Finite Group Databasebeta

← #24918239

Subgroups — #24918239

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#19128211C211yesmaximal, characteristic
2C22⋊Q8×D83C211yesmaximal, characteristic, complemented
3#19128045C211yesmaximal, characteristic, complemented
4#19128041C211yesmaximal, characteristic, complemented
5#19128033C211yesmaximal, characteristic, complemented
6#19128013C211yesmaximal, characteristic, complemented
7#19127942C211yesmaximal, characteristic, complemented
8#19127940C211yesmaximal, characteristic, complemented
9#19127928C211yesmaximal, characteristic, complemented
10#19127864C211yesmaximal, characteristic, complemented
11#19127860C211yesmaximal, characteristic, complemented
12#19127858C211yesmaximal, characteristic, complemented
13#19127838C211yesmaximal, characteristic, complemented
14C2×(C2×D83)⋊C8C211yesmaximal, characteristic, complemented
15#465×C83C211yesmaximal, characteristic, complemented
16C2×#13050589C2211yescharacteristic
17C2×Q8×D83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C2×Dic332C2211yescharacteristic, complemented
20C2×#13050580C2211yescharacteristic, complemented
21C2×#13050578C2211yescharacteristic, complemented
22C22×C83⋊C8C2211yescharacteristic
23#14581774—21no—
24#14581772C2211yescharacteristic
25#14581772C2211yescharacteristic
26#14581771C2211yescharacteristic
27#14581771C2211yescharacteristic
28#14581770—21no—
29#14581769—21no—
30#14581768C2211yescharacteristic
31#14581763C2211yescharacteristic
32#14581760—21no—
33D4×Dic83—21no—
34(C22⋊D83)⋊C4—21no—
35#14581750C2211yescharacteristic
36C4⋊C4×D83—21no—
37C4⋊C4×D83C2211yescharacteristic
38C22⋊C4×D83—21no—
39#14581726C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41#14581723C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581716C2211yescharacteristic, complemented
45#14581715C2211yescharacteristic, complemented
46#14581715C2211yescharacteristic, complemented
47#14581713C2211yescharacteristic, complemented
48(C2×D83)⋊C8C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic
52C2×SD16×C83C2211yescharacteristic, complemented
53C22×C8×C83C2211yescharacteristic
54C22⋊Q8×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56#106×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic, complemented
59C23×D83D411yescharacteristic
60#13050589—41no—
61#13050589—41no—
62C2×Dic166—21no—
63C2×Dic166C2311yescharacteristic
64C2×Dic166C2311yescharacteristic
65Q8×D83—41no—
66Q8×D83—41no—
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C22×Dic83—21no—
70C22×Dic83D411yescharacteristic
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83—41no—
74C2×C4×D83—41no—
75C2×C4×D83D411yescharacteristic
76C2×C4×D83D411yescharacteristic
77C2×C4×D83C2311yescharacteristic
78C2×C4×D83C2311yescharacteristic
79Dic332—41no—
80#13050580—41no—
81#13050578—41no—
82C2×C83⋊C8—41no—
83C2×C83⋊C8C2311yescharacteristic
84C2×C83⋊C8C2311yescharacteristic
85C4⋊Dic83—21no—
86C4⋊Dic83—21no—
87C4⋊Dic83C2311yescharacteristic
88C4⋊Dic83C2311yescharacteristic
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91Dic83⋊C4—21no—
92Dic83⋊C4—21no—
93C22⋊Dic83—21no—
94C22⋊Dic83—21no—
95C22⋊Dic83—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×D4×C83—21no—
103C2×D4×C83C2311yescharacteristic
104C22×C4×C83C2311yescharacteristic
105SD16×C83—41no—
106C2×C8×C83—41no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83—21no—
110C4⋊C4×C83C2311yescharacteristic
111C4⋊C4×C83C2311yescharacteristic
112C22⋊C4×C83—21no—
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—
123D4×C83—21no—
124D4×C83—21no—
125D4×C83—42no—
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—
134C22⋊D83—42no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—41no—
140C2×Dic83—42no—
141C2×Dic83C2×D411yescharacteristic
