Finite Group Databasebeta

← #24918281

Subgroups — #24918281

Order 10624.

466 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128213C211yesmaximal, characteristic, complemented
2#121×D83C211yesmaximal, characteristic, complemented
3#19128033C211yesmaximal, characteristic, complemented
4#19128025C211yesmaximal, characteristic, complemented
5#19128016C211yesmaximal, characteristic, complemented
6#19127955C211yesmaximal, characteristic, complemented
7#19127953C211yesmaximal, characteristic, complemented
8#19127948C211yesmaximal, characteristic, complemented
9#19127946C211yesmaximal, characteristic, complemented
10#19127853C211yesmaximal, characteristic, complemented
11#19127848C211yesmaximal, characteristic, complemented
12#19127844C211yesmaximal, characteristic, complemented
13#19127836C211yesmaximal, characteristic, complemented
14#19127663C211yesmaximal, characteristic, complemented
15#482×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic, complemented
17C2×D4×D83—21no—
18C2×D4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C2×#13050579C2211yescharacteristic, complemented
21C2×#13050578C2211yescharacteristic, complemented
22C2×D664C2211yescharacteristic, complemented
23C2×#13050575C2211yescharacteristic, complemented
24C2×#13050573C2211yescharacteristic, complemented
25C2×#13050573C2211yescharacteristic, complemented
26#14581776—21no—
27#14581772C2211yescharacteristic
28#14581772C2211yescharacteristic, complemented
29#14581767C2211yescharacteristic
30#14581767C2211yescharacteristic
31#14581767C2211yescharacteristic, complemented
32#14581766C2211yescharacteristic
33#14581765—21no—
34#14581764—21no—
35#14581763C2211yescharacteristic
36#14581762—21no—
37C4×D332—21no—
38C4×D332—21no—
39C4⋊C4×D83C2211yescharacteristic
40C22⋊C4×D83—21no—
41#14581725C2211yescharacteristic, complemented
42#14581723C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic, complemented
46#14581712C2211yescharacteristic, complemented
47#14581712C2211yescharacteristic, complemented
48#14581712C2211yescharacteristic, complemented
49C22⋊(C83⋊C8)C2211yescharacteristic, complemented
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic
52#121×C83C2211yescharacteristic
53#117×C83C2211yescharacteristic, complemented
54#107×C83C2211yescharacteristic, complemented
55#106×C83C2211yescharacteristic, complemented
56#102×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic, complemented
58C22⋊C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C23×D83D411yescharacteristic
61#13050590—41no—
62#13050590—41no—
63C2×Dic166C2311yescharacteristic
64C2×D332—21no—
65C2×D332—21no—
66C2×D332—21no—
67C2×D332C2311yescharacteristic
68C2×D332C2311yescharacteristic
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C2×C22⋊D83—21no—
73D4×D83—41no—
74D4×D83—41no—
75D4×D83—44no—
76C22×Dic83D411yescharacteristic
77C2×C4×D83—21no—
78C2×C4×D83—21no—
79C2×C4×D83—41no—
80C2×C4×D83—41no—
81C2×C4×D83D411yescharacteristic
82C2×C4×D83D411yescharacteristic
83C2×C4×D83C2311yescharacteristic
84C2×C4×D83C2311yescharacteristic
85#13050579—41no—
86#13050578—41no—
87D664—41no—
88#13050575—41no—
89#13050573—41no—
90#13050573—41no—
91C2×C83⋊C8C2311yescharacteristic
92C2×C83⋊C8C2311yescharacteristic
93C4⋊Dic83—21no—
94C4⋊Dic83—21no—
95C4⋊Dic83C2311yescharacteristic
96Dic83⋊C4—21no—
97C22⋊Dic83—21no—
98(C2×D83)⋊C4—21no—
99(C2×D83)⋊C4—21no—
100(C2×D83)⋊C4—21no—
101(C2×D83)⋊C4—21no—
102(C2×D83)⋊C4—21no—
103(C2×D83)⋊C4—21no—
104C2×D4×C83—21no—
105C2×D4×C83C2311yescharacteristic
106C22×C4×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C4⋊C4×C83C2311yescharacteristic
