Finite Group Databasebeta

← #24917860

Subgroups — #24917860

Order 10624.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128250C211yesmaximal, characteristic
2#125×D83C211yesmaximal, characteristic, complemented
3#19128068C212yesmaximal, complemented
4#19128064C212yesmaximal, complemented
5#19128058C211yesmaximal, characteristic, complemented
6#19128056C211yesmaximal, characteristic, complemented
7#19128055C211yesmaximal, characteristic, complemented
8#19127701C212yesmaximal, complemented
9#19127699C212yesmaximal, complemented
10C4×#13050573C211yesmaximal, characteristic, complemented
11#488×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C2×#13050589C2211yescharacteristic
14C2×#13050573C2212yes—
15#14581790C2211yescharacteristic
16#14581788C2211yescharacteristic
17#14581784C2211yescharacteristic
18#14581783C2212yes—
19#14581782—22no—
20#14581780C2211yescharacteristic
21#14581765—22no—
22(C22⋊D83)⋊C4—22no—
23C4⋊C4×D83—22no—
24C4⋊C4×D83C2212yes—
25C42×D83C2211yescharacteristic
26#14581717C2212yes—
27#14581716C2212yes—
28#14581715C2212yes—
29#14581714C2212yes—
30#14581713C2212yes—
31#14581712C2212yes—
32#14581702C2212yes—
33C4×C83⋊C8C2211yescharacteristic
34#125×C83C2211yescharacteristic
35#124×C83C2211yescharacteristic
36#103×C83C2212yes—
37#102×C83C2212yes—
38C4×C8×C83C2211yescharacteristic
39#13050591—24no—
40#13050589—24no—
41C2×Dic166C2311yescharacteristic
42C2×D332—21no—
43C2×D332C2311yescharacteristic
44C2×C22⋊D83—21no—
45C2×C22⋊D83—22no—
46C22×Dic83—21no—
47C2×C4×D83—22no—
48C2×C4×D83—21no—
49C2×C4×D83D412yes—
50C2×C4×D83C2311yescharacteristic
51#13050573—24no—
52C2×C83⋊C8C2312yes—
53C4⋊Dic83—22no—
54C4⋊Dic83C2312yes—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59(C2×D83)⋊C4—22no—
60C4×Dic83D412yes—
61C4×Dic83C2311yescharacteristic
62C2×Q8×C83C2311yescharacteristic
63C2×D4×C83C2311yescharacteristic
64C2×C8×C83C2312yes—
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C42×C83C2311yescharacteristic
69C23×C83—21no—
70C22×D83—21no—
71C22×D83C2×D411yescharacteristic
72Q8×C83—22no—
73D4×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2411yescharacteristic
78C22⋊D83—44no—
79C22⋊D83—42no—
80C2×Dic83—21no—
81C2×Dic83—22no—
82C2×Dic83—42no—
83C2×Dic83C2×D412yes—
84C2×Dic83C2×D411yescharacteristic
85D332—22no—
86D332—42no—
87C4×D83—42no—
88C4×D83—44no—
89C4×D83—42no—
90C4×D83#4114yes—
91Dic166—22no—
92C8×C83—22no—
93C83⋊C8—22no—
94C22×C83—41no—
95C22×C83—42no—
96C22×C83C22×D411yescharacteristic
97C2×D83—22no—
98C2×D83—41no—
99C2×D83—42no—
100C2×D83#12412yes—
101C4×C83—22no—
102C4×C83—41no—
103C4×C83—42no—
104C4×C83C2×#4112yes—
105Dic83—22no—
106Dic83—41no—
107Dic83—42no—
108Dic83#12412yes—
109C2×C83—81no—
110C2×C83C2×#12411yescharacteristic
111C2×C83#57512yes—
112D83—22no—
113D83—81no—
114#3142—831nomaximal, Sylow
115C83#314211yescharacteristic, complemented, Sylow
116#532—831no—
117C2×#125—831no—
118#491—832no—
119#490—832no—
120#488—831no—
121#488—831no—
122#488—831no—
123#488D8311yescharacteristic, complemented
124#417—832no—
125#417—832no—
126C4×M16—831no—
127C2×#41—831no—
128C2×#41—831no—
129C2×M16—832no—
130C4⋊Q8—831no—
131#127—831no—
132#125—832no—
133#125—831no—
134#125—1662no—
135#125C2×D8311yescharacteristic
136#124—831no—
137#124C2×D8311yescharacteristic
138#121—1662no—
139C4×D4—1662no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4—1662no—
142C2×C42—831no—
143#103—832no—
144#103—832no—
145#103—832no—
146#103C2×D8312yes—
147#102—832no—
148#102—832no—
149#102—832no—
150#102C2×D8312yes—
151#97—832no—
152C4×C8—831no—
153C4×C8C2×D8311yescharacteristic
154#41—1664no—
155#41—1664no—
156C2×Q8—831no—
157C2×Q8C22×D8311yescharacteristic
158C2×D4—831no—
159C2×D4—1661no—
160C2×D4—1662no—
161C2×D4—1661no—
162C2×D4C22×D8311yescharacteristic
163C22×C4—832no—
164C22×C4—831no—
165C22×C4—1661no—
166C22×C4—1662no—
167C22×C4—1661no—
168M16—1664no—
169C2×C8—832no—
170C2×C8C22×D8312yes—
171C4⋊C4—22no—
172C4⋊C4—832no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4C22×D8312yes—
179C22⋊C4—22no—
180C22⋊C4—1662no—
181C42—832no—
182C42—831no—
183C42C22×D8311yescharacteristic
184C23—21no—
185C23—831no—
186C23—1661no—
187Q8—22no—
188Q8—1662no—
189D4—22no—
190D4—3322no—
191D4—3322no—
192D4—3324no—
193D4—1662no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—3322no—
197C2×C4—3324no—
198C2×C4—3322no—
199C2×C4—3322no—
200C2×C4—831no—
201C2×C4—834no—
202C2×C4—832no—
203C2×C4—1661no—
204C2×C4—1662no—
205C2×C4D4×D8312yes—
206C2×C4C23×D8311yescharacteristic
207C8—22no—
208C8—1662no—
209C22—41no—
210C22—42no—
211C22—3321no—
212C22—3322no—
213C22—832no—
214C22—1662no—
215C22C2×D4×D8311yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—3321no—
220C4—3322no—
221C4—832no—
222C4—1662no—
223C4#41×D8312yes—
224C2—81no—
225C2—6641no—
226C2—1662no—
227C2#124×D8311yescharacteristic
228C2#1912849812yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#488×C83C2
socleC22×C83C22×D4
prime radical#19128250C2
hypercenter#488D83
hypocenterC83#3142
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D83C2semidirect
3. #19128068C2semidirect
4. #19128064C2semidirect
5. #19128058C2semidirect
6. #19128056C2semidirect
7. #19128055C2semidirect
8. #19127701C2semidirect
9. #19127699C2semidirect
10. C4×#13050573C2semidirect
11. #488×C83C2semidirect
115. C83#3142semidirect
123. #488D83semidirect

Maximal subgroups

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

subgroupindex
#191282502
#125×D832
#191280682
#191280642
#191280582
#191280562
#191280552
#191277012
#191276992
C4×#130505732
#488×C832
#314283