Finite Group Databasebeta

← #24919167

Subgroups — #24919167

Order 10624.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581813C211yesmaximal, characteristic, complemented
2C22×#13050579C211yesmaximal, characteristic, complemented
3#19128357C211yesmaximal, characteristic, complemented
4#19128342C211yesmaximal, characteristic, complemented
5#19128082C211yesmaximal, characteristic, complemented
6#19127931C211yesmaximal, characteristic, complemented
7#19127926C211yesmaximal, characteristic, complemented
8#19127915C211yesmaximal, characteristic, complemented
9#19127866C211yesmaximal, characteristic, complemented
10#19127856C211yesmaximal, characteristic, complemented
11#19127847C211yesmaximal, characteristic, complemented
12#19127834C211yesmaximal, characteristic, complemented
13#19127788C211yesmaximal, characteristic, complemented
14C8×C22⋊D83C211yesmaximal, characteristic, complemented
15#465×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×#13050589C2211yescharacteristic, complemented
18C22×Dic166C2211yescharacteristic
19C22×D332C2211yescharacteristic
20#14581813—24no—
21C2×#13050580C2211yescharacteristic, complemented
22C2×#13050580C2211yescharacteristic, complemented
23C2×#13050579—24no—
24C2×#13050579C2211yescharacteristic
25C2×#13050579C2211yescharacteristic
26C2×#13050579C2211yescharacteristic
27C2×#13050579C2211yescharacteristic
28C2×#13050575C2211yescharacteristic
29C2×#13050575C2211yescharacteristic
30C2×C8×D83C2211yescharacteristic, complemented
31#14581781C2211yescharacteristic
32#14581779C2211yescharacteristic, complemented
33#14581778—21no—
34#14581777—21no—
35#14581772C2211yescharacteristic, complemented
36#14581769—21no—
37#14581767—21no—
38#14581767C2211yescharacteristic
39#14581764—21no—
40#14581762—21no—
41#14581757C2211yescharacteristic, complemented
42D4×Dic83—21no—
43#14581754C2211yescharacteristic
44C4×C22⋊D83C2211yescharacteristic
45C22⋊C4×D83—21no—
46Dic83⋊C8C2211yescharacteristic, complemented
47#14581715C2211yescharacteristic, complemented
48#14581715C2211yescharacteristic, complemented
49#14581712C2211yescharacteristic
50#14581712C2211yescharacteristic
51C22⋊(C83⋊C8)C2211yescharacteristic
52(C2×D83)⋊C8C2211yescharacteristic, complemented
53C8×Dic83C2211yescharacteristic, complemented
54C2×SD16×C83C2211yescharacteristic, complemented
55C22×C8×C83C2211yescharacteristic
56C22⋊Q8×C83C2211yescharacteristic
57#121×C83C2211yescharacteristic
58#106×C83C2211yescharacteristic, complemented
59#103×C83C2211yescharacteristic, complemented
60#102×C83C2211yescharacteristic, complemented
61C23×D83—21no—
62#13050591—22no—
63#13050591—41no—
64#13050589—22no—
65#13050589—41no—
66C2×Dic166—22no—
67C2×Dic166—21no—
68C2×Dic166—41no—
69C2×Dic166C2311yescharacteristic
70C2×Dic166C2311yescharacteristic
71C2×D332—22no—
72C2×D332—21no—
73C2×D332—21no—
74C2×D332—21no—
75C2×D332—41no—
76C2×D332C2311yescharacteristic
77C2×D332C2311yescharacteristic
78C2×C22⋊D83—21no—
79C2×C22⋊D83D411yescharacteristic
80C22×Dic83—21no—
81C22×Dic83—21no—
82C2×C4×D83—21no—
83C2×C4×D83—21no—
84C2×C4×D83C2311yescharacteristic
85#13050580—22no—
86#13050580—41no—
87#13050579—22no—
88#13050579—22no—
89#13050579—22no—
90#13050579D414yescomplemented
91#13050575—22no—
92#13050575—41no—
93C2×C83⋊C8C2311yescharacteristic
94C2×C83⋊C8C2311yescharacteristic
95C8×D83—22no—
96C4⋊Dic83—21no—
97Dic83⋊C4—21no—
98Dic83⋊C4D411yescharacteristic
99C22⋊Dic83—21no—
100C22⋊Dic83—21no—
101C22⋊Dic83D411yescharacteristic
102(C2×D83)⋊C4—21no—
103(C2×D83)⋊C4—21no—
104(C2×D83)⋊C4—21no—
105(C2×D83)⋊C4—21no—
106(C2×D83)⋊C4—21no—
107(C2×D83)⋊C4D411yescharacteristic
108C4×Dic83—21no—
