Finite Group Databasebeta

← #181369

Subgroups — #181369

Order 512.

1239 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic
3#34187C211yesmaximal, characteristic
4#8110C214yesmaximal
5#8109C212yesmaximal
6#8106C214yesmaximal
7#8105C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2984—24no—
10#2984C418yescomplemented
11#2983—22no—
12#2983—21no—
13#2983—22no—
14#2983—21no—
15#2982—21no—
16#2982—21no—
17Q8×M16—24no—
18Q8×M16C418yescomplemented
19#2920C2212yes—
20#2920C2214yes—
21#2920C2212yes—
22#2920C2214yes—
23#2356—21no—
24#2356—22no—
25#2356—21no—
26#2355—21no—
27#2355—22no—
28#2355—21no—
29#2313C414yes—
30#2310C412yes—
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#2283C2212yes—
34#2283C2212yes—
35#2274C412yes—
36#1305C2214yes—
37#1305C2214yes—
38#1305C2218yes—
39C23×Q8C2×C411yescharacteristic
40C2×C4×Q8C2×C414yes—
41C2×C4×Q8C2×C414yes—
42C2×C4×Q8D411yescharacteristic
43C2×C4×Q8D412yes—
44C2×C4×Q8D411yescharacteristic
45C2×C4×Q8D411yescharacteristic
46C2×C4×Q8D412yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D414yes—
49C2×C4×Q8Q814yes—
50C2×#117C2×C412yes—
51C2×#117C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56#446—28no—
57#446—216no—
58#446—28no—
59#446—216no—
60C8×Q8—28no—
61C8×Q8—28no—
62#432—24no—
63#432—22no—
64#432—24no—
65#432—22no—
66#432—22no—
67#432—24no—
68#432C2×C414yes—
69#432C2×C418yes—
70#431—24no—
71#431—22no—
72#431—24no—
73#431—22no—
74#431—22no—
75#431—24no—
76#431C2×C414yes—
77#431C2×C418yes—
78#423—22no—
79#423—24no—
80#423C2×C414yes—
81#423C2×C418yes—
82#407C2314yes—
83#407C2314yes—
84C4×M16—22no—
85C4×M16—24no—
86C4×M16C2×C414yes—
87C4×M16C2×C418yes—
88#391—22no—
89#391—22no—
90#391—24no—
91#387—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#384—21no—
96#384—21no—
97#383—22no—
98#383—22no—
99#382—22no—
100#382—22no—
101#382—21no—
102#382—21no—
103#381C2×C414yes—
104#381C2×C414yes—
105#380C2×C412yes—
106#380C2×C412yes—
107#380C2311yescharacteristic
108#380C2311yescharacteristic
109#380C2312yes—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C22⋊C4C2×C414yes—
120C4×C22⋊C4C2×C414yes—
121#376—22no—
122#376—22no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8C4214yes—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22⋊C412yes—
129C22×Q8C4⋊C414yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—24no—
136C2×M16C22×C418yes—
137C4×Q8C22⋊C414yes—
138C4×Q8C22⋊C414yes—
139C4×Q8C22⋊C414yes—
140C4×Q8C22⋊C414yes—
141C4×Q8C22⋊C418yes—
142C4×Q8C22⋊C418yes—
143C4×Q8C4⋊C414yes—
144C4×Q8C4⋊C414yes—
145C4×Q8C4⋊C414yes—
146C4×Q8C4⋊C414yes—
147C4×Q8C4⋊C418yes—
148C4×Q8C4⋊C418yes—
149#117—44no—
150#117—44no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C4214yes—
159C2×C4⋊C4C4218yes—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C4⋊C414yes—
168C2×C4⋊C4C4⋊C418yes—
169C2×C22⋊C4C22×C414yes—
170C2×C22⋊C4C22×C412yes—
171C2×C22⋊C4C22×C412yes—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42C22×C412yes—
177C2×C42C22×C414yes—
178C2×C42C22×C412yes—
179C2×C42C22×C414yes—
180C2×C42C2×D412yes—
