Finite Group Databasebeta

← #24918461

Subgroups — #24918461

Order 10624.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#19128256C211yesmaximal, characteristic, complemented
2#124×D83C211yesmaximal, characteristic
3#19128093C211yesmaximal, characteristic, complemented
4#19128061C211yesmaximal, characteristic, complemented
5#19128060C211yesmaximal, characteristic, complemented
6#19127925C212yesmaximal, complemented
7#19127921C212yesmaximal, complemented
8#19127855C212yesmaximal, complemented
9#19127850C212yesmaximal
10#19127756C211yesmaximal, characteristic, complemented
11#497×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2211yescharacteristic
13C2×D4×D83C2211yescharacteristic
14C2×#13050580C2212yescomplemented
15C2×#13050577C2212yes—
16#14581792C2211yescharacteristic, complemented
17#14581783—21no—
18#14581781C2211yescharacteristic
19#14581780C2211yescharacteristic
20#14581779C2211yescharacteristic
21#14581778—21no—
22#14581778—21no—
23#14581772—21no—
24#14581768C2212yes—
25Q8×Dic83C2212yescomplemented
26#14581757—21no—
27#14581750—21no—
28C22⋊C4×D83—21no—
29C22⋊C4×D83—21no—
30C42×D83C2211yescharacteristic
31Dic83⋊C8C2212yescomplemented
32#14581717C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yes—
35#14581713C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic, complemented
38C2×Q16×C83C2212yescomplemented
39C2×SD16×C83C2212yes—
40C4⋊Q8×C83C2211yescharacteristic, complemented
41#124×C83C2211yescharacteristic
42#97×C83C2211yescharacteristic, complemented
43C23×D83—21no—
44C2×Dic166—21no—
45C2×Dic166—21no—
46C2×Dic166C2311yescharacteristic
47Q8×D83—42no—
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—21no—
50D4×D83—42no—
51C2×C4×D83—21no—
52C2×C4×D83D411yescharacteristic
53C2×C4×D83D411yescharacteristic
54C2×C4×D83C2311yescharacteristic
55#13050580—42no—
56#13050577—42no—
57C2×C83⋊C8C2312yes—
58C4⋊Dic83—21no—
59C4⋊Dic83—22no—
60C4⋊Dic83C2312yes—
61Dic83⋊C4—22no—
62C22⋊Dic83—21no—
63C22⋊Dic83—21no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69C4×Dic83—22no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83C2312yes—
74C2×Q8×C83C2311yescharacteristic
75C2×D4×C83C2311yescharacteristic
76Q16×C83—42no—
77SD16×C83—42no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83—21no—
81C22⋊C4×C83—21no—
82C22⋊C4×C83—21no—
83C42×C83C2311yescharacteristic
84C23×C83—21no—
85C22×D83—21no—
86C22×D83—42no—
87C22×D83—42no—
88C22×D83—42no—
89C22×D83C2×D411yescharacteristic
90Q8×C83—22no—
91Q8×C83—22no—
92Q8×C83—22no—
93D4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105D332—22no—
106C4×D83—22no—
107C4×D83—22no—
108C4×D83—41no—
109C4×D83—42no—
110Dic166—22no—
111Dic166—42no—
112Dic166—42no—
113C8×C83—22no—
114C83⋊C8—22no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83—41no—
118C22×C83C22×D411yescharacteristic
119C2×D83—21no—
120C2×D83—21no—
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—41no—
124C2×D83—84no—
125C2×D83#13712yes—
126C4×C83—21no—
127C4×C83—21no—
128C4×C83—41no—
129C4×C83—42no—
130C4×C83H(5,𝔽2)12yes—
131Dic83—21no—
132Dic83—21no—
133Dic83—21no—
134Dic83—41no—
135Dic83—42no—
136C2×C83—81no—
137C2×C83#53511yescharacteristic
138C2×C83C2×#13711yescharacteristic
139C2×C83#57511yescharacteristic
140D83—22no—
141D83—81no—
142#3215—831nomaximal, Sylow
143C83#321511yescharacteristic, complemented, Sylow
144#533—831no—
145C2×#124—831no—
146#497—831no—
147#497D8311yescharacteristic, complemented
