Finite Group Databasebeta

← #135286

Subgroups — #135286

Order 512.

314 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20820C211yesmaximal, characteristic
2#12602C211yesmaximal, characteristic
3#8601C211yesmaximal, characteristic
4#7175C212yesmaximal, complemented
5#7174C212yesmaximal, complemented
6C2×#405C2211yescharacteristic
7#2314—21no—
8C2×#349—21no—
9C4×M32C2212yes—
10#1846C411yescharacteristic
11#1844C411yescharacteristic
12#1836—21no—
13#1809—21no—
14#1807C411yescharacteristic
15#1807C411yescharacteristic
16#1776—21no—
17C4×#106C2211yescharacteristic
18C4×#106C2211yescharacteristic
19#1564C2212yes—
20#1458—21no—
21#1377—21no—
22#1275—21no—
23C22×M16D411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C2×#117D411yescharacteristic
26C2×M32—22no—
27#431—22no—
28C2×#107—21no—
29C2×#106—21no—
30C2×#106C2×C411yescharacteristic
31C2×#106C2×C411yescharacteristic
32C2×#106C2×C411yescharacteristic
33C2×#106C2×C411yescharacteristic
34#405—22no—
35#405—22no—
36C2×#97D411yescharacteristic
37C2×#97D411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8C2311yescharacteristic
40#388—41no—
41#388—41no—
42#387—41no—
43#387—41no—
44#384—21no—
45#382—21no—
46#380—21no—
47#380—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4—21no—
50#349—24no—
51#346—22no—
52C4×C16—22no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57#335—22no—
58#334—22no—
59#324—41no—
60#323—41no—
61C23×C4C22⋊C411yescharacteristic
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—41no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22×C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73#117—22no—
74#117—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4D811yescharacteristic
82C2×C4⋊C4SD1611yescharacteristic
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4D811yescharacteristic
86C2×C22⋊C4SD1611yescharacteristic
87C2×C42—41no—
88C2×C42—41no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C2×D411yescharacteristic
91M32—24no—
92M32—42no—
93C2×C16—42no—
94#107—42no—
95#106—22no—
96#106—22no—
97#106—22no—
98#106—22no—
99#106—42no—
100C4⋊C8—42no—
101C22⋊C8—42no—
102#97—22no—
103#97—22no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—42no—
107C4×C8#4112yes—
108C4×C8#4112yes—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115C24#10211yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4#10211yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4#10211yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C2×D811yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132M16—44no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—82no—
143C2×C8#10412yes—
144C2×C8#11711yescharacteristic
145C2×C8#11711yescharacteristic
146C2×C8#12312yes—
147C2×C8#13112yes—
148C2×C8#13112yes—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—161no—
160C4⋊C4—161no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—161no—
164C22⋊C4—161no—
165C42—21no—
166C42—21no—
167C42—161no—
168C42—161no—
169C42#12312yes—
170C42#13712yes—
171C16—44no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×#10211yescharacteristic
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4#38111yescharacteristic
204C2×C4#40811yescharacteristic
205C2×C4#40811yescharacteristic
206C2×C4#41911yescharacteristic
207C2×C4C2×#10411yescharacteristic
208C2×C4#42111yescharacteristic
209C2×C4#48012yes—
210C2×C4#48312yes—
211C8—22no—
212C8—22no—
213C8—41no—
214C8—162no—
215C8#43312yes—
216C8#43312yes—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#148011yescharacteristic
225C22#149211yescharacteristic
226C22#182311yescharacteristic
227C4—22no—
228C4—22no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—161no—
233C4—161no—
234C4—161no—
235C4—161no—
236C4#148511yescharacteristic
237C4#149711yescharacteristic
238C4#181811yescharacteristic
239C4#182611yescharacteristic
240C2—21no—
241C2—21no—
242C2—81no—
243C2#870911yescharacteristic
244C2#1287311yescharacteristic
245C2#1287411yescharacteristic
subgroupG/H
center, socleC22#1823
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7175C2semidirect
5. #7174C2semidirect

Maximal subgroups

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

subgroupindex
#208202
#126022
#86012
#71752
#71742