181C2×C42C2×D414yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q812yes—
191C2×C42C2×Q814yes—
192C4⋊C8—28no—
193C4⋊C8—24no—
194C4⋊C8—28no—
195C4⋊C8—24no—
196C4⋊C8—28no—
197C4⋊C8—24no—
198C4⋊C8—28no—
199C4⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8C22×C418yes—
204#97—28no—
205#97—24no—
206#97—28no—
207#97—24no—
208C4×C8—28no—
209C4×C8—24no—
210C4×C8—28no—
211C4×C8—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—22no—
218#95—24no—
219#95—24no—
220#95—21no—
221#95—22no—
222#95—21no—
223C24C23×C411yescharacteristic
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—48no—
231C2×Q8—48no—
232C2×Q8#9514yes—
233C2×Q8#9514yes—
234C2×Q8#9518yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—81no—
245C22×C4C2×C4212yes—
246C22×C4C2×C4214yes—
247C22×C4C2×C4212yes—
248C22×C4C2×C4214yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C4⋊C412yes—
258C22×C4C2×C4⋊C414yes—
259C22×C4C2×C4⋊C412yes—
260C22×C4C2×C4⋊C414yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×Q811yescharacteristic
267M16—48no—
268M16—48no—
269C2×C8—28no—
270C2×C8—28no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4#9518yes—
279C4⋊C4#9518yes—
280C4⋊C4#9518yes—
281C4⋊C4#9518yes—
282C4⋊C4#9518yes—
283C4⋊C4#9514yes—
284C4⋊C4#9514yes—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C42—44no—
289C42—44no—
290C42—84no—
291C42—84no—
292C42C2×C22⋊C414yes—
293C42C2×C22⋊C412yes—
294C42C2×C22⋊C414yes—
295C42C2×C22⋊C412yes—
296C42C2×C22⋊C414yes—
297C42C2×C22⋊C412yes—
298C42C2×C22⋊C414yes—
299C42C2×C22⋊C412yes—
300C42C2×C4⋊C414yes—
301C42C2×C4⋊C414yes—
302C42C2×C4⋊C412yes—
303C42C2×C4⋊C414yes—
304C42C2×C4⋊C414yes—
305C42C2×C4⋊C412yes—
306C42C2×C4⋊C412yes—
307C42C2×C4⋊C412yes—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23C22×C4211yescharacteristic
313C23C22×C22⋊C411yescharacteristic
314C23C22×C4⋊C411yescharacteristic
315Q8—28no—
316Q8—28no—
317Q8—48no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4C2×#9514yes—
352C2×C4C2×#9512yes—
353C2×C4C2×#9512yes—
354C2×C4C2×#9514yes—
355C2×C4C2×#9512yes—
356C2×C4C2×#9514yes—
357C2×C4C2×#9514yes—
358C2×C4C2×#9512yes—
359C2×C4#41312yes—
360C2×C4#41312yes—
361C2×C4C22×C22⋊C411yescharacteristic
362C2×C4C22×C22⋊C411yescharacteristic
363C2×C4C22×C4⋊C411yescharacteristic
364C2×C4C22×C4⋊C411yescharacteristic
365C8—48no—
366C8—48no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—42no—
376C22—42no—
377C22C22×#9511yescharacteristic
378C22C2×#41311yescharacteristic
379C22C2×#41311yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—24no—
385C4—24no—
386C4—24no—
387C4—24no—
388C4—22no—
389C4—22no—
390C4—84no—
391C4—82no—
392C4—82no—
393C4#174214yes—
394C4#288412yes—
395C2—21no—
396C2—21no—
397C2—41no—
398C2—41no—
399C2#2035511yescharacteristic
400C2C2×#174211yescharacteristic
401C2#2036011yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2984C4semidirect
18. Q8×M16C4semidirect

Maximal subgroups

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

subgroupindex
#602482
#341872
#81102
#81092
#81062
#81052