148#489—831no—
149#489—831no—
150#464—832no—
151#464—832no—
152#451—832no—
153#450—832no—
154#432—831no—
155C22×Q8—831no—
156C22×D4—831no—
157C2×Q16—832no—
158C2×Q16C2×D8312yescomplemented
159C2×SD16—832no—
160C2×SD16C2×D8312yes—
161C4⋊Q8—831no—
162C4⋊Q8C2×D8311yescharacteristic, complemented
163#125—1661no—
164#124—831no—
165#124—831no—
166#124—831no—
167#124—1661no—
168#124—1661no—
169#124C2×D8311yescharacteristic
170C22⋊Q8—832no—
171C22⋊Q8—1661no—
172C4×Q8—832no—
173C4×Q8—1661no—
174#117—1661no—
175C2×C22⋊C4—1661no—
176C2×C22⋊C4—1661no—
177C2×C42—831no—
178C4⋊C8—832no—
179#103—832no—
180#103—832no—
181#102—832no—
182#102—832no—
183C22⋊C8—832no—
184#97—831no—
185#97C2×D8311yescharacteristic, complemented
186C24—1661no—
187C2×Q8—3322no—
188C2×Q8—831no—
189C2×Q8—1661no—
190C2×Q8—1661no—
191C2×Q8C22×D8311yescharacteristic
192C2×Q8C22×D8312yes—
193C2×D4—3322no—
194C2×D4—831no—
195C2×D4—1661no—
196C2×D4C22×D8311yescharacteristic
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—1661no—
201Q16—42no—
202Q16—3322no—
203SD16—42no—
204SD16—3322no—
205C2×C8—832no—
206C2×C8C22×D8312yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—832no—
210C4⋊C4—1662no—
211C4⋊C4—1662no—
212C4⋊C4—1661no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1661no—
216C22⋊C4—1661no—
217C22⋊C4—1662no—
218C22⋊C4—1661no—
219C22⋊C4—1661no—
220C22⋊C4—1661no—
221C22⋊C4—1661no—
222C42—831no—
223C42—831no—
224C42—831no—
225C42—1662no—
226C42C22×D8311yescharacteristic
227C23—21no—
228C23—3322no—
229C23—3322no—
230C23—3322no—
231C23—831no—
232C23—1661no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—3322no—
237Q8—3322no—
238Q8—1662no—
239D4—22no—
240D4—3322no—
241D4—1662no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—3321no—
245C2×C4—3322no—
246C2×C4—831no—
247C2×C4—831no—
248C2×C4—831no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4—1661no—
252C2×C4—1662no—
253C2×C4D4×D8311yescharacteristic
254C2×C4D4×D8311yescharacteristic
255C2×C4C23×D8311yescharacteristic
256C8—22no—
257C8—1662no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—3321no—
262C22—3321no—
263C22—3321no—
264C22—832no—
265C22—6644no—
266C22—1661no—
267C22—1661no—
268C22C2×D4×D8311yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—3322no—
274C4—3321no—
275C4—1661no—
276C4—1661no—
277C4—1661no—
278C4#1458183712yes—
279C2—81no—
280C2—6641no—
281C2—1662no—
282C2#1912826411yescharacteristic
283C2#137×D8311yescharacteristic
284C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#497×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×D83C22
hypercenter#497D83
hypocenterC83#3215
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128093C2semidirect
4. #19128061C2semidirect
5. #19128060C2semidirect
6. #19127925C2semidirect
7. #19127921C2semidirect
8. #19127855C2semidirect
10. #19127756C2semidirect
11. #497×C83C2semidirect
14. C2×#13050580C22semidirect
16. #14581792C22semidirect
25. Q8×Dic83C22semidirect
31. Dic83⋊C8C22semidirect
32. #14581717C22semidirect
33. #14581716C22semidirect
37. #14581703C22semidirect
38. C2×Q16×C83C22semidirect
40. C4⋊Q8×C83C22semidirect
42. #97×C83C22semidirect
143. C83#3215semidirect
147. #497D83semidirect
158. C2×Q16C2×D83semidirect
162. C4⋊Q8C2×D83semidirect
185. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#124×D832
#191280932
#191280612
#191280602
#191279252
#191279212
#191278552
#191278502
#191277562
#497×C832
#321583