Finite Group Databasebeta

← #24918633

Subgroups — #24918633

Order 10624.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128235C211yesmaximal, characteristic
2C2×#14581784C211yesmaximal, characteristic, complemented
3#19128103C212yesmaximal, complemented
4#19128056C212yesmaximal, complemented
5#19128037C211yesmaximal, characteristic, complemented
6#19128032C211yesmaximal, characteristic, complemented
7#19128020C211yesmaximal, characteristic
8#19128015C211yesmaximal, characteristic
9#19128009C212yesmaximal, complemented
10#19128002C212yesmaximal, complemented
11#431×C83C211yesmaximal, characteristic, complemented
12#14581786—21no—
13#14581784—21no—
14#14581784—21no—
15#14581784C2211yescharacteristic
16#14581784C2212yes—
17#14581784C2212yes—
18#14581780C2211yescharacteristic
19#14581775—21no—
20#14581771C2211yescharacteristic
21#14581766C2211yescharacteristic
22C4×Dic166C2211yescharacteristic
23C4×D332C2211yescharacteristic
24C4×C22⋊D83—21no—
25#14581749—21no—
26C2×C4⋊Dic83C2211yescharacteristic
27C2×C4⋊Dic83C2211yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30#14581726C2212yes—
31#14581726C2212yes—
32#14581724C2212yes—
33#14581724C2212yes—
34#14581717C2212yes—
35#14581717C2212yes—
36#14581714C2212yes—
37#14581714C2212yes—
38C2×C42×C83C2211yescharacteristic
39C4⋊C8×C83C2212yes—
40C22⋊C8×C83C2211yescharacteristic
41C22⋊C8×C83C2211yescharacteristic
42C4×C8×C83C2212yes—
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C22×Dic83—21no—
47C22×Dic83—21no—
48C2×C4×D83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83—21no—
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2312yes—
54C4⋊Dic83C2312yes—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58Dic83⋊C4—22no—
59Dic83⋊C4—22no—
60Dic83⋊C4—22no—
61Dic83⋊C4—22no—
62Dic83⋊C4—21no—
63C22⋊Dic83—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68C4×Dic83—21no—
69C22×C4×C83D411yescharacteristic
70C22×C4×C83D411yescharacteristic
71C22×C4×C83C2311yescharacteristic
72C2×C8×C83C2312yes—
73C2×C8×C83C2312yes—
74C42×C83D411yescharacteristic
75C42×C83D411yescharacteristic
76C42×C83C2311yescharacteristic
77C42×C83C2311yescharacteristic
78C23×C83C2×D411yescharacteristic
79C22×D83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—41no—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2411yescharacteristic
89C22⋊D83—42no—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—42no—
96C2×Dic83—42no—
97D332—22no—
98C4×D83—42no—
99Dic166—22no—
100C8×C83—22no—
101C8×C83—22no—
102C22×C83—21no—
103C22×C83—21no—
104C22×C83#13512yes—
105C22×C83C22×D411yescharacteristic
106C2×D83—41no—
107C2×D83—41no—
108C2×D83—41no—
109C4×C83—21no—
110C4×C83—21no—
111C4×C83—21no—
112C4×C83—21no—
113C4×C83—21no—
114C4×C832−1+412yes—
115Dic83—41no—
116Dic83—42no—
117Dic83—41no—
118Dic83—41no—
119Dic83—42no—
120C2×C83—22no—
121C2×C83#53611yescharacteristic
122C2×C83C2×#13511yescharacteristic
123C2×C83#57511yescharacteristic
124D83—81no—
125#3232—831nomaximal, Sylow
126C83#323211yescharacteristic, complemented, Sylow
127#529—831no—
128C2×#125—831no—
129#499—832no—
130#488—832no—
131#484—831no—
132#483—831no—
133#481—831no—
134#480—831no—
135#479—832no—
136#478—832no—
137#431D8311yescharacteristic, complemented
138#126—1661no—
139#125—832no—
140#125—832no—
141#125—831no—
142#125—1661no—
143#125—1661no—
144#124—831no—
145#123—1661no—
146C22⋊Q8—831no—
147#121—831no—
148C4×Q8—831no—
149C4×D4—831no—
150C4×D4—1661no—
151#117—1661no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—1661no—
155C2×C4⋊C4—1661no—
156C2×C42C2×D8311yescharacteristic
157#107—832no—
158#107—832no—
159#106—832no—
160#106—832no—
161C4⋊C8C2×D8312yes—
162#103—832no—
163#103—832no—
164#102—832no—
165#102—832no—
166C22⋊C8C2×D8311yescharacteristic
167C22⋊C8C2×D8311yescharacteristic
168C4×C8C2×D8312yes—
169C2×Q8—831no—
170C2×D4—831no—
171C2×D4—1661no—
172C22×C4—1661no—
173C22×C4—1661no—
174C22×C4—1661no—
175C22×C4D33211yescharacteristic
176C22×C4D33211yescharacteristic
177C22×C4C22×D8311yescharacteristic
178C2×C8C22×D8312yes—
179C2×C8C22×D8312yes—
180C4⋊C4—832no—
181C4⋊C4—832no—
182C4⋊C4—831no—
183C4⋊C4—831no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C4⋊C4—1662no—
189C4⋊C4—1662no—
190C4⋊C4—1661no—
191C4⋊C4—1661no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C22⋊C4—1661no—
195C22⋊C4—1661no—
196C22⋊C4—1661no—
197C22⋊C4—1661no—
198C22⋊C4—1661no—
199C42—1661no—
200C42D33211yescharacteristic
201C42D33211yescharacteristic
202C42C22×D8311yescharacteristic
203C42C22×D8311yescharacteristic
204C23—1661no—
205C23C2×D33211yescharacteristic
206Q8—1662no—
207D4—3322no—
208D4—1662no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—3322no—
213C2×C4—3322no—
214C2×C4—3322no—
215C2×C4—1661no—
216C2×C4—1661no—
217C2×C4—1661no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4C2×D33211yescharacteristic
221C2×C4C2×D33211yescharacteristic
222C2×C4C2×D33211yescharacteristic
223C2×C4C2×D33211yescharacteristic
224C2×C4C2×D33211yescharacteristic
225C2×C4C23×D8311yescharacteristic
226C8—22no—
227C8—22no—
228C22—21no—
229C22—21no—
230C22—3321no—
231C22—3321no—
232C22—3321no—
233C22#1458181412yes—
234C22C22×D33211yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—3322no—
241C4—3321no—
242C4—3321no—
243C4—3322no—
244C4—3321no—
245C4#1458183912yes—
246C2—22no—
247C2—6641no—
248C2#1912827711yescharacteristic
249C2C2×#1458181411yescharacteristic
250C2#1912850111yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#431×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenter#431D83
hypocenterC83#3232
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. C2×#14581784C2semidirect
3. #19128103C2semidirect
4. #19128056C2semidirect
5. #19128037C2semidirect
6. #19128032C2semidirect
9. #19128009C2semidirect
10. #19128002C2semidirect
11. #431×C83C2semidirect
126. C83#3232semidirect
137. #431D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
C2×#145817842
#191281032
#191280562
#191280372
#191280322
#191280202
#191280152
#191280092
#191280022
#431×C832
#323283