Finite Group Databasebeta

← #24911129

Subgroups — #24911129

Order 10624.

780 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14581769C212yesmaximal, complemented
2C22×C22⋊Dic83C212yesmaximal
3#19127637C211yesmaximal, characteristic, complemented
4#19127624C212yesmaximal, complemented
5#19127613C211yesmaximal, characteristic, complemented
6#19127576C212yesmaximal, complemented
7#19127573C212yesmaximal, complemented
8#19127559C212yesmaximal, complemented
9#379×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic, complemented
11C23×Dic83C2212yes—
12C23×Dic83C2212yes—
13#14581769—28no—
14C2×C4⋊Dic83C2212yescomplemented
15C2×C4⋊Dic83C2211yescharacteristic, complemented
16C2×Dic83⋊C4C2212yescomplemented
17C2×Dic83⋊C4C2212yescomplemented
18C2×Dic83⋊C4C2212yescomplemented
19C2×C22⋊Dic83—28no—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C83⋊#95C2211yescharacteristic, complemented
25C83⋊#95C2212yescomplemented
26C83⋊#95C2211yescharacteristic, complemented
27C83⋊#95C2212yescomplemented
28C25×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2211yescharacteristic
32C2×C22⋊C4×C83C2211yescharacteristic
33C2×Dic166—24no—
34C2×Dic166—41no—
35C22×Dic83—24no—
36C22×Dic83—24no—
37C22×Dic83—24no—
38C22×Dic83—24no—
39C22×Dic83—42no—
40C22×Dic83—42no—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C4⋊Dic83—24no—
46C4⋊Dic83—41no—
47Dic83⋊C4—24no—
48Dic83⋊C4—24no—
49Dic83⋊C4—42no—
50C22⋊Dic83—24no—
51C22⋊Dic83—24no—
52C22⋊Dic83—24no—
53C22⋊Dic83D418yes—
54C24×C83—24no—
55C24×C83—24no—
56C24×C83—41no—
57C24×C83—41no—
58C24×C83Q814yescomplemented
59C24×C83C2311yescharacteristic
60C24×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C22⋊C4×C83—24no—
65C22⋊C4×C83—24no—
66C22⋊C4×C83—41no—
67C22⋊C4×C83—41no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—216no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—21no—
76C23×C83—21no—
77C23×C83—44no—
78C23×C83—44no—
79C23×C83—48no—
80C23×C83—44no—
81C23×C83C2×D414yes—
82C23×C83C2×Q812yes—
83C23×C83C2×Q814yes—
84C23×C83#4114yes—
85C23×C83C2411yescharacteristic
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—48no—
104C2×Dic83—48no—
105C2×Dic83C2×D414yes—
106C2×Dic83C2×D414yes—
107Dic166—44no—
108C22×C83—24no—
109C22×C83—24no—
110C22×C83—24no—
111C22×C83—24no—
112C22×C83—24no—
113C22×C83—24no—
114C22×C83—24no—
115C22×C83—22no—
116C22×C83—416no—
117C22×C83—41no—
118C22×C83—42no—
119C22×C83—41no—
120C22×C83C22⋊Q818yescomplemented
121C22×C83C22×D412yes—
122C22×C83C22×Q811yescharacteristic
123C22×C83C2×#4112yes—
124C22×C83H(5,𝔽2)11yescharacteristic
125C22×C83H(5,𝔽2)11yescharacteristic
126C4×C83—41no—
127C4×C83—42no—
128C4×C83—41no—
129Dic83—24no—
130Dic83—24no—
131Dic83—42no—
132Dic83—42no—
133C2×C83—28no—
134C2×C83—42no—
135C2×C83C2×C22⋊Q812yes—
136C2×C83#54311yescharacteristic
137C2×C83D4211yescharacteristic
138C2×C83#54612yes—
139C2×C83#55111yescharacteristic
140#2437—831nomaximal, Sylow
141C83#243711yescharacteristic, complemented, Sylow
142C2×C22⋊Q8—832no—
143C22×C22⋊C4—832no—
144#399—831no—
145#396—832no—
146#393—831no—
147#385—832no—
148#385—832no—
149#380—832no—
150#379D8311yescharacteristic
