Finite Group Databasebeta

← #219585

Subgroups — #219585

Order 512.

507 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21733C211yesmaximal, characteristic
2#21724C211yesmaximal, characteristic
3#19663C211yesmaximal, characteristic
4#9767C211yesmaximal, characteristic
5#9755C211yesmaximal, characteristic
6#9298C212yesmaximal
7#9297C212yesmaximal
8#9290C212yesmaximal
9#9288C212yesmaximal
10#8793C211yesmaximal, characteristic
11#8138C211yesmaximal, characteristic
12#3230C2212yes—
13#3209C2212yes—
14#3160C2211yescharacteristic
15C2×#498C2211yescharacteristic
16#2961C2211yescharacteristic
17#2961C2211yescharacteristic
18#2331C2212yes—
19#1995C2211yescharacteristic
20#1995C2211yescharacteristic
21#1993C2212yes—
22#1946—21no—
23#1945—21no—
24#1944—21no—
25#1941—21no—
26#1848C2211yescharacteristic
27#1846C2211yescharacteristic
28#1776—21no—
29C4×#106—21no—
30#1773—21no—
31#1772—21no—
32#1726C2211yescharacteristic
33#1723C414yescomplemented
34#1707C414yescomplemented
35#1549C2212yes—
36#1548C2212yes—
37#1518C2212yes—
38#1518C2212yes—
39#1450C2211yescharacteristic
40#1450C2211yescharacteristic
41#1286C2212yes—
42#1286C2212yes—
43#1275C2212yes—
44#1274C2212yes—
45#533C2311yescharacteristic
46#533C2311yescharacteristic
47C2×C4⋊Q8C2312yes—
48#500C2×C412yes—
49#500C2×C414yes—
50#499—21no—
51#499—21no—
52#498—21no—
53#498—21no—
54#498C2×C412yes—
55#498C2×C412yes—
56#498C2×C412yes—
57#494C2×C414yes—
58#487C2×C414yes—
59#487C2×C414yes—
60#484—42no—
61#477—42no—
62#462—42no—
63#451—42no—
64#446—42no—
65#446—42no—
66#431C2312yes—
67#431C2312yes—
68C2×#108—21no—
69C2×#108—21no—
70#427—21no—
71#427—21no—
72C2×#107—21no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—21no—
76#424—21no—
77#424—21no—
78#423C2311yescharacteristic
79#423C2311yescharacteristic
80#405—21no—
81#405—21no—
82C2×#97D411yescharacteristic
83C2×#97D411yescharacteristic
84C2×#97C2311yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#391—42no—
89#389—22no—
90#387—42no—
91#382C2312yes—
92C4×C4⋊C4—22no—
93#376C2311yescharacteristic
94#340—22no—
95#340—22no—
96#339—21no—
97#339—21no—
98#339—21no—
99#339—21no—
100#333—42no—
101#332—42no—
102#328—42no—
103#328—42no—
104#322—21no—
105#322C2×C412yes—
106C82—21no—
107C82C2×C412yes—
108C22×Q8—22no—
109C2×Q16—44no—
110C2×M16—21no—
111C2×M16—21no—
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8C22×C414yes—
120C4⋊Q8C22×C412yes—
121C4⋊Q8C22×C412yes—
122#125—21no—
123#125—21no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128#117—21no—
129#117—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C42—42no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2411yescharacteristic
138#108—42no—
139#108—42no—
140#107—22no—
141#107—22no—
142#107—44no—
143#107—42no—
144#106—22no—
145#106—22no—
146#106—22no—
147#106—22no—
148#106—22no—
149#106—22no—
150#106—42no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—42no—
156C4⋊C8—42no—
157#103—44no—
158#103—44no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161#97—22no—
162#97—22no—
163#97—41no—
164#97—41no—
165#97Q1614yescomplemented
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—41no—
170C4×C8—41no—
171C4×C8C22×C414yes—
172C4×C8C22×C411yescharacteristic
173C4×C8C22×C411yescharacteristic
174C4×C8#4111yescharacteristic
175C4×C8#4111yescharacteristic
176#95—22no—
177#95—22no—
178#95—22no—
179#95—42no—
180#95—42no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—22no—
184C2×Q8—42no—
185C2×Q8—162no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4H(5,𝔽2)12yes—
192Q16—164no—
193M16—41no—
194M16—41no—
195M16—41no—
196M16—41no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—24no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8C4×D411yescharacteristic
207C2×C8C4×D411yescharacteristic
208C2×C8C4×D411yescharacteristic
209C2×C8C4×D411yescharacteristic
210C2×C8C2×Q1612yes—
211C2×C8C2×Q1612yes—
212C2×C8#13512yes—
213C2×C8#13712yes—
214C2×C8#13712yes—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—162no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—41no—
234C42—41no—
235C42—162no—
236C42C2×Q1612yes—
237C42C23×C411yescharacteristic
238C42C2×#4111yescharacteristic
239C42H(5,𝔽2)12yes—
240C23#53511yescharacteristic
241Q8—82no—
242Q8—84no—
243Q8—82no—
244Q8—162no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4C2×C4×D411yescharacteristic
259C2×C4#51812yes—
260C2×C4#53511yescharacteristic
261C2×C4#54012yes—
262C2×C4C22×Q1611yescharacteristic
263C2×C4C2×#13511yescharacteristic
264C2×C4C2×#13711yescharacteristic
265C2×C4C2×#13711yescharacteristic
266C8—22no—
267C8—44no—
268C8—41no—
269C8—82no—
270C8—81no—
271C8—81no—
272C8C4×Q1612yes—
273C8C4×Q1612yes—
274C8#44112yes—
275C8#44112yes—
276C22—22no—
277C22#232811yescharacteristic
278C22#320911yescharacteristic
279C22#321011yescharacteristic
280C4—22no—
281C4—22no—
282C4—42no—
283C4—81no—
284C4—84no—
285C4—82no—
286C4—82no—
287C4—162no—
288C4C2×C4×Q1611yescharacteristic
289C4C2×#44111yescharacteristic
290C4#320911yescharacteristic
291C4#321011yescharacteristic
292C2—21no—
293C2—21no—
294C2#2227011yescharacteristic
295C2#2228412yes—
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23#535
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
33. #1723C4semidirect
34. #1707C4semidirect
165. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#217332
#217242
#196632
#97672
#97552
#92982
#92972
#92902
#92882
#87932
#81382