Finite Group Databasebeta

← #24919200

Subgroups — #24919200

Order 10624.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581812C211yesmaximal, characteristic, complemented
2C22×#13050580C211yesmaximal, characteristic, complemented
3#19128360C211yesmaximal, characteristic, complemented
4#19128083C211yesmaximal, characteristic, complemented
5#19127940C211yesmaximal, characteristic, complemented
6#19127935C211yesmaximal, characteristic, complemented
7#19127923C211yesmaximal, characteristic, complemented
8#19127920C211yesmaximal, characteristic, complemented
9#19127863C211yesmaximal, characteristic, complemented
10#19127861C211yesmaximal, characteristic, complemented
11#19127840C211yesmaximal, characteristic, complemented
12#19127839C211yesmaximal, characteristic, complemented
13C4×#13050580C211yesmaximal, characteristic, complemented
14D4×C83⋊C8C211yesmaximal, characteristic, complemented
15#548×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×Dic166C2211yescharacteristic
18#14581812—24no—
19C2×#13050580—24no—
20C2×#13050580C2211yescharacteristic
21C2×#13050580C2211yescharacteristic
22C2×#13050580C2211yescharacteristic
23C2×#13050580C2211yescharacteristic
24C2×#13050578C2211yescharacteristic
25C2×#13050578C2211yescharacteristic
26C2×#13050577C2211yescharacteristic
27C2×#13050577C2211yescharacteristic
28C2×#13050575C2211yescharacteristic
29C22×C83⋊C8C2211yescharacteristic
30C22×C83⋊C8C2211yescharacteristic
31#14581780C2211yescharacteristic
32#14581778—21no—
33#14581771C2211yescharacteristic
34#14581770—21no—
35#14581766C2211yescharacteristic
36#14581764—21no—
37C4×Dic166C2211yescharacteristic
38(C22⋊D83)⋊C4—21no—
39#14581725C2211yescharacteristic
40C4⋊(C83⋊C8)C2211yescharacteristic
41#14581716C2211yescharacteristic
42#14581716C2211yescharacteristic
43#14581715C2211yescharacteristic
44#14581715C2211yescharacteristic
45#14581713C2211yescharacteristic
46#14581712C2211yescharacteristic
47C22⋊(C83⋊C8)C2211yescharacteristic
48C22⋊(C83⋊C8)C2211yescharacteristic
49C4×C83⋊C8C2211yescharacteristic
50C2×#41×C83C2211yescharacteristic
51C22×Q8×C83C2211yescharacteristic
52#124×C83—21no—
53#124×C83C2211yescharacteristic
54C22⋊Q8×C83—21no—
55C22⋊Q8×C83C2211yescharacteristic
56#121×C83—21no—
57#121×C83C2211yescharacteristic
58C4×Q8×C83C2211yescharacteristic
59C4×D4×C83—21no—
60C4×D4×C83C2211yescharacteristic
61#13050590—22no—
62#13050590—41no—
63C2×Dic166—22no—
64C2×Dic166—21no—
65C2×Dic166—41no—
66C2×Dic166C2311yescharacteristic
67C2×Dic166C2311yescharacteristic
68C2×D332C2311yescharacteristic
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C22×Dic83—21no—
72C2×C4×D83—21no—
73#13050580—22no—
74#13050580—22no—
75#13050580—22no—
76#13050580D414yescomplemented
77#13050578—22no—
78#13050578—41no—
79#13050577—22no—
80#13050577—41no—
81#13050575—22no—
82C2×C83⋊C8—22no—
83C2×C83⋊C8—22no—
84C2×C83⋊C8C2311yescharacteristic
85C2×C83⋊C8C2311yescharacteristic
86C2×C83⋊C8C2311yescharacteristic
87C2×C83⋊C8C2311yescharacteristic
88C4⋊Dic83C2311yescharacteristic
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95C4×Dic83—21no—
96#41×C83—22no—
97#41×C83—41no—
98C2×Q8×C83—21no—
99C2×Q8×C83—22no—
100C2×Q8×C83—41no—
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83—21no—
