Finite Group Databasebeta

← #24916981

Subgroups — #24916981

Order 10624.

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

normal subgroups only

This group has 490 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×#14581823C211yesmaximal, characteristic, complemented
2#19128268C211yesmaximal, characteristic
3#19128179C211yesmaximal, characteristic, complemented
4#19127921C214yesmaximal, complemented
5#19127899C214yesmaximal, complemented
6#19127727C211yesmaximal, characteristic, complemented
7#19127706C211yesmaximal, characteristic, complemented
8#19127698C211yesmaximal, characteristic, complemented
9C22×SD16×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11#14581823—24no—
12C2×#13050580C2212yescomplemented
13C2×#13050577C2212yescomplemented
14C2×#13050574C2211yescharacteristic
15#14581788C2212yes—
16#14581779C2212yes—
17#14581763—22no—
18#14581761—22no—
19Q8×Dic83—22no—
20Q8×Dic83C2214yescomplemented
21#14581751—22no—
22#14581751C2211yescharacteristic
23#14581751C2211yescharacteristic
24Dic83⋊C8C2214yescomplemented
25#14581717C2212yescomplemented
26#14581714C2212yescomplemented
27#14581713C2214yes—
28C22×Q8×C83C2211yescharacteristic
29C22×D4×C83C2211yescharacteristic
30C2×SD16×C83—24no—
31C2×SD16×C83C2214yes—
32C22×C8×C83C2211yescharacteristic
33#13050590—22no—
34#13050590—22no—
35C2×Dic166C2311yescharacteristic
36C2×D332C2311yescharacteristic
37C2×C22⋊D83—21no—
38C2×C22⋊D83—24no—
39C2×C4×D83—21no—
40#13050580—24no—
41#13050577—24no—
42#13050574—22no—
43C2×C83⋊C8C2312yes—
44C4⋊Dic83—24no—
45C4⋊Dic83D412yes—
46C4⋊Dic83C2312yes—
47C22⋊Dic83—21no—
48C22⋊Dic83—24no—
49C22⋊Dic83D412yes—
50C4×Dic83—21no—
51C4×Dic83—24no—
52C4×Dic83C2312yes—
53C24×C83—21no—
54C2×Q8×C83—21no—
55C2×Q8×C83—22no—
56C2×Q8×C83D414yescomplemented
57C2×Q8×C83C2312yes—
58C2×D4×C83—22no—
59C2×D4×C83—21no—
60C2×D4×C83—22no—
61C2×D4×C83C2312yes—
62C22×C4×C83—21no—
63C22×C4×C83C2311yescharacteristic
64SD16×C83—28no—
65C2×C8×C83—22no—
66C2×C8×C83C2312yes—
67C23×C83—22no—
68C23×C83—42no—
69C23×C83—82no—
70C23×C83C2×D411yescharacteristic
71C22×D83—21no—
72Q8×C83—24no—
73Q8×C83—22no—
74Q8×C83C2×D414yes—
75D4×C83—22no—
76D4×C83—22no—
77D4×C83—42no—
78C2×C4×C83—24no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D412yes—
82C2×C4×C83#4112yes—
83C2×C4×C83C2411yescharacteristic
84C22⋊D83—42no—
85C22⋊D83—84no—
86C2×Dic83—22no—
87C2×Dic83—21no—
88C2×Dic83C2×D414yes—
89D332—21no—
90D332—21no—
91C4×D83—42no—
92Dic166—21no—
93Dic166—21no—
94C8×C83—22no—
95C83⋊C8—22no—
96C22×C83—22no—
97C22×C83—42no—
98C22×C83—44no—
99C22×C83—84no—
100C22×C83#12112yes—
101C22×C83C22×D411yescharacteristic
102C2×D83—41no—
103C2×D83—42no—
104C4×C83—24no—
105C4×C83#12112yes—
106C4×C83C22×D411yescharacteristic
107C4×C83C2×#4111yescharacteristic
108Dic83—24no—
109Dic83—41no—
110Dic83—42no—
111C2×C83—22no—
112C2×C83—82no—
113C2×C83C2×#12111yescharacteristic
114C2×C83#57712yes—
115D83—81no—
