Finite Group Databasebeta

← #24916963

Subgroups — #24916963

Order 10624.

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

normal subgroups only

This group has 526 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×#14581818C211yesmaximal, characteristic, complemented
2#19128268C211yesmaximal, characteristic, complemented
3#19128176C211yesmaximal, characteristic, complemented
4#19127914C214yesmaximal, complemented
5#19127894C214yesmaximal, complemented
6#19127727C211yesmaximal, characteristic, complemented
7#19127706C211yesmaximal, characteristic, complemented
8#19127698C211yesmaximal, characteristic, complemented
9C22×D8×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11#14581818—24no—
12C2×#13050578C2212yescomplemented
13C2×#13050575C2212yescomplemented
14C2×#13050574C2211yescharacteristic
15#14581788C2212yescomplemented
16#14581779C2212yescomplemented
17#14581763—22no—
18#14581761—22no—
19D4×Dic83—22no—
20D4×Dic83C2214yescomplemented
21#14581751C2211yescharacteristic
22#14581751C2211yescharacteristic
23C2×C22⋊Dic83—22no—
24Dic83⋊C8C2214yescomplemented
25#14581717C2212yescomplemented
26#14581714C2212yescomplemented
27#14581713C2214yescomplemented
28C22×D4×C83C2211yescharacteristic
29C22×D4×C83C2211yescharacteristic
30C2×D8×C83—24no—
31C2×D8×C83C2214yes—
32C22×C8×C83C2211yescharacteristic
33#13050590—22no—
34#13050590—22no—
35C2×Dic166C2311yescharacteristic
36C2×D332C2311yescharacteristic
37C2×C22⋊D83—21no—
38C2×C22⋊D83—24no—
39C22×Dic83—24no—
40C2×C4×D83—21no—
41#13050578—24no—
42#13050575—24no—
43#13050574—22no—
44C2×C83⋊C8C2312yes—
45C4⋊Dic83D412yes—
46C4⋊Dic83C2312yes—
47C22⋊Dic83—21no—
48C22⋊Dic83—24no—
49C22⋊Dic83—24no—
50C22⋊Dic83D412yes—
51C4×Dic83—21no—
52C4×Dic83C2312yes—
53C24×C83—21no—
54C24×C83—21no—
55C2×D4×C83—21no—
56C2×D4×C83—22no—
57C2×D4×C83—21no—
58C2×D4×C83—22no—
59C2×D4×C83—22no—
60C2×D4×C83D414yescomplemented
61C2×D4×C83C2312yes—
62C2×D4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64D8×C83—28no—
65C2×C8×C83—22no—
66C2×C8×C83C2312yes—
67C23×C83—22no—
68C23×C83—24no—
69C23×C83—22no—
70C23×C83—42no—
71C23×C83—82no—
72C23×C83—82no—
73C23×C83C2×D411yescharacteristic
74C22×D83—21no—
75D4×C83—24no—
76D4×C83—22no—
77D4×C83—22no—
78D4×C83—22no—
79D4×C83—42no—
80D4×C83C2×D414yes—
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2×D412yes—
83C2×C4×C83#4112yes—
84C2×C4×C83C2411yescharacteristic
85C22⋊D83—42no—
86C22⋊D83—84no—
87C2×Dic83—22no—
88C2×Dic83—21no—
89C2×Dic83—84no—
90C2×Dic83C2×D414yes—
91D332—21no—
92D332—21no—
93C4×D83—42no—
94Dic166—21no—
95Dic166—21no—
96C8×C83—22no—
97C83⋊C8—22no—
98C22×C83—22no—
99C22×C83—24no—
100C22×C83—44no—
101C22×C83—42no—
102C22×C83—44no—
103C22×C83—84no—
104C22×C83—84no—
105C22×C83#12112yes—
106C22×C83C22×D411yescharacteristic
107C2×D83—41no—
108C2×D83—42no—
109C4×C83#12112yes—
110C4×C83C22×D411yescharacteristic
111C4×C83C2×#4111yescharacteristic
112Dic83—24no—
113Dic83—41no—
114Dic83—42no—
115C2×C83—22no—
116C2×C83—44no—
117C2×C83—82no—
118C2×C83C2×#12111yescharacteristic
119C2×C83#57612yes—
120D83—81no—
121#3041—831nomaximal, Sylow
