Finite Group Databasebeta

← #24918245

Subgroups — #24918245

Order 10624.

420 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128214C211yesmaximal, characteristic
2C22⋊Q8×D83C211yesmaximal, characteristic, complemented
3#19128044C211yesmaximal, characteristic, complemented
4#19128040C211yesmaximal, characteristic, complemented
5#19128030C211yesmaximal, characteristic, complemented
6#19128016C211yesmaximal, characteristic, complemented
7#19127940C211yesmaximal, characteristic, complemented
8#19127932C211yesmaximal, characteristic, complemented
9#19127930C211yesmaximal, characteristic, complemented
10#19127866C211yesmaximal, characteristic, complemented
11#19127862C211yesmaximal, characteristic, complemented
12#19127855C211yesmaximal, characteristic, complemented
13#19127841C211yesmaximal, characteristic, complemented
14C2×(C2×D83)⋊C8C211yesmaximal, characteristic, complemented
15#467×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×Q8×D83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C2×#13050580C2211yescharacteristic, complemented
20C2×#13050579C2211yescharacteristic, complemented
21C2×#13050577C2211yescharacteristic, complemented
22C22×C83⋊C8C2211yescharacteristic
23#14581777—21no—
24#14581772C2211yescharacteristic
25#14581771C2211yescharacteristic
26#14581770—21no—
27#14581769—21no—
28#14581768C2211yescharacteristic
29#14581768C2211yescharacteristic
30#14581767C2211yescharacteristic
31#14581766C2211yescharacteristic
32#14581762—21no—
33#14581754—21no—
34(C22⋊D83)⋊C4—21no—
35#14581750C2211yescharacteristic
36C4⋊C4×D83—21no—
37C4⋊C4×D83C2211yescharacteristic
38C22⋊C4×D83—21no—
39#14581725C2211yescharacteristic, complemented
40#14581724C2211yescharacteristic, complemented
41#14581723C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581716C2211yescharacteristic, complemented
45#14581715C2211yescharacteristic, complemented
46#14581714C2211yescharacteristic, complemented
47#14581712C2211yescharacteristic, complemented
48(C2×D83)⋊C8C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic
52C2×Q16×C83C2211yescharacteristic, complemented
53C22×C8×C83C2211yescharacteristic
54C22⋊Q8×C83C2211yescharacteristic
55C22⋊Q8×C83C2211yescharacteristic
56#107×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#103×C83C2211yescharacteristic, complemented
59C23×D83D411yescharacteristic
60#13050591—41no—
61#13050591—41no—
62C2×Dic166—21no—
63C2×Dic166C2311yescharacteristic
64Q8×D83—41no—
65Q8×D83—41no—
66C2×D332—21no—
67C2×D332—21no—
68C2×D332C2311yescharacteristic
69C2×C22⋊D83—21no—
70C22×Dic83D411yescharacteristic
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83—21no—
74C2×C4×D83—41no—
75C2×C4×D83—41no—
76C2×C4×D83D411yescharacteristic
77C2×C4×D83D411yescharacteristic
78C2×C4×D83C2311yescharacteristic
79C2×C4×D83C2311yescharacteristic
80#13050580—41no—
81#13050579—41no—
82#13050577—41no—
83C2×C83⋊C8—41no—
84C2×C83⋊C8C2311yescharacteristic
85C2×C83⋊C8C2311yescharacteristic
86C4⋊Dic83—21no—
87C4⋊Dic83C2311yescharacteristic
88C4⋊Dic83C2311yescharacteristic
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91Dic83⋊C4—21no—
92Dic83⋊C4—21no—
93C22⋊Dic83—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98(C2×D83)⋊C4—21no—
