Finite Group Databasebeta

← #181541

Subgroups — #181541

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2982C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3C2×#2356C211yesmaximal, characteristic
4#8026C212yesmaximal
5#8025C214yesmaximal
6#8014C212yesmaximal
7#8013C214yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2982—24no—
10#2982C418yescomplemented
11C2×C8×Q8—24no—
12C2×#431C2214yes—
13C2×#431C2212yes—
14#2356—24no—
15#2356—24no—
16#2356C418yescomplemented
17#2355C418yescomplemented
18#2313—22no—
19#2310—21no—
20C2×#391—21no—
21C2×#387—22no—
22#2285C2212yes—
23#2283C2214yes—
24C2×#383C2212yes—
25C2×#382C2211yescharacteristic
26#2274—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2212yes—
30#1294C2214yes—
31#1294C2218yes—
32#1294C2214yes—
33C23×Q8D411yescharacteristic
34C2×C4×Q8C2×C414yes—
35C2×C4×Q8C2×C414yes—
36C2×C4×Q8C2×C414yes—
37C2×C4×Q8D414yes—
38C2×C4×Q8D414yes—
39C2×C4×Q8Q814yes—
40C2×#117—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D411yescharacteristic
45C22×C42—22no—
46C22×C42—21no—
47C22×C42C2312yes—
48C22×C42C2311yescharacteristic
49C8×Q8—232no—
50#431—28no—
51#431—24no—
52#431C2×C4116yes—
53#431C2×C418yes—
54C2×C4⋊C8—24no—
55C2×C4⋊C8—22no—
56C2×C22⋊C8C2314yes—
57C2×C4×C8—24no—
58C2×C4×C8—22no—
59#391—28no—
60#391—28no—
61#387—216no—
62#387—216no—
63#384C2×C414yes—
64#383—24no—
65#383—24no—
66#383C2×C418yes—
67#382—22no—
68#382—22no—
69#382C2×C418yes—
70#382C2×C414yes—
71#381—24no—
72#380—22no—
73#380C2314yes—
74C4×C4⋊C4—28no—
75C4×C4⋊C4—28no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2×C418yes—
79C4×C4⋊C4C2×C414yes—
80C4×C4⋊C4C2×C414yes—
81C4×C22⋊C4—24no—
82#376—24no—
83#376—24no—
84#376C2×C418yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95C2311yescharacteristic
90C2×#95C2312yes—
91C2×#95C2311yescharacteristic
92C22×Q8—24no—
93C22×Q8—24no—
94C22×Q8C22⋊C414yes—
95C22×Q8C22⋊C414yes—
96C22×Q8C4⋊C414yes—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2411yescharacteristic
105C22×C8—24no—
106C4×Q8C42116yes—
107C4×Q8C22⋊C4116yes—
108C4×Q8C4⋊C4116yes—
109C4×Q8C4⋊C4116yes—
110#117—44no—
111#117—44no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4C22⋊C414yes—
119C2×C4⋊C4C22⋊C414yes—
120C2×C4⋊C4C22⋊C418yes—
121C2×C4⋊C4C22⋊C418yes—
122C2×C4⋊C4C4⋊C418yes—
123C2×C4⋊C4C4⋊C414yes—
124C2×C4⋊C4C22×C418yes—
125C2×C22⋊C4—24no—
126C2×C42—24no—
127C2×C42—28no—
128C2×C42—24no—
129C2×C42—28no—
130C2×C42—48no—
131C2×C42—44no—
132C2×C42C22×C414yes—
133C2×C42C22×C412yes—
134C2×C42C22×C414yes—
135C2×C42C22×C414yes—
136C2×C42C22×C412yes—
137C2×C42C22×C412yes—
138C2×C42C2×D414yes—
139C2×C42C2×D414yes—
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C2×C42C2×Q814yes—
143C2×C42C2×Q812yes—
144C4⋊C8—216no—
145C4⋊C8—28no—
146C22⋊C8—24no—
147C22⋊C8C22×C418yes—
148C4×C8—216no—
149C4×C8—28no—
150#95—28no—
151#95—28no—
152#95—24no—
153#95—28no—
154#95—28no—