109C4×Dic83C2311yescharacteristic
110C2×Q8×C83C2311yescharacteristic
111C2×D4×C83—21no—
112C2×D4×C83C2311yescharacteristic
113C22×C4×C83C2311yescharacteristic
114SD16×C83—22no—
115C2×C8×C83—22no—
116C2×C8×C83C2311yescharacteristic
117C2×C8×C83C2311yescharacteristic
118C4⋊C4×C83—21no—
119C4⋊C4×C83C2311yescharacteristic
120C4⋊C4×C83C2311yescharacteristic
121C22⋊C4×C83—21no—
122C22⋊C4×C83—21no—
123C23×C83—21no—
124C23×C83C2×D411yescharacteristic
125C22×D83—21no—
126C22×D83—21no—
127C22×D83—42no—
128C22×D83—42no—
129C22×D83—81no—
130C22×D83C2×D411yescharacteristic
131Q8×C83—21no—
132Q8×C83—21no—
133D4×C83—21no—
134D4×C83—21no—
135D4×C83—42no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83C2×D411yescharacteristic
142C2×C4×C83C2411yescharacteristic
143C22⋊D83—22no—
144C22⋊D83—42no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—42no—
148C2×Dic83—42no—
149C2×Dic83C2×D411yescharacteristic
150C2×Dic83C2×D411yescharacteristic
151C2×Dic83C2×D411yescharacteristic
152D332—21no—
153D332—21no—
154D332—21no—
155D332—42no—
156D332—42no—
157D332—81no—
158D332C2×D412yes—
159C4×D83—21no—
160C4×D83—21no—
161C4×D83—42no—
162C4×D83—81no—
163Dic166—21no—
164Dic166—21no—
165Dic166—21no—
166Dic166—42no—
167Dic166C2×D412yes—
168C8×C83—21no—
169C8×C83C2×D412yes—
170C83⋊C8—21no—
171C83⋊C8—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—41no—
175C22×C83—41no—
176C22×C83—41no—
177C22×C83#13512yescomplemented
178C22×C83C22×D411yescharacteristic
179C2×D83—21no—
180C2×D83—22no—
181C2×D83—21no—
182C2×D83—21no—
183C2×D83—41no—
184C2×D83—41no—
185C2×D83—41no—
186C2×D83—41no—
187C2×D83—41no—
188C2×D83—84no—
189C4×C83—21no—
190C4×C83—41no—
191C4×C83—41no—
192C4×C83—41no—
193C4×C83C22×D411yescharacteristic
194C4×C83H(5,𝔽2)11yescharacteristic
195Dic83—22no—
196Dic83—21no—
197Dic83—21no—
198Dic83—21no—
199Dic83—41no—
200C2×C83—22no—
201C2×C83—81no—
202C2×C83D4211yescharacteristic
203C2×C83C2×#13511yescharacteristic
204C2×C83#57711yescharacteristic
205D83—41no—
206D83—42no—
207D83—81no—
208#3282—831nomaximal, Sylow
209C83#328211yescharacteristic, complemented, Sylow
210C2×#135—831no—
211C22×SD16—831no—
212#548—831no—
213#546—831no—
214#495—831no—
215#465—831no—
216#465D8311yescharacteristic, complemented
217#464—831no—
218#463—831no—
219#452—831no—
220#451—831no—
221#449—831no—
222#447—831no—
223C4×SD16—831no—
224C8×D4—831no—
225C2×#41—831no—
226C2×#41—831no—
227C22×Q8—831no—
228C22×D4—831no—
229#135—1664no—
230C2×Q16—831no—
231C2×Q16—831no—
232C2×SD16—831no—
233C2×SD16—831no—
234C2×SD16—831no—
235C2×SD16—831no—
236C2×SD16—1664no—
237C2×SD16C2×D8311yescharacteristic, complemented
238C2×D8—831no—
239C2×D8—831no—
240C22×C8—831no—
241C22×C8C2×D8311yescharacteristic
242#124—831no—
243#124—831no—
244#124—1661no—
245#123—1661no—
246C22⋊Q8—831no—
247C22⋊Q8—1661no—
248C22⋊Q8C2×D8311yescharacteristic
249#121—831no—
250#121—1661no—
251#121—1661no—
252#121C2×D8311yescharacteristic
253#120—1661no—
254C4×Q8—831no—
255C4×D4—831no—
256C4×D4—831no—
257C4×D4—1661no—
258C2×C22⋊C4—1661no—
259#106C2×D8311yescharacteristic, complemented
260C4⋊C8—831no—
261#103—831no—
262#103—831no—
263#103C2×D8311yescharacteristic, complemented
264#102—831no—
265#102—831no—
266#102C2×D8311yescharacteristic, complemented
267C22⋊C8—831no—
268C22⋊C8—831no—
269C4×C8—831no—
270C24—1661no—
271#41—3321no—
272#41—3321no—
273#41—1662no—
274#41—1662no—
275C2×Q8—3321no—
276C2×Q8—831no—
277C2×Q8—831no—
278C2×Q8—1662no—
279C2×Q8—1661no—
280C2×Q8C22×D8311yescharacteristic
281C2×D4—21no—
282C2×D4—3321no—
283C2×D4—831no—
284C2×D4—831no—
285C2×D4—831no—
286C2×D4—1661no—
287C2×D4—1662no—
288C2×D4—1661no—
289C2×D4—1661no—
290C2×D4—1661no—
