complemented

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

C₄⋊C₄×#6923

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

122

—

217

C₄

—

488

—

218

C₄

C₂×#14519831

yes

characteristic

219

C₄

#16840728

yes

characteristic

220

C₄

SD₁₆×#6923

yes

—

221

C₂

—

222

C₂

#21461616

yes

characteristic

223

C₂

#21461624

yes

characteristic

224

C₂

#106×#6923

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#334	C₄
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄²×D₆₁	C₂³
hypercenter	#334	#6923
hypocenter	C₆₁	#8283
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×#6923

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₄²

C₂²×#6923

Fitting

C₆₁×#334

C₄

socle

C₂²×C₆₁

C₄×C₄⋊C₄

prime radical

C₂²×D₆₁

C₄×D₄

cosocle

C₄²×D₆₁

C₂³

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
13. C₈×#92420	C₄	semidirect
14. C₈×#92420	C₄	semidirect
17. #16839513	C₄	semidirect
18. #16839512	C₄	semidirect
19. C₆₁×#334	C₄	semidirect
88. #92420	#106	semidirect
91. C₈×C₆₁	#97	semidirect
94. #92418	#97	semidirect
129. #334	#6923	semidirect
132. C₆₁	#8283	semidirect
207. C₈	#14519831	semidirect

H ≅ G/N

type

13. C₈×#92420

C₄

semidirect

14. C₈×#92420

C₄

semidirect

17. #16839513

C₄

semidirect

18. #16839512

C₄

semidirect

19. C₆₁×#334

C₄

semidirect

88. #92420

#106

semidirect

91. C₈×C₆₁

#97

semidirect

94. #92418

#97

semidirect

129. #334

#6923

semidirect

132. C₆₁

#8283

semidirect

207. C₈

#14519831

semidirect

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

subgroup	index
C₄×#14519845	2
#21461616	2
C₈×D₆₁⋊C₈	2
#21460653	2
#334×D₆₁	2
#8283	61

subgroup

index

C₈×D₆₁⋊C₈

#334×D₆₁

Subgroups — #30598046

Conjugacy classes of subgroups

Complements

Maximal subgroups