Finite Group Databasebeta

← #24912812

Subgroups — #24912812

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
2C22×Dic83⋊C4C211yesmaximal, characteristic, complemented
3#19127631C212yesmaximal, complemented
4#19127623C212yesmaximal, complemented
5#19127586C212yesmaximal, complemented
6#19127578C211yesmaximal, characteristic
7#19127578C211yesmaximal, characteristic
8#19127576C211yesmaximal, characteristic, complemented
9#19127576C211yesmaximal, characteristic, complemented
10#19127575C212yesmaximal, complemented
11#380×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15C22×C4×D83C2211yescharacteristic
16C4×C22⋊D83—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4—24no—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×Dic83⋊C4C2211yescharacteristic
23C2×C22⋊Dic83C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C23×C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2212yes—
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—22no—
39C2×C22⋊D83—22no—
40C2×C22⋊D83Q814yes—
41C22×Dic83—22no—
42C22×Dic83—24no—
43C22×Dic83—22no—
44C22×Dic83—42no—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2312yes—
50C2×C4×D83—22no—
51C2×C4×D83—42no—
52C4⋊Dic83—24no—
53C4⋊Dic83—42no—
54Dic83⋊C4—24no—
55Dic83⋊C4—24no—
56Dic83⋊C4—22no—
57C22⋊Dic83—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—42no—
61C4×Dic83—22no—
62C24×C83C2311yescharacteristic
63C22×C4×C83—42no—
64C22×C4×C83D414yes—
65C22×C4×C83C2312yes—
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—42no—
70C22⋊C4×C83—22no—
71C22⋊C4×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83C2×D412yes—
75C23×C83C2×Q812yes—
76C23×C83#4112yes—
77C23×C83C2411yescharacteristic
78C22×D83—21no—
79C22×D83—22no—
80C22×D83—21no—
81C22×D83—22no—
82C22×D83C2×Q812yes—
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—28no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—44no—
108C2×Dic83C2×Q812yes—
109C2×Dic83#4114yes—
110C4×D83—44no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—24no—
114C22×C83—24no—
115C22×C83C22⋊Q814yes—
116C22×C83C22×D411yescharacteristic
117C22×C83C22×Q811yescharacteristic
118C22×C83C2×#4112yes—
119C22×C83C2×#4111yescharacteristic
120C22×C83H(5,𝔽2)12yes—
121C2×D83—22no—
122C2×D83—24no—
123C2×D83—41no—
124C2×D83—41no—
125C2×D83—42no—
126C4×C83—22no—
127C4×C83—22no—
128C4×C83—42no—
129C4×C83—42no—
130Dic83—22no—
131Dic83—24no—
132Dic83—41no—
133Dic83—41no—
134Dic83—42no—
135C2×C83—24no—
136C2×C83C2×C22⋊Q811yescharacteristic
137C2×C83#52511yescharacteristic
138C2×C83#53411yescharacteristic
139C2×C83#55312yes—
140C2×C83#55412yes—
141D83—42no—
142#2579—831nomaximal, Sylow
143C83#257911yescharacteristic, complemented, Sylow
144C2×C4×D4—831no—
145C22×C4⋊C4—831no—
146#398—832no—
147#396—832no—
148#387—832no—
149#386—831no—
150#386—831no—
151#385—831no—
152#385—831no—
153#385—832no—
154#380D8311yescharacteristic, complemented
155C22×D4—831no—
156C23×C4—831no—
157C23×C4—831no—
158C23×C4—831no—
159C23×C4C2×D8311yescharacteristic
160C4×D4—1664no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—832no—
164C2×C4⋊C4—831no—
165C2×C4⋊C4—832no—
166C2×C4⋊C4—1664no—
167C2×C4⋊C4C2×D8312yes—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—831no—
170C2×C22⋊C4—831no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4C2×D8311yescharacteristic
173C2×C22⋊C4C2×D8311yescharacteristic
174C2×C42—831no—
175#95—832no—
176#95—832no—
177#95—832no—
178#95—832no—
179#95C2×D8312yes—
180C24—831no—
181C24C22×D8311yescharacteristic
182C2×D4—834no—
183C2×D4—1662no—
184C2×D4—1662no—
185C22×C4—42no—
186C22×C4—3322no—
187C22×C4—3322no—
188C22×C4—831no—
189C22×C4—832no—
190C22×C4—831no—
191C22×C4—831no—
192C22×C4—832no—
193C22×C4—1662no—
194C22×C4—1662no—
195C22×C4—1664no—
196C22×C4—1662no—
197C22×C4D33214yes—
198C22×C4C22×D8312yes—
199C22×C4C22×D8312yes—
200C22×C4C22×D8311yescharacteristic
201C22×C4C22×D8311yescharacteristic
202C4⋊C4—42no—
203C4⋊C4—3322no—
204C4⋊C4—1664no—
205C4⋊C4—1664no—
206C4⋊C4—1664no—
207C4⋊C4—1662no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—3322no—
211C22⋊C4—1664no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C42—1662no—
215C23—22no—
216C23—22no—
217C23—832no—
218C23—1661no—
219C23—1662no—
220C23—1661no—
221C23—1662no—
222C23C2×D33212yes—
223C23Q8×D8312yes—
224C23#1305058912yes—
225C23C23×D8311yescharacteristic
226D4—1668no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—3324no—
237C2×C4—3324no—
238C2×C4—832no—
239C2×C4—834no—
240C2×C4—1661no—
241C2×C4—1661no—
242C2×C4—1662no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1668no—
248C2×C4—1661no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4—1661no—
252C2×C4C2×D33212yes—
253C2×C4C2×D33212yes—
254C22—24no—
255C22—24no—
256C22—22no—
257C22—22no—
258C22—3322no—
259C22—3321no—
260C22—3321no—
261C22—1664no—
262C22—1662no—
263C22#1458177214yes—
264C22C22×D33211yescharacteristic
265C22C2×Q8×D8311yescharacteristic
266C22#41×D8312yes—
267C22C2×#1305058911yescharacteristic
268C22#1458183712yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—3321no—
274C4—3321no—
275C4—3322no—
276C4—1662no—
277C4—1664no—
278C2—24no—
279C2—3322no—
280C2C2×#1458177211yescharacteristic
281C2#1912821311yescharacteristic
282C2#1912826311yescharacteristic
283C2#1912839412yes—
284C2#1912840512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#380×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#380D83
hypocenterC83#2579
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C22×Dic83⋊C4C2semidirect
3. #19127631C2semidirect
4. #19127623C2semidirect
5. #19127586C2semidirect
8. #19127576C2semidirect
9. #19127576C2semidirect
10. #19127575C2semidirect
11. #380×C83C2semidirect
143. C83#2579semidirect
154. #380D83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D832
C22×Dic83⋊C42
#191276312
#191276232
#191275862
#191275782
#191275762
#191275752
#380×C832
#257983