Finite Group Databasebeta

← #213830

Subgroups — #213830

Order 512.

1856 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1856 conjugacy classes of subgroups — too many to draw. Only its 206 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#35446C211yesmaximal, characteristic
2#20585C212yesmaximal
3#9499C218yesmaximal, complemented
4#8746C212yesmaximal, complemented
5#8543C212yesmaximal, complemented
6C22×H(5,𝔽2)C2211yescharacteristic
7C22×#123C412yescomplemented
8C22×#123C2211yescharacteristic
9C22×#99—24no—
10#2406—24no—
11#2406C414yescomplemented
12#2405—24no—
13#2405C2214yes—
14#2308C412yescomplemented
15#2308C2214yes—
16#2308C2211yescharacteristic
17#1819C2218yescomplemented
18#1782—24no—
19#1781C2214yes—
20#1781C2214yes—
21#1304C2218yescomplemented
22C2×H(5,𝔽2)—48no—
23C2×H(5,𝔽2)—44no—
24C22×#41C2×C412yes—
25C22×#41D414yes—
26C23×D4C2×C412yes—
27C23×D4D414yes—
28C23×D4C2311yescharacteristic
29C23×D4C2312yes—
30C24×C4C2×C411yescharacteristic, complemented
31C2×#123—24no—
32C2×#123—24no—
33C2×#123—44no—
34C2×#123—48no—
35C2×#123C2×C418yescomplemented
36C22×C4⋊C4—22no—
37C22×C4⋊C4C2×C412yescomplemented
38C22×C22⋊C4—22no—
39C22×C22⋊C4—24no—
40C22×C22⋊C4C2×C411yescharacteristic, complemented
41C22×C22⋊C4C2×C412yescomplemented
42C22×C22⋊C4C2312yes—
43C2×#99—28no—
44C2×#99—28no—
45C2×#99—416no—
46#395—24no—
47#394—24no—
48#394—28no—
49#386—24no—
50#386—24no—
51#386—48no—
52#386C2×C418yescomplemented
53#385—28no—
54#385—24no—
55#381—22no—
56#381—24no—
57#381C2×C412yescomplemented
58#379—22no—
59#379—24no—
60#379C2318yes—
61#379C2312yes—
62#342—48no—
63C25C22⋊C412yes—
64C25C22×C411yescharacteristic
65C25C2×D412yes—
66C25C2411yescharacteristic
67H(5,𝔽2)—416no—
68H(5,𝔽2)—432no—
69H(5,𝔽2)—416no—
70C2×#41—44no—
71C2×#41—48no—
72C2×#41—48no—
73C2×#41—48no—
74C2×#41—48no—
75C22×Q8C22⋊C412yes—
76C22×D4—28no—
77C22×D4—216no—
78C22×D4—24no—
79C22×D4—28no—
80C22×D4—44no—
81C22×D4—44no—
82C22×D4—48no—
83C22×D4—48no—
84C22×D4—44no—
85C22×D4—48no—
86C22×D4C22⋊C414yes—
87C22×D4C22⋊C412yes—
88C22×D4C22×C414yes—
89C22×D4C2×D414yes—
90C22×D4C2×D414yes—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—42no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C414yes—
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102#123—416no—
103#123—416no—
104#123—416no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—48no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—216no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—48no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4—416no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4C22×C414yes—
124#99—416no—
125#99—416no—
126#95—28no—
127#95—24no—
128#95—48no—
129C24—22no—
130C24—24no—
131C24—24no—
132C24—28no—
133C24—42no—
134C24—42no—
135C24—42no—
136C24—82no—
137C24—82no—
138C24—84no—
139C24—84no—
140C24#9918yes—
141C24C2×C22⋊C414yes—
142C24C2×C22⋊C412yes—
143C24C23×C411yescharacteristic
144C24C22×D412yes—
145C24H(5,𝔽2)14yes—
146#41—432no—
147#41—464no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×D4—28no—
151C2×D4—28no—
152C2×D4—28no—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—28no—
156C2×D4—28no—
157C2×D4—28no—
158C2×D4—28no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—432no—
165C2×D4—416no—
166C2×D4—416no—
167C2×D4—416no—
168C2×D4—432no—
169C2×D4—48no—
170C2×D4—48no—
171C2×D4—416no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—48no—
193C22×C4—84no—
194C22×C4—82no—
195C22×C4—84no—
196C22×C4—84no—
197C22×C4—88no—
198C22×C4—168no—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C414yes—
201C4⋊C4—48no—
202C4⋊C4—416no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—48no—
206C22⋊C4—416no—
207C22⋊C4—44no—
208C22⋊C4—416no—
209C22⋊C4—84no—
210C22⋊C4—88no—
211C22⋊C4—88no—
212C22⋊C4—816no—
213C23—28no—
214C23—24no—
215C23—24no—
216C23—28no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—42no—
224C23—44no—
225C23—48no—
226C23—48no—
227C23—44no—
228C23—44no—
229C23—48no—
230C23—88no—
231C23—84no—
232C23—88no—
233C23—88no—
234C23—164no—
235C23—168no—
236C23—164no—
237C23—164no—
238C23C2×#9918yes—
239C23C2×#9914yes—
240C23C22×C22⋊C411yescharacteristic
241C23#51812yes—
242C23#53414yes—
243Q8—48no—
244D4—416no—
245D4—416no—
246D4—416no—
247D4—416no—
248D4—48no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—44no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—416no—
258C2×C4—48no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—88no—
265C2×C4—88no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—88no—
269C2×C4—816no—
270C2×C4—1616no—
271C22—22no—
272C22—28no—
273C22—24no—
274C22—22no—
275C22—24no—
276C22—24no—
277C22—28no—
278C22—44no—
279C22—48no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—48no—
284C22—84no—
285C22—84no—
286C22—88no—
287C22—84no—
288C22—84no—
289C22—88no—
290C22—164no—
291C22—168no—
292C22#230811yescharacteristic
293C22C22×#9912yes—
294C4—42no—
295C4—44no—
296C4—84no—
297C4—82no—
298C4—82no—
299C4—168no—
300C2—22no—
301C2—44no—
302C2—42no—
303C2—44no—
304C2—48no—
305C2#2174611yescharacteristic
306C2#2177712yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9499C2semidirect
4. #8746C2semidirect
5. #8543C2semidirect
7. C22×#123C4semidirect
11. #2406C4semidirect
14. #2308C4semidirect
17. #1819C22semidirect
21. #1304C22semidirect
30. C24×C4C2×C4semidirect
35. C2×#123C2×C4semidirect
37. C22×C4⋊C4C2×C4semidirect
40. C22×C22⋊C4C2×C4semidirect
41. C22×C22⋊C4C2×C4semidirect
52. #386C2×C4semidirect
57. #381C2×C4semidirect

Maximal subgroups

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

subgroupindex
#354462
#205852
#94992
#87462
#85432