Finite Group Databasebeta

← #24918242

Subgroups — #24918242

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#19128208C211yesmaximal, characteristic
2C22⋊Q8×D83C211yesmaximal, characteristic, complemented
3#19128043C211yesmaximal, characteristic, complemented
4#19128039C211yesmaximal, characteristic, complemented
5#19128030C211yesmaximal, characteristic, complemented
6#19128013C211yesmaximal, characteristic, complemented
7#19127941C211yesmaximal, characteristic, complemented
8#19127939C211yesmaximal, characteristic, complemented
9#19127929C211yesmaximal, characteristic, complemented
10#19127927C211yesmaximal, characteristic, complemented
11#19127863C211yesmaximal, characteristic, complemented
12#19127855C211yesmaximal, characteristic, complemented
13#19127838C211yesmaximal, characteristic, complemented
14C22⋊C8×D83C211yesmaximal, characteristic, complemented
15#452×C83C211yesmaximal, characteristic, complemented
16C2×Q8×D83C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050578C2211yescharacteristic, complemented
20C2×C8×D83C2211yescharacteristic
21C2×C8×D83C2211yescharacteristic
22#14581773—21no—
23#14581772C2211yescharacteristic
24#14581771C2211yescharacteristic
25#14581770—21no—
26#14581769—21no—
27#14581768C2211yescharacteristic
28#14581768C2211yescharacteristic
29#14581768C2211yescharacteristic
30#14581763C2211yescharacteristic
31#14581763C2211yescharacteristic
32#14581760—21no—
33C4×C22⋊D83—21no—
34C4×C22⋊D83—21no—
35#14581751C2211yescharacteristic
36C4⋊C4×D83—21no—
37C4⋊C4×D83C2211yescharacteristic
38C22⋊C4×D83—21no—
39#14581726C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41#14581724C2211yescharacteristic, complemented
42#14581723C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581717C2211yescharacteristic, complemented
45#14581716C2211yescharacteristic, complemented
46#14581716C2211yescharacteristic, complemented
47#14581716C2211yescharacteristic, complemented
48#14581713C2211yescharacteristic, complemented
49C22⋊(C83⋊C8)C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic
52C2×#41×C83C2211yescharacteristic
53C2×Q16×C83C2211yescharacteristic, complemented
54C2×SD16×C83C2211yescharacteristic, complemented
55C22⋊Q8×C83C2211yescharacteristic
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic, complemented
58C22⋊C8×C83C2211yescharacteristic
59C23×D83D411yescharacteristic
60C2×Dic166—21no—
61C2×Dic166C2311yescharacteristic
62Q8×D83—41no—
63Q8×D83—41no—
64C2×C22⋊D83—21no—
65C2×C22⋊D83—21no—
66C22×Dic83D411yescharacteristic
67C2×C4×D83—21no—
68C2×C4×D83—21no—
69C2×C4×D83—41no—
70C2×C4×D83—41no—
71C2×C4×D83D411yescharacteristic
72C2×C4×D83D411yescharacteristic
73C2×C4×D83C2311yescharacteristic
74C2×C4×D83C2311yescharacteristic
75#13050580—41no—
76#13050578—41no—
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C8×D83—41no—
80C8×D83—41no—
81C4⋊Dic83—21no—
82C4⋊Dic83C2311yescharacteristic
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89Dic83⋊C4—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83—21no—
92C22⋊Dic83—21no—
93C22⋊Dic83—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4—21no—
98C4×Dic83—21no—
99#41×C83—41no—
100#41×C83—41no—
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83—21no—
104C2×D4×C83C2311yescharacteristic
105C22×C4×C83—21no—
106C22×C4×C83C2311yescharacteristic
107Q16×C83—41no—
108SD16×C83—41no—
109C2×C8×C83C2311yescharacteristic
110C2×C8×C83C2311yescharacteristic
111C4⋊C4×C83—21no—
112C4⋊C4×C83C2311yescharacteristic
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83C2×D411yescharacteristic
116C22×D83—22no—
117C22×D83—22no—
118C22×D83—41no—
119C22×D83C2×D411yescharacteristic
120C22×D83C2×D411yescharacteristic
121Q8×C83—21no—
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—42no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—42no—
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2411yescharacteristic
136C22⋊D83—42no—
137C22⋊D83—42no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—41no—
143C2×Dic83C2×D411yescharacteristic
144C2×Dic83C2×D411yescharacteristic
145C4×D83—21no—
146C4×D83—21no—
147C4×D83—21no—
148C4×D83—21no—
149C4×D83—41no—
150C4×D83—41no—
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—21no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83C22×D411yescharacteristic
167C2×D83—21no—
168C2×D83—21no—
169C2×D83—21no—
170C2×D83—21no—
171C2×D83—44no—
172C2×D83#13512yes—
173C2×D83#13712yes—
174C4×C83—21no—
175C4×C83—41no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83H(5,𝔽2)11yescharacteristic
179C4×C83H(5,𝔽2)11yescharacteristic
