Finite Group Databasebeta

← #24917106

Subgroups — #24917106

Order 10624.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×D83C211yesmaximal, characteristic, complemented
2C2×#14581798C211yesmaximal, characteristic, complemented
3#19128267C212yesmaximal, complemented
4#19127953C212yesmaximal, complemented
5#19127952C212yesmaximal, complemented
6#19127938C211yesmaximal, characteristic, complemented
7#19127937C211yesmaximal, characteristic, complemented
8#19127934C211yesmaximal, characteristic, complemented
9#19127738C211yesmaximal, characteristic, complemented
10#19127704C212yesmaximal, complemented
11#466×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×D4×D83C2212yes—
14C2×D664C2211yescharacteristic
15C2×#13050575C2212yes—
16D8×D83—24no—
17#14581798—24no—
18C2×#13050574C2211yescharacteristic, complemented
19C2×#13050573C2211yescharacteristic
20C2×#13050573C2211yescharacteristic, complemented
21C2×C8×D83C2211yescharacteristic
22C2×C8×D83C2211yescharacteristic, complemented
23#14581788—22no—
24#14581778—22no—
25#14581767C2212yescomplemented
26#14581766C2212yescomplemented
27#14581763C2212yescomplemented
28#14581762—22no—
29#14581760—22no—
30#14581750C2212yes—
31#14581726C2211yescharacteristic, complemented
32#14581723C2212yescomplemented
33#14581714C2212yescomplemented
34#14581713C2212yescomplemented
35#14581712C2212yescomplemented
36C2×D8×C83C2211yescharacteristic
37C22×C8×C83C2211yescharacteristic, complemented
38#121×C83C2212yescomplemented
39#107×C83C2211yescharacteristic, complemented
40#102×C83C2212yescomplemented
41C23×D83—22no—
42#13050590—22no—
43#13050590—22no—
44C2×Dic166D411yescharacteristic
45C2×D332—22no—
46C2×D332D411yescharacteristic
47C2×D332C2312yes—
48C2×C22⋊D83—22no—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83D411yescharacteristic
52C2×C22⋊D83D411yescharacteristic
53D4×D83—24no—
54D4×D83—24no—
55C2×C4×D83C2311yescharacteristic
56C2×C4×D83C2311yescharacteristic
57D664—22no—
58#13050575—24no—
59#13050574—22no—
60#13050573—22no—
61#13050573—22no—
62C2×C83⋊C8C2311yescharacteristic
63C2×C83⋊C8C2311yescharacteristic
64C8×D83—22no—
65C8×D83D414yescomplemented
66C4⋊Dic83C2312yes—
67C22⋊Dic83—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—22no—
71C4×Dic83—22no—
72C2×D4×C83—22no—
73C2×D4×C83C2312yes—
74C22×C4×C83C2311yescharacteristic
75D8×C83—22no—
76C2×C8×C83—22no—
77C2×C8×C83C2311yescharacteristic
78C2×C8×C83C2311yescharacteristic
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—22no—
81C23×C83—22no—
82C23×C83C2×D411yescharacteristic
83C22×D83—22no—
84C22×D83—44no—
85C22×D83—82no—
86C22×D83—82no—
87C22×D83C2×D411yescharacteristic
88C22×D83C2×D411yescharacteristic
89D4×C83—22no—
90D4×C83—22no—
91D4×C83—82no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2411yescharacteristic
97C22⋊D83—22no—
98C22⋊D83—22no—
99C22⋊D83—44no—
100C22⋊D83—82no—
101C22⋊D83—82no—
102C2×Dic83—22no—
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105D332—22no—
106D332—22no—
107D332—22no—
108D332—82no—
109C4×D83—21no—
110C4×D83—21no—
111C4×D83C2×D412yes—
112C4×D83#4112yes—
113Dic166—22no—
114C8×C83—21no—
115C8×C83C2×D412yes—
116C83⋊C8—21no—
117C83⋊C8C2×D412yes—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—42no—
122C22×C83—42no—
123C22×C83—42no—
124C22×C83C22×D411yescharacteristic
125C2×D83—21no—
126C2×D83—21no—
127C2×D83—21no—
128C2×D83—21no—
129C2×D83—21no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—88no—
134C2×D83#12112yes—
135C4×C83—21no—
136C4×C83—42no—
137C4×C83C22×D411yescharacteristic
138C4×C83C2×#4111yescharacteristic
139Dic83—21no—
140Dic83—42no—
141Dic83#12112yes—
142C2×C83—41no—
143C2×C83—82no—
144C2×C83C2×#12111yescharacteristic
145C2×C83#57611yescharacteristic
146C2×C83#57611yescharacteristic
147D83—22no—
148D83—41no—
149D83—82no—
150#3058—831nomaximal, Sylow
