Finite Group Databasebeta

← #6389771

Subgroups — #6389771

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3160C211yesmaximal, characteristic
2#34403C212yesmaximal
3#23165C214yesmaximal, complemented
4#23159C214yesmaximal, complemented
5#22616C214yesmaximal
6#22610C214yesmaximal
7#22254C212yesmaximal, complemented
8#22254C212yesmaximal, complemented
9#21842C212yesmaximal
10#21724C214yesmaximal, complemented
11#20818C211yesmaximal, characteristic
12#20476C211yesmaximal, characteristic
13C2×#533C2212yes—
14#3160C4116yescomplemented
15C2×#500C2211yescharacteristic
16C2×#500C2211yescharacteristic
17C2×#499C2212yes—
18C2×#499C2212yes—
19C2×#498C2212yes—
20C2×#432C2211yescharacteristic
21C2×#427C2212yes—
22C2×#427C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2378—24no—
25#2370C2218yes—
26#2356—24no—
27#2347C2214yes—
28#2346C2214yes—
29#2332—24no—
30#2331C2214yes—
31#2331C2214yes—
32#2320—24no—
33#2283C2212yes—
34#2283C2211yescharacteristic
35#1993C2214yes—
36#1993C2214yes—
37#1989C2218yes—
38#1944C2218yes—
39#1941C2218yes—
40#1921C2218yes—
41#1921C2218yes—
42#1918C2218yes—
43#1918C2218yes—
44#1846C2212yes—
45#1835C2212yes—
46#1825C2214yes—
47#1825C2214yes—
48#1808C2212yes—
49#1807C2212yes—
50#1776C2218yes—
51C4×#107C2212yes—
52C4×#106C2212yes—
53C4×#106C2218yes—
54#1770C2212yes—
55C8×C4⋊C4C2212yes—
56#1752C2212yes—
57C8×C22⋊C4C2212yes—
58#1746C2212yes—
59#1726C2212yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62#533—28no—
63#533C2×C4116yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125C2314yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×#117C2314yes—
72C22×C42C2311yescharacteristic
73#500C2×C418yes—
74#500C2×C418yes—
75#499C2×C4116yes—
76#499C2×C4116yes—
77#498C2×C4116yes—
78#432—24no—
79#427C2×C4116yes—
80#427C2×C4116yes—
81C2×#107C2314yes—
82C2×#107C2312yes—
83C2×#107C2312yes—
84C2×#107C2312yes—
85C2×#107C2312yes—
86C2×#106C2314yes—
87C2×#106C2314yes—
88C2×#106C2314yes—
89C2×C4⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×#97C2312yes—
92C2×#97C2314yes—
93C2×C4×C8C2×C418yes—
94C2×C4×C8C2314yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—28no—
100#391—28no—
101#389—28no—
102#389C2318yes—
103#389C2318yes—
104#388—28no—
105#388—28no—
106#387—28no—
107#387—28no—
108#387—28no—
109#387—28no—
110#383—22no—
111#383—24no—
112#383D412yes—
113#382—24no—
114#382D412yes—
115#382C2314yes—
116#381—22no—
117#381D412yes—
118#380D412yes—
119#380C2314yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4C2318yes—
123C4×C4⋊C4C2318yes—
124C4×C4⋊C4C2314yes—
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2312yes—
128#376—22no—
129#376—24no—
130#376C2312yes—
131#340C2314yes—
132#340C2314yes—
133#340C2318yes—
134#340C2318yes—
135#340C2318yes—
136#340C2318yes—
137#336C2314yes—
138#336C2314yes—
139C22×Q8—22no—
140C22×Q8—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—24no—
145C22×C8—24no—
146C22×C8C22×C414yes—
147C22×C8C22×C414yes—
148C22×C8C2414yes—
149C22×C8C2411yescharacteristic
150C22×C8C2411yescharacteristic
151C22×C8C2411yescharacteristic
152C22×C8C2411yescharacteristic
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8C22×C418yes—
156C4⋊Q8C22×C418yes—
157#125—28no—
158#125C22×C4116yes—