111C4⋊C4×C83C2311yescharacteristic
112C22⋊C4×C83—21no—
113C22⋊C4×C83—21no—
114C42×C83—21no—
115C23×C83—21no—
116C23×C83C2×D411yescharacteristic
117C22×D83—22no—
118C22×D83—21no—
119C22×D83—21no—
120C22×D83—41no—
121C22×D83—41no—
122C22×D83—42no—
123C22×D83—42no—
124C22×D83—42no—
125C22×D83C2×D411yescharacteristic
126C22×D83C2×D411yescharacteristic
127D4×C83—21no—
128D4×C83—21no—
129D4×C83—42no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2411yescharacteristic
137C22⋊D83—42no—
138C22⋊D83—42no—
139C22⋊D83—42no—
140C22⋊D83—42no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—41no—
144C2×Dic83C2×D411yescharacteristic
145C2×Dic83C2×D411yescharacteristic
146D332—21no—
147D332—21no—
148D332—21no—
149D332—21no—
150D332—42no—
151D332—42no—
152D332—42no—
153C4×D83—21no—
154C4×D83—21no—
155C4×D83—21no—
156C4×D83—21no—
157C4×D83—41no—
158C4×D83—41no—
159C4×D83—42no—
160C4×D83—42no—
161Dic166—21no—
162Dic166—21no—
163C8×C83—21no—
164C8×C83—21no—
165C83⋊C8—21no—
166C83⋊C8—21no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—41no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83C22×D411yescharacteristic
174C2×D83—21no—
175C2×D83—21no—
176C2×D83—21no—
177C2×D83—21no—
178C2×D83—21no—
179C2×D83—44no—
180C2×D83—41no—
181C2×D83—41no—
182C2×D83—41no—
183C2×D83—41no—
184C2×D83—41no—
185C2×D83—41no—
186C2×D83—84no—
187C2×D83AGL(1,ℤ/8)12yes—
188C4×C83—21no—
189C4×C83—41no—
190C4×C83—41no—
191C4×C83—41no—
192C4×C83H(5,𝔽2)11yescharacteristic
193C4×C83H(5,𝔽2)11yescharacteristic
194Dic83—21no—
195Dic83—21no—
196Dic83—41no—
197Dic83—41no—
198C2×C83—41no—
199C2×C83—81no—
200C2×C83#53411yescharacteristic
201C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
202C2×C83#57511yescharacteristic
203D83—22no—
204D83—41no—
205D83—81no—
206D83—81no—
207#3194—831nomaximal, Sylow
208C83#319411yescharacteristic, complemented, Sylow
209#525—831no—
210C2×#121—831no—
211#483—831no—
212#482—831no—
213#482D8311yescharacteristic, complemented
214#480—831no—
215#469—831no—
216#469—831no—
217#468—831no—
218#468—831no—
219#450—831no—
220#449—831no—
221#449—831no—
222#447—831no—
223#407—831no—
224C2×#41—831no—
225C22×D4—831no—
226C22×D4—1661no—
227C23×C4—831no—
228C2×SD16—831no—
229C2×SD16—831no—
230C2×D8—831no—
231C2×D8—831no—
232C2×M16—831no—
233C2×M16—831no—
234#123—1661no—
235C22⋊Q8—831no—
236C22⋊Q8—831no—
237#121—831no—
238#121—831no—
239#121—831no—
240#121—831no—
241#121—831no—
242#121—1661no—
243#121—1661no—
244#121C2×D8311yescharacteristic
245#120—1661no—
246C4×D4—1661no—
247C4×D4—1661no—
248#117C2×D8311yescharacteristic, complemented
249C2×C4⋊C4—831no—
250C2×C22⋊C4—1661no—
251#107—831no—
252#107C2×D8311yescharacteristic, complemented
253#106—831no—
254#106C2×D8311yescharacteristic, complemented
255#103—831no—
256#103—831no—
257#102—831no—
258#102—831no—
259#102—831no—
260#102—831no—
261#102C2×D8311yescharacteristic, complemented
262#102C2×D8311yescharacteristic, complemented
263C22⋊C8—831no—
264C22⋊C8—831no—
265C22⋊C8—831no—
266C22⋊C8C2×D8311yescharacteristic, complemented
267C24—831no—
268C24—1661no—
269#41—3321no—
270#41—3321no—
271C2×Q8—831no—
272C2×D4—21no—
273C2×D4—3324no—
274C2×D4—3321no—
275C2×D4—3321no—
276C2×D4—831no—
277C2×D4—831no—
278C2×D4—1661no—
279C2×D4—1661no—
280C2×D4—1661no—
281C2×D4—1661no—
282C2×D4—1661no—
283C2×D4—1661no—
284C2×D4—1661no—
285C2×D4C22×D8311yescharacteristic
286C22×C4—3321no—
287C22×C4—3321no—
288C22×C4—831no—
289C22×C4—831no—
290C22×C4—831no—
291C22×C4—831no—
292C22×C4—831no—
293C22×C4—1661no—
294C22×C4—1661no—
295C22×C4C22×D8311yescharacteristic