99(C2×D83)⋊C4—21no—
100C4×Dic83—21no—
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C22×C4×C83C2311yescharacteristic
104Q16×C83—41no—
105C2×C8×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83—21no—
110C4⋊C4×C83C2311yescharacteristic
111C4⋊C4×C83C2311yescharacteristic
112C22⋊C4×C83—21no—
113C22⋊C4×C83—21no—
114C23×C83C2×D411yescharacteristic
115C22×D83—22no—
116C22×D83—22no—
117C22×D83—21no—
118C22×D83—41no—
119C22×D83C2×D411yescharacteristic
120C22×D83C2×D411yescharacteristic
121Q8×C83—21no—
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2411yescharacteristic
133C22⋊D83—42no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—41no—
138C2×Dic83C2×D411yescharacteristic
139C2×Dic83C2×D411yescharacteristic
140D332—21no—
141D332—21no—
142D332—42no—
143D332—42no—
144C4×D83—21no—
145C4×D83—21no—
146C4×D83—21no—
147C4×D83—21no—
148C4×D83—41no—
149C4×D83—41no—
150C4×D83—42no—
151C4×D83—42no—
152C4×D83—42no—
153Dic166—21no—
154Dic166—21no—
155Dic166—42no—
156C8×C83—21no—
157C8×C83—21no—
158C83⋊C8—21no—
159C83⋊C8—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83#13712yes—
163C22×C83C22×D411yescharacteristic
164C2×D83—21no—
165C2×D83—21no—
166C2×D83—21no—
167C2×D83—21no—
168C2×D83—21no—
169C2×D83—44no—
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83#13512yes—
174C4×C83—21no—
175C4×C83—41no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83H(5,𝔽2)11yescharacteristic
180C4×C83H(5,𝔽2)11yescharacteristic
181Dic83—21no—
182Dic83—21no—
183Dic83—41no—
184Dic83—41no—
185Dic83—41no—
186C2×C83—22no—
187C2×C83#53411yescharacteristic
188C2×C83C2×#13511yescharacteristic
189C2×C83C2×#13711yescharacteristic
190D83—22no—
191D83—41no—
192D83—81no—
193#3190—831nomaximal, Sylow
194C83#319011yescharacteristic, complemented, Sylow
195#525—831no—
196C2×C22⋊Q8—831no—
197#485—831no—
198#485—831no—
199#483—831no—
200#480—831no—
201#467—831no—
202#467D8311yescharacteristic, complemented
203#465—831no—
204#465—831no—
205#452—831no—
206#452—831no—
207#451—831no—
208#448—831no—
209C2×C22⋊C8—831no—
210C2×#41—831no—
211C22×Q8—831no—
212C23×C4—831no—
213C2×Q16—831no—
214C2×Q16C2×D8311yescharacteristic, complemented
215C2×SD16—831no—
216C2×SD16—831no—
217C22×C8—831no—
218C22×C8C2×D8311yescharacteristic
219#123—1661no—
220C22⋊Q8—831no—
221C22⋊Q8—831no—
222C22⋊Q8—831no—
223C22⋊Q8—831no—
224C22⋊Q8—1661no—
225C22⋊Q8—1661no—
226C22⋊Q8C2×D8311yescharacteristic
227C22⋊Q8C2×D8311yescharacteristic
228#121—831no—
229#121—831no—
230#120—1661no—
231C4×D4—1661no—
232C4×D4—1661no—
233#117—831no—
234C2×C4⋊C4—831no—
235C2×C4⋊C4—1661no—
236C2×C22⋊C4—1661no—
237#107—831no—
238#107C2×D8311yescharacteristic, complemented
239#106—831no—
240#106—831no—
241#103—831no—
242#103—831no—
243#103—831no—
244#103—831no—
245#103C2×D8311yescharacteristic, complemented
246#103C2×D8311yescharacteristic, complemented
247#102—831no—
248#102—831no—
249C22⋊C8—831no—
250C22⋊C8—831no—
251C22⋊C8—831no—
252C22⋊C8—831no—
253C24—831no—
254#41—3321no—
255#41—3321no—
256C2×Q8—3321no—
257C2×Q8—3321no—
258C2×Q8—831no—
259C2×Q8—1661no—
260C2×Q8C22×D8311yescharacteristic
261C2×Q8C22×D8311yescharacteristic
262C2×D4—831no—
263C2×D4—1661no—
264C2×D4—1661no—
265C2×D4—1661no—
266C22×C4—3321no—
