Finite Group Databasebeta

← #517608

Subgroups — #517608

Order 512.

5476 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#519C211yesmaximal, characteristic, complemented
2C22×#393C211yesmaximal, characteristic
3C2×#2342C213yesmaximal
4#14336C211yesmaximal, characteristic, complemented
5#14301C212yesmaximal, complemented
6#14188C216yesmaximal, complemented
7#14172C216yesmaximal, complemented
8#13878C213yesmaximal, complemented
9#13868C213yesmaximal
10C4×#393C212yesmaximal, complemented
11#13799C213yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C2×#519—224no—
14C22×C4×Q8C2212yescomplemented
15C22×C4×Q8C2216yescomplemented
16C22×#117C2213yes—
17C22×#117C2213yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2213yes—
20C2×#393C4116yescomplemented
21C2×#393C2211yescharacteristic
22C2×#393C2213yes—
23C2×#393C2213yes—
24C2×#393C2211yescharacteristic
25#2342—224no—
26C2×#388C2216yes—
27C2×#388C2216yes—
28C2×#385C2216yes—
29C2×#381C2213yes—
30C2×#381C2213yes—
31C2×#380C2213yes—
32C2×#379C2213yes—
33C2×C4×C22⋊C4C2216yes—
34C2×C4×C22⋊C4C2213yes—
35C22×#95C2213yes—
36#1446C2212yescomplemented
37#1446C2216yescomplemented
38#1444C2216yescomplemented
39#1442C2216yescomplemented
40#1439C2216yes—
41#1439C2216yes—
42#1437C2216yes—
43#1437C2216yes—
44#1436C22112yescomplemented
45#1436C22112yescomplemented
46#1434C2216yescomplemented
47C22⋊#95C2216yes—
48C4×#95C2216yescomplemented
49C26C2311yescharacteristic
50C23×Q8—212no—
51C23×Q8—212no—
52C23×Q8C2×C418yescomplemented
53C23×Q8C2311yescharacteristic
54C23×Q8C2313yes—
55C24×C4C2312yes—
56C24×C4C2313yes—
57C24×C4C2313yes—
58#519—2128no—
59C2×C4×Q8—248no—
60C2×C4×Q8—224no—
61C2×C4×Q8—224no—
62C2×#117—212no—
63C2×#117—224no—
64C2×#117—224no—
65C2×#117D4124yes—
66C22×C4⋊C4C2316yes—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2316yes—
69C22×C22⋊C4—212no—
70C22×C22⋊C4—212no—
71C22×C22⋊C4—212no—
72C22×C22⋊C4C2×C4124yes—
73C22×C22⋊C4C2311yescharacteristic
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2313yes—
79C22×C22⋊C4C2313yes—
80C22×C22⋊C4C2313yes—
81C22×C22⋊C4C2313yes—
82C22×C22⋊C4C2316yes—
83C22×C22⋊C4C2311yescharacteristic
84C22×C42C2316yes—
85C22×C42C2316yes—
86C22×C42C2316yes—
87#393C2×C418yes—
88#393C2×C4124yes—
89#393C2×C4124yescomplemented
90#393C2×C418yescomplemented
91#388—224no—
92#388—224no—
93#385—224no—
94#381—212no—
95#381—212no—
96#380—212no—
97#379—212no—
98C4×C22⋊C4—212no—
99C4×C22⋊C4—224no—
100C2×#95C2×C4124yescomplemented
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C23112yes—
108C2×#95C23112yes—
109C2×#95C2313yes—
110C2×#95C2313yes—
111C2×#95C2316yes—
112C25—24no—
113C25—26no—
114C25—26no—
115C25—46no—
116C25C22×C414yes—
117C25C2411yescharacteristic
118C25C2411yescharacteristic
119C25C2411yescharacteristic
120C22×Q8—232no—
121C22×Q8—296no—
122C22×Q8—296no—
123C22×Q8—212no—
124C22×Q8—26no—
125C22×Q8—26no—
