Finite Group Databasebeta

← #24912646

Subgroups — #24912646

Order 10624.

584 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 147 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×#14581772C211yesmaximal, characteristic, complemented
2C2×#14581754C211yesmaximal, characteristic
3#19127636C211yesmaximal, characteristic, complemented
4#19127621C211yesmaximal, characteristic, complemented
5#19127614C211yesmaximal, characteristic, complemented
6#19127588C212yesmaximal, complemented
7#19127579C212yesmaximal
8#19127575C212yesmaximal, complemented
9#19127560C212yesmaximal, complemented
10#95×D83C211yesmaximal, characteristic, complemented
11#400×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×D332C2211yescharacteristic
14C22×C4×D83C2212yes—
15C22×C4×D83C2211yescharacteristic
16C22×C4×D83C2211yescharacteristic
17#14581772—24no—
18#14581754—24no—
19C2×C4⋊Dic83C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34#95×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C23×D83C2311yescharacteristic
39C2×Dic166—22no—
40C2×Dic166—42no—
41C2×D332—22no—
42C2×D332—22no—
43C2×D332Q814yescomplemented
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2312yes—
48C2×C4×D83—24no—
49C2×C4×D83—22no—
50C2×C4×D83—24no—
51C2×C4×D83—42no—
52C2×C4×D83—42no—
53C2×C4×D83—42no—
54C2×C4×D83D414yes—
55C4⋊Dic83—24no—
56Dic83⋊C4—24no—
57Dic83⋊C4—42no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—24no—
62(C2×D83)⋊C4—42no—
63C4×Dic83—22no—
64C22×C4×C83C2312yes—
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—22no—
70C4⋊C4×C83—22no—
71C23×C83C2411yescharacteristic
72C22×D83—22no—
73C22×D83—22no—
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—21no—
77C22×D83—22no—
78C22×D83C2×D412yes—
79C22×D83C2×Q812yes—
80C22×D83C2×Q812yes—
81C22×D83#4112yes—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83C2×D412yes—
94C2×C4×C83C2×Q812yes—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83C2×D412yes—
104C2×Dic83#4114yes—
105D332—28no—
106C4×D83—44no—
107C4×D83—48no—
108C4×D83—44no—
109Dic166—44no—
110C22×C83C22×D411yescharacteristic
111C22×C83C22×Q811yescharacteristic
112C22×C83C2×#4111yescharacteristic
113C22×C83C2×#4112yes—
114C22×C83H(5,𝔽2)12yes—
115C2×D83—22no—
116C2×D83—22no—
117C2×D83—24no—
118C2×D83—24no—
119C2×D83—22no—
120C2×D83—24no—
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—42no—
124C2×D83C22⋊Q814yes—
125C4×C83—22no—
126C4×C83—22no—
127C4×C83—42no—
128C4×C83—41no—
129C4×C83—42no—
130Dic83—22no—
131Dic83—24no—
132Dic83—41no—
133Dic83—42no—
134C2×C83C2×C22⋊Q811yescharacteristic
135C2×C83#53211yescharacteristic
136C2×C83#53411yescharacteristic
137C2×C83#55112yes—
138C2×C83#55412yes—
139D83—24no—
140D83—42no—
141#2564—831nomaximal, Sylow
142C83#256411yescharacteristic, complemented, Sylow
143C2×C22⋊Q8—831no—
144C2×C4×D4—831no—
145#400D8311yescharacteristic, complemented
146#399—831no—
147#395—831no—
148#393—831no—
149#387—832no—
150#386—832no—
151#385—832no—
152#380—832no—
153C2×#95—831no—
154C22×Q8—831no—
155C22×D4—831no—
156C23×C4—831no—
157C23×C4—831no—
158C23×C4—832no—
159C22⋊Q8—1664no—
160C4×D4—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—832no—
164C2×C4⋊C4C2×D8311yescharacteristic
165C2×C4⋊C4C2×D8311yescharacteristic
166C2×C22⋊C4—832no—
167C2×C22⋊C4—831no—
168C2×C22⋊C4—831no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C42—831no—
172#95—832no—
173#95—831no—
174#95—832no—
175#95—832no—
176#95C2×D8311yescharacteristic
177#95C2×D8312yes—
178#95C2×D8312yes—
179C24—831no—
180C24—831no—
181C2×Q8—3322no—
182C2×Q8—1662no—
183C2×D4—834no—
184C2×D4—1662no—
185C2×D4—1662no—
186C22×C4—3322no—
187C22×C4—3322no—
188C22×C4—3322no—
189C22×C4—834no—
190C22×C4—832no—
191C22×C4—831no—
192C22×C4—831no—
193C22×C4—832no—
194C22×C4—1662no—
195C22×C4—1664no—
196C22×C4—1664no—
197C22×C4C22×D8312yes—
198C22×C4C22×D8312yes—
199C22×C4C22×D8311yescharacteristic
200C22×C4C22×D8311yescharacteristic
201C22×C4C22×D8311yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—3322no—
205C4⋊C4—1664no—
206C4⋊C4—1664no—
207C22⋊C4—3322no—
208C22⋊C4—1664no—
209C22⋊C4—1662no—
210C22⋊C4—1664no—
211C22⋊C4—1662no—
212C42—1662no—
213C23—832no—
214C23—832no—
215C23—832no—
216C23—832no—
217C23—1661no—
218C23—1662no—
219C23—1661no—
220C23—1662no—
221C23—1662no—
222C23—1662no—
223C23C23×D8311yescharacteristic
224Q8—3324no—
225D4—1668no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—3324no—
238C2×C4—3324no—
239C2×C4—3328no—
240C2×C4—832no—
241C2×C4—834no—
242C2×C4—1661no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4D4×D8312yes—
251C2×C4Q8×D8312yes—
252C22—3322no—
253C22—3321no—
254C22—3321no—
255C22—834no—
256C22—1664no—
257C22—1662no—
258C22—1664no—
259C22—1664no—
260C22—1662no—
261C22—1662no—
262C22C2×D4×D8311yescharacteristic
263C22C2×Q8×D8311yescharacteristic
264C22#41×D8311yescharacteristic
265C22#41×D8312yes—
266C22#1458183812yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—3321no—
273C4—3322no—
274C4—1664no—
275C4—1662no—
276C2—3322no—
277C2—1664no—
278C2C22⋊Q8×D8311yescharacteristic
279C2#1912825011yescharacteristic
280C2#1912826211yescharacteristic
281C2#1912837612yes—
282C2#1912840712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC22×D332C22
hypercenter#400D83
hypocenterC83#2564
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#14581772C2semidirect
3. #19127636C2semidirect
4. #19127621C2semidirect
5. #19127614C2semidirect
6. #19127588C2semidirect
8. #19127575C2semidirect
9. #19127560C2semidirect
10. #95×D83C2semidirect
11. #400×C83C2semidirect
43. C2×D332Q8semidirect
142. C83#2564semidirect
145. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817722
C2×#145817542
#191276362
#191276212
#191276142
#191275882
#191275792
#191275752
#191275602
#95×D832
#400×C832
#256483