267C22×C4—3321no—
268C22×C4—831no—
269C22×C4—831no—
270C22×C4—831no—
271C22×C4—831no—
272C22×C4—831no—
273C22×C4—1661no—
274C22×C4—1661no—
275C22×C4—1661no—
276C22×C4C22×D8311yescharacteristic
277Q16—41no—
278Q16—3321no—
279SD16—3321no—
280SD16—3321no—
281C2×C8—41no—
282C2×C8—3321no—
283C2×C8—831no—
284C2×C8—831no—
285C2×C8C22×D8311yescharacteristic
286C2×C8C22×D8311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—831no—
290C4⋊C4—831no—
291C4⋊C4—1661no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
294C4⋊C4—1661no—
295C4⋊C4—1661no—
296C4⋊C4C22×D8311yescharacteristic
297C4⋊C4C22×D8311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1661no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C42—1661no—
308C23—3321no—
309C23—831no—
310C23—831no—
311C23—1662no—
312C23—1661no—
313C23—1662no—
314C23D4×D8311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—3322no—
320Q8—1661no—
321Q8—1661no—
322D4—3322no—
323D4—3322no—
324D4—3322no—
325D4—1661no—
326D4—1661no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3321no—
334C2×C4—3321no—
335C2×C4—3322no—
336C2×C4—3322no—
337C2×C4—3322no—
338C2×C4—3321no—
339C2×C4—831no—
340C2×C4—831no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4D4×D8311yescharacteristic
349C2×C4C23×D8311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1661no—
353C8—1661no—
354C22—21no—
355C22—21no—
356C22—3324no—
357C22—3321no—
358C22—3321no—
359C22—3321no—
360C22—832no—
361C22—1661no—
362C22—1661no—
363C22—1661no—
364C22—1661no—
365C22—1661no—
366C22#1458182412yes—
367C22C2×D4×D8311yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3321no—
374C4—3321no—
375C4—3321no—
376C4—1661no—
377C4—1661no—
378C4#1458183711yescharacteristic
379C4#1458183811yescharacteristic
380C2—22no—
381C2—3321no—
382C2—6641no—
383C2—1662no—
384C2#1912825911yescharacteristic
385C2#135×D8311yescharacteristic
386C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#467×C83C2
socleC22×C83C22×D4
prime radical#19128214C2
hypercenter#467D83
hypocenterC83#3190
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D83C2semidirect
3. #19128044C2semidirect
4. #19128040C2semidirect
5. #19128030C2semidirect
6. #19128016C2semidirect
7. #19127940C2semidirect
8. #19127932C2semidirect
9. #19127930C2semidirect
10. #19127866C2semidirect
11. #19127862C2semidirect
12. #19127855C2semidirect
13. #19127841C2semidirect
14. C2×(C2×D83)⋊C8C2semidirect
15. #467×C83C2semidirect
19. C2×#13050580C22semidirect
20. C2×#13050579C22semidirect
21. C2×#13050577C22semidirect
39. #14581725C22semidirect
40. #14581724C22semidirect
41. #14581723C22semidirect
42. #14581717C22semidirect
43. #14581716C22semidirect
44. #14581716C22semidirect
45. #14581715C22semidirect
46. #14581714C22semidirect
47. #14581712C22semidirect
52. C2×Q16×C83C22semidirect
56. #107×C83C22semidirect
57. #103×C83C22semidirect
58. #103×C83C22semidirect
194. C83#3190semidirect
202. #467D83semidirect
214. C2×Q16C2×D83semidirect
238. #107C2×D83semidirect
245. #103C2×D83semidirect
246. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
C22⋊Q8×D832
#191280442
#191280402
#191280302
#191280162
#191279402
#191279322
#191279302
#191278662
#191278622
#191278552
#191278412
C2×(C2×D83)⋊C82
#467×C832
#319083