Finite Group Databasebeta

← #24918644

Subgroups — #24918644

Order 10624.

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

normal subgroups only

This group has 332 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#19128258C211yesmaximal, characteristic, complemented
2#19128235C211yesmaximal, characteristic
3#19128103C212yesmaximal, complemented
4#19128056C212yesmaximal, complemented
5#19128043C212yesmaximal, complemented
6#19128026C212yesmaximal
7#19127997C211yesmaximal, characteristic, complemented
8#19127992C211yesmaximal, characteristic, complemented
9#19127987C211yesmaximal, characteristic, complemented
10#19127982C211yesmaximal, characteristic, complemented
11#431×C83C211yesmaximal, characteristic, complemented
12#14581792C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581786—21no—
15#14581784C2211yescharacteristic
16#14581784C2212yes—
17#14581780C2211yescharacteristic
18#14581775—21no—
19#14581771—21no—
20#14581771—21no—
21#14581771C2211yescharacteristic
22#14581766C2211yescharacteristic
23C4×Dic166—21no—
24C4×Dic166—21no—
25C4×Dic166C2211yescharacteristic
26C4×D332C2211yescharacteristic
27C4×C22⋊D83—21no—
28#14581751C2212yes—
29#14581749—21no—
30#14581726C2212yes—
31#14581726C2212yes—
32#14581724C2212yes—
33#14581724C2212yes—
34#14581717C2212yes—
35#14581717C2212yes—
36#14581714C2212yes—
37#14581714C2212yes—
38C2×C42×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40C4⋊C8×C83C2211yescharacteristic
41C22⋊C8×C83C2212yes—
42C4×C8×C83C2212yes—
43C2×Dic166—21no—
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C2×C4×D83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83—21no—
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2312yes—
54C4⋊Dic83C2312yes—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61C22⋊Dic83—22no—
62C22⋊Dic83—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67C4×Dic83—21no—
68C4×Dic83—22no—
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—21no—
82C2×C4×C83—21no—
83C2×C4×C83—41no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2411yescharacteristic
90C22⋊D83—42no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—21no—
96D332—21no—
97D332—21no—
98C4×D83—42no—
99Dic166—21no—
100Dic166—21no—
101Dic166—42no—
102Dic166—42no—
103C8×C83—22no—
104C8×C83—22no—
105C22×C83—21no—
106C22×C83—21no—
107C22×C83—21no—
108C22×C83C22×D411yescharacteristic
109C2×D83—41no—
110C2×D83—41no—
111C2×D83—41no—
112C4×C83—21no—
113C4×C83—21no—
114C4×C83—21no—
115C4×C83—21no—
116C4×C83#13512yes—
117C4×C832−1+411yescharacteristic
118C4×C832−1+411yescharacteristic
119Dic83—41no—
120Dic83—42no—
121Dic83—41no—
122Dic83—42no—
123Dic83—41no—
124C2×C83—41no—
125C2×C83#53611yescharacteristic
126C2×C83C2×#13511yescharacteristic
127C2×C83#57511yescharacteristic
128D83—81no—
129#3233—831nomaximal, Sylow
130C83#323311yescharacteristic, complemented, Sylow
131#533—831no—
132#529—831no—
133#499—832no—
134#488—832no—
135#485—832no—
136#482—832no—
137#477—831no—
138#476—831no—
139#475—831no—
140#474—831no—
141#431D8311yescharacteristic, complemented
142C4⋊Q8—831no—
143C4⋊Q8—831no—
144#126—1661no—
145#125—831no—
146#125—832no—
147#124—831no—
148#123—1661no—
149C22⋊Q8—831no—
150C22⋊Q8—1661no—
151C22⋊Q8—1661no—
152#121—831no—
153C4×Q8—831no—
154C4×Q8—1661no—
155C4×Q8—1661no—
156C4×D4—831no—
157C4×D4—1661no—
158#117—832no—
159#117—1661no—
160C2×C42C2×D8311yescharacteristic
161#107—832no—
162#107—832no—
163#106—832no—
164#106—832no—
165C4⋊C8C2×D8311yescharacteristic
166C4⋊C8C2×D8311yescharacteristic
167#103—832no—
168#103—832no—
169#102—832no—
170#102—832no—
171C22⋊C8C2×D8312yes—
172C4×C8C2×D8312yes—
173C2×Q8—831no—
174C2×Q8—1661no—
175C2×Q8—1661no—
176C2×D4—831no—
177C2×D4—1661no—
178C22×C4—1661no—
179C22×C4D33211yescharacteristic
180C22×C4D33211yescharacteristic
181C22×C4C22×D8311yescharacteristic
182C2×C8C22×D8312yes—
183C2×C8C22×D8312yes—
184C4⋊C4—832no—
185C4⋊C4—832no—
186C4⋊C4—831no—
187C4⋊C4—831no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4—1661no—
191C4⋊C4—1661no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C4⋊C4—1661no—
195C4⋊C4—1661no—
196C22⋊C4—1662no—
197C22⋊C4—1661no—
198C22⋊C4—1661no—
199C22⋊C4—1661no—
200C22⋊C4—1661no—
201C22⋊C4—1661no—
202C42—1662no—
203C42—1661no—
204C42D33211yescharacteristic
205C42D33211yescharacteristic
206C42C22×D8311yescharacteristic
207C42C22×D8311yescharacteristic
208C23—1661no—
209C23C2×D33211yescharacteristic
210Q8—3322no—
211Q8—3322no—
212Q8—1661no—
213Q8—1661no—
214D4—3322no—
215D4—1661no—
216D4—1661no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—3322no—
222C2×C4—1661no—
223C2×C4—1661no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1661no—
227C2×C4C2×D33211yescharacteristic
228C2×C4C2×D33211yescharacteristic
229C2×C4C2×D33211yescharacteristic
230C2×C4C2×D33211yescharacteristic
231C2×C4C2×D33211yescharacteristic
232C2×C4C23×D8311yescharacteristic
233C8—22no—
234C8—22no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—3321no—
239C22—3321no—
240C22—3321no—
241C22C22×D33211yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—3321no—
247C4—3322no—
248C4—3321no—
249C4—3322no—
250C4—3321no—
251C4#1458181412yes—
252C4#1458183911yescharacteristic
253C4#1458183911yescharacteristic
254C2—41no—
255C2—6641no—
256C2#1912827711yescharacteristic
257C2C2×#1458181411yescharacteristic
258C2#1912850111yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#431×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenter#431D83
hypocenterC83#3233
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
1. #19128258C2semidirect
3. #19128103C2semidirect
4. #19128056C2semidirect
5. #19128043C2semidirect
7. #19127997C2semidirect
8. #19127992C2semidirect
9. #19127987C2semidirect
10. #19127982C2semidirect
11. #431×C83C2semidirect
130. C83#3233semidirect
141. #431D83semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191282352
#191281032
#191280562
#191280432
#191280262
#191279972
#191279922
#191279872
#191279822
#431×C832
#323383