Finite Group Databasebeta

← #24911931

Subgroups — #24911931

Order 10624.

708 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 169 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×#14581777C214yesmaximal, complemented
2C2×#14581767C211yesmaximal, characteristic
3C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
4#19127636C214yesmaximal, complemented
5#19127603C211yesmaximal, characteristic, complemented
6#19127602C211yesmaximal, characteristic, complemented
7#19127570C211yesmaximal, characteristic, complemented
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
9#384×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×D332C2212yes—
12C22×C4×D83C2212yes—
13C22×C4×D83C2211yescharacteristic
14#14581777—216no—
15#14581767—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2214yes—
18C2×Dic83⋊C4C2212yes—
19C4⋊C4×D83—24no—
20C2×(C2×D83)⋊C4C2214yes—
21C2×(C2×D83)⋊C4C2214yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C83⋊#95C2212yes—
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2214yes—
29C2×C42×C83C2211yescharacteristic
30C23×D83C2312yes—
31C23×D83C2311yescharacteristic
32C2×D332—24no—
33C2×D332—28no—
34C2×D332—22no—
35C2×D332—22no—
36C2×D332—22no—
37C22×Dic83C2312yes—
38C22×Dic83C2311yescharacteristic
39C22×Dic83C2312yes—
40C2×C4×D83—22no—
41C2×C4×D83—22no—
42C2×C4×D83—28no—
43C2×C4×D83—24no—
44C2×C4×D83D414yes—
45C4⋊Dic83—22no—
46Dic83⋊C4—28no—
47Dic83⋊C4—24no—
48(C2×D83)⋊C4—28no—
49(C2×D83)⋊C4—24no—
50(C2×D83)⋊C4—28no—
51(C2×D83)⋊C4D418yescomplemented
52C4×Dic83—22no—
53C22×C4×C83C2312yes—
54C22×C4×C83C2312yes—
55C22×C4×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57C4⋊C4×C83—22no—
58C4⋊C4×C83—28no—
59C4⋊C4×C83—22no—
60C42×C83—22no—
61C23×C83C2411yescharacteristic
62C22×D83—21no—
63C22×D83—21no—
64C22×D83—22no—
65C22×D83—24no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—24no—
70C22×D83C2×D414yes—
71C22×D83C2×D412yes—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—24no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—24no—
78C2×C4×C83C2×D414yes—
79C2×C4×C83C2×D412yes—
80C2×C4×C83#4112yes—
81C2×C4×C83#4112yes—
82C2×Dic83—22no—
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83—22no—
86C2×Dic83—24no—
87C2×Dic83—22no—
88C2×Dic83C2×D414yes—
89C2×Dic83C2×D412yes—
90D332—48no—
91D332—44no—
92C4×D83—28no—
93C4×D83—48no—
94C22×C83C22×D411yescharacteristic
95C22×C83C22×D412yes—
96C22×C83C2×#4111yescharacteristic
97C22×C83C2×#4111yescharacteristic
98C22×C832−1+412yes—
99C2×D83—22no—
100C2×D83—22no—
101C2×D83—24no—
102C2×D83—24no—
103C2×D83—42no—
104C2×D83—42no—
105C2×D83—42no—
106C2×D83—44no—
107C2×D83—44no—
108C2×D83#12714yes—
109C4×C83—24no—
110C4×C83—42no—
111C4×C83—42no—
112C4×C83#12314yescomplemented
113Dic83—22no—
114Dic83—24no—
115Dic83—42no—
116C2×C83C2×#12311yescharacteristic
117C2×C83C2×#12711yescharacteristic
118C2×C83#53611yescharacteristic
119C2×C83#54712yes—
120C2×C83#54712yes—
121D83—24no—
122D83—82no—
123#2501—831nomaximal, Sylow
