Finite Group Databasebeta

← #24917246

Subgroups — #24917246

Order 10624.

388 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 131 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, complemented
2C4×#13050591C211yesmaximal, characteristic
3#19128006C212yesmaximal, complemented
4#19128000C212yesmaximal, complemented
5#19127996C211yesmaximal, characteristic, complemented
6#19127992C211yesmaximal, characteristic, complemented
7#19127978C211yesmaximal, characteristic, complemented
8#19127736C212yesmaximal, complemented
9C4⋊C8×D83C211yesmaximal, characteristic, complemented
10#19127696C212yesmaximal
11#475×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C2×C8×D83C2212yes—
14#14581792C2211yescharacteristic
15#14581783C2212yes—
16#14581772—21no—
17#14581768—21no—
18Q8×Dic83—21no—
19Q8×Dic83C2211yescharacteristic
20#14581757—21no—
21C4×D332—21no—
22C4×D332C2211yescharacteristic
23#14581754—21no—
24#14581754C2211yescharacteristic
25#14581750—21no—
26#14581750C2211yescharacteristic
27#14581750C2212yes—
28C42×D83—21no—
29C42×D83C2211yescharacteristic
30#14581725C2212yes—
31#14581725C2212yes—
32#14581724C2212yes—
33C4⋊(C83⋊C8)C2211yescharacteristic
34Dic83⋊C8C2212yes—
35#14581716C2212yes—
36#14581714C2212yes—
37#14581712C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39C4×Q8×C83C2211yescharacteristic
40#106×C83C2212yes—
41C4⋊C8×C83C2211yescharacteristic
42#103×C83C2212yes—
43#13050591—22no—
44#13050591Q814yescomplemented
45C2×Dic166—21no—
46C2×D332—21no—
47C2×D332C2311yescharacteristic
48C2×C4×D83—21no—
49C2×C4×D83—21no—
50C2×C4×D83—21no—
51C2×C4×D83D411yescharacteristic
52C2×C4×D83D411yescharacteristic
53C2×C4×D83C2311yescharacteristic
54C2×C83⋊C8C2312yes—
55C8×D83—24no—
56C4⋊Dic83—21no—
57C4⋊Dic83—22no—
58C4⋊Dic83—21no—
59C4⋊Dic83C2312yes—
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—22no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66C4×Dic83—21no—
67C4×Dic83—21no—
68C4×Dic83—22no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C2×Q8×C83—21no—
73C2×Q8×C83C2311yescharacteristic
74C2×C8×C83C2312yes—
75C4⋊C4×C83—21no—
76C4⋊C4×C83—21no—
77C4⋊C4×C83C2312yes—
78C4⋊C4×C83C2311yescharacteristic
79C42×C83—21no—
80C42×C83C2311yescharacteristic
81C22×D83—21no—
82C22×D83C2×D411yescharacteristic
83Q8×C83—21no—
84Q8×C83—42no—
85Q8×C83C2×Q812yes—
86C2×C4×C83—21no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2411yescharacteristic
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83C2×D411yescharacteristic
95C2×Dic83C2×D411yescharacteristic
96C2×Dic83C2×D411yescharacteristic
97D332—24no—
98D332—21no—
99D332C2×Q812yes—
100C4×D83—22no—
101C4×D83—24no—
102C4×D83—41no—
103C4×D83—42no—
104C4×D83—42no—
105C4×D83C2×Q812yes—
106C4×D83#4112yes—
107Dic166—42no—
108C8×C83—22no—
109C83⋊C8—22no—
110C22×C83C22×D411yescharacteristic
111C2×D83—21no—
112C2×D83—21no—
113C2×D83—22no—
114C2×D83—41no—
115C2×D83—41no—
116C2×D83C22⋊Q812yes—
117C4×C83—22no—
118C4×C83—21no—
119C4×C83—42no—
120C4×C83—41no—
121C4×C83#13512yes—
122C4×C83C22×Q811yescharacteristic
123C4×C83C2×#4111yescharacteristic
124Dic83—21no—
125Dic83—21no—
126Dic83—41no—
127Dic83—42no—
128Dic83C22⋊Q812yes—
129C2×C83C2×C22⋊Q811yescharacteristic
130C2×C83C2×#13511yescharacteristic
