Finite Group Databasebeta

← #24918014

Subgroups — #24918014

Order 10624.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#19128256C211yesmaximal, characteristic
2#125×D83C211yesmaximal, characteristic
3#19128111C211yesmaximal, characteristic, complemented
4#19128110C211yesmaximal, characteristic, complemented
5#19128107C211yesmaximal, characteristic, complemented
6#19128102C212yesmaximal, complemented
7#19128101C212yesmaximal, complemented
8#19127742C212yesmaximal
9#19127740C212yesmaximal
10#19127653C211yesmaximal, characteristic
11#501×C83C211yesmaximal, characteristic, complemented
12C2×#13050573C2212yes—
13#14581792C2211yescharacteristic
14#14581784C2211yescharacteristic
15#14581783C2211yescharacteristic
16#14581783C2212yes—
17#14581783C2211yescharacteristic
18#14581782—21no—
19#14581782—21no—
20#14581772—21no—
21#14581768—21no—
22Q8×Dic83—21no—
23#14581757—21no—
24#14581750C2212yes—
25C4⋊C4×D83—21no—
26C4⋊C4×D83—21no—
27C4⋊C4×D83C2211yescharacteristic
28C4⋊C4×D83C2211yescharacteristic
29C42×D83C2211yescharacteristic
30#14581726C2212yes—
31#14581725C2212yes—
32#14581725C2212yes—
33#14581724C2212yes—
34#14581723C2212yes—
35#14581723C2212yes—
36#14581703C2211yescharacteristic
37C8×Dic83C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39#125×C83C2211yescharacteristic
40#107×C83C2212yes—
41#106×C83C2212yes—
42#97×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×C4×D83—21no—
45C2×C4×D83—21no—
46C2×C4×D83D411yescharacteristic
47C2×C4×D83D411yescharacteristic
48C2×C4×D83C2311yescharacteristic
49#13050573Q818yescomplemented
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83—21no—
54C4⋊Dic83—21no—
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2311yescharacteristic
57C4⋊Dic83C2312yes—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60Dic83⋊C4—22no—
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64Dic83⋊C4—21no—
65(C2×D83)⋊C4—22no—
66C4×Dic83—22no—
67C4×Dic83D411yescharacteristic
68C4×Dic83D411yescharacteristic
69C4×Dic83C2311yescharacteristic
70C2×Q8×C83—21no—
71C2×C8×C83C2312yes—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2311yescharacteristic
76C4⋊C4×C83C2312yes—
77C4⋊C4×C83C2311yescharacteristic
78C42×C83C2311yescharacteristic
79C22×D83C2×D411yescharacteristic
80Q8×C83—42no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C2×Dic83—22no—
88C2×Dic83—21no—
89C2×Dic83—21no—
90C2×Dic83C2×D411yescharacteristic
91C2×Dic83C2×D411yescharacteristic
92C2×Dic83C2×D411yescharacteristic
93C4×D83—41no—
94C4×D83—41no—
95C4×D83—42no—
96C4×D83—42no—
97C4×D83C2×Q814yes—
98Dic166—42no—
99C8×C83C2×Q814yes—
100C83⋊C8C2×Q814yes—
101C22×C83C22×D411yescharacteristic
102C2×D83—21no—
103C2×D83—21no—
104C2×D83C4⋊Q812yes—
105C4×C83—21no—
106C4×C83—21no—
107C4×C83—41no—
108C4×C83—41no—
109C4×C83—42no—
110C4×C83C22×Q812yes—
111Dic83—21no—
112Dic83—21no—
113Dic83—42no—
114Dic83—41no—
115Dic83—41no—
116Dic83C4⋊Q812yes—
117C2×C83C2×C4⋊Q811yescharacteristic
118C2×C83#57511yescharacteristic
119C2×C83#57511yescharacteristic
120D83—22no—
121#3166—831nomaximal, Sylow
122C83#316611yescharacteristic, complemented, Sylow
123#533—831no—
124C2×#125—831no—
125#501—831no—
126#501—831no—
127#501—831no—
128#501D8311yescharacteristic, complemented
129#499—832no—
130#499—832no—
131#428—832no—
132#428—832no—
133C4×M16—831no—
134C2×M16—832no—
135C4⋊Q8—831no—
136C4⋊Q8C2×D8311yescharacteristic
137#125—831no—
138#125—831no—
139#125—832no—
140#125—831no—
141#125—1661no—
142#125—1661no—
143#125C2×D8311yescharacteristic
144C22⋊Q8—1661no—
145C22⋊Q8—1661no—
146C4×Q8—1661no—
147C4×Q8—1661no—
148#117—832no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4—831no—
151C2×C4⋊C4—1661no—
152C2×C4⋊C4—1661no—
153C2×C42—831no—
154#107—832no—
155#107—832no—
156#107—832no—
157#107C2×D8312yes—
158#106—832no—
159#106—832no—
160#106—832no—
161#106C2×D8312yes—
162#97—831no—
163#97C2×D8311yescharacteristic
164C4×C8—832no—
165C2×Q8—21no—
166C2×Q8—1661no—
167C22×C4—831no—
168C22×C4—831no—
169C22×C4—831no—
170C22×C4—1661no—
171C22×C4—1661no—
172M16—838no—
173C2×C8—832no—
174C2×C8C22×D8312yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—831no—
179C4⋊C4—831no—
180C4⋊C4—832no—
181C4⋊C4—1661no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4—1661no—
187C4⋊C4—1662no—
188C4⋊C4—1662no—
189C4⋊C4—1661no—
190C4⋊C4—1661no—
191C4⋊C4—1661no—
192C4⋊C4C22×D8311yescharacteristic
193C4⋊C4C22×D8312yes—
194C4⋊C4C22×D8311yescharacteristic
195C22⋊C4—1662no—
196C42—831no—
197C42—831no—
198C42—831no—
199C42—1662no—
200C42C22×D8311yescharacteristic
201C23—831no—
202Q8—42no—
203Q8—3322no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—3321no—
208C2×C4—3321no—
209C2×C4—3322no—
210C2×C4—3322no—
211C2×C4—831no—
212C2×C4—834no—
213C2×C4—831no—
214C2×C4—831no—
215C2×C4—1662no—
216C2×C4—1661no—
217C2×C4—1661no—
218C2×C4D4×D8311yescharacteristic
219C2×C4D4×D8311yescharacteristic
220C2×C4C23×D8311yescharacteristic
221C8—834no—
222C8Q8×D8314yes—
223C22—832no—
224C22—1661no—
225C22—1661no—
226C22C2×D4×D8311yescharacteristic
227C4—21no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—3321no—
233C4—3321no—
234C4—3322no—
235C4—832no—
236C4—1661no—
237C4—1661no—
238C4C2×Q8×D8312yes—
239C2—1662no—
240C2C4⋊Q8×D8311yescharacteristic
241C2#1912849711yescharacteristic
242C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#501×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenter#501D83
hypocenterC83#3166
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #19128111C2semidirect
4. #19128110C2semidirect
5. #19128107C2semidirect
6. #19128102C2semidirect
7. #19128101C2semidirect
11. #501×C83C2semidirect
49. #13050573Q8semidirect
122. C83#3166semidirect
128. #501D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#125×D832
#191281112
#191281102
#191281072
#191281022
#191281012
#191277422
#191277402
#191276532
#501×C832
#316683