Finite Group Databasebeta

← #24917101

Subgroups — #24917101

Order 10624.

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#13050578C211yesmaximal, characteristic, complemented
2C2×#14581800C211yesmaximal, characteristic, complemented
3#19128277C211yesmaximal, characteristic, complemented
4#19128268C211yesmaximal, characteristic
5#19127946C214yesmaximal, complemented
6#19127929C214yesmaximal, complemented
7#19127737C211yesmaximal, characteristic, complemented
8#19127711C211yesmaximal, characteristic, complemented
9#418×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C22×Dic166C2211yescharacteristic
12C2×#13050578—24no—
13C2×#13050578C2212yes—
14C2×#13050578C2212yes—
15#14581800—24no—
16C2×#13050573C2212yescomplemented
17C22×C83⋊C8C2211yescharacteristic
18C2×C8×D83C2212yescomplemented
19#14581792—21no—
20#14581788—21no—
21#14581780—21no—
22#14581779—21no—
23#14581776—22no—
24#14581772C2214yescomplemented
25#14581763C2214yes—
26#14581761—22no—
27#14581751C2211yescharacteristic
28#14581725C2212yescomplemented
29#14581723C2212yescomplemented
30#14581717C2214yescomplemented
31C22×D4×C83C2211yescharacteristic
32C2×M16×C83C2211yescharacteristic
33#117×C83C2211yescharacteristic
34#102×C83C2214yes—
35#13050590—22no—
36#13050590—22no—
37C2×Dic166—22no—
38C2×Dic166—22no—
39C2×Dic166—21no—
40C2×Dic166D411yescharacteristic
41C2×Dic166C2312yes—
42C2×D332D411yescharacteristic
43C2×C22⋊D83—24no—
44C2×C22⋊D83D412yes—
45C22×Dic83—21no—
46C2×C4×D83C2312yes—
47#13050578—24no—
48#13050578—24no—
49#13050573—24no—
50C2×C83⋊C8D414yescomplemented
51C2×C83⋊C8C2311yescharacteristic
52C2×C83⋊C8C2311yescharacteristic
53C8×D83—24no—
54C4⋊Dic83—24no—
55C4⋊Dic83C2312yes—
56C22⋊Dic83—21no—
57C22⋊Dic83—24no—
58(C2×D83)⋊C4—24no—
59C4×Dic83—21no—
60C24×C83—21no—
61C2×D4×C83—22no—
62C2×D4×C83—21no—
63C2×D4×C83—22no—
64C2×D4×C83C2312yes—
65C22×C4×C83C2311yescharacteristic
66M16×C83—22no—
67C2×C8×C83C2312yes—
68C4⋊C4×C83C2312yes—
69C22⋊C4×C83—21no—
70C42×C83—21no—
71C23×C83—22no—
72C23×C83—42no—
73C23×C83—82no—
74C23×C83C2×D411yescharacteristic
75C22×D83C2×D412yes—
76D4×C83—22no—
77D4×C83—22no—
78D4×C83—42no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D412yes—
82C2×C4×C83#4112yes—
83C2×C4×C83C2411yescharacteristic
84C22⋊D83—24no—
85C22⋊D83—84no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—42no—
89C2×Dic83C2×D412yes—
90D332—22no—
91C4×D83—22no—
92C4×D83—22no—
93Dic166—22no—
94Dic166—22no—
95Dic166—22no—
96Dic166—42no—
97C8×C83—22no—
98C83⋊C8C2×D412yes—
99C83⋊C8C2×D412yes—
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×SD16—831no—
123C2×#131—831no—
124#536—831no—
125#535—831no—
126#468—834no—
127#465—834no—
128#427—831no—
129#419—831no—
130#418D8311yescharacteristic, complemented
131C2×#41—831no—
132C22×Q8—831no—
133C22×D4C2×D8311yescharacteristic
134C2×SD16—832no—
135C2×SD16—832no—
136C2×SD16—1664no—
137#131—1664no—
138C2×M16—832no—
139C2×M16C2×D8311yescharacteristic
140C22×C8—831no—
141C22×C8—832no—
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#117C2×D8311yescharacteristic
152#107—832no—
153#106—832no—
154#103—834no—
155#102C2×D8314yes—
156C24—21no—
157#41—1662no—
158#41—1662no—
159C2×Q8—831no—
160C2×Q8—832no—
161C2×Q8—1662no—
162C2×Q8—1662no—
163C2×Q8—1661no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—832no—
168C2×D4—831no—
169C2×D4—1664no—
170C2×D4C22×D8312yes—
171C22×C4—832no—
172C22×C4—1661no—
173C22×C4C22×D8311yescharacteristic
174SD16—1664no—
175SD16—1664no—
176M16—22no—
177M16—1664no—
178C2×C8—834no—
179C2×C8—831no—
180C2×C8—831no—
181C2×C8—1664no—
182C2×C8C22×D8312yes—
183C4⋊C4—832no—
184C4⋊C4—1664no—
185C4⋊C4C22×D8312yes—
186C22⋊C4—21no—
187C22⋊C4—1664no—
188C22⋊C4—1664no—
189C22⋊C4—1661no—
190C42—21no—
191C42—1661no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—832no—
196C23D4×D8311yescharacteristic
197Q8—3322no—
198Q8—1662no—
199Q8—1662no—
200Q8—1662no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—6644no—
205D4—1664no—
206D4—1662no—
207C2×C4—22no—
208C2×C4—3322no—
209C2×C4—832no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4D4×D8311yescharacteristic
215C2×C4D4×D8312yes—
216C2×C4#1305059012yes—
217C2×C4C23×D8311yescharacteristic
218C8—22no—
219C8—832no—
220C8—832no—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—1662no—
226C22—1664no—
227C22#1458176512yes—
228C22C2×D4×D8311yescharacteristic
229C4—42no—
230C4—3322no—
231C4—3322no—
232C4—1662no—
233C4#1458176512yes—
234C4C2×D4×D8311yescharacteristic
235C4C2×#1305059011yescharacteristic
236C2—22no—
237C2—82no—
238C2—3322no—
239C2C2×#1458176511yescharacteristic
240C2#1912851112yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#418×C83C2
socleC22×C83C22×D4
prime radical#19128268C2
hypercenter#418D83
hypocenterC83#3057
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#13050578C2semidirect
2. C2×#14581800C2semidirect
3. #19128277C2semidirect
5. #19127946C2semidirect
6. #19127929C2semidirect
7. #19127737C2semidirect
8. #19127711C2semidirect
9. #418×C83C2semidirect
16. C2×#13050573C22semidirect
18. C2×C8×D83C22semidirect
24. #14581772C22semidirect
28. #14581725C22semidirect
29. #14581723C22semidirect
30. #14581717C22semidirect
50. C2×C83⋊C8D4semidirect
121. C83#3057semidirect
130. #418D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505782
C2×#145818002
#191282772
#191282682
#191279462
#191279292
#191277372
#191277112
#418×C832
#305783