Finite Group Databasebeta

← #24919822

Subgroups — #24919822

Order 10624.

388 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128411C211yesmaximal, characteristic, complemented
2#19128356C211yesmaximal, characteristic
3#19128099C211yesmaximal, characteristic, complemented
4#19128030C212yesmaximal, complemented
5#19128018C212yesmaximal, complemented
6#19127991C211yesmaximal, characteristic, complemented
7#19127986C211yesmaximal, characteristic, complemented
8#19127927C212yesmaximal, complemented
9C8×D332C211yesmaximal, characteristic, complemented
10#106×D83C212yesmaximal, complemented
11C4×SD16×C83C211yesmaximal, characteristic, complemented
12C2×#13050590—21no—
13C2×C8×D83C2212yes—
14#14581792C2211yescharacteristic
15#14581792C2211yescharacteristic
16#14581784—21no—
17#14581774—22no—
18#14581771—21no—
19#14581770—22no—
20#14581768C2212yes—
21#14581763C2212yes—
22C4×D332—21no—
23C4×D332C2211yescharacteristic
24C4⋊C4×D83C2212yes—
25C4⋊C4×D83C2212yes—
26#14581724C2212yes—
27#14581724C2212yes—
28#14581723C2212yes—
29#14581723C2212yes—
30C4⋊(C83⋊C8)C2211yescharacteristic
31#14581716C2212yes—
32#14581713C2212yes—
33(C2×D83)⋊C8C2212yes—
34C2×SD16×C83C2211yescharacteristic
35C4×Q8×C83C2211yescharacteristic
36C4×D4×C83C2211yescharacteristic
37#106×C83C2211yescharacteristic
38#103×C83C2211yescharacteristic
39#102×C83C2211yescharacteristic
40C4×C8×C83C2211yescharacteristic
41#13050590—44no—
42C2×Dic166—21no—
43C2×Dic166—21no—
44C2×D332D411yescharacteristic
45C2×C22⋊D83—22no—
46C2×C4×D83—22no—
47C2×C4×D83—22no—
48C2×C4×D83C2312yes—
49C2×C83⋊C8C2312yes—
50C8×D83—24no—
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83D411yescharacteristic
54C4⋊Dic83C2312yes—
55C4⋊Dic83C2312yes—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59C22⋊Dic83—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4D412yes—
62C2×Q8×C83C2311yescharacteristic
63C2×D4×C83C2311yescharacteristic
64C22×C4×C83—21no—
65SD16×C83—22no—
66C2×C8×C83C2311yescharacteristic
67C2×C8×C83C2311yescharacteristic
68C4⋊C4×C83—21no—
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C42×C83—21no—
73C42×C83C2311yescharacteristic
74C23×C83—21no—
75C22×D83C2×D412yes—
76Q8×C83—21no—
77Q8×C83—21no—
78D4×C83—21no—
79D4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—42no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C22⋊D83—44no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83C2×D412yes—
91D332SD1614yescomplemented
92C4×D83—22no—
93C4×D83—22no—
94C4×D83—44no—
95C4×D83—82no—
96Dic166—42no—
97Dic166—81no—
98C8×C83—21no—
99C8×C83#4112yescomplemented
100C83⋊C8—22no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83C22×D411yescharacteristic
105C2×D83—22no—
106C2×D83—22no—
107C2×D83C2×SD1614yes—
108C4×C83—21no—
109C4×C83—41no—
110C4×C83—41no—
111C4×C83—41no—
112C4×C83C2×SD1612yes—
113C4×C83C2×#4111yescharacteristic
114C4×C832−1+411yescharacteristic
115Dic83—22no—
116Dic83—42no—
117Dic83—42no—
118C2×C83—81no—
119C2×C83#54711yescharacteristic
120C2×C83C22×SD1611yescharacteristic
121C2×C83#57711yescharacteristic
122D83—24no—
123#3335—831nomaximal, Sylow
