Finite Group Databasebeta

← #181133

Subgroups — #181133

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#34192C211yesmaximal, characteristic
3#34059C211yesmaximal, characteristic
4#7945C214yesmaximal
5#7944C214yesmaximal
6#7941C212yesmaximal
7#7940C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2984C418yescomplemented
10#2983—22no—
11#2983—21no—
12#2982—21no—
13Q8×M16—24no—
14#2920C2212yes—
15#2920C2214yes—
16#2356C412yescomplemented
17#2356C414yescomplemented
18#2351C412yescomplemented
19#2345—22no—
20#2345—21no—
21Q8×C22⋊C4—21no—
22#2320—24no—
23#2320—21no—
24#2320—22no—
25#2316—21no—
26#2313—22no—
27#2310—21no—
28#2295C418yescomplemented
29#2283C2211yescharacteristic
30#2283C2212yes—
31#2283C2211yescharacteristic
32#2283C2212yes—
33#2283C2214yes—
34#2276—21no—
35C4×#117C2212yes—
36#1284C2218yes—
37#1284C2214yes—
38#1284C2214yes—
39C23×Q8D411yescharacteristic
40C2×C4×Q8—22no—
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C414yes—
45C2×C4×Q8C2×C414yes—
46C2×C4×Q8D414yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D412yes—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8Q814yes—
51C2×#117—22no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D411yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2312yes—
56#446—28no—
57#446—216no—
58C8×Q8—28no—
59#432—24no—
60#432—22no—
61#432C2×C414yes—
62#432C2×C418yes—
63#431—22no—
64#431—24no—
65#431C2×C414yes—
66#431C2×C418yes—
67#423—22no—
68#423—24no—
69#407C2314yes—
70C4×M16—22no—
71C4×M16—24no—
72#391—28no—
73#391—22no—
74#391—24no—
75#389—28no—
76#389—24no—
77#389—22no—
78#387—216no—
79#387—24no—
80#387—24no—
81#387—24no—
82#383—24no—
83#383—22no—
84#383—21no—
85#383C2×C412yes—
86#383C2×C414yes—
87#382—24no—
88#382—22no—
89#382—21no—
90#382C2×C412yes—
91#382C2×C414yes—
92#381—24no—
93#381C2311yescharacteristic
94#381C2312yes—
95#381C2311yescharacteristic
96#380—24no—
97#380—21no—
98#380—22no—
99#380—21no—
100#380—22no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4C2×C414yes—
107C4×C4⋊C4C2×C418yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C414yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4C2314yes—
116#376—22no—
117#376—21no—
118#376—22no—
119#376C2×C418yes—
120#376C2×C412yes—
121#376C2×C414yes—
122C43C2×C414yes—
123C22×Q8—44no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C412yes—
127C22×Q8C22⋊C414yes—
128C22×Q8C4⋊C414yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2411yescharacteristic
134C2×M16—24no—
135C4×Q8—48no—
136C4×Q8C4214yes—
137C4×Q8C4214yes—
138C4×Q8C4218yes—
139C4×Q8C22⋊C414yes—
140C4×Q8C22⋊C414yes—
141C4×Q8C22⋊C418yes—
142C4×Q8C4⋊C414yes—
143C4×Q8C4⋊C414yes—
144C4×Q8C4⋊C414yes—
145C4×Q8C4⋊C414yes—
146C4×Q8C4⋊C418yes—
147C4×Q8C4⋊C418yes—
148#117—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C418yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C4⋊C414yes—
165C2×C4⋊C4C4⋊C418yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42C22×C418yes—
177C2×C42C22×C412yes—
178C2×C42C22×C414yes—
179C2×C42C22×C412yes—
180C2×C42C22×C414yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D414yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q812yes—
