Finite Group Databasebeta

← #225543

Subgroups — #225543

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22763C212yesmaximal
2#22741C211yesmaximal, characteristic
3#22740C211yesmaximal, characteristic
4C2×C8×Q16C211yesmaximal, characteristic
5#9619C212yesmaximal
6#8896C212yesmaximal
7#8895C212yesmaximal
8#8817C211yesmaximal, characteristic
9#8649C211yesmaximal, characteristic
10#8649C211yesmaximal, characteristic
11#8282C211yesmaximal, characteristic
12C2×C8×Q8C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14C2×C4×Q16C2211yescharacteristic
15#2355C2212yes—
16Q8×C4⋊C4C2212yes—
17#1949—22no—
18#1948C2212yes—
19#1947—22no—
20#1947C2212yes—
21#1935—22no—
22#1918—22no—
23#1878—22no—
24#1835—21no—
25#1834—21no—
26#1834C2212yes—
27#1834C2211yescharacteristic
28#1834C2211yescharacteristic
29#1834C2212yes—
30#1803C2212yes—
31#1803C2212yes—
32C4×#107—21no—
33C4×#107C2211yescharacteristic
34C4×#106—21no—
35#1770C2212yes—
36#1763—22no—
37C8×Q16C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#335C2211yescharacteristic
40C2×Q8⋊C8C2211yescharacteristic
41C2×Q8⋊C8C2211yescharacteristic
42C2×C82C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1279C2212yes—
46C22×Q16D411yescharacteristic
47C2×C4×Q8—22no—
48C2×C4×Q8C2311yescharacteristic
49C2×C4×Q8C2311yescharacteristic
50C8×Q8—22no—
51C8×Q8—22no—
52C8×Q8C2×C414yes—
53C8×Q8C2×C414yes—
54C4×Q16C2×C414yes—
55C4×Q16Q814yes—
56C2×#107—22no—
57C2×#107D412yes—
58C2×#107D411yescharacteristic
59C2×#106—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—22no—
67C2×#103D411yescharacteristic
68C2×#103D411yescharacteristic
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#389—22no—
77#389—42no—
78#387—42no—
79#384—22no—
80#384C2312yes—
81#384C2312yes—
82#382—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2312yes—
87#340—21no—
88#340—21no—
89#340—22no—
90#340—22no—
91#340—21no—
92#340—21no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—42no—
98#340D411yescharacteristic
99#340D411yescharacteristic
100#336—22no—
101#335—22no—
102#335C2×C414yes—
103Q8⋊C8C2×C414yes—
104Q8⋊C8C2×C414yes—
105C82C2×C414yes—
106C22×Q8C2×D411yescharacteristic
107C22×Q8C2×D411yescharacteristic
108C2×Q16—22no—
109C2×Q16—22no—
110C2×Q16C4⋊C414yes—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—44no—
124C4×Q8—82no—
125C4×Q8C22×C412yes—
126C4×Q8C22×C412yes—
127C4×Q8C2×Q812yes—
128C4×Q8C2×Q812yes—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—42no—
147C2×C42—42no—
148C2×C42C2411yescharacteristic
149#107—42no—
150#107—44no—
151#107C4⋊C414yes—
152#106—44no—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8C22×C412yes—
160C4⋊C8C22×C412yes—
161#103—44no—
162#103—82no—
163#103C4⋊C414yes—
164#103C4⋊C414yes—
165#97—82no—
166C4×C8—24no—
167C4×C8—24no—
168C4×C8—21no—
169C4×C8C22×C412yes—
170C4×C8C22×C412yes—
171C4×C8C22×C412yes—
172C4×C8C2×Q812yes—
173C4×C8#4112yes—
174#95—22no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8C2×C4⋊C412yes—
188C2×Q8C2×C4⋊C412yes—
189C2×Q8#13512yes—
190C2×Q8#13512yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4H(5,𝔽2)11yescharacteristic
202Q16—28no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—22no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8C2×C4⋊C412yes—
219C2×C8C2×C4⋊C412yes—
220C2×C8C4×D412yes—
221C2×C8C4×D412yes—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—84no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—162no—
244C4⋊C4C2×C4⋊C412yes—
245C4⋊C4C2×C4⋊C412yes—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—42no—
251C42—42no—
252C42—84no—
253C42—162no—
254C42C23×C411yescharacteristic
255C42C22×Q811yescharacteristic
256C42C2×#4111yescharacteristic
257C422−1+411yescharacteristic
258C23D4211yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—22no—
264Q8—41no—
265Q8—41no—
266Q8#42714yes—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4C22×C4⋊C411yescharacteristic
291C2×C4C2×C4×D411yescharacteristic
292C2×C4#52011yescharacteristic
293C2×C4#54711yescharacteristic
294C2×C4D4×Q811yescharacteristic
295C2×C4#55411yescharacteristic
296C2×C4C2×#13511yescharacteristic
297C2×C4C2×#13511yescharacteristic
298C2×C4#57512yes—
299C2×C4#57611yescharacteristic
300C2×C4#57811yescharacteristic
301C2×C4#57812yes—
302C8—22no—
303C8—22no—
304C8—22no—
305C8—22no—
306C8—42no—
307C8—42no—
308C8—42no—
309C8—82no—
310C22D4×C4⋊C411yescharacteristic
311C22#327712yes—
312C22#328011yescharacteristic
313C22#328511yescharacteristic
314C22#329012yes—
315C4—24no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—82no—
322C4—82no—
323C4—162no—
324C4—162no—
325C4C2×#42711yescharacteristic
326C4#291311yescharacteristic
327C4#294311yescharacteristic
328C4#295011yescharacteristic
329C4#333611yescharacteristic
330C4#333911yescharacteristic
331C4#339012yes—
332C2#1990612yes—
333C2#1994112yes—
334C2#2265611yescharacteristic
335C2#2267511yescharacteristic
336C2#2269011yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
37. C8×Q16C4semidirect

Maximal subgroups

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

subgroupindex
#227632
#227412
#227402
C2×C8×Q162
#96192
#88962
#88952
#88172
#86492
#82822