Finite Group Databasebeta

← #6386217

Subgroups — #6386217

Order 512.

2848 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2C2×#2985C212yesmaximal, complemented
3#22511C218yesmaximal, complemented
4#22509C218yesmaximal, complemented
5#21824C214yesmaximal, complemented
6#21823C211yesmaximal, characteristic, complemented
7#21819C211yesmaximal, characteristic, complemented
8#20717C212yesmaximal
9#20716C214yesmaximal
10#3458—216no—
11C2×#544C414yes—
12C2×#544C414yes—
13C2×#544C414yes—
14C2×#543C414yes—
15C2×#533C2211yescharacteristic
16C2×#533C2212yes—
17C2×#533C2218yes—
18C2×#533C2211yescharacteristic
19C22×#117C2212yes—
20C22×#117C2211yescharacteristic
21#2985—216no—
22C2×#446C2218yescomplemented
23C2×#446C2214yescomplemented
24C2×C8×Q8C2214yescomplemented
25C2×#433C2218yes—
26C2×#432C2212yes—
27C2×#431C2212yes—
28#1985C2218yescomplemented
29#1985C2218yescomplemented
30#1985C2218yescomplemented
31#1984C2214yescomplemented
32#1984C2214yescomplemented
33#1984C2218yescomplemented
34#1984C2218yescomplemented
35#1818C2214yes—
36#1818C2214yes—
37#1817C2214yes—
38#1798C2214yes—
39#1798C2214yes—
40#1798C2218yes—
41#1797C2218yes—
42#1779C2214yes—
43#1750C2218yes—
44#1750C2214yes—
45#1750C2214yes—
46#1749C2214yes—
47C4×C22⋊C8C2214yes—
48C24×C4C2311yescharacteristic
49#544—232no—
50#544—232no—
51#544—232no—
52#543—232no—
53#533—232no—
54#533—232no—
55#533—216no—
56#533—28no—
57#533D4116yes—
58C2×C4⋊Q8C2×C418yes—
59C2×C4⋊Q8C2×C418yes—
60C2×C4⋊Q8C2×C414yes—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×#125C2×C418yes—
64C2×#125C2×C418yes—
65C2×#125C2×C414yes—
66C2×#125C2×C412yes—
67C2×#125C2×C412yes—
68C2×C22⋊Q8C2×C418yes—
69C2×C22⋊Q8C2×C414yes—
70C2×C22⋊Q8C2×C418yes—
71C2×C22⋊Q8C2×C418yes—
72C2×C22⋊Q8C2×C418yes—
73C2×C22⋊Q8C2×C418yes—
74C2×C22⋊Q8C2×C414yes—
75C2×C4×Q8C2314yes—
76C2×C4×Q8C2318yes—
77C2×C4×Q8C2314yes—
78C2×#117—216no—
79C2×#117—28no—
80C2×#117C2×C414yes—
81C2×#117C2×C414yes—
82C2×#117C2×C412yes—
83C2×#117C2×C412yes—
84C2×#117C2314yes—
85C2×#117C2314yes—
86C2×#117C2312yes—
87C2×#117C2312yes—
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C414yes—
90C22×C22⋊C4C2×C412yes—
91C22×C22⋊C4C2×C414yes—
92C22×C42C2314yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#446—216no—
96#446—232no—
97C8×Q8—216no—
98#433—232no—
99#432—28no—
100#431—28no—
101C2×C4⋊C8C2318yes—
102C2×C4⋊C8C2318yes—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×C22⋊C8C2318yes—
106C2×C22⋊C8C2318yes—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×#97C2314yes—
110C2×#97C2318yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2312yes—
113C2×C4×C8C2318yes—
114#336C2318yes—
115#336C2318yes—
116#336C2318yes—
117#336C2318yes—
118#336C2314yes—
119#336C2314yes—
120#336C2314yes—
121#336C2314yes—
122C25C22×C411yescharacteristic
123C22×Q8C22×C418yes—
124C22×Q8C22×C414yes—
125C22×Q8C22×C414yes—
126C23×C4—24no—
127C23×C4—28no—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C412yes—
131C23×C4C22×C418yes—
132C23×C4C22×C412yes—
133C23×C4C2411yescharacteristic
134C23×C4C2412yes—
135C22×C8C2414yes—
136C22×C8C2414yes—
137C22×C8C2418yes—
138C4⋊Q8—232no—
139C4⋊Q8—232no—
140C4⋊Q8—216no—
141C4⋊Q8—28no—
142C4⋊Q8C22⋊C4116yes—
143#125—232no—
144#125—232no—
145#125—216no—
146#125—28no—
147#125C22⋊C4116yes—
148C22⋊Q8—232no—
149C22⋊Q8—216no—
150C22⋊Q8—232no—
151C22⋊Q8—232no—
152C22⋊Q8—232no—
153C22⋊Q8—232no—
154C22⋊Q8C22⋊C4132yes—
155C4×Q8—28no—
156C4×Q8—232no—
157C4×Q8—28no—
158C4×Q8C2×D4116yes—
159C4×Q8C2×D4116yes—
160#117—28no—
161#117—216no—
162#117—216no—
163#117—28no—
164#117—28no—
165#117—28no—
166#117—216no—
167#117—28no—
168#117C2×D4116yes—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—216no—
