Finite Group Databasebeta

← #6386218

Subgroups — #6386218

Order 512.

2848 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2C2×#2987C211yesmaximal, characteristic, complemented
3C2×#2986C211yesmaximal, characteristic, complemented
4#22511C218yesmaximal, complemented
5#22509C218yesmaximal, complemented
6#21824C214yesmaximal, complemented
7#21821C212yesmaximal, complemented
8#20717C212yesmaximal
9#20716C214yesmaximal
10#3458—28no—
11#3458—28no—
12C2×#544C414yes—
13C2×#544C414yes—
14C2×#544C414yes—
15C2×#543C414yes—
16C2×#533C2211yescharacteristic
17C2×#533C2212yes—
18C2×#533C2218yes—
19C2×#533C2211yescharacteristic
20C22×#117C2212yes—
21C22×#117C2211yescharacteristic
22#2987—28no—
23#2986—28no—
24C2×#446C2214yescomplemented
25C2×#446C2218yescomplemented
26C2×C8×Q8C2214yescomplemented
27C2×#433C2214yes—
28C2×#433C2214yes—
29C2×#432C2212yes—
30C2×#431C2212yes—
31#1985C2214yescomplemented
32#1985C2214yescomplemented
33#1985C2218yescomplemented
34#1985C2218yescomplemented
35#1984C2218yescomplemented
36#1984C2218yescomplemented
37#1984C2218yescomplemented
38#1818C2218yes—
39#1817C2214yes—
40#1798C2218yes—
41#1798C2214yes—
42#1798C2214yes—
43#1797C2218yes—
44#1779C2214yes—
45#1750C2214yes—
46#1750C2214yes—
47#1750C2218yes—
48#1749C2214yes—
49C4×C22⋊C8C2214yes—
50C24×C4C2311yescharacteristic
51#544—232no—
52#544—232no—
53#544—232no—
54#543—232no—
55#533—232no—
56#533—232no—
57#533—28no—
58#533—28no—
59#533—24no—
60#533—24no—
61#533D418yes—
62#533D418yes—
63C2×C4⋊Q8C2×C418yes—
64C2×C4⋊Q8C2×C418yes—
65C2×C4⋊Q8C2×C414yes—
66C2×C4⋊Q8C2×C412yes—
67C2×C4⋊Q8C2×C412yes—
68C2×#125C2×C418yes—
69C2×#125C2×C418yes—
70C2×#125C2×C414yes—
71C2×#125C2×C412yes—
72C2×#125C2×C412yes—
73C2×C22⋊Q8C2×C418yes—
74C2×C22⋊Q8C2×C414yes—
75C2×C22⋊Q8C2×C418yes—
76C2×C22⋊Q8C2×C418yes—
77C2×C22⋊Q8C2×C418yes—
78C2×C22⋊Q8C2×C418yes—
79C2×C22⋊Q8C2×C414yes—
80C2×C4×Q8C2314yes—
81C2×C4×Q8C2318yes—
82C2×C4×Q8C2314yes—
83C2×#117—28no—
84C2×#117—28no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117C2×C414yes—
88C2×#117C2×C414yes—
89C2×#117C2×C412yes—
90C2×#117C2×C412yes—
91C2×#117C2314yes—
92C2×#117C2314yes—
93C2×#117C2312yes—
94C2×#117C2312yes—
95C22×C4⋊C4C2×C412yes—
96C22×C4⋊C4C2×C414yes—
97C22×C22⋊C4C2×C412yes—
98C22×C22⋊C4C2×C414yes—
99C22×C42C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#446—216no—
103#446—232no—
104C8×Q8—216no—
105#433—216no—
106#433—216no—
107#432—28no—
108#431—28no—
109C2×C4⋊C8C2318yes—
110C2×C4⋊C8C2318yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2314yes—
114C2×C22⋊C8C2318yes—
115C2×C22⋊C8C2318yes—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118C2×#97C2314yes—
119C2×#97C2314yes—
120C2×#97C2314yes—
121C2×C4×C8C2314yes—
122C2×C4×C8C2314yes—
123C2×C4×C8C2314yes—
124#336C2318yes—
125#336C2318yes—
126#336C2318yes—
127#336C2318yes—
128#336C2318yes—
129#336C2318yes—
130C25C22×C411yescharacteristic
131C22×Q8C22×C418yes—
132C22×Q8C22×C414yes—
133C22×Q8C22×C414yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—28no—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C22×C418yes—
141C23×C4C22×C412yes—
142C23×C4C2411yescharacteristic
143C23×C4C2412yes—
144C22×C8C2414yes—
145C22×C8C2418yes—
146C22×C8C2414yes—
147C4⋊Q8—232no—
148C4⋊Q8—232no—
149C4⋊Q8—216no—
150C4⋊Q8—28no—
151C4⋊Q8C22⋊C4116yes—
152#125—232no—
153#125—232no—
154#125—216no—
155#125—28no—
156#125C22⋊C4116yes—
157C22⋊Q8—232no—
158C22⋊Q8—216no—
159C22⋊Q8—232no—
160C22⋊Q8—232no—
161C22⋊Q8—232no—
162C22⋊Q8—232no—
163C22⋊Q8C22⋊C4132yes—
164C4×Q8—28no—
165C4×Q8—216no—
166C4×Q8—216no—
167C4×Q8—28no—
168C4×Q8C2×D4116yes—
169C4×Q8C2×D4116yes—
170#117—24no—
171#117—24no—
172#117—216no—
173#117—216no—
174#117—28no—
175#117—28no—
176#117—28no—
177#117—28no—
178#117—28no—
179#117—28no—
180#117C2×D418yes—
181#117C2×D418yes—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C22×C418yes—