124C83#250111yescharacteristic, complemented, Sylow
125C2×#123—834no—
126C2×#121—831no—
127C22×C4⋊C4—831no—
128#399—834no—
129#390—831no—
130#390—831no—
131#384—831no—
132#384D8311yescharacteristic, complemented
133C4×C22⋊C4—831no—
134C22×D4—832no—
135C22×D4—831no—
136C23×C4—831no—
137C23×C4—832no—
138#123—16616no—
139#121—1664no—
140C2×C4⋊C4—831no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—834no—
143C2×C4⋊C4—1664no—
144C2×C4⋊C4C2×D8311yescharacteristic
145C2×C4⋊C4C2×D8311yescharacteristic
146C2×C4⋊C4C2×D8314yes—
147C2×C22⋊C4—832no—
148C2×C22⋊C4—832no—
149C2×C22⋊C4—834no—
150C2×C22⋊C4—834no—
151C2×C42—831no—
152C2×C42C2×D8311yescharacteristic
153#95—832no—
154C24—831no—
155C24—832no—
156C2×D4—1664no—
157C2×D4—1668no—
158C2×D4—1662no—
159C2×D4—1662no—
160C2×D4—1662no—
161C22×C4—831no—
162C22×C4—832no—
163C22×C4—834no—
164C22×C4—832no—
165C22×C4—1664no—
166C22×C4—1668no—
167C22×C4—1662no—
168C22×C4—1662no—
169C22×C4C22×D8312yes—
170C22×C4C22×D8312yes—
171C22×C4C22×D8312yes—
172C22×C4C22×D8311yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—28no—
176C4⋊C4—1664no—
177C4⋊C4—1668no—
178C4⋊C4—1662no—
179C22⋊C4—838no—
180C22⋊C4—1664no—
181C22⋊C4—1668no—
182C22⋊C4—1668no—
183C42—22no—
184C42—1662no—
185C23—834no—
186C23—832no—
187C23—1664no—
188C23—1662no—
189C23—1662no—
190C23—1662no—
191C23—1664no—
192C23—1661no—
193C23—1661no—
194C23—1662no—
195C23C23×D8311yescharacteristic
196D4—3328no—
197D4—3324no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—3328no—
205C2×C4—834no—
206C2×C4—832no—
207C2×C4—1664no—
208C2×C4—1662no—
209C2×C4—1661no—
210C2×C4—1662no—
211C2×C4—1661no—
212C2×C4—1662no—
213C2×C4—1668no—
214C2×C4D4×D8314yes—
215C2×C4D4×D8312yes—
216C2×C4#1305059012yes—
217C2×C4#1305059112yes—
218C22—3324no—
219C22—3324no—
220C22—3322no—
221C22—3322no—
222C22—3322no—
223C22—834no—
224C22—1664no—
225C22—1664no—
226C22—1662no—
227C22—1662no—
228C22C2×D4×D8312yes—
229C22C2×D4×D8311yescharacteristic
230C22C2×#1305059011yescharacteristic
231C22C2×#1305059111yescharacteristic
232C22#1458184012yes—
233C4—24no—
234C4—42no—
235C4—42no—
236C4—3322no—
237C4—1664no—
238C4—1662no—
239C4#1458177714yescomplemented
240C2—6642no—
241C2—1664no—
242C2C2×#1458177711yescharacteristic
243C2#127×D8311yescharacteristic
244C2#1912827611yescharacteristic
245C2#1912835212yes—
246C2#1912835412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#384×C83C2
prime radicalC2×#14581767C2
hypercenter#384D83
hypocenterC83#2501
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#14581777C2semidirect
3. C2×C4⋊C4×D83C2semidirect
4. #19127636C2semidirect
5. #19127603C2semidirect
6. #19127602C2semidirect
7. #19127570C2semidirect
8. C4×(C2×D83)⋊C4C2semidirect
9. #384×C83C2semidirect
51. (C2×D83)⋊C4D4semidirect
112. C4×C83#123semidirect
124. C83#2501semidirect
132. #384D83semidirect
239. C4#14581777semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×#145817672
C2×C4⋊C4×D832
#191276362
#191276032
#191276022
#191275702
C4×(C2×D83)⋊C42
#384×C832
#250183