116#3043—831nomaximal, Sylow
117C83#304311yescharacteristic, complemented, Sylow
118C2×#137—831no—
119C22×SD16D8311yescharacteristic, complemented
120#535—831no—
121#519—831no—
122#464—834no—
123#460—834no—
124#424—831no—
125#418—831no—
126#416—831no—
127C2×#41—831no—
128C22×Q8C2×D8311yescharacteristic
129C22×D4C2×D8311yescharacteristic
130#137—1664no—
131C2×Q16—832no—
132C2×SD16—24no—
133C2×SD16—832no—
134C2×SD16C2×D8314yes—
135C2×M16—831no—
136C22×C8C2×D8311yescharacteristic
137#127—832no—
138#124—832no—
139#121—1662no—
140#120—1662no—
141C4×Q8—834no—
142C4×Q8—1662no—
143#117—831no—
144#117—831no—
145#117—1662no—
146C4⋊C8—834no—
147#103—832no—
148#102—834no—
149#102—832no—
150C24—21no—
151#41—1662no—
152#41—1662no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—831no—
156C2×Q8C22⋊D8314yescomplemented
157C2×Q8C22×D8312yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—831no—
162C2×D4—1664no—
163C2×D4—1661no—
164C2×D4C22×D8312yes—
165C22×C4—21no—
166C22×C4—1661no—
167C22×C4C22×D8311yescharacteristic
168Q16—1664no—
169SD16—28no—
170SD16—1664no—
171M16—1662no—
172C2×C8—22no—
173C2×C8—832no—
174C2×C8C22×D8312yes—
175C4⋊C4—832no—
176C4⋊C4—832no—
177C4⋊C4—1664no—
178C22⋊C4—832no—
179C22⋊C4—1664no—
180C22⋊C4—1661no—
181C42—832no—
182C42—1664no—
183C42—1661no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—1661no—
188C23D4×D8311yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—1661no—
192Q8—1661no—
193Q8C2×C22⋊D8314yes—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—3322no—
198D4—6644no—
199D4—1661no—
200D4—1661no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—3322no—
204C2×C4—834no—
205C2×C4—1662no—
206C2×C4—1661no—
207C2×C4D4×D8311yescharacteristic
208C2×C4C2×C22⋊D8312yes—
209C2×C4#1305058912yes—
210C2×C4C23×D8311yescharacteristic
211C8—22no—
212C8—1662no—
213C22—22no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—3321no—
218C22—3322no—
219C22#1458176412yes—
220C22C2×D4×D8311yescharacteristic
221C4—24no—
222C4—3321no—
223C4—3322no—
224C4—1664no—
225C4#1458176412yes—
226C4C22×C22⋊D8311yescharacteristic
227C4C2×#1305058911yescharacteristic
228C2—22no—
229C2—82no—
230C2—6641no—
231C2C2×#1458176411yescharacteristic
232C2#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#3043
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×#14581823C2semidirect
3. #19128179C2semidirect
4. #19127921C2semidirect
5. #19127899C2semidirect
6. #19127727C2semidirect
7. #19127706C2semidirect
8. #19127698C2semidirect
9. C22×SD16×C83C2semidirect
12. C2×#13050580C22semidirect
13. C2×#13050577C22semidirect
20. Q8×Dic83C22semidirect
24. Dic83⋊C8C22semidirect
25. #14581717C22semidirect
26. #14581714C22semidirect
56. C2×Q8×C83D4semidirect
117. C83#3043semidirect
119. C22×SD16D83semidirect
156. C2×Q8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
#191282682
#191281792
#191279212
#191278992
#191277272
#191277062
#191276982
C22×SD16×C832
#304383