Finite Group Databasebeta

← #24918666

Subgroups — #24918666

Order 10624.

324 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128253C211yesmaximal, characteristic
2C2×#14581783C211yesmaximal, characteristic
3#19128108C212yesmaximal, complemented
4#19128068C212yesmaximal, complemented
5#19128037C211yesmaximal, characteristic
6#19128034C211yesmaximal, characteristic
7#19128031C211yesmaximal, characteristic
8#19128011C212yesmaximal, complemented
9#19128010C212yesmaximal, complemented
10#19127755C211yesmaximal, characteristic
11#483×C83C211yesmaximal, characteristic
12#14581791C2211yescharacteristic
13#14581790C2211yescharacteristic
14#14581783—21no—
15#14581783—21no—
16#14581783C2212yes—
17#14581783C2212yes—
18#14581782—21no—
19#14581771C2211yescharacteristic
20#14581769—21no—
21Q8×Dic83C2211yescharacteristic
22#14581757—21no—
23#14581757C2211yescharacteristic
24#14581748—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83C2211yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×C4×Dic83C2211yescharacteristic
29#14581726C2212yes—
30#14581725C2212yes—
31#14581723C2212yes—
32Dic83⋊C8C2212yes—
33#14581717C2212yes—
34#14581716C2212yes—
35#14581715C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yes—
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#106×C83C2212yes—
41#103×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166C2311yescharacteristic
45C22×Dic83—21no—
46C22×Dic83D411yescharacteristic
47C22×Dic83D411yescharacteristic
48C2×C83⋊C8C2312yes—
49C4⋊Dic83—21no—
50C4⋊Dic83—21no—
51C4⋊Dic83—22no—
52C4⋊Dic83—22no—
53C4⋊Dic83C2311yescharacteristic
54C4⋊Dic83C2312yes—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60Dic83⋊C4—22no—
61Dic83⋊C4—22no—
62C22⋊Dic83—21no—
63C4×Dic83—21no—
64C4×Dic83—21no—
65C4×Dic83D411yescharacteristic
66C4×Dic83D411yescharacteristic
67C4×Dic83C2311yescharacteristic
68C4×Dic83C2311yescharacteristic
69C2×Q8×C83C2311yescharacteristic
70C22×C4×C83—21no—
71C22×C4×C83C2311yescharacteristic
72C2×C8×C83C2312yes—
73C4⋊C4×C83—21no—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C4⋊C4×C83C2311yescharacteristic
77C22⋊C4×C83—21no—
78C23×C83C2×D411yescharacteristic
79Q8×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—42no—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C2×Dic83—22no—
88C2×Dic83—21no—
89C2×Dic83—21no—
90C2×Dic83—22no—
91C2×Dic83—41no—
92C2×Dic83—42no—
93C2×Dic83C2×D411yescharacteristic
94C2×Dic83C2×D411yescharacteristic
95C2×Dic83C2×D411yescharacteristic
96C2×Dic83C2×D411yescharacteristic
97Dic166—22no—
98Dic166—42no—
99C8×C83—22no—
100C83⋊C8—22no—
101C22×C83—21no—
102C22×C83—21no—
103C22×C83#13712yes—
104C22×C83C22×D411yescharacteristic
105C4×C83—21no—
106C4×C83—41no—
107C4×C83—42no—
108C4×C83—41no—
109C4×C832−1+412yes—
110Dic83—21no—
111Dic83—21no—
112Dic83—21no—
113Dic83—21no—
114Dic83—41no—
115Dic83—41no—
116Dic83—42no—
117C2×C83—22no—
118C2×C83#53611yescharacteristic
119C2×C83C2×#13711yescharacteristic
120C2×C83#57511yescharacteristic
121#3236—831nomaximal, Sylow
122C83#323611yescharacteristic, complemented, Sylow
123#533—831no—
124C2×#125—831no—
125#501—832no—
126#491—832no—
127#484—831no—
128#484—831no—
129#483—831no—
130#483D8311yescharacteristic
131#479—832no—
132#479—832no—
133#432—831no—
134C4⋊Q8—831no—
135C4⋊Q8—831no—
136#125—832no—
137#125—832no—
138#125—1661no—
139#125—1661no—
140#125—1661no—
141C22⋊Q8—831no—
142C22⋊Q8—1661no—
143C22⋊Q8C2×D8311yescharacteristic
144C4×Q8—831no—
145C4×Q8—831no—
146C4×Q8—1661no—
147#117—1661no—
148C2×C4⋊C4—831no—
149C2×C4⋊C4—1661no—
150C2×C4⋊C4—1661no—
151C2×C4⋊C4C2×D8311yescharacteristic
152C2×C42—831no—
153#107—832no—
154#107—832no—
155#106—832no—
156#106C2×D8312yes—
157C4⋊C8—832no—
158#103—832no—
159#103—832no—
160#103—832no—
161#103C2×D8312yes—
162C22⋊C8—831no—
163C22⋊C8C2×D8311yescharacteristic
164#97—832no—
165C2×Q8—831no—
166C2×Q8—1661no—
167C2×Q8C22×D8311yescharacteristic
168C22×C4—21no—
169C22×C4—831no—
170C22×C4—831no—
171C22×C4—1661no—
172C22×C4C22×D8311yescharacteristic
173C2×C8—832no—
174C2×C8C22×D8312yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—832no—
178C4⋊C4—831no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C4⋊C4—1662no—
183C4⋊C4—1661no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4—1661no—
187C4⋊C4—1661no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4C22×D8312yes—
191C4⋊C4C22×D8311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—1661no—
194C42—831no—
195C42—831no—
196C42—831no—
197C42—831no—
198C42—1661no—
199C42—1661no—
200C23D4×D8311yescharacteristic
201Q8—22no—
202Q8—3322no—
203Q8—1662no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3322no—
210C2×C4—3321no—
211C2×C4—831no—
212C2×C4—831no—
213C2×C4—831no—
214C2×C4—831no—
215C2×C4—1661no—
216C2×C4—1661no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4D4×D8311yescharacteristic
220C2×C4C23×D8311yescharacteristic
221C8—22no—
222C8—1662no—
223C22—21no—
224C22—21no—
225C22#1458182112yes—
226C22C2×D4×D8311yescharacteristic
227C4—21no—
228C4—42no—
229C4—41no—
230C4—41no—
231C4—3322no—
232C4—3321no—
233C4—3321no—
234C4—1661no—
235C4—1661no—
236C4—1661no—
237C4—1661no—
238C4#1458183912yes—
239C2—22no—
240C2#1912827311yescharacteristic
241C2C2×#1458182111yescharacteristic
242C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#483×C83C2
socleC22×C83C22×D4
prime radicalC23×C83C2×D4
hypercenter#483D83
hypocenterC83#3236
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #19128108C2semidirect
4. #19128068C2semidirect
8. #19128011C2semidirect
9. #19128010C2semidirect
122. C83#3236semidirect

Maximal subgroups

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

subgroupindex
#191282532
C2×#145817832
#191281082
#191280682
#191280372
#191280342
#191280312
#191280112
#191280102
#191277552
#483×C832
#323683