122C83#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—831no—
124C22×D8D8311yescharacteristic, complemented
125#535—831no—
126#518—831no—
127#463—834no—
128#459—834no—
129#424—831no—
130#418—831no—
131#416—831no—
132C2×#41—831no—
133C22×D4C2×D8311yescharacteristic
134C22×D4C2×D8311yescharacteristic
135AGL(1,ℤ/8)—1664no—
136C2×SD16—832no—
137C2×D8—24no—
138C2×D8—832no—
139C2×D8C2×D8314yes—
140C2×M16—831no—
141C22×C8C2×D8311yescharacteristic
142#127—832no—
143#124—832no—
144#121—1662no—
145#120—1662no—
146C4×D4—834no—
147C4×D4—1662no—
148#117—831no—
149#117—831no—
150C2×C22⋊C4—1662no—
151C4⋊C8—834no—
152#103—832no—
153#102—834no—
154#102—832no—
155C24—21no—
156C24—21no—
157#41—1662no—
158#41—1662no—
159C2×Q8—831no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—831no—
166C2×D4—1664no—
167C2×D4—1661no—
168C2×D4C22⋊D8314yescomplemented
169C2×D4C22×D8312yes—
170C2×D4C22×D8312yes—
171C22×C4—1661no—
172C22×C4—1664no—
173C22×C4C22×D8311yescharacteristic
174SD16—1664no—
175D8—28no—
176D8—1664no—
177M16—1662no—
178C2×C8—22no—
179C2×C8—832no—
180C2×C8C22×D8312yes—
181C4⋊C4—832no—
182C4⋊C4—832no—
183C22⋊C4—832no—
184C22⋊C4—1664no—
185C22⋊C4—1664no—
186C22⋊C4—1661no—
187C42—832no—
188C42—1661no—
189C23—22no—
190C23—24no—
191C23—22no—
192C23—42no—
193C23—82no—
194C23—82no—
195C23—1661no—
196C23D4×D8311yescharacteristic
197Q8—1661no—
198Q8—1661no—
199D4—24no—
200D4—22no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—3322no—
205D4—6644no—
206D4—1661no—
207D4—1661no—
208D4C2×C22⋊D8314yes—
209C2×C4—3322no—
210C2×C4—834no—
211C2×C4—6644no—
212C2×C4—1662no—
213C2×C4—1661no—
214C2×C4D4×D8311yescharacteristic
215C2×C4C2×C22⋊D8312yes—
216C2×C4#1305058912yes—
217C2×C4C23×D8311yescharacteristic
218C8—22no—
219C8—1662no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—42no—
224C22—44no—
225C22—84no—
226C22—84no—
227C22—3321no—
228C22—3322no—
229C22#1458176412yes—
230C22C2×D4×D8311yescharacteristic
231C4—3321no—
232C4—3322no—
233C4—1664no—
234C4#1458176412yes—
235C4C22×C22⋊D8311yescharacteristic
236C4C2×#1305058911yescharacteristic
237C2—22no—
238C2—44no—
239C2—82no—
240C2—6641no—
241C2C2×#1458176411yescharacteristic
242C2#1912850212yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC22×D8×C83C2
socleC22×C83C22×D4
hypercenterC22×D8D83
hypocenterC83#3041
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#14581818C2semidirect
2. #19128268C2semidirect
3. #19128176C2semidirect
4. #19127914C2semidirect
5. #19127894C2semidirect
6. #19127727C2semidirect
7. #19127706C2semidirect
8. #19127698C2semidirect
9. C22×D8×C83C2semidirect
12. C2×#13050578C22semidirect
13. C2×#13050575C22semidirect
15. #14581788C22semidirect
16. #14581779C22semidirect
20. D4×Dic83C22semidirect
24. Dic83⋊C8C22semidirect
25. #14581717C22semidirect
26. #14581714C22semidirect
27. #14581713C22semidirect
60. C2×D4×C83D4semidirect
122. C83#3041semidirect
124. C22×D8D83semidirect
168. C2×D4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
#191282682
#191281762
#191279142
#191278942
#191277272
#191277062
#191276982
C22×D8×C832
#304183