131C2×C83#57511yescharacteristic
132D83—22no—
133D83—42no—
134#3076—831nomaximal, Sylow
135C83#307611yescharacteristic, complemented, Sylow
136#533—831no—
137C4×#41—831no—
138#479—832no—
139#478—832no—
140#477—831no—
141#476—831no—
142#475D8311yescharacteristic, complemented
143#474—831no—
144#427—832no—
145C2×C4⋊C8—831no—
146#416—832no—
147C2×#41—831no—
148C22×C8—832no—
149C4⋊Q8—831no—
150C4⋊Q8C2×D8311yescharacteristic
151#125—832no—
152C22⋊Q8—1661no—
153C22⋊Q8—1661no—
154C4×Q8—831no—
155C4×Q8—1661no—
156C4×Q8—1661no—
157C4×Q8C2×D8311yescharacteristic
158C4×D4—831no—
159C4×D4—831no—
160C4×D4—1661no—
161C4×D4—1661no—
162#117—831no—
163#117—832no—
164#117—1661no—
165C2×C42—831no—
166C2×C42—1661no—
167#107—832no—
168#107—832no—
169#106—832no—
170#106C2×D8312yes—
171C4⋊C8—832no—
172C4⋊C8—831no—
173C4⋊C8C2×D8311yescharacteristic
174#103—832no—
175#103C2×D8312yes—
176#102—832no—
177#102—832no—
178#41—834no—
179#41—1662no—
180C2×Q8—21no—
181C2×Q8—1661no—
182C2×Q8C22×D8311yescharacteristic
183C2×D4—831no—
184C2×D4—1661no—
185C22×C4—831no—
186C22×C4—831no—
187C22×C4—831no—
188C22×C4—1661no—
189C22×C4—1661no—
190C22×C4—1661no—
191C2×C8—832no—
192C2×C8—1664no—
193C2×C8C22×D8312yes—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—832no—
197C4⋊C4—831no—
198C4⋊C4—1661no—
199C4⋊C4—1662no—
200C4⋊C4—1662no—
201C4⋊C4—1661no—
202C4⋊C4C22×D8312yes—
203C4⋊C4C22×D8311yescharacteristic
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C22⋊C4—1662no—
207C22⋊C4—1661no—
208C42—21no—
209C42—831no—
210C42—831no—
211C42—831no—
212C42—1661no—
213C42—1661no—
214C42—1662no—
215C42C22×D8311yescharacteristic
216C23—831no—
217C23—1661no—
218Q8—21no—
219Q8—42no—
220Q8—3322no—
221Q8Q8×D8312yes—
222D4—832no—
223D4—1664no—
224D4—1661no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—3321no—
229C2×C4—3322no—
230C2×C4—3322no—
231C2×C4—831no—
232C2×C4—832no—
233C2×C4—832no—
234C2×C4—831no—
235C2×C4—831no—
236C2×C4—1664no—
237C2×C4—1661no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4D4×D8311yescharacteristic
241C2×C4D4×D8311yescharacteristic
242C2×C4C23×D8311yescharacteristic
243C8—22no—
244C8—1662no—
245C22—3321no—
246C22—3321no—
247C22—832no—
248C22—1662no—
249C22—1661no—
250C22—1661no—
251C22C2×D4×D8311yescharacteristic
252C4—22no—
253C4—21no—
254C4—42no—
255C4—41no—
256C4—3321no—
257C4—3322no—
258C4—832no—
259C4—1661no—
260C4—1661no—
261C4#1458181312yes—
262C4C2×Q8×D8311yescharacteristic
263C4#41×D8311yescharacteristic
264C2—3322no—
265C2—1662no—
266C2C22⋊Q8×D8311yescharacteristic
267C2C2×#1458181311yescharacteristic
268C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#475×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050591C22
hypercenter#475D83
hypocenterC83#3076
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128006C2semidirect
4. #19128000C2semidirect
5. #19127996C2semidirect
6. #19127992C2semidirect
7. #19127978C2semidirect
8. #19127736C2semidirect
9. C4⋊C8×D83C2semidirect
11. #475×C83C2semidirect
44. #13050591Q8semidirect
135. C83#3076semidirect
142. #475D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
C4×#130505912
#191280062
#191280002
#191279962
#191279922
#191279782
#191277362
C4⋊C8×D832
#191276962
#475×C832
#307683