Finite Group Databasebeta

← #24916319

Subgroups — #24916319

Order 10624.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D83C211yesmaximal, characteristic
2#19128177C211yesmaximal, characteristic, complemented
3#19128173C211yesmaximal, characteristic
4#19127803C211yesmaximal, characteristic, complemented
5#19127802C211yesmaximal, characteristic, complemented
6#19127800C211yesmaximal, characteristic, complemented
7#19127792C211yesmaximal, characteristic, complemented
8C4×#13050577C211yesmaximal, characteristic, complemented
9C4×#13050578C211yesmaximal, characteristic, complemented
10#19127786C211yesmaximal, characteristic, complemented
11#19127738C211yesmaximal, characteristic, complemented
12#19127708C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic
14#19127658C211yesmaximal, characteristic, complemented
15#440×C83C211yesmaximal, characteristic, complemented
16C2×Q8×D83C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050579C2211yescharacteristic
19C2×#13050578C2211yescharacteristic
20C2×#13050577C2211yescharacteristic
21SD16×D83C418yescomplemented
22C2×#13050573C2211yescharacteristic
23C2×C8×D83C2211yescharacteristic
24Q8×Dic83C2211yescharacteristic, complemented
25C4×Dic166—21no—
26C4×Dic166C2211yescharacteristic, complemented
27#14581757—21no—
28#14581757C2211yescharacteristic, complemented
29C4×D332C2211yescharacteristic
30D4×Dic83C2211yescharacteristic
31#14581754C2211yescharacteristic
32C4×C22⋊D83—21no—
33(C22⋊D83)⋊C4—21no—
34#14581750—21no—
35#14581750C2211yescharacteristic
36#14581750C2211yescharacteristic
37#14581749—21no—
38#14581749C2211yescharacteristic
39C2×(C2×D83)⋊C4—21no—
40C22⋊C4×D83—21no—
41#14581726C2211yescharacteristic, complemented
42#14581723C2211yescharacteristic, complemented
43#14581723C2211yescharacteristic, complemented
44#14581717C2211yescharacteristic, complemented
45#14581716C2211yescharacteristic, complemented
46#14581715C2211yescharacteristic, complemented
47#14581714C2211yescharacteristic
48#14581713C2211yescharacteristic
49#14581712C2211yescharacteristic
50#14581702C2211yescharacteristic, complemented
51C4×C83⋊C8C2211yescharacteristic, complemented
52C8×Dic83C2211yescharacteristic, complemented
53C2×SD16×C83C2211yescharacteristic
54C4×Q8×C83C2211yescharacteristic, complemented
55C4×D4×C83C2211yescharacteristic
56#107×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic
59#97×C83C2211yescharacteristic, complemented
60C23×D83—21no—
61C2×Dic166—21no—
62C2×Dic166C2311yescharacteristic
63Q8×D83—22no—
64Q8×D83C2×C414yescomplemented
65C2×D332C2311yescharacteristic
66C2×C22⋊D83—21no—
67D4×D83—22no—
68D4×D83C2×C414yes—
69C22×Dic83—21no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83C2311yescharacteristic
74#13050579C2×C414yescomplemented
75#13050578C2×C414yescomplemented
76#13050577C2×C414yescomplemented
77#13050573—22no—
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C8×D83C2×C414yescomplemented
81C4⋊Dic83—21no—
82C4⋊Dic83C2311yescharacteristic
83C4⋊Dic83C2311yescharacteristic
84Dic83⋊C4—21no—
85Dic83⋊C4—21no—
86Dic83⋊C4D411yescharacteristic
87Dic83⋊C4D411yescharacteristic
88C22⋊Dic83—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4D411yescharacteristic
94(C2×D83)⋊C4D411yescharacteristic
95C4×Dic83—21no—
96C4×Dic83—21no—
97C4×Dic83—21no—
98C4×Dic83—21no—
99C4×Dic83—21no—
100C4×Dic83C2311yescharacteristic