126C22×Q8—26no—
127C22×Q8—26no—
128C22×Q8—212no—
129C22×Q8C22×C414yes—
130C22×Q8C22×C4112yes—
131C23×C4—212no—
132C23×C4—212no—
133C23×C4—26no—
134C23×C4—26no—
135C23×C4—24no—
136C23×C4—212no—
137C23×C4—46no—
138C23×C4—412no—
139C23×C4C22×C4112yes—
140C23×C4C22×C4112yes—
141C23×C4C2×D4112yes—
142C23×C4#41112yes—
143C23×C4C2416yes—
144C23×C4C2412yes—
145C23×C4C2413yes—
146C23×C4C2413yes—
147C23×C4C2416yes—
148C23×C4C2413yes—
149C23×C4C2413yes—
150C23×C4C2412yes—
151C4×Q8—2192no—
152C4×Q8—264no—
153#117—296no—
154#117—296no—
155C2×C4⋊C4—224no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—224no—
158C2×C4⋊C4—212no—
159C2×C4⋊C4—412no—
160C2×C4⋊C4C2×D4124yes—
161C2×C22⋊C4—212no—
162C2×C22⋊C4—296no—
163C2×C22⋊C4—26no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—26no—
168C2×C22⋊C4—26no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—26no—
171C2×C22⋊C4—26no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—212no—
177C2×C22⋊C4—212no—
178C2×C22⋊C4—46no—
179C2×C22⋊C4—416no—
180C2×C22⋊C4—412no—
181C2×C22⋊C4C22×C4112yes—
182C2×C22⋊C4C22×C4112yes—
183C2×C22⋊C4C22×C4112yes—
184C2×C22⋊C4C22×C4112yes—
185C2×C22⋊C4C2×D4112yes—
186C2×C22⋊C4C2×D4112yes—
187C2×C42—224no—
188C2×C42—212no—
189C2×C42—224no—
190C2×C42—212no—
191C2×C42—412no—
192C2×C42C2×D4124yes—
193#95—212no—
194#95—212no—
195#95—212no—
196#95—212no—
197#95—212no—
198#95—212no—
199#95—412no—
200#95—412no—
201#95C22×C4112yes—
202#95C22×C4112yes—
203#95C22×C4124yes—
204C24—22no—
205C24—26no—
206C24—23no—
207C24—23no—
208C24—26no—
209C24—28no—
210C24—23no—
211C24—23no—
212C24—22no—
213C24—26no—
214C24—224no—
215C24—412no—
216C24—424no—
217C24—424no—
218C24—44no—
219C24—412no—
220C24—812no—
221C24—812no—
222C24C23×C412yes—
223C24C23×C412yes—
224C24C23×C412yes—
225C24C22×D416yes—
226C24C2×#4116yes—
227C242−1+414yes—
228C24C2511yescharacteristic
229C2×Q8—224no—
230C2×Q8—28no—
231C2×Q8—248no—
232C2×Q8—224no—
233C2×Q8—224no—
234C2×Q8—2256no—
235C2×Q8—248no—
236C2×Q8—224no—
237C2×Q8—224no—
238C22×C4—248no—
239C22×C4—224no—
240C22×C4—224no—
241C22×C4—248no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—22no—
247C22×C4—26no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—212no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—212no—
256C22×C4—212no—
257C22×C4—212no—
258C22×C4—224no—
259C22×C4—26no—
260C22×C4—26no—
261C22×C4—212no—
262C22×C4—26no—
263C22×C4—22no—
264C22×C4—26no—
265C22×C4—212no—
266C22×C4—26no—
267C22×C4—424no—
268C22×C4—424no—
269C22×C4—48no—
270C22×C4—812no—
271C22×C4—812no—
272C22×C4—812no—
273C22×C4C4×D4148yes—
274C22×C4C23×C416yes—
275C22×C4C23×C416yes—
276C22×C4C23×C416yes—
277C22×C4C23×C416yes—
278C22×C4C22×D4112yes—
279C22×C4C22×D416yes—
280C22×C4C22×D416yes—
281C22×C4C22×D4112yes—
282C22×C4C2×#4116yes—
283C22×C4C2×#4116yes—
284C4⋊C4—248no—
285C4⋊C4—248no—
286C4⋊C4—812no—
287C22⋊C4—224no—
288C22⋊C4—224no—
289C22⋊C4—224no—
