Finite Group Databasebeta

← #24918837

Subgroups — #24918837

Order 10624.

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

normal subgroups only

This group has 348 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#19128009C212yesmaximal, complemented
4#19127997C211yesmaximal, characteristic, complemented
5#19127992C211yesmaximal, characteristic, complemented
6#19127959C212yesmaximal, complemented
7#19127947C212yesmaximal
8#19127924C212yesmaximal, complemented
9#19127912C211yesmaximal, characteristic, complemented
10#19127907C211yesmaximal, characteristic, complemented
11#423×C83C211yesmaximal, characteristic, complemented
12C2×Dic332C2212yes—
13C2×#13050579C2212yes—
14#14581792C2211yescharacteristic
15#14581792C2211yescharacteristic
16#14581786—21no—
17#14581784—21no—
18#14581784C2211yescharacteristic
19#14581780C2211yescharacteristic
20#14581775—21no—
21#14581771—21no—
22#14581771C2212yes—
23#14581771C2211yescharacteristic
24#14581766C2211yescharacteristic
25C4×Dic166—21no—
26C4×Dic166C2211yescharacteristic
27C4×Dic166C2212yes—
28C4×D332C2211yescharacteristic
29C4×C22⋊D83—21no—
30#14581751—21no—
31#14581749—21no—
32#14581726C2212yes—
33#14581724C2212yes—
34#14581717C2212yes—
35#14581717C2212yes—
36#14581717C2212yes—
37#14581714C2212yes—
38C2×M16×C83C2212yes—
39C2×C42×C83C2211yescharacteristic
40C4⋊C8×C83C2211yescharacteristic
41C4⋊C8×C83C2211yescharacteristic
42C4⋊C8×C83C2212yes—
43C2×Dic166—21no—
44C2×Dic166C2311yescharacteristic
45C2×Dic166C2312yes—
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C2×C4×D83—21no—
49Dic332—42no—
50#13050579—42no—
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83C2311yescharacteristic
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2312yes—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63C22⋊Dic83—22no—
64C22⋊Dic83—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69C4×Dic83—21no—
70C4×Dic83—22no—
71C22×C4×C83D411yescharacteristic
72C22×C4×C83D411yescharacteristic
73C22×C4×C83C2311yescharacteristic
74M16×C83—42no—
75C2×C8×C83C2312yes—
76C2×C8×C83C2312yes—
77C42×C83D411yescharacteristic
78C42×C83D411yescharacteristic
79C42×C83C2311yescharacteristic
80C42×C83C2311yescharacteristic
81C23×C83C2×D411yescharacteristic
82C22×D83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—41no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—42no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99D332—21no—
100D332—21no—
101C4×D83—42no—
102Dic166—22no—
103Dic166—21no—
104Dic166—21no—
105Dic166—22no—
106Dic166—42no—
107C8×C83—22no—
108C8×C83—22no—
109C22×C83—21no—
110C22×C83—21no—
111C22×C83—21no—
112C22×C83C22×D411yescharacteristic
113C2×D83—41no—
114C2×D83—41no—
115C2×D83—41no—
116C4×C83—21no—
117C4×C83—21no—
118C4×C83—21no—
119C4×C83—21no—
120C4×C83#13712yes—
121C4×C83H(5,𝔽2)11yescharacteristic
122C4×C832−1+411yescharacteristic
123Dic83—42no—
124Dic83—41no—
125Dic83—41no—
126Dic83—42no—
127Dic83—41no—
128C2×C83—41no—
129C2×C83#53711yescharacteristic
130C2×C83C2×#13711yescharacteristic
131C2×C83#57511yescharacteristic
132D83—81no—
133#3252—831nomaximal, Sylow
134C83#325211yescharacteristic, complemented, Sylow
135#533—831no—
136#529—831no—
137#479—832no—
138#477—831no—
139#476—831no—
140#470—832no—
141#468—832no—
142#464—832no—
143#462—831no—
144#461—831no—
145#423D8311yescharacteristic, complemented
146C2×Q16—832no—
147C2×SD16—832no—
148C2×M16C2×D8312yes—
149C4⋊Q8—831no—
150C4⋊Q8—831no—
151#126—1661no—
152#125—831no—
153#125—1661no—
154#124—831no—
155#123—1661no—
156C22⋊Q8—831no—
157C22⋊Q8—832no—
158C22⋊Q8—1661no—
159#121—831no—
160C4×Q8—831no—
161C4×Q8—832no—
162C4×Q8—1661no—
163C4×D4—831no—
164C4×D4—1661no—
165#117—1661no—
166#117—1661no—
167C2×C42C2×D8311yescharacteristic
168#107—832no—
169#106—832no—
170C4⋊C8C2×D8311yescharacteristic
171C4⋊C8C2×D8311yescharacteristic
172C4⋊C8C2×D8312yes—
173#103—832no—
174#103—832no—
175#103—832no—
176#102—832no—
177C2×Q8—831no—
178C2×Q8—832no—
179C2×Q8—1661no—
180C2×D4—831no—
181C2×D4—1661no—
182C22×C4—1661no—
183C22×C4D33211yescharacteristic
184C22×C4D33211yescharacteristic
185C22×C4C22×D8311yescharacteristic
186Q16—3322no—
187SD16—3322no—
188M16—42no—
189C2×C8C22×D8312yes—
190C2×C8C22×D8312yes—
191C4⋊C4—832no—
192C4⋊C4—831no—
193C4⋊C4—831no—
194C4⋊C4—1661no—
195C4⋊C4—1661no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4—1662no—
200C4⋊C4—1661no—
201C4⋊C4—1662no—
202C4⋊C4—1661no—
203C22⋊C4—1662no—
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C22⋊C4—1661no—
207C22⋊C4—1661no—
208C22⋊C4—1661no—
209C42—1661no—
210C42—1662no—
211C42D33211yescharacteristic
212C42D33211yescharacteristic
213C42C22×D8311yescharacteristic
214C42C22×D8311yescharacteristic
215C23—1661no—
216C23C2×D33211yescharacteristic
217Q8—3322no—
218Q8—1662no—
219Q8—1661no—
220Q8—1661no—
221Q8—1662no—
222D4—3322no—
223D4—1661no—
224D4—1661no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—3322no—
230C2×C4—1661no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1661no—
235C2×C4C2×D33211yescharacteristic
236C2×C4C2×D33211yescharacteristic
237C2×C4C2×D33211yescharacteristic
238C2×C4C2×D33211yescharacteristic
239C2×C4C2×D33211yescharacteristic
240C2×C4C23×D8311yescharacteristic
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—3321no—
247C22—3321no—
248C22—3321no—
249C22C22×D33211yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—3321no—
255C4—3321no—
256C4—3322no—
257C4—3321no—
258C4—3322no—
259C4#1458182512yes—
260C4#1458183711yescharacteristic
261C4#1458184011yescharacteristic
262C2—41no—
263C2—6641no—
264C2#1912828211yescharacteristic
265C2C2×#1458182511yescharacteristic
266C2#1912850111yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#423×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenter#423D83
hypocenterC83#3252
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
3. #19128009C2semidirect
4. #19127997C2semidirect
5. #19127992C2semidirect
6. #19127959C2semidirect
8. #19127924C2semidirect
9. #19127912C2semidirect
10. #19127907C2semidirect
11. #423×C83C2semidirect
134. C83#3252semidirect
145. #423D83semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191282352
#191280092
#191279972
#191279922
#191279592
#191279472
#191279242
#191279122
#191279072
#423×C832
#325283