296SD16—3321no—
297SD16—3321no—
298D8—3321no—
299D8—3321no—
300M16—3321no—
301M16—3321no—
302C2×C8—831no—
303C2×C8—831no—
304C2×C8C22×D8311yescharacteristic
305C2×C8C22×D8311yescharacteristic
306C4⋊C4—831no—
307C4⋊C4—1661no—
308C4⋊C4—1661no—
309C4⋊C4—1661no—
310C4⋊C4C22×D8311yescharacteristic
311C4⋊C4C22×D8311yescharacteristic
312C4⋊C4C22×D8311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—1661no—
316C22⋊C4—1661no—
317C22⋊C4—1661no—
318C22⋊C4—1661no—
319C22⋊C4—1661no—
320C22⋊C4—1661no—
321C22⋊C4—1661no—
322C42—21no—
323C23—21no—
324C23—3321no—
325C23—3321no—
326C23—3322no—
327C23—3322no—
328C23—3322no—
329C23—831no—
330C23—831no—
331C23—1662no—
332C23—1661no—
333C23—1661no—
334C23D4×D8311yescharacteristic
335Q8—1661no—
336Q8—1661no—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—3322no—
341D4—3322no—
342D4—3322no—
343D4—3322no—
344D4—3322no—
345D4—3322no—
346D4—3322no—
347D4—1661no—
348D4—1661no—
349D4—1661no—
350D4—1661no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—3322no—
357C2×C4—3321no—
358C2×C4—3321no—
359C2×C4—3322no—
360C2×C4—3321no—
361C2×C4—831no—
362C2×C4—831no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4—1661no—
368C2×C4—1661no—
369C2×C4D4×D8311yescharacteristic
370C2×C4C23×D8311yescharacteristic
371C8—21no—
372C8—21no—
373C8—1661no—
374C8—1661no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—3324no—
382C22—3321no—
383C22—3321no—
384C22—3321no—
385C22—3321no—
386C22—3321no—
387C22—3321no—
388C22—832no—
389C22—6644no—
390C22—1661no—
391C22—1661no—
392C22—1661no—
393C22—1661no—
394C22—1661no—
395C22C2×D4×D8311yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—3321no—
401C4—3321no—
402C4—1661no—
403C4—1661no—
404C4#1458183811yescharacteristic
405C4#1458183811yescharacteristic
406C2—41no—
407C2—81no—
408C2—3321no—
409C2—6641no—
410C2—6641no—
411C2—1662no—
412C2#1912826211yescharacteristic
413C2AGL(1,ℤ/8)×D8311yescharacteristic
414C2#1912849811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#482×C83C2
socleC22×C83C22×D4
hypercenter#482D83
hypocenterC83#3194
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #19128213C2semidirect
2. #121×D83C2semidirect
3. #19128033C2semidirect
4. #19128025C2semidirect
5. #19128016C2semidirect
6. #19127955C2semidirect
7. #19127953C2semidirect
8. #19127948C2semidirect
9. #19127946C2semidirect
10. #19127853C2semidirect
11. #19127848C2semidirect
12. #19127844C2semidirect
13. #19127836C2semidirect
14. #19127663C2semidirect
15. #482×C83C2semidirect
16. C2×#13050590C22semidirect
20. C2×#13050579C22semidirect
21. C2×#13050578C22semidirect
22. C2×D664C22semidirect
23. C2×#13050575C22semidirect
24. C2×#13050573C22semidirect
25. C2×#13050573C22semidirect
28. #14581772C22semidirect
31. #14581767C22semidirect
41. #14581725C22semidirect
42. #14581723C22semidirect
43. #14581717C22semidirect
44. #14581715C22semidirect
45. #14581714C22semidirect
46. #14581712C22semidirect
47. #14581712C22semidirect
48. #14581712C22semidirect
49. C22⋊(C83⋊C8)C22semidirect
53. #117×C83C22semidirect
54. #107×C83C22semidirect
55. #106×C83C22semidirect
56. #102×C83C22semidirect
57. #102×C83C22semidirect
58. C22⋊C8×C83C22semidirect
208. C83#3194semidirect
213. #482D83semidirect
248. #117C2×D83semidirect
252. #107C2×D83semidirect
254. #106C2×D83semidirect
261. #102C2×D83semidirect
262. #102C2×D83semidirect
266. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282132
#121×D832
#191280332
#191280252
#191280162
#191279552
#191279532
#191279482
#191279462
#191278532
#191278482
#191278442
#191278362
#191276632
#482×C832
#319483