291C2×D4C22×D8311yescharacteristic
292C22×C4—831no—
293C22×C4—1661no—
294C22×C4—1661no—
295C22×C4—1661no—
296C22×C4—1661no—
297C22×C4C22×D8311yescharacteristic
298Q16—3321no—
299Q16—1662no—
300SD16—22no—
301SD16—834no—
302SD16—1662no—
303SD16—1662no—
304SD16—1662no—
305D8—3321no—
306D8—1662no—
307C2×C8—22no—
308C2×C8—831no—
309C2×C8—831no—
310C2×C8—1662no—
311C2×C8C22×D8311yescharacteristic
312C2×C8C22×D8311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—831no—
315C4⋊C4—1661no—
316C4⋊C4—1661no—
317C4⋊C4C22×D8311yescharacteristic
318C4⋊C4C22×D8311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—831no—
322C22⋊C4—831no—
323C22⋊C4—1661no—
324C22⋊C4—1661no—
325C22⋊C4—1661no—
326C22⋊C4—1661no—
327C22⋊C4—1661no—
328C22⋊C4—1661no—
329C22⋊C4—1661no—
330C42—831no—
331C42—1661no—
332C23—21no—
333C23—3322no—
334C23—3322no—
335C23—831no—
336C23—6641no—
337C23—1661no—
338C23—1661no—
339C23D4×D8311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3322no—
343Q8—832no—
344Q8—1661no—
345Q8—1661no—
346Q8—1661no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—3322no—
351D4—3322no—
352D4—3322no—
353D4—832no—
354D4—6641no—
355D4—1662no—
356D4—1661no—
357D4—1661no—
358D4—1661no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—3322no—
365C2×C4—3322no—
366C2×C4—3322no—
367C2×C4—831no—
368C2×C4—831no—
369C2×C4—831no—
370C2×C4—6641no—
371C2×C4—1661no—
372C2×C4—1661no—
373C2×C4—1661no—
374C2×C4—1661no—
375C2×C4D4×D8311yescharacteristic
376C2×C4C23×D8311yescharacteristic
377C8—21no—
378C8—1661no—
379C8—1661no—
380C8D4×D8312yes—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3321no—
387C22—3321no—
388C22—3321no—
389C22—3321no—
390C22—3321no—
391C22—6644no—
392C22—1662no—
393C22—1661no—
394C22—1661no—
395C22—1661no—
396C22#1458181312yescomplemented
397C22C2×D4×D8311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—3321no—
403C4—1661no—
404C4—1662no—
405C4—1661no—
406C4—1661no—
407C4C2×D4×D8311yescharacteristic
408C4#1458183811yescharacteristic
409C2—22no—
410C2—81no—
411C2—3322no—
412C2—3321no—
413C2—6641no—
414C2#1912833511yescharacteristic
415C2C2×#1458181311yescharacteristic
416C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#465×C83C2
socleC22×C83C22×D4
hypercenter#465D83
hypocenterC83#3282
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
1. C2×#14581813C2semidirect
2. C22×#13050579C2semidirect
3. #19128357C2semidirect
4. #19128342C2semidirect
5. #19128082C2semidirect
6. #19127931C2semidirect
7. #19127926C2semidirect
8. #19127915C2semidirect
9. #19127866C2semidirect
10. #19127856C2semidirect
11. #19127847C2semidirect
12. #19127834C2semidirect
13. #19127788C2semidirect
14. C8×C22⋊D83C2semidirect
15. #465×C83C2semidirect
17. C2×#13050589C22semidirect
21. C2×#13050580C22semidirect
22. C2×#13050580C22semidirect
30. C2×C8×D83C22semidirect
32. #14581779C22semidirect
35. #14581772C22semidirect
41. #14581757C22semidirect
46. Dic83⋊C8C22semidirect
47. #14581715C22semidirect
48. #14581715C22semidirect
52. (C2×D83)⋊C8C22semidirect
53. C8×Dic83C22semidirect
54. C2×SD16×C83C22semidirect
58. #106×C83C22semidirect
59. #103×C83C22semidirect
60. #102×C83C22semidirect
90. #13050579D4semidirect
177. C22×C83#135semidirect
209. C83#3282semidirect
216. #465D83semidirect
237. C2×SD16C2×D83semidirect
259. #106C2×D83semidirect
263. #103C2×D83semidirect
266. #102C2×D83semidirect
396. C22#14581813semidirect

Maximal subgroups

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

subgroupindex
C2×#145818132
C22×#130505792
#191283572
#191283422
#191280822
#191279312
#191279262
#191279152
#191278662
#191278562
#191278472
#191278342
#191277882
C8×C22⋊D832
#465×C832
#328283