Finite Group Databasebeta

← #24913600

Subgroups — #24913600

Order 10624.

534 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581786C211yesmaximal, characteristic, complemented
2C2×#14581780C211yesmaximal, characteristic, complemented
3#19127623C212yesmaximal, complemented
4#19127622C211yesmaximal, characteristic, complemented
5#19127619C211yesmaximal, characteristic, complemented
6#19127618C212yesmaximal, complemented
7#19127614C211yesmaximal, characteristic, complemented
8#19127611C211yesmaximal, characteristic
9#19127603C212yesmaximal, complemented
10#19127592C212yesmaximal, complemented
11C4×C4⋊C4×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×D332C2212yes—
14C22×D332C2211yescharacteristic
15#14581786—24no—
16#14581780—24no—
17C2×C4⋊Dic83C2212yes—
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×(C2×D83)⋊C4C2212yes—
29C2×(C2×D83)⋊C4C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2212yes—
33C2×C42×C83C2212yes—
34C2×C42×C83C2211yescharacteristic
35#95×C83C2211yescharacteristic
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C23×D83C2312yes—
39C2×Dic166—22no—
40C2×Dic166—42no—
41C2×D332—22no—
42C2×D332—24no—
43C2×D332—42no—
44C2×D332—42no—
45C2×D332—42no—
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2311yescharacteristic
50C4⋊Dic83—42no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—42no—
61(C2×D83)⋊C4—42no—
62(C2×D83)⋊C4—42no—
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2312yes—
68C4⋊C4×C83—24no—
69C42×C83—42no—
70C42×C83D414yescomplemented
71C23×C83C2411yescharacteristic
72C22×D83—21no—
73C22×D83—22no—
74C22×D83—22no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—22no—
80C22×D83—21no—
81C22×D83—22no—
82C22×D83—21no—
83C22×D83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83C2×D412yes—
94C2×C4×C83C2×D412yes—
95C2×C4×C83C2×D412yes—
96C2×C4×C83#4114yes—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—21no—
107D332—44no—
108D332—48no—
109Dic166—44no—
110C22×C83C22×D411yescharacteristic
111C22×C83C2×#4112yes—
112C22×C83H(5,𝔽2)11yescharacteristic
113C22×C83H(5,𝔽2)12yes—
114C22×C832−1+411yescharacteristic
115C2×D83—41no—
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—41no—
119C2×D83—41no—
120C2×D83—42no—
121C2×D83—42no—
122C2×D83—42no—
123C2×D83—42no—
124C2×D83—42no—
125C2×D83—42no—
126C2×D83—42no—
127C4×C83—22no—
128C4×C83—22no—
129C4×C83—22no—
130C4×C83—24no—
131C4×C83—42no—
132Dic83—41no—
133Dic83—42no—
134Dic83—41no—
135Dic83—41no—
136C2×C83#53211yescharacteristic
137C2×C83#53511yescharacteristic
138C2×C83#53711yescharacteristic
139C2×C83#53812yes—
140C2×C83#54212yes—
141D83—82no—
142D83—81no—
143#2662—831nomaximal, Sylow
144C83#266211yescharacteristic, complemented, Sylow
145C2×#126—831no—
146C2×#124—831no—
147#396—832no—
148#395—831no—
149#394—832no—
150#394—831no—
151#393—831no—
152#392—831no—
153#390—832no—
154#388—832no—
155C4×C4⋊C4D8311yescharacteristic, complemented
156C22×Q8—831no—
157C22×D4—831no—
158C22×D4—832no—
159#126—1664no—
160#124—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—831no—
163C2×C4⋊C4—831no—
164C2×C4⋊C4—831no—
165C2×C4⋊C4C2×D8312yes—
166C2×C22⋊C4—832no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C22⋊C4—831no—
172C2×C22⋊C4—831no—
173C2×C22⋊C4—831no—
174C2×C22⋊C4—832no—
175C2×C42C2×D8312yes—
176C2×C42C2×D8311yescharacteristic
177#95—832no—
178#95—832no—
179#95C2×D8311yescharacteristic
180#95C2×D8311yescharacteristic
181C24—831no—
182C24—832no—
183C2×Q8—3322no—
184C2×Q8—1662no—
185C2×D4—3322no—
186C2×D4—3322no—
187C2×D4—3322no—
188C2×D4—1662no—
189C2×D4—1664no—
190C22×C4—831no—
191C22×C4—831no—
192C22×C4—832no—
193C22×C4—831no—
194C22×C4C22×D8312yes—
195C22×C4C22×D8311yescharacteristic
196C22×C4C22×D8311yescharacteristic
197C22×C4C22×D8311yescharacteristic
198C22×C4C22×D8312yes—
199C4⋊C4—24no—
200C4⋊C4—3322no—
201C4⋊C4—1662no—
202C4⋊C4—1662no—
203C4⋊C4—1662no—
204C22⋊C4—3322no—
205C22⋊C4—3322no—
206C22⋊C4—3322no—
207C22⋊C4—1664no—
208C22⋊C4—1664no—
209C22⋊C4—1662no—
210C22⋊C4—1662no—
211C22⋊C4—1662no—
212C22⋊C4—1664no—
213C42—42no—
214C42D33214yescomplemented
215C23—1661no—
216C23—1662no—
217C23—1662no—
218C23—1662no—
219C23—1662no—
220C23—1662no—
221C23—1661no—
222C23—1662no—
223C23—1662no—
224C23—1662no—
225C23—1662no—
226C23—1661no—
227C23C23×D8311yescharacteristic
228Q8—3324no—
229D4—3324no—
230D4—3328no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—1662no—
241C2×C4—1661no—
242C2×C4—1661no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4—1661no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1661no—
250C2×C4C2×D33212yes—
251C2×C4C2×D33212yes—
252C2×C4C2×D33212yes—
253C2×C4#1305059014yes—
254C22—3322no—
255C22—3322no—
256C22—3322no—
257C22—3321no—
258C22—3322no—
259C22—3322no—
260C22—3321no—
261C22—3321no—
262C22—3322no—
263C22—3322no—
264C22—3322no—
265C22—3322no—
266C22C22×D33211yescharacteristic
267C22C2×#1305059012yes—
268C22#1458183711yescharacteristic
269C22#1458183812yes—
270C22#1458184011yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—24no—
275C4—42no—
276C4—3321no—
277C4—3322no—
278C4—3321no—
279C4—3321no—
280C2—6642no—
281C2—6641no—
282C2#1912825111yescharacteristic
283C2#1912827011yescharacteristic
284C2#1912828211yescharacteristic
285C2#1912828412yes—
286C2#1912832312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC4×C4⋊C4×C83C2
prime radical#19127611C2
hypercenterC4×C4⋊C4D83
hypocenterC83#2662
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14581786C2semidirect
2. C2×#14581780C2semidirect
3. #19127623C2semidirect
4. #19127622C2semidirect
5. #19127619C2semidirect
6. #19127618C2semidirect
7. #19127614C2semidirect
9. #19127603C2semidirect
10. #19127592C2semidirect
11. C4×C4⋊C4×C83C2semidirect
70. C42×C83D4semidirect
144. C83#2662semidirect
155. C4×C4⋊C4D83semidirect
214. C42D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145817862
C2×#145817802
#191276232
#191276222
#191276192
#191276182
#191276142
#191276112
#191276032
#191275922
C4×C4⋊C4×C832
#266283