Finite Group Databasebeta

← #6532208

Subgroups — #6532208

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35335C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic
3#27622C214yesmaximal
4#27619C214yesmaximal
5#26291C214yesmaximal, complemented
6#26290C214yesmaximal, complemented
7#26289C214yesmaximal, complemented
8#26288C214yesmaximal, complemented
9#21838C212yesmaximal
10#21823C211yesmaximal, characteristic
11#21364C212yesmaximal
12C2×#533C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#432C2212yes—
15#2859—21no—
16#2859—21no—
17#2660—22no—
18#2584C2214yes—
19#2584C2214yes—
20#2583—24no—
21#2382—21no—
22#2382—21no—
23#2365—22no—
24#2355C2214yes—
25Q8×C4⋊C4C2214yes—
26#2323—24no—
27#2317C2212yes—
28#2317C2212yes—
29#2314C2214yes—
30#2285C2212yes—
31C4×#117C2211yescharacteristic
32#2101C2214yes—
33#2101C2214yes—
34#2100C2214yes—
35#2099C2214yes—
36#2085C2214yes—
37#2085C2214yes—
38#2084C2214yes—
39#2083C2214yes—
40#1992C2214yes—
41#1992C2214yes—
42#1989C2214yes—
43#1989C2214yes—
44#1985C2214yes—
45#1985C2214yes—
46#1948C2212yes—
47#1948C2212yes—
48#1947C2212yes—
49#1947C2212yes—
50#1921C2214yes—
51#1921C2214yes—
52#1920C2214yes—
53#1920C2214yes—
54#1919C2214yes—
55#1919C2214yes—
56#1918C2214yes—
57#1918C2214yes—
58#1854C2214yes—
59#1854C2214yes—
60#1824C2214yes—
61#1823C2214yes—
62#1818C2214yes—
63#1776C2214yes—
64#1776C2214yes—
65#533—28no—
66C2×C4⋊Q8D412yes—
67C2×#125—21no—
68C2×#125—21no—
69C2×#125—28no—
70C2×#125D411yescharacteristic
71C2×#125D411yescharacteristic
72C2×#125C2314yes—
73C2×C22⋊Q8D414yes—
74C2×C4×Q8—24no—
75C2×C4×Q8C2314yes—
76C2×#117—22no—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2312yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C42C2311yescharacteristic
83#432—28no—
84C2×#107C2314yes—
85C2×#107C2314yes—
86C2×#106C2314yes—
87C2×#106C2314yes—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C22⋊C8C2314yes—
91C2×#97C2314yes—
92#398—24no—
93#398—24no—
94#395—24no—
95#395—24no—
96#391—24no—
97#389—24no—
98#389—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389C2314yes—
105#389C2314yes—
106#389C2314yes—
107#389C2314yes—
108#388—24no—
109#388—24no—
110#387—24no—
111#387—24no—
112#387—24no—
113#387—24no—
114#385—24no—
115#385—24no—
116#384—22no—
117#384—22no—
118#384C2314yes—
119#384C2312yes—
120#384C2312yes—
121#382—22no—
122#382—22no—
123#382—22no—
124#382C2314yes—
125#380—22no—
126#380C2314yes—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4C2314yes—
130C4×C4⋊C4C2314yes—
131C4×C4⋊C4C2312yes—
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2314yes—
134C4×C22⋊C4—22no—
135#376—21no—
136C43—21no—
137#340C2314yes—
138#340C2314yes—
139#340C2314yes—
140#340C2314yes—
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#340C2314yes—
146#340C2314yes—
147#340C2314yes—
148#340C2314yes—
149#336C2314yes—
150#336C2314yes—
151#336C2314yes—
152#336C2314yes—
153C22×Q8C2×D414yes—
154C23×C4—22no—
155C23×C4C2×D412yes—
156C23×C4C2411yescharacteristic
157C22×C8C2414yes—
158C22×C8C2414yes—
159C4⋊Q8—28no—
160#125—24no—
161#125—24no—
162#125—28no—
163#125—28no—
164#125—44no—
165#125—44no—
166C22⋊Q8—216no—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—88no—
170#117—28no—
171#117—22no—
172#117—24no—
173#117—22no—
174#117—84no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C4⋊C4C2411yescharacteristic
199C2×C4⋊C4C2412yes—
200C2×C4⋊C4C2411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4C2×D414yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4118yes—
215C2×C42C2414yes—
216C2×C42C2412yes—
217#107—28no—
218#107—28no—
219#106—28no—
220#106—28no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C22⋊C8—28no—
224#97—28no—
225#95—22no—
226#95—22no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237C24C22×D411yescharacteristic
238C2×Q8—28no—
239C2×Q8—44no—
240C2×Q8—44no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4#12318yes—
256C22×C4C22×D414yes—
257C22×C4C22×D414yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D412yes—
260C22×C4C22×D412yes—
261C22×C4C2×#4114yes—
262C22×C4C2511yescharacteristic
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—28no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C42−1+412yes—
296C4⋊C42−1+414yes—
297C4⋊C42−1+412yes—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—28no—
301C22⋊C4—84no—
302C42—24no—
303C42—24no—
304C42—28no—
305C42—48no—
306C42—82no—
307C42—82no—
308C42—84no—
309C42—84no—
310C42#12314yes—
311C42#12314yes—
312C42C2×#4114yes—
313C42C2×#4114yes—
314C23—22no—
315C23—22no—
316C23—21no—
317C23—21no—
318C23C2×#12312yes—
319C23C23×D411yescharacteristic
320Q8—416no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—88no—
353C2×C4C2×#12312yes—
354C2×C4C2×#12314yes—
355C2×C4C2×#12314yes—
356C2×C4C22×#4112yes—
357C2×C4C2×2−1+411yescharacteristic
358C2×C4C2×2−1+411yescharacteristic
359C2×C4C2×2−1+412yes—
360C8—44no—
361C8—44no—
362C22—22no—
363C22—24no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22C22×#12311yescharacteristic
368C22#348611yescharacteristic
369C22#348611yescharacteristic
370C22#358512yes—
371C22#358612yes—
372C4—22no—
373C4—24no—
374C4—24no—
375C4—44no—
376C4—44no—
377C4—42no—
378C4—42no—
379C4—44no—
380C4—84no—
381C4—84no—
382C4—84no—
383C4#350912yes—
384C4#350912yes—
385C2—42no—
386C2#3567411yescharacteristic
387C2#6051712yes—
388C2#6187611yescharacteristic
389C2#6187711yescharacteristic
390C2#6187812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26291C2semidirect
6. #26290C2semidirect
7. #26289C2semidirect
8. #26288C2semidirect

Maximal subgroups

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

subgroupindex
#353352
#343742
#276222
#276192
#262912
#262902
#262892
#262882
#218382
#218232
#213642