142C2×Dic83C2×D411yescharacteristic
143C4×D83—21no—
144C4×D83—21no—
145C4×D83—21no—
146C4×D83—21no—
147C4×D83—41no—
148C4×D83—41no—
149C4×D83—42no—
150C4×D83—42no—
151Dic166—21no—
152Dic166—21no—
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—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83#13712yes—
166C22×C83C22×D411yescharacteristic
167C2×D83—21no—
168C2×D83—21no—
169C2×D83—21no—
170C2×D83—21no—
171C2×D83—21no—
172C2×D83—44no—
173C2×D83#13512yes—
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—22no—
187C2×C83—81no—
188C2×C83#53411yescharacteristic
189C2×C83C2×#13511yescharacteristic
190C2×C83C2×#13711yescharacteristic
191D83—22no—
192D83—41no—
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#465D8311yescharacteristic, complemented
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×Q16—831no—
215C2×SD16—831no—
216C2×SD16C2×D8311yescharacteristic, complemented
217C22×C8—831no—
218C22×C8C2×D8311yescharacteristic
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#121C2×D8311yescharacteristic
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#106C2×D8311yescharacteristic, complemented
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⋊C8—831no—
253C24—831no—
254#41—3321no—
255#41—3321no—
256C2×Q8—3321no—
257C2×Q8—3321no—
258C2×Q8—831no—
259C2×Q8—831no—
260C2×Q8—1661no—
261C2×Q8C22×D8311yescharacteristic
262C2×D4—21no—
263C2×D4—1661no—
264C2×D4—1661no—
265C2×D4C22×D8311yescharacteristic
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—3321no—
278Q16—3321no—
279SD16—41no—
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—831no—
289C4⋊C4—831no—
290C4⋊C4—1661no—
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—21no—
309C23—3321no—
310C23—831no—
311C23—831no—
312C23—1662no—
313C23—1662no—
314C23D4×D8311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—3322no—
318Q8—1661no—
319Q8—1661no—
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—3321no—
333C2×C4—3321no—
334C2×C4—3322no—
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—41no—
357C22—41no—
358C22—41no—
359C22—3324no—
360C22—832no—
361C22—1661no—
362C22—1661no—
363C22—1661no—
364C22—1661no—
365C22—1661no—
366C22#1458182112yes—
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#1458183811yescharacteristic
380C2—22no—
381C2—81no—
382C2—3321no—
383C2—1662no—
384C2#1912825911yescharacteristic
385C2#135×D8311yescharacteristic
386C2C2×#1458182111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#465×C83C2
socleC22×C83C22×D4
prime radical#19128211C2
hypercenter#465D83
hypocenterC83#3190
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D83C2semidirect
3. #19128045C2semidirect
4. #19128041C2semidirect
5. #19128033C2semidirect
6. #19128013C2semidirect
7. #19127942C2semidirect
8. #19127940C2semidirect
9. #19127928C2semidirect
10. #19127864C2semidirect
11. #19127860C2semidirect
12. #19127858C2semidirect
13. #19127838C2semidirect
14. C2×(C2×D83)⋊C8C2semidirect
15. #465×C83C2semidirect
19. C2×Dic332C22semidirect
20. C2×#13050580C22semidirect
21. C2×#13050578C22semidirect
39. #14581726C22semidirect
40. #14581725C22semidirect
41. #14581723C22semidirect
42. #14581717C22semidirect
43. #14581717C22semidirect
44. #14581716C22semidirect
45. #14581715C22semidirect
46. #14581715C22semidirect
47. #14581713C22semidirect
52. C2×SD16×C83C22semidirect
56. #106×C83C22semidirect
57. #103×C83C22semidirect
58. #102×C83C22semidirect
194. C83#3190semidirect
204. #465D83semidirect
216. C2×SD16C2×D83semidirect
240. #106C2×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
#191282112
C22⋊Q8×D832
#191280452
#191280412
#191280332
#191280132
#191279422
#191279402
#191279282
#191278642
#191278602
#191278582
#191278382
C2×(C2×D83)⋊C82
#465×C832
#319083