104C2×D4×C83—21no—
105C2×D4×C83D411yescharacteristic
106C2×D4×C83C2311yescharacteristic
107C22×C4×C83—21no—
108C22×C4×C83—21no—
109C22×C4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111C4⋊C4×C83—21no—
112C4⋊C4×C83—21no—
113C4⋊C4×C83D411yescharacteristic
114C4⋊C4×C83C2311yescharacteristic
115C22⋊C4×C83—21no—
116C22⋊C4×C83—21no—
117C22⋊C4×C83—21no—
118C22⋊C4×C83—21no—
119C22⋊C4×C83D411yescharacteristic
120C22⋊C4×C83D411yescharacteristic
121C42×C83—21no—
122C42×C83C2311yescharacteristic
123C23×C83—21no—
124C23×C83C2×D411yescharacteristic
125C23×C83C2×D411yescharacteristic
126C22×D83—21no—
127Q8×C83—21no—
128Q8×C83—21no—
129Q8×C83—21no—
130Q8×C83—42no—
131Q8×C83C2×D412yes—
132D4×C83—22no—
133D4×C83—21no—
134D4×C83—21no—
135D4×C83—42no—
136D4×C83—42no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
146C2×C4×C83C2×D411yescharacteristic
147C2×C4×C83C2×D411yescharacteristic
148C2×C4×C83C2×D411yescharacteristic
149C2×C4×C83C2×D411yescharacteristic
150C2×C4×C83C2411yescharacteristic
151C22⋊D83—42no—
152C22⋊D83—42no—
153C2×Dic83—21no—
154C2×Dic83—21no—
155C2×Dic83—21no—
156C2×Dic83—42no—
157D332—21no—
158D332—21no—
159C4×D83—42no—
160Dic166—21no—
161Dic166—21no—
162Dic166—21no—
163Dic166—42no—
164Dic166C2×D412yes—
165C83⋊C8—21no—
166C83⋊C8—21no—
167C83⋊C8—21no—
168C83⋊C8C2×D412yes—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—41no—
175C22×C83—41no—
176C22×C83—41no—
177C22×C83#13512yescomplemented
178C22×C83C22×D411yescharacteristic
179C2×D83—41no—
180C2×D83—41no—
181C2×D83—41no—
182C4×C83—21no—
183C4×C83—21no—
184C4×C83—22no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—41no—
188C4×C83—41no—
189C4×C83C22×D411yescharacteristic
190C4×C83H(5,𝔽2)11yescharacteristic
191Dic83—22no—
192Dic83—41no—
193Dic83—41no—
194C2×C83—22no—
195C2×C83—41no—
196C2×C83—81no—
197C2×C83D4211yescharacteristic
198C2×C83C2×#13511yescharacteristic
199C2×C83#57811yescharacteristic
200D83—81no—
201#3285—831nomaximal, Sylow
202C83#328511yescharacteristic, complemented, Sylow
203C2×#135—831no—
204C22×Q16—831no—
205#548—831no—
206#548D8311yescharacteristic, complemented
207#495—831no—
208#467—831no—
209#466—831no—
210#464—831no—
211#464—831no—
212#452—831no—
213#452—831no—
214#448—831no—
215#448—831no—
216C4×Q16—831no—
217C8×D4—831no—
218C2×#41—831no—
219C2×#41C2×D8311yescharacteristic
220C22×Q8—831no—
221C22×Q8C2×D8311yescharacteristic
222#135—1664no—
223C2×Q16—831no—
224C2×Q16—831no—
225C2×Q16—831no—
226C2×Q16—831no—
227C2×Q16—1664no—
228C2×SD16—831no—
229C2×SD16—831no—
230C2×SD16—831no—
231C2×SD16—831no—
232C2×D8—831no—
233C22×C8—831no—
234C22×C8—831no—
235#124—21no—
236#124—831no—
237#124—1661no—
238#124C2×D8311yescharacteristic
239C22⋊Q8—21no—
240C22⋊Q8—831no—
241C22⋊Q8—1661no—
242C22⋊Q8C2×D8311yescharacteristic
243#121—21no—
244#121—831no—
245#121—1661no—
246#121C2×D8311yescharacteristic
247C4×Q8—831no—
248C4×Q8C2×D8311yescharacteristic
249C4×D4—21no—
250C4×D4—1661no—
251C4×D4C2×D8311yescharacteristic
252#107—831no—
253C4⋊C8—831no—
254#103—831no—
255#103—831no—
256#103—831no—
257#103—831no—
258#102—831no—
259#102—831no—
260C22⋊C8—831no—
261C22⋊C8—831no—
262C4×C8—831no—
263#41—22no—
264#41—41no—
265#41—3321no—
266#41—1662no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—3321no—
