Finite Group Databasebeta

← #24918664

Subgroups — #24918664

Order 10624.

324 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#19128256C211yesmaximal, characteristic
2#125×D83C211yesmaximal, characteristic
3#19128111C211yesmaximal, characteristic, complemented
4#19128110C211yesmaximal, characteristic, complemented
5#19128069C211yesmaximal, characteristic, complemented
6#19128035C212yesmaximal
7#19128030C212yesmaximal
8#19128010C212yesmaximal, complemented
9#19128006C212yesmaximal, complemented
10#19127756C211yesmaximal, characteristic
11#491×C83C211yesmaximal, characteristic, complemented
12#14581792C2211yescharacteristic
13#14581784C2211yescharacteristic
14#14581783—21no—
15#14581783C2211yescharacteristic
16#14581783C2211yescharacteristic
17#14581782—21no—
18#14581782—21no—
19#14581772—21no—
20#14581768C2212yes—
21Q8×Dic83C2212yes—
22#14581757—21no—
23#14581750—21no—
24C4⋊C4×D83—21no—
25C4⋊C4×D83—21no—
26C4⋊C4×D83C2211yescharacteristic
27C4⋊C4×D83C2211yescharacteristic
28C42×D83C2211yescharacteristic
29#14581726C2212yes—
30#14581725C2212yes—
31#14581724C2212yes—
32#14581723C2212yes—
33Dic83⋊C8C2212yes—
34#14581716C2212yes—
35#14581716C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic
38C4⋊Q8×C83C2211yescharacteristic
39#125×C83C2211yescharacteristic
40#103×C83C2212yes—
41#103×C83C2212yes—
42#97×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×C4×D83—21no—
45C2×C4×D83—21no—
46C2×C4×D83D411yescharacteristic
47C2×C4×D83D411yescharacteristic
48C2×C4×D83C2311yescharacteristic
49C2×C83⋊C8C2312yes—
50C4⋊Dic83—21no—
51C4⋊Dic83—22no—
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83C2312yes—
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2311yescharacteristic
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—22no—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64(C2×D83)⋊C4—22no—
65C4×Dic83—22no—
66C4×Dic83D411yescharacteristic
67C4×Dic83D411yescharacteristic
68C4×Dic83C2311yescharacteristic
69C2×Q8×C83C2312yes—
70C2×C8×C83C2312yes—
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2311yescharacteristic
76C4⋊C4×C83C2311yescharacteristic
77C42×C83C2311yescharacteristic
78C22×D83C2×D411yescharacteristic
79Q8×C83—24no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—22no—
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83C2×D411yescharacteristic
90C2×Dic83C2×D411yescharacteristic
91C2×Dic83C2×D411yescharacteristic
92C4×D83—22no—
93C4×D83—22no—
94C4×D83—41no—
95C4×D83—42no—
96C4×D83—42no—
97Dic166—42no—
98C8×C83—22no—
99C83⋊C8—22no—
100C22×C83C22×D411yescharacteristic
101C2×D83—21no—
102C2×D83—21no—
103C2×D83#13712yes—
104C4×C83—21no—
105C4×C83—21no—
106C4×C83—42no—
107C4×C83—41no—
108C4×C83—41no—
109C4×C832−1+412yes—
110Dic83—21no—
111Dic83—21no—
112Dic83—21no—
113Dic83—41no—
114Dic83—42no—
115Dic83—41no—
116C2×C83#53611yescharacteristic
117C2×C83C2×#13711yescharacteristic
118C2×C83#57511yescharacteristic
119D83—22no—
120#3236—831nomaximal, Sylow
121C83#323611yescharacteristic, complemented, Sylow
122#533—831no—
123C2×#125—831no—
124#501—831no—
125#501—831no—
126#491—831no—
127#491D8311yescharacteristic, complemented
128#484—832no—
129#483—832no—
130#479—832no—
131#479—832no—
132#432—831no—
133C4⋊Q8—831no—
134C4⋊Q8C2×D8311yescharacteristic
135#125—831no—
136#125—831no—
137#125—831no—
138#125—1661no—
139#125—1661no—
140#125—1661no—
141#125C2×D8311yescharacteristic
142C22⋊Q8—832no—
143C22⋊Q8—1661no—
144C4×Q8—832no—
145C4×Q8—1661no—
146#117—1661no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4—831no—
149C2×C4⋊C4—1661no—
150C2×C4⋊C4—1661no—
151C2×C42—831no—
152#107—832no—
153#107—832no—
154#106—832no—
155#106—832no—
156C4⋊C8—832no—
157#103—832no—
158#103—832no—
159#103C2×D8312yes—
160#103C2×D8312yes—
161C22⋊C8—832no—
162#97—831no—
163#97C2×D8311yescharacteristic
164C2×Q8—1661no—
165C2×Q8C22×D8312yes—
166C22×C4—831no—
167C22×C4—831no—
168C22×C4—831no—
169C22×C4—1661no—
170C22×C4—1661no—
171C2×C8—832no—
172C2×C8C22×D8312yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—831no—
178C4⋊C4—831no—
179C4⋊C4—832no—
180C4⋊C4—1661no—
181C4⋊C4—1661no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4—1661no—
191C4⋊C4C22×D8311yescharacteristic
192C4⋊C4C22×D8311yescharacteristic
193C22⋊C4—1662no—
194C42—831no—
195C42—831no—
196C42—831no—
197C42—1662no—
198C42C22×D8311yescharacteristic
199C23—831no—
200Q8—24no—
201Q8—3322no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—3321no—
206C2×C4—3322no—
207C2×C4—3322no—
208C2×C4—831no—
209C2×C4—831no—
210C2×C4—831no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1661no—
214C2×C4—1661no—
215C2×C4—1662no—
216C2×C4D4×D8311yescharacteristic
217C2×C4D4×D8311yescharacteristic
218C2×C4C23×D8311yescharacteristic
219C8—22no—
220C8—1662no—
221C22—832no—
222C22—1661no—
223C22—1661no—
224C22C2×D4×D8311yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—3321no—
231C4—3321no—
232C4—3322no—
233C4—1661no—
234C4—1661no—
235C4—1661no—
236C4#1458183912yes—
237C2—1662no—
238C2#1912827111yescharacteristic
239C2#137×D8311yescharacteristic
240C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#491×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenter#491D83
hypocenterC83#3236
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
3. #19128111C2semidirect
4. #19128110C2semidirect
5. #19128069C2semidirect
8. #19128010C2semidirect
9. #19128006C2semidirect
11. #491×C83C2semidirect
121. C83#3236semidirect
127. #491D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#125×D832
#191281112
#191281102
#191280692
#191280352
#191280302
#191280102
#191280062
#191277562
#491×C832
#323683