Finite Group Databasebeta

← #382601

Subgroups — #382601

Order 512.

1465 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34248C211yesmaximal, characteristic
2C2×#1828C212yesmaximal
3#20633C212yesmaximal
4#11222C214yesmaximal, complemented
5#11205C214yesmaximal, complemented
6#10761C212yesmaximal
7C2×#518C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C22×#99C2214yes—
11#2349—24no—
12D4×C4⋊C4—24no—
13#2317C412yescomplemented
14#2315C412yescomplemented
15#2315C2212yes—
16#2315C2214yes—
17#2296C412yescomplemented
18C2×#385C412yescomplemented
19C2×#385C2212yes—
20#1828—28no—
21#1799—24no—
22#1799—24no—
23C2×#342C2214yes—
24C2×#342C2218yes—
25C2×#342C2214yes—
26C22⋊#95C2214yes—
27C23×D4C2311yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4C2311yescharacteristic
30#518—24no—
31#518—44no—
32#518D414yes—
33#518Q814yes—
34C2×C4×D4—24no—
35C2×C4×D4—24no—
36C2×C4×D4—44no—
37C2×C4×D4D414yes—
38C2×C4×D4D414yes—
39C2×#117C2×C412yes—
40C2×#117C2×C412yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2×C411yescharacteristic
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2318yes—
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2312yes—
51C22×C42C2×C411yescharacteristic
52C2×#99—28no—
53C2×#99—28no—
54C2×#99—28no—
55#389—22no—
56#389—22no—
57#389—24no—
58#385—22no—
59#385—22no—
60#385—22no—
61#385—22no—
62#385—24no—
63#385—24no—
64#385—48no—
65#385C2×C414yes—
66#385C2×C414yes—
67#385C2×C414yes—
68#385C2×C414yes—
69#384—22no—
70#382—22no—
71#380—24no—
72#380—24no—
73#380—24no—
74#380—24no—
75C4×C4⋊C4—24no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—44no—
79#342—28no—
80#342—216no—
81#342—44no—
82C25C22×C411yescharacteristic
83C25C2411yescharacteristic
84C22×D4—24no—
85C22×D4—24no—
86C22×D4C2×D414yes—
87C22×D4C2×D412yes—
88C22×D4C2×Q812yes—
89C22×D4#4114yes—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—28no—
96C23×C4—42no—
97C23×C4—82no—
98C23×C4—82no—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C412yes—
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C4×D4—216no—
109C4×D4—216no—
110C4×D4—48no—
111C4×D4—48no—
112C4×D4—416no—
113#117—24no—
114#117—44no—
115#117C4⋊C418yescomplemented
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—84no—
126C2×C4⋊C4—84no—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C412yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4C22⋊C414yes—
147C2×C22⋊C4C4⋊C418yescomplemented
148C2×C22⋊C4C22×C414yes—
149C2×C22⋊C4C22×C414yes—
150C2×C22⋊C4C2×D414yes—
151C2×C22⋊C4C2×D414yes—
152C2×C22⋊C4C2×Q814yes—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—42no—
156C2×C42C22⋊C414yes—
157#99—416no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—44no—
163#95—44no—
164#95—88no—
165C24—22no—
166C24—22no—
167C24—81no—
168C24—81no—
169C24—81no—
170C24—81no—
171C24C2×C22⋊C412yes—
172C24C2×C4⋊C412yes—
173C24#11712yes—
174C24C23×C411yescharacteristic
175C24C22×D412yes—
176C24C22×D411yescharacteristic
177C24C22×Q811yescharacteristic
178C24C2×#4112yes—
179C24H(5,𝔽2)14yes—
180C242−1+414yes—
181C2×D4—28no—
182C2×D4—28no—
183C2×D4—28no—
184C2×D4—216no—
185C2×D4—28no—
186C2×D4#12118yes—
187C2×D4C22⋊Q818yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—28no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—48no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—48no—
214C22×C4—48no—
215C22×C4—84no—
216C22×C4—81no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—84no—
224C22×C4—82no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—84no—
231C22×C4—88no—
232C22×C4—88no—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C414yes—
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C4⋊C412yes—
237C22×C4#11712yes—
238C4⋊C4—28no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—48no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—164no—
247C4⋊C4—164no—
248C22⋊C4—24no—
249C22⋊C4—28no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—28no—
253C22⋊C4—28no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4C2×C4⋊C418yes—
262C42—28no—
263C42—44no—
264C42—44no—
265C23—22no—
266C23—22no—
267C23—24no—
268C23—24no—
269C23—42no—
270C23—44no—
271C23—41no—
272C23—42no—
273C23—44no—
274C23—41no—
275C23—42no—
276C23—42no—
277C23—82no—
278C23—82no—
279C23—81no—
280C23—81no—
281C23—82no—
282C23—84no—
283C23—82no—
284C23—82no—
285C23—84no—
286C23#38014yes—
287C23C22×C22⋊C411yescharacteristic
288C23C22×C4⋊C411yescharacteristic
289C23C2×#11711yescharacteristic
290C23#51812yes—
291C23#51912yes—
292C23C2×#12112yes—
293C23C2×C22⋊Q812yes—
294C23#53414yes—
295C23#53614yes—
296C23#53714yes—
297C23#53918yes—
298C23#54414yes—
299D4—216no—
300D4—216no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—88no—
329C2×C4—88no—
330C2×C4—88no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—88no—
334C2×C4—88no—
335C2×C4—816no—
336C2×C4—88no—
337C2×C4—162no—
338C2×C4—162no—
339C2×C4—168no—
340C2×C4#38014yes—
341C22—21no—
342C22—22no—
343C22—24no—
344C22—21no—
345C22—24no—
346C22—22no—
347C22—24no—
348C22—42no—
349C22—42no—
350C22—48no—
351C22—44no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—84no—
362C22—84no—
363C22C2×#38011yescharacteristic
364C22#230811yescharacteristic
365C22#231011yescharacteristic
366C22#231312yes—
367C22#231612yes—
368C22#261814yes—
369C22#262714yes—
370C4—24no—
371C4—48no—
372C4—42no—
373C4—44no—
374C4—44no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—168no—
379C2—21no—
380C2—24no—
381C2—21no—
382C2—22no—
383C2—22no—
384C2—24no—
385C2—44no—
386C2#1433911yescharacteristic
387C2#2177611yescharacteristic
388C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11222C2semidirect
5. #11205C2semidirect
13. #2317C4semidirect
14. #2315C4semidirect
17. #2296C4semidirect
18. C2×#385C4semidirect
115. #117C4⋊C4semidirect
147. C2×C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342482
C2×#18282
#206332
#112222
#112052
#107612