Finite Group Databasebeta

← #24908471

Subgroups — #24908471

Order 10624.

392 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×#13050580C211yesmaximal, characteristic, complemented
2C2×#14581771C211yesmaximal, characteristic
3C2×#14581716C211yesmaximal, characteristic, complemented
4C2×#14581715C211yesmaximal, characteristic, complemented
5C2×C22⋊(C83⋊C8)C211yesmaximal, characteristic
6#19127463C211yesmaximal, characteristic, complemented
7C2×C22⋊Q8×C83C211yesmaximal, characteristic
8C22×Dic166C2211yescharacteristic
9C2×#13050580—22no—
10C2×#13050580—22no—
11C2×#13050580—22no—
12C22×C83⋊C8C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14#14581771—22no—
15#14581771—22no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4—21no—
18C2×C22⋊Dic83—21no—
19#14581716—22no—
20#14581715—22no—
21C22⋊(C83⋊C8)—22no—
22C22×Q8×C83C2211yescharacteristic
23C23×C4×C83C2211yescharacteristic
24C22⋊Q8×C83—22no—
25C22⋊Q8×C83—22no—
26C2×C4⋊C4×C83—21no—
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83—21no—
29C2×Dic166—21no—
30C2×Dic166—21no—
31C2×Dic166—21no—
32C2×Dic166—21no—
33C2×Dic166—21no—
34C2×Dic166D412yes—
35C22×Dic83—21no—
36C22×Dic83—21no—
37#13050580—28no—
38C2×C83⋊C8—21no—
39C2×C83⋊C8—21no—
40C2×C83⋊C8—21no—
41C2×C83⋊C8—21no—
42C2×C83⋊C8—21no—
43C2×C83⋊C8D412yes—
44C4⋊Dic83—21no—
45C4⋊Dic83D412yes—
46Dic83⋊C4—24no—
47C22⋊Dic83—24no—
48C24×C83D411yescharacteristic
49C2×Q8×C83—21no—
50C2×Q8×C83—21no—
51C2×Q8×C83—21no—
52C2×Q8×C83—21no—
53C2×Q8×C83—21no—
54C2×Q8×C83D412yes—
55C22×C4×C83—21no—
56C22×C4×C83—21no—
57C22×C4×C83—21no—
58C22×C4×C83—21no—
59C22×C4×C83—21no—
60C22×C4×C83—21no—
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83D411yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—24no—
66C4⋊C4×C83—21no—
67C4⋊C4×C83D412yes—
68C22⋊C4×C83—24no—
69C23×C83—21no—
70C23×C83—21no—
71C23×C83—21no—
72C23×C83—21no—
73C23×C83—21no—
74C23×C83—21no—
75C23×C83Q1612yescomplemented
76C23×C83C2×D411yescharacteristic
77Q8×C83—22no—
78Q8×C83—22no—
79Q8×C83—41no—
80Q8×C83—41no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—41no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83—41no—
90C2×C4×C83—41no—
91C2×C4×C83—41no—
92C2×C4×C83—41no—
93C2×C4×C83—41no—
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83#4111yescharacteristic
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—41no—
103C2×Dic83—41no—
104C2×Dic83—41no—
105C2×Dic83—41no—
106Dic166—22no—
107Dic166—22no—
108Dic166—41no—
109Dic166—41no—
110C83⋊C8—22no—
111C83⋊C8—41no—
112C22×C83—22no—
113C22×C83—22no—
114C22×C83—22no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83—41no—
118C22×C83—41no—
119C22×C83#12011yescharacteristic
120C22×C83#12011yescharacteristic
121C22×C83#12111yescharacteristic
122C22×C83C2×Q1611yescharacteristic
123C22×C83#13511yescharacteristic
124C22×C83#13711yescharacteristic
125C22×C83#13711yescharacteristic
126C4×C83—41no—
127C4×C83—42no—
128C4×C83—42no—
129C4×C83#12011yescharacteristic
130C4×C83#12111yescharacteristic
131C4×C83#12111yescharacteristic
132C4×C83#12711yescharacteristic
133Dic83—42no—
134Dic83—42no—
135C2×C83—42no—
136C2×C83#39211yescharacteristic
137C2×C83#45111yescharacteristic
138C2×C83#45111yescharacteristic
139C2×C83#45211yescharacteristic
140C2×C83#45211yescharacteristic
141C2×C83#46711yescharacteristic
142C2×C83#47011yescharacteristic
