Finite Group Databasebeta

← #24912645

Subgroups — #24912645

Order 10624.

584 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 161 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×C4×C22⋊D83C211yesmaximal, characteristic
2#19127639C211yesmaximal, characteristic, complemented
3#19127636C211yesmaximal, characteristic, complemented
4#19127620C211yesmaximal, characteristic, complemented
5#19127612C211yesmaximal, characteristic, complemented
6#19127582C212yesmaximal, complemented
7#19127578C212yesmaximal
8#19127572C212yesmaximal, complemented
9#19127559C212yesmaximal, complemented
10C2×C83⋊#95C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C23×Dic83C2212yes—
14C22×C4×D83C2211yescharacteristic
15C4×C22⋊D83—24no—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2212yes—
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2211yescharacteristic
31C22×Q8×C83C2211yescharacteristic
32C23×C4×C83C2211yescharacteristic
33C22⋊Q8×C83—24no—
34C2×C4⋊C4×C83C2212yes—
35C2×C4⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—22no—
39C2×C22⋊D83—22no—
40C2×C22⋊D83Q814yescomplemented
41C22×Dic83—24no—
42C22×Dic83—24no—
43C22×Dic83—42no—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2312yes—
49C2×C4×D83—22no—
50C2×C4×D83—42no—
51Dic83⋊C4—22no—
52Dic83⋊C4—24no—
53Dic83⋊C4—42no—
54C22⋊Dic83—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—42no—
58C4×Dic83—22no—
59C24×C83C2311yescharacteristic
60C2×Q8×C83—22no—
61C2×Q8×C83—42no—
62C22×C4×C83—42no—
63C22×C4×C83D414yes—
64C22×C4×C83C2312yes—
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C4⋊C4×C83—22no—
69C4⋊C4×C83—24no—
70C22⋊C4×C83—24no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83C2×D412yes—
74C23×C83C2×Q812yes—
75C23×C83#4112yes—
76C23×C83C2411yescharacteristic
77C22×D83—21no—
78C22×D83—22no—
79C22×D83—21no—
80C22×D83—22no—
81C22×D83C2×Q812yes—
82Q8×C83—44no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—44no—
92C2×C4×C83C2×D412yes—
93C2×C4×C83C2×D412yes—
94C22⋊D83—28no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—48no—
107C2×Dic83C2×Q812yes—
108C2×Dic83#4114yes—
109C4×D83—44no—
110C22×C83—22no—
111C22×C83—22no—
112C22×C83—24no—
113C22×C83—24no—
114C22×C83C22⋊Q814yes—
115C22×C83C22×D411yescharacteristic
116C22×C83C22×Q811yescharacteristic
117C22×C83C2×#4112yes—
118C22×C83C2×#4111yescharacteristic
119C22×C83H(5,𝔽2)12yes—
120C2×D83—22no—
121C2×D83—24no—
122C2×D83—41no—
123C2×D83—41no—
124C2×D83—42no—
125C4×C83—22no—
126C4×C83—22no—
127C4×C83—42no—
128C4×C83—42no—
129Dic83—22no—
130Dic83—24no—
131Dic83—41no—
132Dic83—41no—
133Dic83—42no—
134C2×C83—24no—
135C2×C83C2×C22⋊Q811yescharacteristic
136C2×C83#53211yescharacteristic
137C2×C83#53411yescharacteristic
138C2×C83#55112yes—
139C2×C83#55412yes—
140D83—42no—
141#2564—831nomaximal, Sylow
142C83#256411yescharacteristic, complemented, Sylow
143C2×C22⋊Q8D8311yescharacteristic, complemented
144C2×C4×D4—831no—
145#400—831no—
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×Q8C2×D8311yescharacteristic
155C22×D4—831no—
156C23×C4—831no—
157C23×C4—832no—
158C23×C4C2×D8311yescharacteristic
159C22⋊Q8—24no—
160C4×D4—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—831no—
163C2×C4⋊C4—832no—
164C2×C4⋊C4C2×D8312yes—
165C2×C4⋊C4C2×D8311yescharacteristic
166C2×C22⋊C4—832no—
167C2×C22⋊C4—831no—
168C2×C22⋊C4—831no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4C2×D8312yes—
171C2×C42—831no—
172#95—832no—
173#95—831no—
174#95—831no—
175#95—832no—
176#95—832no—
177#95—832no—
178#95—832no—
179C24—831no—
180C24C22×D8311yescharacteristic
181C2×Q8—22no—
182C2×Q8—42no—
183C2×D4—834no—
184C2×D4—1662no—
185C2×D4—1662no—
186C22×C4—42no—
187C22×C4—3322no—
188C22×C4—3322no—
189C22×C4—831no—
190C22×C4—832no—
191C22×C4—832no—
192C22×C4—831no—
193C22×C4—831no—
194C22×C4—1664no—
195C22×C4—1664no—
196C22×C4—1662no—
197C22×C4C22⋊D8314yes—
198C22×C4C22×D8312yes—
199C22×C4C22×D8312yes—
200C22×C4C22×D8311yescharacteristic
201C22×C4C22×D8311yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—3322no—
205C4⋊C4—1664no—
206C4⋊C4—1662no—
207C22⋊C4—24no—
208C22⋊C4—3322no—
209C22⋊C4—1664no—
210C22⋊C4—1662no—
211C22⋊C4—1662no—
212C42—1662no—
213C23—22no—
214C23—22no—
215C23—832no—
216C23—1661no—
217C23—1662no—
218C23—1661no—
219C23—1662no—
220C23C2×C22⋊D8312yes—
221C23Q8×D8312yes—
222C23#1305059112yes—
223C23C23×D8311yescharacteristic
224Q8—44no—
225D4—1668no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—3324no—
236C2×C4—3328no—
237C2×C4—832no—
238C2×C4—834no—
239C2×C4—1661no—
240C2×C4—1661no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1661no—
247C2×C4—1662no—
248C2×C4—1661no—
249C2×C4—1662no—
250C2×C4C2×C22⋊D8312yes—
251C2×C4C2×C22⋊D8312yes—
252C22—24no—
253C22—24no—
254C22—22no—
255C22—22no—
256C22—3322no—
257C22—3321no—
258C22—3321no—
259C22—1664no—
260C22—1662no—
261C22#1458176814yes—
262C22C22×C22⋊D8311yescharacteristic
263C22C2×Q8×D8311yescharacteristic
264C22#41×D8312yes—
265C22C2×#1305059111yescharacteristic
266C22#1458183712yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—3322no—
272C4—3321no—
273C4—3321no—
274C4—1664no—
275C4—1662no—
276C2—24no—
277C2—3322no—
278C2C2×#1458176811yescharacteristic
279C2#1912824711yescharacteristic
280C2#1912826011yescharacteristic
281C2#1912837812yes—
282C2#1912840512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC2×C22⋊Q8D83
hypocenterC83#2564
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19127639C2semidirect
3. #19127636C2semidirect
4. #19127620C2semidirect
5. #19127612C2semidirect
6. #19127582C2semidirect
8. #19127572C2semidirect
9. #19127559C2semidirect
10. C2×C83⋊#95C2semidirect
11. C2×C22⋊Q8×C83C2semidirect
40. C2×C22⋊D83Q8semidirect
142. C83#2564semidirect
143. C2×C22⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D832
#191276392
#191276362
#191276202
#191276122
#191275822
#191275782
#191275722
#191275592
C2×C83⋊#952
C2×C22⋊Q8×C832
#256483