159C22⋊Q8—216no—
160C22⋊Q8—216no—
161C4×Q8—216no—
162C4×Q8—216no—
163#117—28no—
164#117C22×C4116yes—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2418yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4C2×D414yes—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—28no—
181C2×C42—28no—
182C2×C42C22×C414yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4114yes—
186C2×C42C2414yes—
187C2×C42C2411yescharacteristic
188C2×C42C2411yescharacteristic
189#107—28no—
190#107C22×C418yes—
191#107C22×C418yes—
192#107C22×C418yes—
193#107C22×C418yes—
194#106—28no—
195#106C22×C4116yes—
196#106C22×C4116yes—
197C4⋊C8—24no—
198C22⋊C8—24no—
199#97—28no—
200#97—24no—
201C4×C8—28no—
202C4×C8C22×C414yes—
203C4×C8C22×C414yes—
204C4×C8C22×C414yes—
205C4×C8C22×C414yes—
206#95—24no—
207#95—24no—
208#95—28no—
209#95—24no—
210#95—28no—
211#95—24no—
212#95—28no—
213#95—28no—
214#95C2×D414yes—
215#95C2×D414yes—
216#95C2×D414yes—
217#95C2×D414yes—
218C24C22×D411yescharacteristic
219C2×Q8—28no—
220C2×Q8—28no—
221C2×Q8—84no—
222C2×Q8—84no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—28no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C4×D414yes—
231C22×C4C4×D414yes—
232C22×C4C23×C412yes—
233C22×C4C22×D418yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4112yes—
240C22×C4C2511yescharacteristic
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—28no—
244C2×C8—28no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8C23×C412yes—
249C2×C8C23×C412yes—
250C2×C8C23×C412yes—
251C2×C8C23×C412yes—
252C2×C82−1+414yes—
253C2×C82−1+414yes—
254C4⋊C4—216no—
255C4⋊C4—216no—
256C4⋊C4—28no—
257C4⋊C4—28no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—44no—
263C4⋊C4—88no—
264C4⋊C4—88no—
265C4⋊C4C23×C418yes—
266C4⋊C4C23×C418yes—
267C22⋊C4—216no—
268C22⋊C4—44no—
269C22⋊C4—84no—
270C42—216no—
271C42—24no—
272C42—24no—
273C42—84no—
274C42—88no—
275C42—88no—
276C42C4×D414yes—
277C42C4×D414yes—
278C42C23×C412yes—
279C42C23×C412yes—
280C42C2×#4112yes—
281C42C2×#4112yes—
282C23—22no—
283C23—22no—
284C23—22no—
285C23C2×C4×D412yes—
286C23C23×D411yescharacteristic
287Q8—88no—
288Q8—88no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4C2×C4×D412yes—
311C2×C4C2×C4×D412yes—
312C2×C4C2×C4×D412yes—
313C2×C4C2×C4×D412yes—
314C2×C4C2×C4×D412yes—
315C2×C4C24×C411yescharacteristic
316C2×C4C22×#4111yescharacteristic
317C2×C4C2×2−1+412yes—
318C2×C4C2×2−1+412yes—
319C8—22no—
320C8—22no—
321C8—44no—
322C8#51914yes—
323C8#51914yes—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—44no—
328C22C22×C4×D411yescharacteristic
329C22#349912yes—
330C22#358412yes—
331C22#358412yes—
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—48no—
338C4—48no—
339C4—48no—
340C4—88no—
341C4C2×#51911yescharacteristic
342C4C2×#51911yescharacteristic
343C4#350912yes—
344C2—42no—
345C2#3420911yescharacteristic
346C2#6016711yescharacteristic
347C2#6016711yescharacteristic
348C2#6248914yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23165C2semidirect
4. #23159C2semidirect
7. #22254C2semidirect
8. #22254C2semidirect
10. #21724C2semidirect
14. #3160C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
#344032
#231652
#231592
#226162
#226102
#222542
#218422
#217242
#208182
#204762