Finite Group Databasebeta

← #24913225

Subgroups — #24913225

Order 10624.

612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581781C211yesmaximal, characteristic, complemented
2C2×#14581766C211yesmaximal, characteristic
3C2×#14581751C211yesmaximal, characteristic, complemented
4#19127623C214yesmaximal, complemented
5#19127616C214yesmaximal, complemented
6#19127608C211yesmaximal, characteristic, complemented
7#19127602C211yesmaximal, characteristic, complemented
8#19127570C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12#14581781—24no—
13#14581766—24no—
14#14581751—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2214yes—
18C2×C22⋊Dic83C2212yes—
19C2×(C2×D83)⋊C4C2214yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2214yes—
22C2×C4×Dic83C2211yescharacteristic
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2214yes—
25C22×Q8×C83C2211yescharacteristic
26C23×C4×C83C2211yescharacteristic
27C22⋊Q8×C83—24no—
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2212yes—
31C23×D83C2312yes—
32C2×D332—22no—
33C2×D332—22no—
34C2×D332—22no—
35C2×C22⋊D83—24no—
36C2×C22⋊D83—44no—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C4⋊Dic83—22no—
41C4⋊Dic83—24no—
42Dic83⋊C4—44no—
43C22⋊Dic83—24no—
44(C2×D83)⋊C4—24no—
45(C2×D83)⋊C4—28no—
46(C2×D83)⋊C4—44no—
47C4×Dic83—22no—
48C4×Dic83D414yescomplemented
49C24×C83C2311yescharacteristic
50C2×Q8×C83—22no—
51C2×Q8×C83—22no—
52C2×Q8×C83—22no—
53C22×C4×C83—22no—
54C22×C4×C83—22no—
55C22×C4×C83D414yes—
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2311yescharacteristic
60C4⋊C4×C83—22no—
61C4⋊C4×C83—24no—
62C22⋊C4×C83—24no—
63C23×C83—21no—
64C23×C83—21no—
65C23×C83—22no—
66C23×C83—22no—
67C23×C83C2×D412yes—
68C23×C83C2411yescharacteristic
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72C22×D83—24no—
73C22×D83—24no—
74Q8×C83—44no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—28no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—24no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D412yes—
85C2×C4×C83C2×D412yes—
86C2×C4×C83#4112yes—
87C22⋊D83—48no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—24no—
94C2×Dic83—22no—
95C2×Dic83C2×D414yes—
96D332—44no—
97C22×C83—22no—
98C22×C83—22no—
99C22×C83—22no—
100C22×C83—42no—
101C22×C83—42no—
102C22×C83C22×D411yescharacteristic
103C22×C83C22×D411yescharacteristic
104C22×C83C2×#4111yescharacteristic
105C22×C83H(5,𝔽2)12yes—
106C22×C832−1+412yes—
107C2×D83—42no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—44no—
111C2×D83—44no—
112C4×C83—22no—
113C4×C83—42no—
114C4×C83—42no—
115C4×C83#12114yes—
116Dic83—24no—
117Dic83—42no—
118Dic83—42no—
119C2×C83—42no—
120C2×C83C2×#12111yescharacteristic
121C2×C83#53511yescharacteristic
122C2×C83#53611yescharacteristic
123C2×C83#53712yes—
124C2×C83#54212yes—
125D83—82no—
126#2620—831nomaximal, Sylow
127C83#262011yescharacteristic, complemented, Sylow
128C2×#124—831no—
129C2×C22⋊Q8D8311yescharacteristic, complemented
130C2×#121—831no—
131C2×#117—831no—
132#396—834no—
133#394—834no—
134#391—831no—
135#390—831no—
136#384—831no—
137C22×Q8C2×D8311yescharacteristic
138C22×D4—832no—
139C22×D4—831no—
140C23×C4C2×D8311yescharacteristic
141#124—1664no—
142C22⋊Q8—24no—
143#121—1664no—
144#117—1664no—
145C2×C4⋊C4—834no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4C2×D8311yescharacteristic
149C2×C4⋊C4C2×D8312yes—
150C2×C22⋊C4—834no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4—834no—
154C2×C22⋊C4C2×D8312yes—
155C2×C42—831no—
156C2×C42—831no—
157#95—834no—
158C24—832no—
159C24C22×D8311yescharacteristic
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×D4—3324no—
164C2×D4—1664no—
165C2×D4—1662no—
166C2×D4—1662no—
167C2×D4—1662no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—832no—
171C22×C4—832no—
172C22×C4—832no—
173C22×C4C22⋊D8314yes—
174C22×C4C22×D8312yes—
175C22×C4C22×D8312yes—
176C22×C4C22×D8311yescharacteristic
177C22×C4C22×D8311yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—3324no—
181C4⋊C4—1664no—
182C4⋊C4—1662no—
183C22⋊C4—24no—
184C22⋊C4—3324no—
185C22⋊C4—1668no—
186C22⋊C4—1664no—
187C22⋊C4—1664no—
188C42—834no—
189C42—1662no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23—1662no—
195C23—1662no—
196C23—1662no—
197C23—1664no—
198C23—1664no—
199C23C2×C22⋊D8312yes—
200C23C23×D8311yescharacteristic
201Q8—44no—
202D4—3328no—
203D4—3324no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—834no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1664no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4D4×D8312yes—
221C2×C4C2×C22⋊D8312yes—
222C2×C4C2×C22⋊D8312yes—
223C2×C4#1305058912yes—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—42no—
229C22—3324no—
230C22—3322no—
231C22—3322no—
232C22—3322no—
233C22—3324no—
234C22C2×D4×D8311yescharacteristic
235C22C22×C22⋊D8311yescharacteristic
236C22C2×#1305058911yescharacteristic
237C22#1458183812yes—
238C22#1458184012yes—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—3322no—
243C4—3322no—
244C4—1664no—
245C4#1458176414yes—
246C2—42no—
247C2—6642no—
248C2C2×#1458176411yescharacteristic
249C2#1912826511yescharacteristic
250C2#1912827511yescharacteristic
251C2#1912828112yes—
252C2#1912832412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC2×#14581766C2
hypercenterC2×C22⋊Q8D83
hypocenterC83#2620
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14581781C2semidirect
3. C2×#14581751C2semidirect
4. #19127623C2semidirect
5. #19127616C2semidirect
6. #19127608C2semidirect
7. #19127602C2semidirect
8. #19127570C2semidirect
9. C2×C22⋊Q8×C83C2semidirect
48. C4×Dic83D4semidirect
127. C83#2620semidirect
129. C2×C22⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817812
C2×#145817662
C2×#145817512
#191276232
#191276162
#191276082
#191276022
#191275702
C2×C22⋊Q8×C832
#262083