Finite Group Databasebeta

← #24914301

Subgroups — #24914301

Order 10624.

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

normal subgroups only

This group has 512 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×#14581779C211yesmaximal, characteristic
2C2×#14581773C211yesmaximal, characteristic
3#19127626C212yesmaximal, complemented
4#19127626C212yesmaximal, complemented
5#19127622C212yesmaximal, complemented
6#19127586C212yesmaximal, complemented
7#19127576C212yesmaximal, complemented
8#19127561C212yesmaximal, complemented
9#386×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C23×Dic83C2211yescharacteristic
12#14581779—24no—
13#14581773—24no—
14C2×C4⋊Dic83C2212yes—
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C4×Dic83C2211yescharacteristic
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C22×D4×C83C2211yescharacteristic
29C23×C4×C83C2211yescharacteristic
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2211yescharacteristic
32#95×C83C2212yes—
33C2×Dic166—22no—
34C2×Dic166—41no—
35C2×Dic166—41no—
36C22×Dic83—22no—
37C22×Dic83—41no—
38C22×Dic83—41no—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43C4⋊Dic83—24no—
44C4⋊Dic83—42no—
45C4⋊Dic83—42no—
46Dic83⋊C4—24no—
47C22⋊Dic83—22no—
48C22⋊Dic83—24no—
49C22⋊Dic83—24no—
50C22⋊Dic83—24no—
51C4×Dic83—22no—
52C24×C83C2311yescharacteristic
53C24×C83C2311yescharacteristic
54C2×D4×C83—41no—
55C2×D4×C83—41no—
56C2×D4×C83D414yes—
57C22×C4×C83—24no—
58C22×C4×C83—41no—
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2312yes—
62C22⋊C4×C83—24no—
63C22⋊C4×C83—41no—
64C23×C83—22no—
65C23×C83—22no—
66C23×C83—21no—
67C23×C83—21no—
68C23×C83—21no—
69C23×C83—21no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83C2×D412yes—
73C23×C83C2×D412yes—
74C23×C83C2411yescharacteristic
75D4×C83—44no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—44no—
84C2×C4×C83C2×D412yes—
85C2×C4×C83#4114yes—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—44no—
99Dic166—44no—
100C22×C83—22no—
101C22×C83—22no—
102C22×C83—24no—
103C22×C83—24no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83—41no—
109C22×C83—41no—
110C22×C83C22×D411yescharacteristic
111C22×C83C2×#4112yes—
112C22×C83H(5,𝔽2)11yescharacteristic
113C22×C83H(5,𝔽2)11yescharacteristic
114C22×C832−1+412yes—
115C4×C83—22no—
116C4×C83—24no—
117C4×C83—42no—
118Dic83—42no—
119Dic83—42no—
120Dic83—42no—
121Dic83—42no—
122C2×C83—42no—
123C2×C83—42no—
124C2×C83#53411yescharacteristic
125C2×C83#54212yes—
126C2×C83#54712yes—
127C2×C83#55111yescharacteristic
128C2×C83#56311yescharacteristic
129#2717—831nomaximal, Sylow
130C83#271711yescharacteristic, complemented, Sylow
131C2×#124—831no—
132C2×#123—831no—
133#397—832no—
134#397—832no—
135#395—832no—
136#387—832no—
137#386D8311yescharacteristic
138#385—832no—
139#380—832no—
140C22×Q8—831no—
141C22×D4C2×D8311yescharacteristic
142C23×C4—831no—
143C23×C4C2×D8311yescharacteristic
144#124—1664no—
145#123—1664no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4—832no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4—832no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4—831no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4—832no—
154C2×C22⋊C4C2×D8312yes—
155C2×C22⋊C4C2×D8311yescharacteristic
156C2×C42—831no—
157#95—832no—
158#95—832no—
159#95—832no—
160#95—832no—
161#95—832no—
162#95C2×D8312yes—
163C24C22×D8311yescharacteristic
164C24C22×D8311yescharacteristic
165C2×Q8—3321no—
166C2×Q8—3321no—
167C2×Q8—1662no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4D33214yes—
171C22×C4—24no—
172C22×C4—41no—
173C22×C4—3321no—
174C22×C4—3321no—
175C22×C4—832no—
176C22×C4—832no—
177C22×C4—832no—
178C22×C4—832no—
179C22×C4—1662no—
180C22×C4C22×D8312yes—
181C22×C4C22×D8311yescharacteristic
182C22×C4C22×D8312yes—
183C4⋊C4—3322no—
184C4⋊C4—3322no—
185C4⋊C4—1664no—
186C4⋊C4—1664no—
187C22⋊C4—24no—
188C22⋊C4—41no—
189C22⋊C4—1662no—
190C22⋊C4—1664no—
191C22⋊C4—1664no—
192C22⋊C4—1664no—
193C42—1662no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23C2×D33212yes—
203C23C2×D33212yes—
204C23C23×D8311yescharacteristic
205Q8—3324no—
206D4—44no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—3324no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4C2×D33212yes—
229C2×C4#1305058914yes—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—22no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22C22×D33211yescharacteristic
241C22C2×#1305058912yes—
242C22#1458183711yescharacteristic
243C22#1458183711yescharacteristic
244C22#1458183912yes—
245C4—24no—
246C4—22no—
247C4—42no—
248C4—3322no—
249C4—3322no—
250C4—3322no—
251C4—3322no—
252C2—42no—
253C2—42no—
254C2#1912826311yescharacteristic
255C2#1912831812yes—
256C2#1912835512yes—
257C2#1912838211yescharacteristic
258C2#1912845411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC22×D4×C83C22
hypercenter#386D83
hypocenterC83#2717
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #19127626C2semidirect
4. #19127626C2semidirect
5. #19127622C2semidirect
6. #19127586C2semidirect
7. #19127576C2semidirect
8. #19127561C2semidirect
130. C83#2717semidirect

Maximal subgroups

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

subgroupindex
C2×#145817792
C2×#145817732
#191276262
#191276222
#191275862
#191275762
#191275612
#386×C832
#271783