Finite Group Databasebeta

← #24914487

Subgroups — #24914487

Order 10624.

592 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581781C211yesmaximal, characteristic, complemented
2#19127617C212yesmaximal, complemented
3#19127610C212yesmaximal, complemented
4#19127609C212yesmaximal, complemented
5#19127608C211yesmaximal, characteristic, complemented
6#19127601C212yesmaximal, complemented
7#19127592C212yesmaximal, complemented
8#19127589C212yesmaximal, complemented
9C2×#124×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12C22×C22⋊D83C2212yes—
13#14581781—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×C4×Dic83C2211yescharacteristic
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C22×Q8×C83C2211yescharacteristic
28C22×D4×C83C2211yescharacteristic
29#124×C83—24no—
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2212yes—
32C2×C42×C83C2211yescharacteristic
33C23×D83C2312yes—
34C2×D332—22no—
35C2×D332—41no—
36C2×D332—41no—
37C2×C22⋊D83—24no—
38C2×C22⋊D83—42no—
39C2×C22⋊D83—42no—
40C2×C22⋊D83—42no—
41C2×C22⋊D83—42no—
42C2×C22⋊D83—42no—
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C4⋊Dic83—41no—
47Dic83⋊C4—42no—
48C22⋊Dic83—24no—
49C22⋊Dic83—42no—
50(C2×D83)⋊C4—24no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—42no—
54C4×Dic83—22no—
55C4×Dic83—41no—
56C24×C83C2312yes—
57C2×Q8×C83—41no—
58C2×Q8×C83—41no—
59C2×Q8×C83D414yescomplemented
60C2×D4×C83—22no—
61C2×D4×C83—41no—
62C2×D4×C83—41no—
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2311yescharacteristic
67C22⋊C4×C83—24no—
68C22⋊C4×C83—24no—
69C42×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83C2411yescharacteristic
78C22×D83—22no—
79C22×D83—22no—
80C22×D83—22no—
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—22no—
85Q8×C83—44no—
86D4×C83—44no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83C2×D412yes—
97C2×C4×C83C2×D412yes—
98C2×C4×C83C2×D412yes—
99C22⋊D83—48no—
100C22⋊D83—48no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83#4114yes—
110D332—44no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83C22×D411yescharacteristic
119C22×C83C2×#4112yes—
120C22×C83H(5,𝔽2)11yescharacteristic
121C22×C83H(5,𝔽2)11yescharacteristic
122C22×C83H(5,𝔽2)12yes—
123C2×D83—42no—
124C2×D83—42no—
125C2×D83—42no—
126C2×D83—42no—
127C2×D83—42no—
128C2×D83—42no—
129C2×D83—42no—
130C4×C83—22no—
131C4×C83—22no—
132C4×C83—22no—
133C4×C83—42no—
134Dic83—24no—
135Dic83—42no—
136Dic83—42no—
137C2×C83—82no—
138C2×C83#53411yescharacteristic
139C2×C83#53812yes—
140C2×C83#55012yes—
141C2×C83#55511yescharacteristic
142C2×C83H(3,𝔽4)11yescharacteristic
143D83—82no—
144#2733—831nomaximal, Sylow
145C83#273311yescharacteristic, complemented, Sylow
146C2×#124—831no—
147C2×#124D8311yescharacteristic, complemented
148#394—832no—
149#392—832no—
150#392—832no—
151#391—831no—
152#390—832no—
153#388—832no—
154#388—832no—
155C22×Q8C2×D8311yescharacteristic
156C22×D4—832no—
157C22×D4—832no—
158C22×D4—831no—
159C22×D4C2×D8311yescharacteristic
160#124—24no—
161#124—1664no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—831no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—832no—
166C2×C22⋊C4—832no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4C2×D8312yes—
171C2×C22⋊C4C2×D8312yes—
172C2×C42—831no—
173C2×C42—831no—
174C2×C42C2×D8311yescharacteristic
175#95—832no—
176#95—832no—
177#95—832no—
178C24—832no—
179C24C22×D8312yes—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C22⋊D8314yescomplemented
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3321no—
187C2×D4—3321no—
188C2×D4—3322no—
189C2×D4—3322no—
190C2×D4—3322no—
191C2×D4—3322no—
192C2×D4—3322no—
193C2×D4—1664no—
194C2×D4—1662no—
195C22×C4—832no—
196C22×C4—832no—
197C22×C4—832no—
198C22×C4C22×D8312yes—
199C22×C4C22×D8311yescharacteristic
200C22×C4C22×D8311yescharacteristic
201C22×C4C22×D8311yescharacteristic
202C4⋊C4—3322no—
203C4⋊C4—3321no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—3322no—
207C22⋊C4—3322no—
208C22⋊C4—1664no—
209C22⋊C4—1664no—
210C22⋊C4—1664no—
211C22⋊C4—1664no—
212C42—22no—
213C42—3321no—
214C42—1662no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—1662no—
223C23—1662no—
224C23—1662no—
225C23—1662no—
226C23—1662no—
227C23—1662no—
228C23—1662no—
229C23C23×D8311yescharacteristic
230Q8—44no—
231D4—44no—
232D4—3324no—
233D4—3328no—
234D4—3328no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—834no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4C2×C22⋊D8312yes—
254C2×C4C2×C22⋊D8312yes—
255C2×C4C2×C22⋊D8312yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—3322no—
264C22—3322no—
265C22—3322no—
266C22—3322no—
267C22—3322no—
268C22—3322no—
269C22—3322no—
270C22C22×C22⋊D8311yescharacteristic
271C22#41×D8312yes—
272C22#1458183711yescharacteristic
273C22#1458183711yescharacteristic
274C22#1458183812yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—3322no—
280C4—3322no—
281C4—1664no—
282C2—82no—
283C2—6642no—
284C2#1912826011yescharacteristic
285C2#1912828512yes—
286C2#1912837012yes—
287C2#1912841311yescharacteristic
288C2#1912844311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#124×C83C2
hypercenterC2×#124D83
hypocenterC83#2733
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14581781C2semidirect
2. #19127617C2semidirect
3. #19127610C2semidirect
4. #19127609C2semidirect
5. #19127608C2semidirect
6. #19127601C2semidirect
7. #19127592C2semidirect
8. #19127589C2semidirect
9. C2×#124×C83C2semidirect
59. C2×Q8×C83D4semidirect
145. C83#2733semidirect
147. C2×#124D83semidirect
182. C2×Q8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817812
#191276172
#191276102
#191276092
#191276082
#191276012
#191275922
#191275892
C2×#124×C832
#273383