290C22⋊C4—224no—
291C22⋊C4—224no—
292C22⋊C4—424no—
293C22⋊C4—44no—
294C22⋊C4—412no—
295C22⋊C4—48no—
296C22⋊C4—812no—
297C22⋊C4#120116yes—
298C42—248no—
299C42—248no—
300C42—812no—
301C23—22no—
302C23—26no—
303C23—212no—
304C23—212no—
305C23—212no—
306C23—212no—
307C23—26no—
308C23—26no—
309C23—212no—
310C23—212no—
311C23—26no—
312C23—22no—
313C23—26no—
314C23—448no—
315C23—41no—
316C23—43no—
317C23—46no—
318C23—46no—
319C23—41no—
320C23—43no—
321C23—46no—
322C23—46no—
323C23—46no—
324C23—46no—
325C23—416no—
326C23—824no—
327C23—816no—
328C23—848no—
329C23C2×C4×D4112yes—
330C23#51914yescomplemented
331C23C2×#12014yes—
332C23#525112yes—
333C23C24×C411yescharacteristic
334C23C23×D413yes—
335C23C22×#4113yes—
336C23C2×2−1+412yes—
337C23#58516yes—
338Q8—296no—
339Q8—232no—
340C2×C4—212no—
341C2×C4—212no—
342C2×C4—26no—
343C2×C4—26no—
344C2×C4—26no—
345C2×C4—26no—
346C2×C4—212no—
347C2×C4—212no—
348C2×C4—212no—
349C2×C4—212no—
350C2×C4—296no—
351C2×C4—26no—
352C2×C4—26no—
353C2×C4—212no—
354C2×C4—212no—
355C2×C4—212no—
356C2×C4—212no—
357C2×C4—212no—
358C2×C4—412no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—412no—
362C2×C4—46no—
363C2×C4—46no—
364C2×C4—42no—
365C2×C4—46no—
366C2×C4—46no—
367C2×C4—46no—
368C2×C4—412no—
369C2×C4—46no—
370C2×C4—46no—
371C2×C4—412no—
372C2×C4—816no—
373C2×C4—824no—
374C2×C4C2×C4×D4112yes—
375C2×C4C2×C4×D4112yes—
376C2×C4C2×#12018yes—
377C22—212no—
378C22—212no—
379C22—28no—
380C22—224no—
381C22—212no—
382C22—46no—
383C22—42no—
384C22—44no—
385C22—412no—
386C22—46no—
387C22—46no—
388C22—42no—
389C22—832no—
390C22—82no—
391C22—86no—
392C22C4×#12018yes—
393C22C22×C4×D413yes—
394C22C2×#51911yescharacteristic
395C22#342813yes—
396C22C22×#12011yescharacteristic
397C22C2×#52513yes—
398C22#348916yes—
399C22#349516yes—
400C22#350016yes—
401C22#351016yes—
402C4—212no—
403C4—212no—
404C4—28no—
405C4—46no—
406C4—46no—
407C4—412no—
408C4—82no—
409C4—86no—
410C2—28no—
411C2—44no—
412C2—82no—
413C2C2×C4×#12011yescharacteristic
414C2#3420313yes—
415C2#3421113yes—
416C2#3607812yes—
417C2#3607916yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#519C2semidirect
4. #14336C2semidirect
5. #14301C2semidirect
6. #14188C2semidirect
7. #14172C2semidirect
8. #13878C2semidirect
10. C4×#393C2semidirect
11. #13799C2semidirect
14. C22×C4×Q8C22semidirect
15. C22×C4×Q8C22semidirect
20. C2×#393C4semidirect
36. #1446C22semidirect
37. #1446C22semidirect
38. #1444C22semidirect
39. #1442C22semidirect
44. #1436C22semidirect
45. #1436C22semidirect
46. #1434C22semidirect
48. C4×#95C22semidirect
52. C23×Q8C2×C4semidirect
89. #393C2×C4semidirect
90. #393C2×C4semidirect
100. C2×#95C2×C4semidirect
330. C23#519semidirect

Maximal subgroups

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

subgroupindex
C22×#5192
C22×#3932
C2×#23422
#143362
#143012
#141882
#141722
#138782
#138682
C4×#3932
#137992