Finite Group Databasebeta

← #359970

Subgroups — #359970

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3C2×#1820C211yesmaximal, characteristic
4#14684C212yesmaximal
5#11222C214yesmaximal
6#10918C214yesmaximal
7#10736C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2343—28no—
12#2315C2212yes—
13#2313—22no—
14#2296—24no—
15#2296C418yescomplemented
16C2×#389—22no—
17C2×#385C2212yes—
18C2×#385C2212yes—
19#2283—22no—
20C2×#381C2212yes—
21C2×#381C2212yes—
22#2273C2212yes—
23C2×C4×C4⋊C4—22no—
24#1820—24no—
25C2×#342C2212yes—
26C2×#342C2218yes—
27C2×#342C2212yes—
28#1434—28no—
29C22⋊#95C2218yes—
30C24×C4C2312yes—
31C24×C4C2311yescharacteristic
32C2×#117—24no—
33C2×#117—22no—
34C2×#117—24no—
35C2×#117—28no—
36C2×#117C2×C414yes—
37C2×#117C2×C414yes—
38C2×#117D414yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C22⋊C4C2314yes—
44C22×C22⋊C4C2314yes—
45C22×C22⋊C4C2311yescharacteristic
46C22×C22⋊C4C2311yescharacteristic
47C22×C22⋊C4C2312yes—
48C22×C42—21no—
49C22×C42—24no—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—22no—
53C22×C42—24no—
54C22×C42C2×C414yes—
55C22×C42D412yes—
56#389—216no—
57#387—216no—
58#385—28no—
59#385—24no—
60#385—24no—
61#385C2×C418yes—
62#385C2×C418yes—
63#383—24no—
64#382—24no—
65#381—22no—
66#381—24no—
67#381—22no—
68#381—24no—
69#381—24no—
70#380—24no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—216no—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—28no—
75#376—24no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—216no—
79C2×#95—24no—
80C2×#95D418yes—
81#342—24no—
82#342—24no—
83#342—216no—
84C25C2411yescharacteristic
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—24no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—48no—
94C23×C4C22×C412yes—
95C23×C4C22×C414yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D414yes—
100C23×C4C2×D418yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4#4114yes—
104#117—216no—
105#117—216no—
106#117—216no—
107#117—216no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—48no—
117C2×C4⋊C4C22⋊C418yes—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4C22×C412yes—
133C2×C22⋊C4C22×C412yes—
134C2×C22⋊C4C22×C414yes—
135C2×C22⋊C4C2×D412yes—
136C2×C22⋊C4C2×D412yes—
137C2×C22⋊C4#4118yes—
138C2×C42—28no—
139C2×C42—24no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—24no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—216no—
146C2×C42—28no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—48no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42C22⋊C418yes—
153#95—28no—
154#95—28no—
155#95—28no—
156#95—28no—
157#95—28no—
158#95—28no—
159#95—432no—
160#95—48no—
161C24—44no—
162C24C23×C411yescharacteristic
163C24C22×D411yescharacteristic
164C24C22×D412yes—
165C24C22×D411yescharacteristic
166C24C2×#4114yes—
167C24C2×#4112yes—
168C242−1+414yes—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—216no—
178C22×C4—216no—
179C22×C4—216no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—28no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—28no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—28no—
190C22×C4—216no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—416no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—48no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—416no—
204C22×C4—416no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C414yes—
210C22×C4C2×C22⋊C414yes—
211C22×C4C4×D418yes—
212C22×C4#12012yes—
213C22×C4#12012yes—
214C22×C4#12118yes—
215C22×C4#12312yes—
216C22×C4#12312yes—
217C4⋊C4—216no—
218C4⋊C4—216no—
219C4⋊C4—416no—
220C4⋊C4—416no—
221C4⋊C4—88no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—28no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—48no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—48no—
234C22⋊C4#11718yes—
235C42—216no—
236C42—28no—
237C42—28no—
238C42—48no—
239C42—48no—
240C42—88no—
241C23—24no—
242C23—28no—
243C23—42no—
244C23—42no—
245C23—44no—
246C23—48no—
247C23—48no—
248C23C22×C22⋊C411yescharacteristic
249C23C2×#11712yes—
250C23C2×C4×D412yes—
251C23#51912yes—
252C23C2×#12011yescharacteristic
253C23C2×#12112yes—
254C23C2×#12311yescharacteristic
255C23#52512yes—
256C23#52512yes—
257C23#53612yes—
258C23#53612yes—
259C23#54718yes—
260C23#55218yes—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—216no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—216no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—28no—
272C2×C4—216no—
273C2×C4—28no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—416no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—416no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—416no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—816no—
294C2×C4—816no—
295C2×C4—88no—
296C2×C4—84no—
297C2×C4#38614yes—
298C2×C4#38614yes—
299C22—24no—
300C22—216no—
301C22—24no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—48no—
311C22#228011yescharacteristic
312C22C2×#38611yescharacteristic
313C22#231011yescharacteristic
314C22#234414yes—
315C22#240612yes—
316C22#241314yes—
317C22#248412yes—
318C22#302514yes—
319C4—24no—
320C4—24no—
321C4—48no—
322C4—48no—
323C4—44no—
324C4—44no—
325C4—88no—
326C4—164no—
327C2—21no—
328C2—24no—
329C2—21no—
330C2—24no—
331C2—22no—
332C2#1387011yescharacteristic
333C2#2095912yes—
334C2#2411314yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2296C4semidirect

Maximal subgroups

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

subgroupindex
#341772
C2×#22962
C2×#18202
#146842
#112222
#109182
#107362