171C2×C4⋊C4C22×C414yes—
172C2×C4⋊C4C22×C418yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C22×C418yes—
176C2×C4⋊C4C22×C418yes—
177C2×C4⋊C4C22×C418yes—
178C2×C4⋊C4C22×C418yes—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C22×C414yes—
182C2×C4⋊C4C22×C418yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—216no—
185C2×C22⋊C4C22×C414yes—
186C2×C22⋊C4C22×C418yes—
187C2×C22⋊C4C22×C414yes—
188C2×C22⋊C4C22×C414yes—
189C2×C22⋊C4C22×C414yes—
190C2×C42—24no—
191C2×C42—216no—
192C2×C42C22×C412yes—
193C2×C42C22×C418yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D418yes—
197C2×C42C2412yes—
198C2×C42C2414yes—
199C2×C42C2412yes—
200C2×C42C2414yes—
201C4⋊C8—28no—
202C4⋊C8—28no—
203C4⋊C8—216no—
204C4⋊C8—216no—
205C22⋊C8—28no—
206C22⋊C8—28no—
207C22⋊C8—216no—
208C22⋊C8—216no—
209#97—28no—
210#97—216no—
211C4×C8—24no—
212C4×C8—24no—
213C4×C8—216no—
214C24—22no—
215C24—24no—
216C24—44no—
217C24C23×C411yescharacteristic
218C24C23×C412yes—
219C2×Q8—216no—
220C2×Q8—28no—
221C2×Q8—48no—
222C2×Q8—48no—
223C2×Q8—416no—
224C2×Q8C2×C22⋊C4116yes—
225C22×C4—232no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—216no—
236C22×C4—44no—
237C22×C4—416no—
238C22×C4—44no—
239C22×C4C2×C22⋊C418yes—
240C22×C4C23×C418yes—
241C22×C4C23×C414yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C23×C418yes—
244C22×C4C23×C412yes—
245C22×C4C23×C412yes—
246C22×C4C23×C412yes—
247C22×C4C22×D414yes—
248C22×C42−1+418yes—
249C22×C4C2511yescharacteristic
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—28no—
253C2×C8—28no—
254C2×C8—28no—
255C2×C8—28no—
256C2×C8—28no—
257C4⋊C4—216no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—216no—
262C4⋊C4—216no—
263C4⋊C4—216no—
264C4⋊C4—216no—
265C4⋊C4—28no—
266C4⋊C4C2×C22⋊C4116yes—
267C4⋊C4C2×C22⋊C4116yes—
268C4⋊C4C2×C22⋊C4132yes—
269C22⋊C4—28no—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—28no—
273C22⋊C4C2×C22⋊C4116yes—
274C42—216no—
275C42—24no—
276C42—24no—
277C42—28no—
278C42C2×C22⋊C414yes—
279C42C2×C22⋊C414yes—
280C42C22×D414yes—
281C42C22×D4116yes—
282C42C22×D414yes—
283C23—28no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—28no—
288C23—24no—
289C23—88no—
290C23—84no—
291C23C22×C22⋊C412yes—
292C23#51914yes—
293C23C24×C411yescharacteristic
294Q8—416no—
295Q8—88no—
296Q8—816no—
297C2×C4—216no—
298C2×C4—28no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—216no—
305C2×C4—28no—
306C2×C4—216no—
307C2×C4—28no—
308C2×C4—24no—
309C2×C4—216no—
310C2×C4—28no—
311C2×C4—48no—
312C2×C4—816no—
313C2×C4—84no—
314C2×C4C22×C22⋊C412yes—
315C2×C4C22×C22⋊C4116yes—
316C2×C4C22×C22⋊C414yes—
317C2×C4C22×C22⋊C414yes—
318C2×C4#51914yes—
319C2×C4C23×D412yes—
320C2×C4C2×2−1+414yes—
321C8—44no—
322C8—44no—
323C8—48no—
324C22—28no—
325C22—22no—
326C22—24no—
327C22—24no—
328C22—44no—
329C22—44no—
330C22—44no—
331C22—88no—
332C22C23×C22⋊C411yescharacteristic
333C22C2×#51912yes—
334C22#357314yes—
335C4—22no—
336C4—24no—
337C4—24no—
338C4—24no—
339C4—216no—
340C4—48no—
341C4—44no—
342C4#348614yes—
343C2—44no—
344C2—42no—
345C2#3417311yescharacteristic
346C2#6006512yes—
347C2#6025214yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2985C2semidirect
3. #22511C2semidirect
4. #22509C2semidirect
5. #21824C2semidirect
6. #21823C2semidirect
7. #21819C2semidirect
22. C2×#446C22semidirect
23. C2×#446C22semidirect
24. C2×C8×Q8C22semidirect
28. #1985C22semidirect
29. #1985C22semidirect
30. #1985C22semidirect
31. #1984C22semidirect
32. #1984C22semidirect
33. #1984C22semidirect
34. #1984C22semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#29852
#225112
#225092
#218242
#218232
#218192
#207172
#207162