Finite Group Databasebeta

← #24917098

Subgroups — #24917098

Order 10624.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#14581798C211yesmaximal, characteristic, complemented
2#19128275C211yesmaximal, characteristic, complemented
3#19128268C211yesmaximal, characteristic
4#19127944C214yesmaximal, complemented
5#19127927C214yesmaximal, complemented
6#19127739C211yesmaximal, characteristic, complemented
7#19127710C211yesmaximal, characteristic, complemented
8#19127706C211yesmaximal, characteristic, complemented
9C22×SD16×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11#14581798—24no—
12C2×#13050574C2211yescharacteristic
13C2×#13050573C2212yescomplemented
14C2×C8×D83C2212yescomplemented
15#14581790—21no—
16#14581788—21no—
17#14581781—21no—
18#14581779—21no—
19#14581775—22no—
20#14581768C2214yescomplemented
21#14581763C2214yes—
22#14581761—22no—
23#14581751C2211yescharacteristic
24#14581751C2211yescharacteristic
25#14581726C2212yescomplemented
26#14581724C2212yescomplemented
27#14581716C2214yescomplemented
28#14581713C2214yes—
29C22×Q8×C83C2211yescharacteristic
30C22×D4×C83C2211yescharacteristic
31C2×SD16×C83—24no—
32C2×SD16×C83C2212yes—
33C2×SD16×C83C2212yes—
34C22×C8×C83C2211yescharacteristic
35#13050590—22no—
36#13050590—22no—
37C2×Dic166D411yescharacteristic
38C2×D332D411yescharacteristic
39C2×C22⋊D83—24no—
40C2×C22⋊D83D412yes—
41C2×C4×D83C2312yes—
42#13050574—22no—
43#13050573—24no—
44C2×C83⋊C8C2312yes—
45C8×D83—24no—
46C4⋊Dic83C2312yes—
47C4⋊Dic83C2312yes—
48Dic83⋊C4—24no—
49C22⋊Dic83—21no—
50C22⋊Dic83—21no—
51C22⋊Dic83—24no—
52(C2×D83)⋊C4—24no—
53C4×Dic83—21no—
54C4×Dic83—21no—
55C24×C83—21no—
56C2×Q8×C83—21no—
57C2×Q8×C83—22no—
58C2×Q8×C83—22no—
59C2×Q8×C83C2312yes—
60C2×D4×C83—22no—
61C2×D4×C83—21no—
62C2×D4×C83—22no—
63C2×D4×C83C2312yes—
64C22×C4×C83—21no—
65C22×C4×C83C2311yescharacteristic
66SD16×C83—24no—
67SD16×C83—24no—
68C2×C8×C83D414yescomplemented
69C2×C8×C83C2311yescharacteristic
70C2×C8×C83C2311yescharacteristic
71C23×C83—22no—
72C23×C83—42no—
73C23×C83—82no—
74C23×C83C2×D411yescharacteristic
75C22×D83C2×D412yes—
76Q8×C83—22no—
77Q8×C83—22no—
78Q8×C83—42no—
79D4×C83—22no—
80D4×C83—22no—
81D4×C83—42no—
82C2×C4×C83—22no—
83C2×C4×C83—42no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D412yes—
86C2×C4×C83#4112yes—
87C2×C4×C83C2411yescharacteristic
88C22⋊D83—24no—
89C22⋊D83—84no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83C2×D412yes—
93D332—22no—
94C4×D83—22no—
95C4×D83—22no—
96Dic166—22no—
97C8×C83C2×D412yes—
98C8×C83C2×D412yes—
99C83⋊C8—22no—
100C22×C83—22no—
101C22×C83—42no—
102C22×C83—44no—
103C22×C83—84no—
104C22×C83#12112yes—
105C22×C83C22×D411yescharacteristic
106C2×D83—22no—
107C2×D83—24no—
108C4×C83—42no—
109C4×C83#12112yes—
110C4×C83C22×D411yescharacteristic
111C4×C83C2×#4111yescharacteristic
112Dic83—22no—
113Dic83—42no—
114Dic83—42no—
115C2×C83—22no—
116C2×C83—82no—
117C2×C83C2×#12111yescharacteristic
118C2×C83#57712yes—
119D83—42no—
120#3057—831nomaximal, Sylow
121C83#305711yescharacteristic, complemented, Sylow