151C25C2×D8311yescharacteristic
152C22×Q8—831no—
153C23×C4—832no—
154C23×C4—832no—
155C22⋊Q8—1668no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4—831no—
158C2×C4⋊C4—832no—
159C2×C4⋊C4—832no—
160C2×C4⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4—832no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—1668no—
166C2×C22⋊C4C2×D8312yes—
167C2×C22⋊C4C2×D8312yes—
168C2×C22⋊C4C2×D8311yescharacteristic
169C2×C22⋊C4C2×D8311yescharacteristic
170#95—831no—
171#95—832no—
172#95—832no—
173#95—831no—
174C24—24no—
175C24—24no—
176C24—41no—
177C24—41no—
178C24Dic16614yescomplemented
179C24C22×D8311yescharacteristic
180C24C22×D8312yes—
181C2×Q8—3321no—
182C2×Q8—1664no—
183C22×C4—3322no—
184C22×C4—3322no—
185C22×C4—832no—
186C22×C4—832no—
187C22×C4—832no—
188C22×C4—832no—
189C22×C4—1664no—
190C22×C4—1664no—
191C22×C4—1664no—
192C22×C4—1664no—
193C22×C4C22×D8311yescharacteristic
194C22×C4C22×D8312yes—
195C22×C4C22×D8311yescharacteristic
196C4⋊C4—3322no—
197C4⋊C4—3321no—
198C4⋊C4—1664no—
199C4⋊C4—1664no—
200C4⋊C4—1664no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—838no—
206C22⋊C4—1664no—
207C22⋊C4—1664no—
208C22⋊C4—1664no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—216no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—44no—
222C23D4×D8314yes—
223C23C2×Dic16612yes—
224C23C2×Dic16614yes—
225C23#1305058914yes—
226C23C23×D8311yescharacteristic
227Q8—3324no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—3328no—
236C2×C4—3328no—
237C2×C4—834no—
238C2×C4—834no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C22—24no—
250C22—24no—
251C22—24no—
252C22—24no—
253C22—24no—
254C22—24no—
255C22—22no—
256C22—24no—
257C22—416no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22#1458176918yescomplemented
262C22C2×D4×D8312yes—
263C22C22×Dic16611yescharacteristic
264C22C2×#1305058912yes—
265C22#1458183711yescharacteristic
266C22#1458183711yescharacteristic
267C4—41no—
268C4—42no—
269C4—41no—
270C4—3322no—
271C4—3322no—
272C4—1664no—
273C4—1664no—
274C2—28no—
275C2—42no—
276C2C2×#1458176912yes—
277C2#1912832711yescharacteristic
278C2#1912833311yescharacteristic
279C2#1912833812yes—
280C2#1912838211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#379×C83C2
prime radicalC25×C83C22
hypercenter#379D83
hypocenterC83#2437
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#14581769C2semidirect
3. #19127637C2semidirect
4. #19127624C2semidirect
5. #19127613C2semidirect
6. #19127576C2semidirect
7. #19127573C2semidirect
8. #19127559C2semidirect
10. C22×Dic166C22semidirect
14. C2×C4⋊Dic83C22semidirect
15. C2×C4⋊Dic83C22semidirect
16. C2×Dic83⋊C4C22semidirect
17. C2×Dic83⋊C4C22semidirect
18. C2×Dic83⋊C4C22semidirect
24. C83⋊#95C22semidirect
25. C83⋊#95C22semidirect
26. C83⋊#95C22semidirect
27. C83⋊#95C22semidirect
58. C24×C83Q8semidirect
120. C22×C83C22⋊Q8semidirect
141. C83#2437semidirect
178. C24Dic166semidirect
261. C22#14581769semidirect

Maximal subgroups

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

subgroupindex
C2×#145817692
C22×C22⋊Dic832
#191276372
#191276242
#191276132
#191275762
#191275732
#191275592
#379×C832
#243783