180Dic83—21no—
181Dic83—21no—
182Dic83—41no—
183Dic83—41no—
184Dic83—41no—
185Dic83—41no—
186C2×C83—41no—
187C2×C83—81no—
188C2×C83#53411yescharacteristic
189C2×C83C2×#13511yescharacteristic
190C2×C83C2×#13711yescharacteristic
191D83—22no—
192D83—22no—
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#467—831no—
203#465—831no—
204#465—831no—
205#452—831no—
206#452D8311yescharacteristic, complemented
207#451—831no—
208#448—831no—
209C2×C22⋊C8—831no—
210C2×#41C2×D8311yescharacteristic
211C22×Q8—831no—
212C23×C4—831no—
213C2×Q16—831no—
214C2×Q16C2×D8311yescharacteristic, complemented
215C2×SD16—831no—
216C2×SD16C2×D8311yescharacteristic, complemented
217C22×C8—831no—
218C22×C8—831no—
219#123—1661no—
220C22⋊Q8—831no—
221C22⋊Q8—831no—
222C22⋊Q8—831no—
223C22⋊Q8—831no—
224C22⋊Q8—831no—
225C22⋊Q8—1661no—
226C22⋊Q8—1661no—
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#107—831no—
239#106—831no—
240#106—831no—
241#103—831no—
242#103—831no—
243#103—831no—
244#103—831no—
245#103—831no—
246#103C2×D8311yescharacteristic, complemented
247#102—831no—
248#102C2×D8311yescharacteristic, complemented
249C22⋊C8—831no—
250C22⋊C8—831no—
251C22⋊C8—831no—
252C22⋊C8C2×D8311yescharacteristic
253C24—831no—
254#41—41no—
255#41—41no—
256C2×Q8—3321no—
257C2×Q8—3321no—
258C2×Q8—831no—
259C2×Q8—1661no—
260C2×Q8C22×D8311yescharacteristic
261C2×Q8C22×D8311yescharacteristic
262C2×D4—21no—
263C2×D4—1661no—
264C2×D4—1661no—
265C2×D4C22×D8311yescharacteristic
266C22×C4—21no—
267C22×C4—3321no—
268C22×C4—3321no—
269C22×C4—831no—
270C22×C4—831no—
271C22×C4—831no—
272C22×C4—831no—
273C22×C4—831no—
274C22×C4—1661no—
275C22×C4—1661no—
276C22×C4C22×D8311yescharacteristic
277Q16—41no—
278Q16—3321no—
279SD16—41no—
280SD16—3321no—
281C2×C8—3321no—
282C2×C8—3321no—
283C2×C8—831no—
284C2×C8—831no—
285C2×C8C22×D8311yescharacteristic
286C2×C8C22×D8311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—831no—
289C4⋊C4—831no—
290C4⋊C4—831no—
291C4⋊C4—1661no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
294C4⋊C4—1661no—
295C4⋊C4—1661no—
296C4⋊C4—1661no—
297C4⋊C4C22×D8311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—1661no—
300C22⋊C4—1661no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C42—1661no—
308C23—21no—
309C23—3321no—
310C23—831no—
311C23—831no—
312C23—1662no—
313C23—1662no—
314C23D4×D8311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—3322no—
320Q8—1661no—
321Q8—1661no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—3322no—
326D4—3322no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—3321no—
334C2×C4—3321no—
335C2×C4—3322no—
336C2×C4—3322no—
337C2×C4—3321no—
338C2×C4—831no—
339C2×C4—831no—
340C2×C4—1661no—
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—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3324no—
361C22—832no—
362C22—832no—
363C22—1661no—
364C22—1661no—
365C22—1661no—
366C22—1661no—
367C22C2×D4×D8311yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—3321no—
373C4—3321no—
374C4—3321no—
375C4—3321no—
376C4—1661no—
377C4—1661no—
378C4#1458183711yescharacteristic
379C4#1458183711yescharacteristic
380C2—41no—
381C2—81no—
382C2—1662no—
383C2—1662no—
384C2#1912826111yescharacteristic
385C2#135×D8311yescharacteristic
386C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#452×C83C2
socleC22×C83C22×D4
prime radical#19128208C2
hypercenter#452D83
hypocenterC83#3190
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D83C2semidirect
3. #19128043C2semidirect
4. #19128039C2semidirect
5. #19128030C2semidirect
6. #19128013C2semidirect
7. #19127941C2semidirect
8. #19127939C2semidirect
9. #19127929C2semidirect
10. #19127927C2semidirect
11. #19127863C2semidirect
12. #19127855C2semidirect
13. #19127838C2semidirect
14. C22⋊C8×D83C2semidirect
15. #452×C83C2semidirect
18. C2×#13050580C22semidirect
19. C2×#13050578C22semidirect
39. #14581726C22semidirect
40. #14581725C22semidirect
41. #14581724C22semidirect
42. #14581723C22semidirect
43. #14581717C22semidirect
44. #14581717C22semidirect
45. #14581716C22semidirect
46. #14581716C22semidirect
47. #14581716C22semidirect
48. #14581713C22semidirect
53. C2×Q16×C83C22semidirect
54. C2×SD16×C83C22semidirect
56. #103×C83C22semidirect
57. #102×C83C22semidirect
194. C83#3190semidirect
206. #452D83semidirect
214. C2×Q16C2×D83semidirect
216. C2×SD16C2×D83semidirect
246. #103C2×D83semidirect
248. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282082
C22⋊Q8×D832
#191280432
#191280392
#191280302
#191280132
#191279412
#191279392
#191279292
#191279272
#191278632
#191278552
#191278382
C22⋊C8×D832
#452×C832
#319083