155#95—28no—
156#95—24no—
157#95—28no—
158#95—21no—
159#95—22no—
160#95—21no—
161#95—24no—
162#95C22×C412yes—
163#95C22×C414yes—
164#95C22×C412yes—
165C24C22×D411yescharacteristic
166C2×Q8—28no—
167C2×Q8—28no—
168C2×Q8—28no—
169C2×Q8—28no—
170C2×Q8—216no—
171C2×Q8#95116yes—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4C2×C22⋊C414yes—
191C22×C4C2×C22⋊C414yes—
192C22×C4C2×C22⋊C414yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4C2×C22⋊C414yes—
195C22×C4C2×C22⋊C412yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C4⋊C414yes—
199C22×C4C2×C4⋊C412yes—
200C22×C4C2×C4⋊C414yes—
201C22×C4C2×C4⋊C412yes—
202C22×C4#13514yes—
203C22×C4#13514yes—
204C22×C4C23×C411yescharacteristic
205C22×C4C23×C411yescharacteristic
206C22×C4C23×C411yescharacteristic
207C22×C4C22×D411yescharacteristic
208C22×C4C22×D411yescharacteristic
209C22×C4C22×Q811yescharacteristic
210C2×C8—28no—
211C2×C8—48no—
212C4⋊C4—48no—
213C4⋊C4—48no—
214C4⋊C4—48no—
215C4⋊C4—48no—
216C4⋊C4—84no—
217C4⋊C4#95132yes—
218C4⋊C4#95116yes—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C42—416no—
222C42—48no—
223C42—48no—
224C42—88no—
225C42C2×C4218yes—
226C42C2×C4214yes—
227C42C2×C22⋊C418yes—
228C42C2×C22⋊C414yes—
229C42C2×C4⋊C418yes—
230C42C2×C4⋊C418yes—
231C42C2×C4⋊C414yes—
232C42C2×C4⋊C414yes—
233C23—22no—
234C23—22no—
235C23C22×C22⋊C411yescharacteristic
236C23C22×C22⋊C411yescharacteristic
237C23C22×C4⋊C411yescharacteristic
238C23C2×#13512yes—
239C23C2×#13512yes—
240Q8—232no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—28no—
246C2×C4—28no—
247C2×C4—24no—
248C2×C4—28no—
249C2×C4—24no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—48no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—48no—
266C2×C4—42no—
267C2×C4—48no—
268C2×C4—44no—
269C2×C4—48no—
270C2×C4—84no—
271C2×C4—88no—
272C2×C4C2×#9518yes—
273C2×C4C2×#9514yes—
274C2×C4C2×#9518yes—
275C2×C4C2×#9514yes—
276C2×C4#41614yes—
277C2×C4#41614yes—
278C2×C4#41614yes—
279C2×C4#41614yes—
280C2×C4#42718yes—
281C2×C4C22×C4211yescharacteristic
282C2×C4C22×C22⋊C411yescharacteristic
283C2×C4C22×C4⋊C411yescharacteristic
284C2×C4C22×C4⋊C411yescharacteristic
285C8—48no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—24no—
291C22—24no—
292C22C22×#9511yescharacteristic
293C22#288412yes—
294C22C2×#41611yescharacteristic
295C22C2×#41611yescharacteristic
296C22C2×#41612yes—
297C22#290112yes—
298C22C2×#42712yes—
299C4—28no—
300C4—24no—
301C4—28no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—44no—
307C4—82no—
308C4—84no—
309C4—88no—
310C4#173518yes—
311C2—21no—
312C2—21no—
313C2—22no—
314C2#1761614yes—
315C2#2034311yescharacteristic
316C2C2×#173511yescharacteristic
317C2#2035511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2982C4semidirect
16. #2356C4semidirect
17. #2355C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
#341872
C2×#23562
#80262
#80252
#80142
#80132