271C2×Q8—831no—
272C2×Q8—831no—
273C2×Q8—1662no—
274C2×Q8—1661no—
275C2×Q8C22×D8311yescharacteristic
276C2×Q8C22×D8311yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—831no—
280C2×D4—1661no—
281C2×D4—1661no—
282C2×D4C22⋊D8311yescharacteristic
283C2×D4C22×D8311yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—1661no—
287C22×C4—1661no—
288C22×C4C22×D8311yescharacteristic
289C22×C4C22×D8311yescharacteristic
290Q16—834no—
291Q16—1662no—
292Q16—1662no—
293Q16—1662no—
294SD16—3321no—
295SD16—3321no—
296SD16—1662no—
297SD16—1662no—
298D8—1662no—
299C2×C8—831no—
300C2×C8—831no—
301C2×C8—831no—
302C2×C8—831no—
303C2×C8—1662no—
304C2×C8—1662no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—831no—
308C4⋊C4—1661no—
309C4⋊C4—1661no—
310C4⋊C4C22⋊D8311yescharacteristic
311C4⋊C4C22×D8311yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—1661no—
317C22⋊C4—1661no—
318C22⋊C4—1661no—
319C22⋊C4—1661no—
320C22⋊C4C22⋊D8311yescharacteristic
321C22⋊C4C22⋊D8311yescharacteristic
322C42—21no—
323C42—1661no—
324C42C22×D8311yescharacteristic
325C23—21no—
326C23—1661no—
327C23C2×C22⋊D8311yescharacteristic
328C23C2×C22⋊D8311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—3322no—
334Q8—832no—
335Q8—1661no—
336Q8—1661no—
337Q8—1661no—
338Q8D4×D8312yes—
339D4—22no—
340D4—21no—
341D4—21no—
342D4—42no—
343D4—42no—
344D4—3322no—
345D4—3322no—
346D4—1661no—
347D4—1661no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—3322no—
358C2×C4—3322no—
359C2×C4—1661no—
360C2×C4—1661no—
361C2×C4—1661no—
362C2×C4C2×C22⋊D8311yescharacteristic
363C2×C4C2×C22⋊D8311yescharacteristic
364C2×C4C2×C22⋊D8311yescharacteristic
365C2×C4C2×C22⋊D8311yescharacteristic
366C2×C4C23×D8311yescharacteristic
367C8—832no—
368C8—1661no—
369C8—1661no—
370C8—1661no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—3321no—
380C22—3321no—
381C22—3321no—
382C22#1458181212yescomplemented
383C22C22×C22⋊D8311yescharacteristic
384C4—21no—
385C4—21no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—3321no—
392C4—3321no—
393C4—1662no—
394C4C2×D4×D8311yescharacteristic
395C4#1458183711yescharacteristic
396C2—22no—
397C2—41no—
398C2—81no—
399C2—6641no—
400C2D4×C22⋊D8311yescharacteristic
401C2C2×#1458181211yescharacteristic
402C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#548×C83C2
socleC22×C83C22×D4
hypercenter#548D83
hypocenterC83#3285
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#14581812C2semidirect
2. C22×#13050580C2semidirect
3. #19128360C2semidirect
4. #19128083C2semidirect
5. #19127940C2semidirect
6. #19127935C2semidirect
7. #19127923C2semidirect
8. #19127920C2semidirect
9. #19127863C2semidirect
10. #19127861C2semidirect
11. #19127840C2semidirect
12. #19127839C2semidirect
13. C4×#13050580C2semidirect
14. D4×C83⋊C8C2semidirect
15. #548×C83C2semidirect
76. #13050580D4semidirect
177. C22×C83#135semidirect
202. C83#3285semidirect
206. #548D83semidirect
382. C22#14581812semidirect

Maximal subgroups

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

subgroupindex
C2×#145818122
C22×#130505802
#191283602
#191280832
#191279402
#191279352
#191279232
#191279202
#191278632
#191278612
#191278402
#191278392
C4×#130505802
D4×C83⋊C82
#548×C832
#328583