192C2×C42C2×Q814yes—
193C4⋊C8—28no—
194C4⋊C8—24no—
195C4⋊C8—24no—
196C4⋊C8—28no—
197C22⋊C8—24no—
198C22⋊C8C22×C418yes—
199#97—24no—
200#97—28no—
201C4×C8—28no—
202C4×C8—24no—
203#95—24no—
204#95—22no—
205#95—24no—
206#95—28no—
207#95—22no—
208#95—24no—
209#95—28no—
210#95—22no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—28no—
215#95—21no—
216#95—24no—
217#95—21no—
218#95—22no—
219#95C22×C414yes—
220#95C22×C412yes—
221#95C22×C412yes—
222C24C22×D411yescharacteristic
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—44no—
226C2×Q8—48no—
227C2×Q8—48no—
228C2×Q8#9514yes—
229C2×Q8#9514yes—
230C2×Q8#9518yes—
231C22×C4—24no—
232C22×C4—28no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4—82no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C414yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C414yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C4⋊C412yes—
253C22×C4C2×C4⋊C412yes—
254C22×C4C2×C4⋊C414yes—
255C22×C4C2×C4⋊C414yes—
256C22×C4C23×C411yescharacteristic
257C22×C4C23×C411yescharacteristic
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×Q811yescharacteristic
262M16—48no—
263C2×C8—28no—
264C4⋊C4—48no—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—82no—
268C4⋊C4—84no—
269C4⋊C4—88no—
270C4⋊C4#9518yes—
271C4⋊C4#9518yes—
272C4⋊C4#9518yes—
273C4⋊C4#9518yes—
274C4⋊C4#9518yes—
275C4⋊C4#9514yes—
276C4⋊C4#9514yes—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—48no—
281C42—48no—
282C42—48no—
283C42—48no—
284C42—44no—
285C42—82no—
286C42—84no—
287C42—84no—
288C42C2×C4212yes—
289C42C2×C4214yes—
290C42C2×C4214yes—
291C42C2×C4212yes—
292C42C2×C22⋊C414yes—
293C42C2×C22⋊C412yes—
294C42C2×C22⋊C412yes—
295C42C2×C22⋊C414yes—
296C42C2×C4⋊C412yes—
297C42C2×C4⋊C414yes—
298C42C2×C4⋊C414yes—
299C42C2×C4⋊C412yes—
300C42C2×C4⋊C414yes—
301C42C2×C4⋊C414yes—
302C42C2×C4⋊C412yes—
303C42C2×C4⋊C412yes—
304C23—22no—
305C23—22no—
306C23—41no—
307C23C22×C22⋊C411yescharacteristic
308C23C22×C22⋊C411yescharacteristic
309C23C22×C4⋊C411yescharacteristic
310Q8—28no—
311Q8—28no—
312Q8—48no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4C2×#9512yes—
347C2×C4C2×#9514yes—
348C2×C4C2×#9514yes—
349C2×C4C2×#9512yes—
350C2×C4C2×#9512yes—
351C2×C4C2×#9514yes—
352C2×C4C2×#9514yes—
353C2×C4C2×#9512yes—
354C2×C4#42114yes—
355C2×C4C22×C4211yescharacteristic
356C2×C4C22×C22⋊C411yescharacteristic
357C2×C4C22×C4⋊C411yescharacteristic
358C2×C4C22×C4⋊C411yescharacteristic
359C8—48no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22C22×#9511yescharacteristic
369C22#288412yes—
370C22C2×#42112yes—
371C4—22no—
372C4—22no—
373C4—24no—
374C4—24no—
375C4—22no—
376C4—24no—
377C4—24no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—88no—
383C4—84no—
384C4—82no—
385C4—82no—
386C4#173314yes—
387C2—22no—
388C2—22no—
389C2—41no—
390C2#2033811yescharacteristic
391C2C2×#173311yescharacteristic
392C2#2035511yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ 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
9. #2984C4semidirect
16. #2356C4semidirect
17. #2356C4semidirect
18. #2351C4semidirect
28. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#602482
#341922
#340592
#79452
#79442
#79412
#79402