151C83#305811yescharacteristic, complemented, Sylow
152C22×D8—831no—
153C2×#131—831no—
154#535—832no—
155#469—832no—
156#469—832no—
157#466—831no—
158#466—831no—
159#466—831no—
160#466D8311yescharacteristic, complemented
161#427—831no—
162#418—832no—
163C2×#41—831no—
164C22×D4—832no—
165C2×D8—832no—
166C2×D8—831no—
167C2×D8—1664no—
168C2×D8C2×D8311yescharacteristic
169#131—1664no—
170C2×M16—831no—
171C2×M16—831no—
172C2×M16—831no—
173C22×C8—831no—
174C22×C8—831no—
175C22×C8C2×D8311yescharacteristic, complemented
176#127—1662no—
177#124—1662no—
178#121—832no—
179#121—832no—
180#121—832no—
181#121C2×D8312yescomplemented
182#120—1662no—
183#120—1662no—
184#117—832no—
185#107—831no—
186#107C2×D8311yescharacteristic, complemented
187#106—832no—
188#102—832no—
189#102—832no—
190#102—832no—
191#102C2×D8312yescomplemented
192C24—1662no—
193#41—1662no—
194#41—1662no—
195C2×Q8—831no—
196C2×D4—22no—
197C2×D4—831no—
198C2×D4—831no—
199C2×D4—831no—
200C2×D4—832no—
201C2×D4—1662no—
202C2×D4—1664no—
203C2×D4—1664no—
204C2×D4—1662no—
205C2×D4—1662no—
206C2×D4—1662no—
207C2×D4C22×D8312yes—
208C22×C4—831no—
209C22×C4—831no—
210C22×C4C22×D8311yescharacteristic
211D8—22no—
212D8—1664no—
213D8—1662no—
214M16—1662no—
215M16—1662no—
216M16—1662no—
217C2×C8—22no—
218C2×C8—831no—
219C2×C8—834no—
220C2×C8—831no—
221C2×C8—1662no—
222C2×C8C22×D8311yescharacteristic
223C2×C8C22×D8311yescharacteristic
224C4⋊C4—832no—
225C4⋊C4C22×D8312yes—
226C22⋊C4—22no—
227C22⋊C4—1662no—
228C22⋊C4—1662no—
229C22⋊C4—1662no—
230C22⋊C4—1662no—
231C42—1662no—
232C23—22no—
233C23—3324no—
234C23—831no—
235C23—831no—
236C23—6642no—
237C23—6642no—
238C23—1662no—
239C23D4×D8311yescharacteristic
240Q8—1662no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—3324no—
245D4—6642no—
246D4—6642no—
247D4—6642no—
248D4—1662no—
249D4—1662no—
250D4—1662no—
251D4—1662no—
252D4—1662no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—831no—
257C2×C4—832no—
258C2×C4—832no—
259C2×C4—831no—
260C2×C4—1662no—
261C2×C4—1661no—
262C2×C4—1661no—
263C2×C4D4×D8311yescharacteristic
264C2×C4C23×D8311yescharacteristic
265C8—21no—
266C8—832no—
267C8—1661no—
268C8D4×D8312yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—3322no—
276C22—3322no—
277C22—3322no—
278C22—832no—
279C22—6648no—
280C22—1661no—
281C22—1661no—
282C22—1661no—
283C22—1661no—
284C22—1661no—
285C22C2×D4×D8311yescharacteristic
286C4—21no—
287C4—42no—
288C4—3322no—
289C4—832no—
290C4—1661no—
291C4C2×D4×D8311yescharacteristic
292C4#41×D8311yescharacteristic
293C2—41no—
294C2—82no—
295C2—3321no—
296C2—6642no—
297C2—1662no—
298C2#121×D8311yescharacteristic
299C2#1912850211yescharacteristic
300C2#1912850311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#466×C83C2
socleC22×C83C22×D4
hypercenter#466D83
hypocenterC83#3058
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D83C2semidirect
2. C2×#14581798C2semidirect
3. #19128267C2semidirect
4. #19127953C2semidirect
5. #19127952C2semidirect
6. #19127938C2semidirect
7. #19127937C2semidirect
8. #19127934C2semidirect
9. #19127738C2semidirect
10. #19127704C2semidirect
11. #466×C83C2semidirect
18. C2×#13050574C22semidirect
20. C2×#13050573C22semidirect
22. C2×C8×D83C22semidirect
25. #14581767C22semidirect
26. #14581766C22semidirect
27. #14581763C22semidirect
31. #14581726C22semidirect
32. #14581723C22semidirect
33. #14581714C22semidirect
34. #14581713C22semidirect
35. #14581712C22semidirect
37. C22×C8×C83C22semidirect
38. #121×C83C22semidirect
39. #107×C83C22semidirect
40. #102×C83C22semidirect
65. C8×D83D4semidirect
151. C83#3058semidirect
160. #466D83semidirect
175. C22×C8C2×D83semidirect
181. #121C2×D83semidirect
186. #107C2×D83semidirect
191. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D832
C2×#145817982
#191282672
#191279532
#191279522
#191279382
#191279372
#191279342
#191277382
#191277042
#466×C832
#305883