101C2×Q8×C83C2311yescharacteristic
102C2×D4×C83C2311yescharacteristic
103C22×C4×C83—21no—
104SD16×C83C2×C414yescomplemented
105C2×C8×C83C2311yescharacteristic
106C2×C8×C83C2311yescharacteristic
107C4⋊C4×C83—21no—
108C4⋊C4×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C42×C83—21no—
112C42×C83C2311yescharacteristic
113C23×C83—21no—
114C22×D83—21no—
115C22×D83—24no—
116C22×D83—81no—
117C22×D83—81no—
118C22×D83C2×D411yescharacteristic
119Q8×C83—21no—
120Q8×C83C22×C412yes—
121D4×C83—21no—
122D4×C83C22×C412yes—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—81no—
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2411yescharacteristic
130C22⋊D83—24no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—81no—
135C2×Dic83C2×D411yescharacteristic
136C2×Dic83C2×D411yescharacteristic
137C2×Dic83C2×D411yescharacteristic
138D332—21no—
139D332C22×C412yes—
140C4×D83—24no—
141C4×D83—81no—
142C4×D83—81no—
143C4×D83C22×C412yes—
144C4×D83#4112yes—
145Dic166—21no—
146Dic166—24no—
147Dic166C22×C412yes—
148C8×C83—21no—
149C8×C83C22×C412yes—
150C83⋊C8—21no—
151C83⋊C8C22×C412yes—
152C22×C83—22no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83C22×D411yescharacteristic
156C2×D83—21no—
157C2×D83—21no—
158C2×D83—22no—
159C2×D83—41no—
160C2×D83—41no—
161C2×D83—84no—
162C2×D83C4×D412yes—
163C4×C83—22no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83C23×C411yescharacteristic
169C4×C83C2×#4111yescharacteristic
170Dic83—22no—
171Dic83—21no—
172Dic83—21no—
173Dic83—41no—
174Dic83—41no—
175Dic83C4×D412yes—
176C2×C83—42no—
177C2×C83C2×C4×D411yescharacteristic
178C2×C83#57711yescharacteristic
179C2×C83#57711yescharacteristic
180D83—22no—
181D83—42no—
182#2949—831nomaximal, Sylow
183C83#294911yescharacteristic, complemented, Sylow
184C22×SD16—831no—
185#519—831no—
186#518—831no—
187#440—831no—
188#440—831no—
189#440—831no—
190#440D8311yescharacteristic, complemented
191C4×SD16—831no—
192C4×SD16—831no—
193C4×SD16—831no—
194C4×SD16—831no—
195#427—831no—
196#419—831no—
197#418—831no—
198#405—831no—
199C22×Q8—831no—
200C22×D4—831no—
201C2×SD16—831no—
202C2×SD16—831no—
203C2×SD16—838no—
204C2×SD16—831no—
205C2×SD16C2×D8311yescharacteristic
206C2×M16—831no—
207C22×C8—831no—
208C4×Q8—831no—
209C4×Q8—831no—
210C4×Q8—831no—
211C4×Q8—1661no—
212C4×Q8—1661no—
213C4×Q8C2×D8311yescharacteristic, complemented
214C4×D4—831no—
215C4×D4—831no—
216C4×D4—831no—
217C4×D4—1661no—
218C4×D4—1661no—
219C4×D4C2×D8311yescharacteristic
220#117—831no—
221#117—831no—
222#117—831no—
223#117—1661no—
224#117—1661no—
225C2×C22⋊C4—1661no—
226C2×C22⋊C4—1661no—
227#107—831no—
228#107C2×D8311yescharacteristic, complemented
229#106—831no—
230#106—831no—
231#103—831no—
232#103—831no—
233#103—831no—
234#103C2×D8311yescharacteristic, complemented
235#102—831no—
236#102—831no—
237#102—831no—
238#102C2×D8311yescharacteristic
239#97—831no—
240#97C2×D8311yescharacteristic, complemented
241C4×C8—831no—
242C4×C8—831no—
243C24—1661no—
244C2×Q8—834no—
245C2×Q8—831no—
246C2×Q8—1661no—
247C2×Q8—1662no—
248C2×Q8C22×D8311yescharacteristic
249C2×D4—834no—
250C2×D4—831no—
251C2×D4—1662no—
252C2×D4—1661no—
253C2×D4C22×D8311yescharacteristic
254C22×C4—21no—
255C22×C4—831no—
256C22×C4—1661no—
257C22×C4—1661no—
258C22×C4—1661no—
259C22×C4—1661no—
260SD16—834no—
261SD16—834no—
262SD16—834no—
263SD16C4×D8314yescomplemented
264M16—1662no—