122C22×SD16D8311yescharacteristic, complemented
123C2×#131—831no—
124#536—831no—
125#535—831no—
126#468—834no—
127#465—834no—
128#427—831no—
129#419—831no—
130#418—831no—
131C2×#41—831no—
132C22×Q8C2×D8311yescharacteristic
133C22×D4C2×D8311yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D8312yes—
136C2×SD16C2×D8312yes—
137#131—1664no—
138C2×M16—831no—
139C2×M16—832no—
140C22×C8—832no—
141C22×C8C2×D8311yescharacteristic
142C4⋊Q8—1661no—
143#127—1661no—
144#124—1661no—
145#124—1661no—
146#123—1662no—
147C22⋊Q8—834no—
148#121—834no—
149#120—1662no—
150#117—831no—
151#117—831no—
152#107—832no—
153#106—832no—
154#103—834no—
155#102—834no—
156C24—21no—
157#41—1662no—
158#41—1662no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—831no—
163C2×Q8C22×D8312yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—832no—
168C2×D4—831no—
169C2×D4—1664no—
170C2×D4C22×D8312yes—
171C22×C4—21no—
172C22×C4—832no—
173C22×C4C22×D8311yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—1662no—
177M16—1664no—
178C2×C8—832no—
179C2×C8—1664no—
180C2×C8C22⋊D8314yescomplemented
181C2×C8C22×D8311yescharacteristic
182C2×C8C22×D8311yescharacteristic
183C4⋊C4—832no—
184C4⋊C4—832no—
185C4⋊C4—1664no—
186C22⋊C4—1664no—
187C22⋊C4—1664no—
188C22⋊C4—1661no—
189C22⋊C4—1661no—
190C42—1661no—
191C42—1661no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—832no—
196C23D4×D8311yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—1662no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—6644no—
205D4—1664no—
206D4—1662no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—832no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4D4×D8311yescharacteristic
215C2×C4C2×C22⋊D8312yes—
216C2×C4#1305058912yes—
217C2×C4C23×D8311yescharacteristic
218C8—1662no—
219C8C2×C22⋊D8312yes—
220C8C2×C22⋊D8312yes—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—1662no—
226C22—1664no—
227C22#1458176312yes—
228C22C2×D4×D8311yescharacteristic
229C4—42no—
230C4—3322no—
231C4—3322no—
232C4—1662no—
233C4#1458176312yes—
234C4C22×C22⋊D8311yescharacteristic
235C4C2×#1305058911yescharacteristic
236C2—22no—
237C2—82no—
238C2—3322no—
239C2C2×#1458176311yescharacteristic
240C2#1912850712yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC22×SD16×C83C2
socleC22×C83C22×D4
prime radical#19128268C2
hypercenterC22×SD16D83
hypocenterC83#3057
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#14581798C2semidirect
2. #19128275C2semidirect
4. #19127944C2semidirect
5. #19127927C2semidirect
6. #19127739C2semidirect
7. #19127710C2semidirect
8. #19127706C2semidirect
9. C22×SD16×C83C2semidirect
13. C2×#13050573C22semidirect
14. C2×C8×D83C22semidirect
20. #14581768C22semidirect
25. #14581726C22semidirect
26. #14581724C22semidirect
27. #14581716C22semidirect
68. C2×C8×C83D4semidirect
121. C83#3057semidirect
122. C22×SD16D83semidirect
180. C2×C8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817982
#191282752
#191282682
#191279442
#191279272
#191277392
#191277102
#191277062
C22×SD16×C832
#305783