Finite Group Databasebeta

← #24918433

Subgroups — #24918433

Order 10624.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#19128237C211yesmaximal, characteristic, complemented
2C2×#14581781C211yesmaximal, characteristic, complemented
3#19128086C212yesmaximal, complemented
4#19128057C212yesmaximal, complemented
5#19127925C212yesmaximal, complemented
6#19127919C212yesmaximal, complemented
7#19127851C211yesmaximal, characteristic, complemented
8#19127840C211yesmaximal, characteristic, complemented
9#19127837C211yesmaximal, characteristic
10#19127751C211yesmaximal, characteristic, complemented
11#451×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C2×#13050577C2212yes—
14C2×#13050577C2212yescomplemented
15C2×D664C2212yes—
16#14581787—21no—
17#14581783C2211yescharacteristic, complemented
18#14581781—21no—
19#14581781—21no—
20#14581781C2212yes—
21#14581781C2212yes—
22#14581781C2211yescharacteristic, complemented
23#14581776—21no—
24#14581766C2211yescharacteristic
25Q8×Dic83C2211yescharacteristic, complemented
26#14581754C2211yescharacteristic, complemented
27(C22⋊D83)⋊C4—21no—
28#14581750—21no—
29C2×(C2×D83)⋊C4—21no—
30C2×(C2×D83)⋊C4—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yes—
35#14581712C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yescomplemented
38C22×Q8×C83C2211yescharacteristic
39C2×Q16×C83C2212yescomplemented
40C22⋊Q8×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44C2×D332—21no—
45C2×D332—42no—
46C2×D332C2311yescharacteristic
47C2×D332C2312yes—
48C2×C22⋊D83—21no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51C2×C4×D83—21no—
52#13050577—42no—
53#13050577—42no—
54D664—42no—
55C2×C83⋊C8C2312yes—
56C4⋊Dic83—21no—
57C4⋊Dic83—21no—
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—21no—
68C4×Dic83—21no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83—21no—
74C2×Q8×C83—42no—
75C2×Q8×C83C2312yes—
76C2×Q8×C83C2311yescharacteristic
77C22×C4×C83—21no—
78C22×C4×C83C2311yescharacteristic
79Q16×C83—42no—
80C2×C8×C83C2312yes—
81C4⋊C4×C83—21no—
82C4⋊C4×C83C2311yescharacteristic
83C22⋊C4×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—21no—
86C22×D83—22no—
87C22×D83—42no—
88C22×D83—42no—
89C22×D83—42no—
90Q8×C83—22no—
91Q8×C83—22no—
92Q8×C83—22no—
93Q8×C83—42no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—22no—
98C2×C4×C83—42no—
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—42no—
102C2×Dic83—22no—
103C2×Dic83—21no—
104C2×Dic83—41no—
105C2×Dic83C2×D411yescharacteristic
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108C2×Dic83C2×D411yescharacteristic
109D332—22no—
110D332—22no—
111D332—22no—
112D332—42no—
113C4×D83—42no—
114C8×C83—22no—
115C83⋊C8—22no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83#13512yes—
119C22×C83C22×D411yescharacteristic
120C2×D83—42no—
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—42no—
124C2×D83—41no—
125C2×D83—42no—
126C2×D83—84no—
127C4×C83—21no—
128C4×C83—42no—
129C4×C83—41no—
130C4×C83—41no—
131C4×C83H(5,𝔽2)12yes—
132Dic83—21no—
133Dic83—21no—
134Dic83—21no—
135Dic83—21no—
136Dic83—41no—
137C2×C83—22no—
138C2×C83#53511yescharacteristic
139C2×C83C2×#13511yescharacteristic
140C2×C83#57511yescharacteristic
141D83—81no—
142D83—82no—
143#3211—831nomaximal, Sylow
144C83#321111yescharacteristic, complemented, Sylow
145#529—831no—
146C2×#124—831no—
147#495—832no—
148#488—832no—
149#464—832no—
150#463—832no—
