Finite Group Databasebeta

← #6528503

Subgroups — #6528503

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic
3#29112C212yesmaximal
4#29106C212yesmaximal
5#27966C212yesmaximal
6#27964C212yesmaximal
7#27831C218yesmaximal, complemented
8#27745C218yesmaximal, complemented
9#21838C212yesmaximal
10#20740C212yesmaximal
11#20534C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2212yes—
15#2851C2212yes—
16#2851C2212yes—
17#2646—22no—
18#2571C2214yes—
19#2570—24no—
20#2565—24no—
21#2565C2218yes—
22#2552C2214yes—
23C2×#398—22no—
24#2317—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#387—22no—
28#2285C2212yes—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2101C2218yes—
33#2100C2218yes—
34#2074C2214yes—
35#2074C2214yes—
36#2026C2214yes—
37#2026C2214yes—
38#1992C2214yes—
39#1992C2214yes—
40#1992C2214yes—
41#1991C2214yes—
42#1989C2218yes—
43#1935C2218yes—
44#1933C2218yes—
45#1836C2214yes—
46#1835C2214yes—
47#1824C2214yes—
48#1823C2214yes—
49#1823C2214yes—
50#1803C2214yes—
51#1803C2214yes—
52#1787C2214yes—
53#1771C2218yes—
54#1763C2218yes—
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57C23×Q8C2311yescharacteristic
58C2×#125—22no—
59C2×#125—28no—
60C2×#125C2314yes—
61C2×#125C2312yes—
62C2×#125C2312yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—28no—
65C2×C4×Q8C2314yes—
66C2×#117—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—28no—
80C2×C4⋊C8C2314yes—
81C2×#103C2318yes—
82C2×#103C2318yes—
83C2×C22⋊C8C2314yes—
84C2×#97C2314yes—
85C2×#97C2314yes—
86C2×C4×C8C2314yes—
87#400—24no—
88#400—24no—
89#398—28no—
90#398—28no—
91#398—24no—
92#398—24no—
93#391—22no—
94#391—22no—
95#391C2314yes—
96#391C2314yes—
97#389—24no—
98#389—24no—
99#389C2314yes—
100#389C2314yes—
101#387—24no—
102#387—24no—
103#387—28no—
104#387—28no—
105#385—24no—
106#385—24no—
107#384—24no—
108#384—22no—
109#384C2314yes—
110#384C2312yes—
111#384C2312yes—
112#382—24no—
113#382—22no—
114#382—22no—
115#382C2314yes—
116#382C2312yes—
117#382C2312yes—
118#380—22no—
119#380C2314yes—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C4×C4⋊C4C2314yes—
123C4×C4⋊C4C2314yes—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D412yes—
128#340C2318yes—
129#340C2318yes—
130#340C2318yes—
131#340C2318yes—
132#340C2318yes—
133#340C2318yes—
134#336C2314yes—
135#336C2314yes—
136C22×Q8—24no—
137C22×Q8—21no—
138C22×Q8—24no—
139C22×Q8—42no—
140C22×Q8C2412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8C2418yes—
151#125—28no—
152#125—28no—
153#125—44no—
154#125—44no—
155#125—48no—
156C4×Q8—28no—
157C4×Q8—28no—
158C4×Q8—48no—
159#117—48no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2412yes—
186C2×C4⋊C4C2412yes—
187C2×C4⋊C4C2412yes—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2414yes—
190C2×C22⋊C4—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—44no—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42#4118yes—
200C2×C42C2412yes—
201C2×C42C2412yes—
202C2×C42C2412yes—
203C4⋊C8—28no—
204#103—216no—
205#103—48no—
206C22⋊C8—28no—
207#97—28no—
208#97—44no—
209C4×C8—44no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95C2×D414yes—
220#95C2×D414yes—
221C24C22×D411yescharacteristic
222C2×Q8—24no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—416no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q82−1+414yes—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4#12414yes—
256C22×C4#12414yes—
257C22×C4C22×D414yes—
258C22×C4C22×D414yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D412yes—
261C22×C4C22×D412yes—
262C22×C4C2×#4114yes—
263C22×C4C2511yescharacteristic
264C2×C8—28no—
265C2×C8—28no—
266C2×C8—28no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—416no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C22⋊C4—48no—
290C42—24no—
291C42—24no—
292C42—42no—
293C42—42no—
294C42—48no—
295C42—48no—
296C42#12418yes—
297C42C2×#4114yes—
298C42C2×#4114yes—
299C23—22no—
300C23—21no—
301C23—21no—
302C23C2×#12412yes—
303C23#57514yes—
304C23C23×D411yescharacteristic
305Q8—28no—
306Q8—48no—
307Q8—44no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—28no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4C2×#12414yes—
345C2×C4C2×#12412yes—
346C2×C4C2×#12414yes—
347C2×C4C22×#4112yes—
348C2×C4C2×2−1+412yes—
349C2×C4#58512yes—
350C8—48no—
351C22—24no—
352C22—22no—
353C22—24no—
354C22—22no—
355C22#314214yes—
356C22C22×#12411yescharacteristic
357C22#348511yescharacteristic
358C22#348611yescharacteristic
359C22C2×#57512yes—
360C22#358612yes—
361C4—22no—
362C4—24no—
363C4—24no—
364C4—44no—
365C4—44no—
366C4—44no—
367C4—84no—
368C4—84no—
369C4—82no—
370C4—82no—
371C4—82no—
372C4#349514yes—
373C2—24no—
374C2#3564411yescharacteristic
375C2C2×#314211yescharacteristic
376C2#6062011yescharacteristic
377C2#6171112yes—
378C2#6183212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #27831C2semidirect
8. #27745C2semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#25652
#291122
#291062
#279662
#279642
#278312
#277452
#218382
#207402
#205342