265C2×C8—831no—
266C2×C8—834no—
267C2×C8—831no—
268C2×C8C22×D8311yescharacteristic
269C2×C8C22×D8311yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—831no—
272C4⋊C4—831no—
273C4⋊C4—831no—
274C4⋊C4—831no—
275C4⋊C4—1661no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4C22×D8311yescharacteristic
279C4⋊C4C22×D8311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—831no—
282C22⋊C4—831no—
283C22⋊C4—1661no—
284C22⋊C4—1661no—
285C22⋊C4—1661no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C42—21no—
289C42—831no—
290C42—1661no—
291C42—1661no—
292C42—1661no—
293C42—1661no—
294C42—1661no—
295C42C22×D8311yescharacteristic
296C23—21no—
297C23—831no—
298C23—6641no—
299C23—6641no—
300C23—1664no—
301C23—1661no—
302Q8—21no—
303Q8—832no—
304Q8—1664no—
305Q8—1661no—
306Q8C2×C4×D8312yes—
307D4—21no—
308D4—832no—
309D4—1664no—
310D4—1661no—
311D4C2×C4×D8312yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—831no—
317C2×C4—832no—
318C2×C4—832no—
319C2×C4—831no—
320C2×C4—831no—
321C2×C4—6641no—
322C2×C4—6641no—
323C2×C4—6641no—
324C2×C4—1664no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4D4×D8311yescharacteristic
329C2×C4D4×D8311yescharacteristic
330C2×C4C23×D8311yescharacteristic
331C8—21no—
332C8—832no—
333C8—1661no—
334C8C2×C4×D8312yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—3321no—
339C22—3321no—
340C22—832no—
341C22—6644no—
342C22—1662no—
343C22—1661no—
344C22—1661no—
345C22C2×D4×D8311yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—3321no—
352C4—3321no—
353C4—832no—
354C4—1662no—
355C4—1661no—
356C4—1661no—
357C4C22×C4×D8311yescharacteristic
358C4#41×D8311yescharacteristic
359C2—42no—
360C2—3322no—
361C2—1662no—
362C2C4×D4×D8311yescharacteristic
363C2#1912851011yescharacteristic
364C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutatorC4×C83C23×C4
FrattiniC2×C4C23×D83
Fitting#440×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×D83C22
cosocleC2×C4×C83C24
hypercenter#440D83
hypocenterC83#2949
LCSG ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #19128177C2semidirect
4. #19127803C2semidirect
5. #19127802C2semidirect
6. #19127800C2semidirect
7. #19127792C2semidirect
8. C4×#13050577C2semidirect
9. C4×#13050578C2semidirect
10. #19127786C2semidirect
11. #19127738C2semidirect
12. #19127708C2semidirect
14. #19127658C2semidirect
15. #440×C83C2semidirect
21. SD16×D83C4semidirect
24. Q8×Dic83C22semidirect
26. C4×Dic166C22semidirect
28. #14581757C22semidirect
41. #14581726C22semidirect
42. #14581723C22semidirect
43. #14581723C22semidirect
44. #14581717C22semidirect
45. #14581716C22semidirect
46. #14581715C22semidirect
50. #14581702C22semidirect
51. C4×C83⋊C8C22semidirect
52. C8×Dic83C22semidirect
54. C4×Q8×C83C22semidirect
56. #107×C83C22semidirect
57. #103×C83C22semidirect
59. #97×C83C22semidirect
64. Q8×D83C2×C4semidirect
74. #13050579C2×C4semidirect
75. #13050578C2×C4semidirect
76. #13050577C2×C4semidirect
80. C8×D83C2×C4semidirect
104. SD16×C83C2×C4semidirect
183. C83#2949semidirect
190. #440D83semidirect
213. C4×Q8C2×D83semidirect
228. #107C2×D83semidirect
234. #103C2×D83semidirect
240. #97C2×D83semidirect
263. SD16C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D832
#191281772
#191281732
#191278032
#191278022
#191278002
#191277922
C4×#130505772
C4×#130505782
#191277862
#191277382
#191277082
#191277042
#191276582
#440×C832
#294983