186C2×C4⋊C4C22×C414yes—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C22×C418yes—
189C2×C4⋊C4C22×C418yes—
190C2×C4⋊C4C22×C418yes—
191C2×C4⋊C4C22×C418yes—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C22×C414yes—
194C2×C4⋊C4C22×C414yes—
195C2×C4⋊C4C22×C418yes—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—216no—
198C2×C22⋊C4C22×C414yes—
199C2×C22⋊C4C22×C418yes—
200C2×C22⋊C4C22×C414yes—
201C2×C22⋊C4C22×C414yes—
202C2×C22⋊C4C22×C414yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—28no—
206C2×C42—28no—
207C2×C42C22×C412yes—
208C2×C42C22×C418yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D414yes—
212C2×C42C2×D414yes—
213C2×C42C2412yes—
214C2×C42C2414yes—
215C2×C42C2412yes—
216C2×C42C2414yes—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—28no—
220C4⋊C8—216no—
221C4⋊C8—216no—
222C22⋊C8—28no—
223C22⋊C8—28no—
224C22⋊C8—216no—
225C22⋊C8—216no—
226#97—28no—
227#97—28no—
228#97—28no—
229C4×C8—28no—
230C4×C8—28no—
231C4×C8—28no—
232C24—22no—
233C24—24no—
234C24—44no—
235C24C23×C411yescharacteristic
236C24C23×C412yes—
237C2×Q8—216no—
238C2×Q8—28no—
239C2×Q8—48no—
240C2×Q8—48no—
241C2×Q8—416no—
242C2×Q8C2×C22⋊C4116yes—
243C22×C4—216no—
244C22×C4—216no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—216no—
256C22×C4—44no—
257C22×C4—416no—
258C22×C4—44no—
259C22×C4C2×C22⋊C418yes—
260C22×C4C23×C418yes—
261C22×C4C23×C414yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C23×C418yes—
264C22×C4C23×C412yes—
265C22×C4C23×C412yes—
266C22×C4C23×C412yes—
267C22×C4C22×D412yes—
268C22×C4C22×D412yes—
269C22×C42−1+418yes—
270C22×C4C2511yescharacteristic
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—28no—
274C2×C8—28no—
275C2×C8—28no—
276C2×C8—28no—
277C2×C8—28no—
278C4⋊C4—216no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—216no—
283C4⋊C4—216no—
284C4⋊C4—216no—
285C4⋊C4—216no—
286C4⋊C4—28no—
287C4⋊C4C2×C22⋊C4116yes—
288C4⋊C4C2×C22⋊C4116yes—
289C4⋊C4C2×C22⋊C4132yes—
290C22⋊C4—28no—
291C22⋊C4—216no—
292C22⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4C2×C22⋊C4116yes—
295C42—216no—
296C42—24no—
297C42—22no—
298C42—22no—
299C42—24no—
300C42—24no—
301C42C2×C22⋊C414yes—
302C42C2×C22⋊C414yes—
303C42C22×D414yes—
304C42C22×D418yes—
305C42C22×D418yes—
306C42C22×D414yes—
307C23—28no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—28no—
312C23—24no—
313C23—88no—
314C23—84no—
315C23C22×C22⋊C412yes—
316C23#51914yes—
317C23C24×C411yescharacteristic
318Q8—416no—
319Q8—88no—
320Q8—816no—
321C2×C4—216no—
322C2×C4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—216no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—216no—
336C2×C4—28no—
337C2×C4—48no—
338C2×C4—816no—
339C2×C4—84no—
340C2×C4C22×C22⋊C412yes—
341C2×C4C22×C22⋊C4116yes—
342C2×C4C22×C22⋊C414yes—
343C2×C4C22×C22⋊C414yes—
344C2×C4#51914yes—
345C2×C4C23×D411yescharacteristic
346C2×C4C23×D411yescharacteristic
347C2×C4C2×2−1+414yes—
348C8—44no—
349C8—44no—
350C8—48no—
351C22—28no—
352C22—22no—
353C22—24no—
354C22—24no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—88no—
359C22C23×C22⋊C411yescharacteristic
360C22C2×#51912yes—
361C22#357314yes—
362C4—22no—
363C4—24no—
364C4—24no—
365C4—24no—
366C4—216no—
367C4—48no—
368C4—44no—
369C4#348612yes—
370C4#348612yes—
371C2—44no—
372C2—42no—
373C2#3417311yescharacteristic
374C2#6006512yes—
375C2#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×#2987C2semidirect
3. C2×#2986C2semidirect
4. #22511C2semidirect
5. #22509C2semidirect
6. #21824C2semidirect
7. #21821C2semidirect
24. C2×#446C22semidirect
25. C2×#446C22semidirect
26. C2×C8×Q8C22semidirect
31. #1985C22semidirect
32. #1985C22semidirect
33. #1985C22semidirect
34. #1985C22semidirect
35. #1984C22semidirect
36. #1984C22semidirect
37. #1984C22semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#29872
C2×#29862
#225112
#225092
#218242
#218212
#207172
#207162