Finite Group Databasebeta

← #215161

Subgroups — #215161

Order 512.

1376 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1376 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C2×#2561C211yesmaximal, characteristic
2#21904C212yesmaximal, complemented
3#9652C214yesmaximal, complemented
4#9651C214yesmaximal, complemented
5#8308C212yesmaximal, complemented
6#8298C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C23×C22⋊C4C2211yescharacteristic
9#2561—24no—
10#2561—24no—
11C2×#397C412yes—
12C2×#393C412yes—
13C2×#388C2214yes—
14C2×#381C412yes—
15C2×#379C412yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1924—24no—
18#1923—24no—
19#1797C2214yescomplemented
20#1779C2214yes—
21#1749C2212yescomplemented
22C4×C22⋊C8C2212yescomplemented
23C2×#324C2212yescomplemented
24C2×#324C2214yescomplemented
25C2×#324C2212yescomplemented
26C2×#323C2212yes—
27C2×#323C2214yes—
28C2×#323C2212yes—
29C26C2×C411yescharacteristic
30C23×Q8C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C2×C22⋊Q8—24no—
34C2×C22⋊Q8—42no—
35C2×C22⋊Q8—42no—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2312yes—
38C22×C22⋊C4—24no—
39C22×C22⋊C4—42no—
40C22×C22⋊C4—42no—
41C22×C22⋊C4C2×C414yes—
42C22×C22⋊C4C2×C412yes—
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C414yes—
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2312yes—
47C22×C42C2312yes—
48C2×C4⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8C2314yes—
52C2×C22⋊C8C2314yes—
53C2×#97—22no—
54C2×C4×C8—22no—
55#397—22no—
56#397—24no—
57#397—22no—
58#393—22no—
59#393—24no—
60#393—22no—
61#388—24no—
62#388—24no—
63#388—24no—
64#388—24no—
65#381—24no—
66#381—22no—
67#381—22no—
68#379—24no—
69#379—22no—
70#379—22no—
71C4×C22⋊C4D414yes—
72C4×C22⋊C4D414yes—
73C2×#95C2×C414yes—
74C2×#95C2×C414yes—
75C2×#95C2×C412yes—
76#336—24no—
77#336—22no—
78#336—22no—
79#324—42no—
80#324—44no—
81#324—42no—
82#323—42no—
83#323—44no—
84#323—42no—
85C25—22no—
86C25—24no—
87C25—41no—
88C25—41no—
89C25—42no—
90C25—44no—
91C25—82no—
92C25C22×C411yescharacteristic
93C25C22×C412yes—
94C22×Q8—22no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—82no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4—44no—
104C23×C4—82no—
105C23×C4C22×C412yes—
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C414yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D414yes—
111C23×C4C2411yescharacteristic
112C22×C8—24no—
113C22×C8—24no—
114C22⋊Q8—416no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—416no—
142C2×C22⋊C4—84no—
143C2×C22⋊C4C22⋊C414yes—
144C2×C22⋊C4C22⋊C412yes—
145C2×C22⋊C4C22⋊C412yes—
146C2×C42—82no—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×D414yes—
150C4⋊C8—84no—
151C22⋊C8—24no—
152C22⋊C8—24no—
153C22⋊C8—24no—
154C22⋊C8—24no—
155C22⋊C8—84no—
156C22⋊C8—84no—
157#97—82no—
158C4×C8—82no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95C22⋊C412yes—
164#95C22⋊C414yes—
165#95C22⋊C412yes—
166C24—21no—
167C24—21no—
168C24—24no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—22no—
175C24—22no—
176C24—22no—
177C24—42no—
178C24—48no—
179C24—44no—
180C24—44no—
181C24—48no—
182C24—44no—
183C24—44no—
184C24—44no—
185C24—44no—
186C24—44no—
187C24—84no—
188C24—84no—
189C24—84no—
190C24—816no—
191C24—816no—
192C24—84no—
193C24—84no—
194C24C2×C22⋊C412yes—
195C24#13114yes—
196C24C23×C411yescharacteristic
197C2×Q8—44no—
198C2×Q8—84no—
199C2×Q8—84no—
200C2×Q8—84no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4C2×C22⋊C414yes—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C22×D412yes—
236C22×C4H(5,𝔽2)12yes—
237C22×C4H(5,𝔽2)12yes—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C22⋊C4—28no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—48no—
251C22⋊C4—48no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—84no—
259C22⋊C4—84no—
260C42—42no—
261C42—42no—
262C42—44no—
263C23—22no—
264C23—22no—
265C23—24no—
266C23—24no—
267C23—22no—
268C23—24no—
269C23—24no—
270C23—22no—
271C23—24no—
272C23—24no—
273C23—24no—
274C23—48no—
275C23—48no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—48no—
280C23—42no—
281C23—41no—
282C23—41no—
283C23—416no—
284C23—42no—
285C23—42no—
286C23—44no—
287C23—44no—
288C23—42no—
289C23—42no—
290C23—816no—
291C23—84no—
292C23—84no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23—82no—
297C23—88no—
298C23—816no—
299C23—816no—
300C23—832no—
301C23—816no—
302C23—88no—
303C23#41314yes—
304C23C22×C22⋊C411yescharacteristic
305C23#51811yescharacteristic
306C23#51811yescharacteristic
307C23C2×#13112yes—
308C23#56812yes—
309Q8—88no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—88no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—88no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#53514yes—
342C8—84no—
343C8—84no—
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—28no—
351C22—24no—
352C22—48no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—48no—
359C22—44no—
360C22—84no—
361C22—84no—
362C22—84no—
363C22—84no—
364C22—832no—
365C22—84no—
366C22—82no—
367C22—82no—
368C22#230911yescharacteristic
369C22#288012yes—
370C22#288312yes—
371C22C2×#41312yes—
372C22#297612yes—
373C22#297711yescharacteristic
374C22#297811yescharacteristic
375C4—24no—
376C4—44no—
377C4—44no—
378C4—42no—
379C4—82no—
380C4—82no—
381C4—82no—
382C2—22no—
383C2—21no—
384C2—21no—
385C2—44no—
386C2—84no—
387C2#1742214yes—
388C2#2179211yescharacteristic
389C2#2179511yescharacteristic
390C2#2179811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21904C2semidirect
3. #9652C2semidirect
4. #9651C2semidirect
5. #8308C2semidirect
19. #1797C22semidirect
21. #1749C22semidirect
22. C4×C22⋊C8C22semidirect
23. C2×#324C22semidirect
24. C2×#324C22semidirect
25. C2×#324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
#219042
#96522
#96512
#83082
#82982