151#451D8311yescharacteristic, complemented
152#450—831no—
153#448—831no—
154#447—831no—
155#431—831no—
156C22×Q8C2×D8311yescharacteristic
157C22×D4—831no—
158C2×Q16C2×D8312yescomplemented
159C2×SD16—832no—
160C2×SD16—832no—
161C2×D8—832no—
162#126—1661no—
163#125—831no—
164#124—832no—
165#124—832no—
166#124—831no—
167#124—1661no—
168#124—1661no—
169#123—1661no—
170C22⋊Q8C2×D8311yescharacteristic
171#121—831no—
172C4×Q8—831no—
173C4×D4—831no—
174C4×D4—1661no—
175#117—1661no—
176C2×C22⋊C4—1661no—
177C2×C22⋊C4—1661no—
178C2×C42—831no—
179C4⋊C8—832no—
180#103—832no—
181#103C2×D8312yescomplemented
182#102—832no—
183#102—832no—
184C22⋊C8—831no—
185C22⋊C8C2×D8311yescharacteristic
186C4×C8—832no—
187C24—1661no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8C22×D8312yes—
191C2×Q8C22×D8311yescharacteristic
192C2×D4—3322no—
193C2×D4—832no—
194C2×D4—831no—
195C2×D4—1661no—
196C2×D4—1661no—
197C22×C4—21no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—1661no—
201C22×C4C22×D8311yescharacteristic
202Q16—42no—
203SD16—3322no—
204SD16—3322no—
205D8—3322no—
206C2×C8—832no—
207C2×C8C22×D8312yes—
208C4⋊C4—21no—
209C4⋊C4—831no—
210C4⋊C4—1661no—
211C4⋊C4—1661no—
212C4⋊C4—1661no—
213C4⋊C4C22×D8311yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1662no—
216C22⋊C4—1662no—
217C22⋊C4—1662no—
218C22⋊C4—1662no—
219C22⋊C4—1661no—
220C22⋊C4—1661no—
221C22⋊C4—1661no—
222C22⋊C4—1661no—
223C42—831no—
224C42—831no—
225C42—831no—
226C42—831no—
227C42—1661no—
228C23—3322no—
229C23—3322no—
230C23—3322no—
231C23—1661no—
232C23—1662no—
233C23D4×D8311yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—42no—
238D4—3322no—
239D4—3322no—
240D4—1662no—
241D4—1662no—
242D4—1662no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—3322no—
249C2×C4—3321no—
250C2×C4—831no—
251C2×C4—831no—
252C2×C4—831no—
253C2×C4—831no—
254C2×C4—1661no—
255C2×C4—1662no—
256C2×C4D4×D8311yescharacteristic
257C2×C4C23×D8311yescharacteristic
258C8—22no—
259C8—1662no—
260C22—21no—
261C22—21no—
262C22—3321no—
263C22—3322no—
264C22—3321no—
265C22—3322no—
266C22—3322no—
267C22—3321no—
268C22—6644no—
269C22#1458181512yes—
270C22C2×D4×D8311yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—3321no—
276C4—1661no—
277C4—1661no—
278C4—1661no—
279C4—1661no—
280C4#1458183712yes—
281C2—22no—
282C2—6642no—
283C2—6641no—
284C2#1912826611yescharacteristic
285C2C2×#1458181511yescharacteristic
286C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#451×C83C2
socleC22×C83C22×D4
prime radical#19127837C2
hypercenter#451D83
hypocenterC83#3211
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #19128237C2semidirect
2. C2×#14581781C2semidirect
3. #19128086C2semidirect
4. #19128057C2semidirect
5. #19127925C2semidirect
6. #19127919C2semidirect
7. #19127851C2semidirect
8. #19127840C2semidirect
10. #19127751C2semidirect
11. #451×C83C2semidirect
14. C2×#13050577C22semidirect
17. #14581783C22semidirect
22. #14581781C22semidirect
25. Q8×Dic83C22semidirect
26. #14581754C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581716C22semidirect
37. C8×Dic83C22semidirect
39. C2×Q16×C83C22semidirect
41. #103×C83C22semidirect
144. C83#3211semidirect
151. #451D83semidirect
158. C2×Q16C2×D83semidirect
181. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282372
C2×#145817812
#191280862
#191280572
#191279252
#191279192
#191278512
#191278402
#191278372
#191277512
#451×C832
#321183