124C83#333511yescharacteristic, complemented, Sylow
125#554—831no—
126#547—831no—
127#498—831no—
128#483—832no—
129#481—832no—
130#476—831no—
131#475—831no—
132#465—832no—
133C4×SD16D8311yescharacteristic, complemented
134C8×D4—831no—
135C2×#106—832no—
136C2×#41—1661no—
137C2×SD16C2×D8311yescharacteristic
138C22×C8—832no—
139C4⋊Q8—831no—
140C4⋊Q8—831no—
141#125—1661no—
142#123—1662no—
143C22⋊Q8—832no—
144C22⋊Q8—1662no—
145C22⋊Q8—1661no—
146#121—832no—
147C4×Q8C2×D8311yescharacteristic
148C4×D4—831no—
149C4×D4—1661no—
150C4×D4C2×D8311yescharacteristic
151C2×C4⋊C4—832no—
152C2×C4⋊C4—832no—
153#106—832no—
154#106—832no—
155#106—832no—
156#106—832no—
157#106C2×D8311yescharacteristic
158C4⋊C8—831no—
159#103—832no—
160#103C2×D8311yescharacteristic
161#102—832no—
162#102C2×D8311yescharacteristic
163C22⋊C8—832no—
164C4×C8C2×D8311yescharacteristic
165#41—3324no—
166C2×Q8—1661no—
167C2×Q8—1661no—
168C2×Q8C22×D8311yescharacteristic
169C2×D4—831no—
170C2×D4—1662no—
171C2×D4C22×D8311yescharacteristic
172C22×C4—21no—
173C22×C4—832no—
174C22×C4—1662no—
175C22×C4—1662no—
176SD16—22no—
177C2×C8—832no—
178C2×C8—1664no—
179C2×C8C22×D8311yescharacteristic
180C2×C8C22×D8311yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—831no—
183C4⋊C4—832no—
184C4⋊C4—832no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4C22×D8311yescharacteristic
191C4⋊C4C22×D8311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—832no—
194C22⋊C4—1662no—
195C22⋊C4—1662no—
196C42—21no—
197C42C22×D8311yescharacteristic
198C23—21no—
199C23—832no—
200Q8—21no—
201Q8—21no—
202Q8—3322no—
203Q8—6641no—
204D4—21no—
205D4—21no—
206D4—3324no—
207D4—834no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—3324no—
213C2×C4—832no—
214C2×C4—6642no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4D4×D8311yescharacteristic
220C2×C4D4×D8311yescharacteristic
221C2×C4C23×D8311yescharacteristic
222C8—21no—
223C8—1662no—
224C8#1305059012yescomplemented
225C22—41no—
226C22—41no—
227C22—41no—
228C22—834no—
229C22—1662no—
230C22—1662no—
231C22C2×D4×D8311yescharacteristic
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—3322no—
237C4—3322no—
238C4—1662no—
239C4SD16×D8312yes—
240C4C2×#1305059011yescharacteristic
241C4#1458183911yescharacteristic
242C2—81no—
243C2—1664no—
244C2#1912835611yescharacteristic
245C2C2×SD16×D8311yescharacteristic
246C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×SD16×C83C2
socleC22×C83C22×D4
prime radical#19128356C2
hypercenterC4×SD16D83
hypocenterC83#3335
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #19128411C2semidirect
3. #19128099C2semidirect
4. #19128030C2semidirect
5. #19128018C2semidirect
6. #19127991C2semidirect
7. #19127986C2semidirect
8. #19127927C2semidirect
9. C8×D332C2semidirect
10. #106×D83C2semidirect
11. C4×SD16×C83C2semidirect
91. D332SD16semidirect
99. C8×C83#41semidirect
124. C83#3335semidirect
133. C4×SD16D83semidirect
224. C8#13050590semidirect

Maximal subgroups

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

subgroupindex
#191284112
#191283562
#191280992
#191280302
#191280182
#191279912
#191279862
#191279272
C8×D3322
#106×D832
C4×SD16×C832
#333583