143#2001—831nomaximal, Sylow
144C83#200111yescharacteristic, complemented, Sylow
145C22×Q16—831no—
146C2×C22⋊Q8—831no—
147C2×C22⋊Q8D8311yescharacteristic
148C2×#103—831no—
149C2×#103—831no—
150C2×C22⋊C8—831no—
151#340—831no—
152C22×Q8—831no—
153C22×Q8C2×D8311yescharacteristic
154C23×C4C2×D8311yescharacteristic
155C2×Q16—1662no—
156C2×Q16—1662no—
157C2×Q16—1662no—
158C22×C8—831no—
159C22×C8—831no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—1662no—
163C22⋊Q8—1662no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—831no—
166C2×C4⋊C4—1661no—
167C2×C4⋊C4C2×D8311yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—1661no—
170#103—1662no—
171#103—1662no—
172C22⋊C8—1662no—
173C24C22⋊D8311yescharacteristic
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—832no—
180C2×Q8—1661no—
181C2×Q8—1661no—
182C2×Q8—1661no—
183C2×Q8—1661no—
184C2×Q8—1661no—
185C2×Q8C22⋊D8312yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—1661no—
195C22×C4—1661no—
196C22×C4C22⋊D8311yescharacteristic
197C22×C4C22×D8311yescharacteristic
198Q16—1668no—
199C2×C8—832no—
200C2×C8—1661no—
201C2×C8—1661no—
202C2×C8—1661no—
203C2×C8—1661no—
204C2×C8—1661no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4—832no—
208C4⋊C4—1664no—
209C4⋊C4—1661no—
210C4⋊C4C22⋊D8312yes—
211C22⋊C4—24no—
212C22⋊C4—1664no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23#1305058012yescomplemented
220C23C2×C22⋊D8311yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—41no—
224Q8—41no—
225Q8—3321no—
226Q8—3321no—
227Q8—1662no—
228Q8—1662no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—3321no—
243C2×C4—3321no—
244C2×C4—3321no—
245C2×C4—3321no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4D4×D8311yescharacteristic
249C2×C4D4×D8311yescharacteristic
250C2×C4D4×D8311yescharacteristic
251C2×C4C2×C22⋊D8311yescharacteristic
252C2×C4C2×C22⋊D8311yescharacteristic
253C2×C4#1305058911yescharacteristic
254C8—3321no—
255C8—1662no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22#1458176011yescharacteristic
264C22#1458176111yescharacteristic
265C22#1458176411yescharacteristic
266C22C2×#1305058011yescharacteristic
267C22#1458181211yescharacteristic
268C22#1458182211yescharacteristic
269C22#1458182311yescharacteristic
270C4—42no—
271C4—41no—
272C4—42no—
273C4—3322no—
274C4—3322no—
275C4#1458176011yescharacteristic
276C4#1458176311yescharacteristic
277C4#1458176411yescharacteristic
278C4#1458178811yescharacteristic
279C2—42no—
280C2#1912760911yescharacteristic
281C2#1912785611yescharacteristic
282C2#1912785711yescharacteristic
283C2#1912786111yescharacteristic
284C2#1912786311yescharacteristic
285C2#1912794011yescharacteristic
286C2#1912795711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D83
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
FittingC2×C22⋊Q8×C83C2
socleC23×C83C2×D4
prime radicalC24×C83D4
hypercenterC2×C22⋊Q8D83
hypocenterC83#2001
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13050580C2semidirect
3. C2×#14581716C2semidirect
4. C2×#14581715C2semidirect
6. #19127463C2semidirect
75. C23×C83Q16semidirect
144. C83#2001semidirect
219. C23#13050580semidirect

Maximal subgroups

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

subgroupindex
C22×#130505802
C2×#145817712
C2×#145817162
C2×#145817152
C2×C22⋊(C83⋊C8)2
#191274632
C2×C22⋊Q8×C832
#200183