Finite Group Databasebeta

← #24911101

Subgroups — #24911101

Order 10624.

1348 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C23×D332C211yesmaximal, characteristic, complemented
2C2×#14581788C212yesmaximal, complemented
3C2×#14581760C214yesmaximal, complemented
4#19127610C214yesmaximal, complemented
5#19127602C212yesmaximal, complemented
6C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
7C2×#121×C83C211yesmaximal, characteristic, complemented
8C24×D83C2212yes—
9C22×D332—28no—
10C22×D332—24no—
11C22×D332C2212yes—
12C22×D332C2212yes—
13C22×C22⋊D83C2218yescomplemented
14#14581788—28no—
15#14581760—216no—
16C2×C22⋊Dic83C2212yes—
17C2×(C2×D83)⋊C4C2218yescomplemented
18C2×C4×Dic83C2212yescomplemented
19C83⋊#95C2212yescomplemented
20C22×D4×C83C2212yes—
21C22×D4×C83C2211yescharacteristic, complemented
22C23×C4×C83C2211yescharacteristic
23#121×C83—24no—
24C2×C4⋊C4×C83C2211yescharacteristic, complemented
25C2×C22⋊C4×C83C2212yescomplemented
26C23×D83—24no—
27C23×D83—28no—
28C23×D83—28no—
29C23×D83—44no—
30C23×D83C2314yes—
31C2×D332—232no—
32C2×D332—28no—
33C2×D332—24no—
34C2×D332—24no—
35C2×D332—24no—
36C2×D332D418yescomplemented
37C2×C22⋊D83—216no—
38C2×C22⋊D83—216no—
39C2×C22⋊D83—48no—
40C22×Dic83C2314yes—
41C22⋊Dic83D418yescomplemented
42(C2×D83)⋊C4—216no—
43C4×Dic83—24no—
44C24×C83C2312yes—
45C24×C83C2311yescharacteristic
46C2×D4×C83—28no—
47C2×D4×C83—24no—
48C2×D4×C83—24no—
49C2×D4×C83—22no—
50C22×C4×C83—24no—
51C22×C4×C83D414yescomplemented
52C22×C4×C83C2312yes—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—22no—
56C22⋊C4×C83—24no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83—22no—
60C23×C83—24no—
61C23×C83—24no—
62C23×C83—24no—
63C23×C83C2×D414yes—
64C23×C83C2×D412yes—
65C23×C83C2411yescharacteristic
66C22×D83—216no—
67C22×D83—28no—
68C22×D83—24no—
69C22×D83—24no—
70C22×D83—28no—
71C22×D83—432no—
72C22×D83—48no—
73C22×D83—48no—
74C22×D83C2×D418yes—
75D4×C83—48no—
76D4×C83—44no—
77C2×C4×C83—22no—
78C2×C4×C83—28no—
79C2×C4×C83—22no—
80C2×C4×C83—24no—
81C2×C4×C83C2×D414yes—
82C2×C4×C83C2×D412yes—
83C2×C4×C83C2×D412yes—
84C22⋊D83—432no—
85C2×Dic83—24no—
86C2×Dic83—24no—
87C2×Dic83C2×D418yescomplemented
88D332—216no—
89D332—216no—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83—28no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83—48no—
97C22×C83#12714yescomplemented
98C22×C83C22×D412yes—
99C22×C83C22×D412yes—
100C22×C83C22×D411yescharacteristic
101C22×C83H(5,𝔽2)12yes—
102C2×D83—28no—
103C2×D83—216no—
104C2×D83—432no—
105C2×D83—44no—
106C2×D83—44no—
107C2×D83—48no—
108C4×C83—22no—
109C4×C83—42no—
110C4×C83#12014yescomplemented
111Dic83—28no—
112C2×C83—24no—
113C2×C83—82no—
114C2×C83C2×#12011yescharacteristic
115C2×C83C2×#12711yescharacteristic
116C2×C83#53511yescharacteristic
117C2×C83D4214yes—
118D83—48no—
119#2433—831nomaximal, Sylow
120C83#243311yescharacteristic, complemented, Sylow
121C23×D4—831no—
122C2×#127—832no—
123C2×#121D8311yescharacteristic, complemented
124C2×#120—834no—
125#392—834no—
126#390—832no—
127C4×C22⋊C4—831no—
128C25—832no—
129C22×D4—838no—
130C22×D4—832no—
131C22×D4—832no—
132C22×D4—1668no—
133C22×D4—1664no—
134C22×D4C2×D8312yes—
135C22×D4C2×D8311yescharacteristic, complemented
136C23×C4C2×D8311yescharacteristic
137#127—1668no—
138#121—24no—
139#120—16616no—
140C2×C4⋊C4C2×D8311yescharacteristic, complemented
141C2×C22⋊C4—832no—
142C2×C22⋊C4—838no—
143C2×C22⋊C4C2×D8312yescomplemented
144C2×C42—832no—
145#95—832no—
146C24—3324no—
147C24—834no—
148C24—1668no—
149C24—1668no—
150C24—1664no—
151C24C22×D8312yes—
152C24C22×D8311yescharacteristic
153C2×D4—28no—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—3328no—
158C2×D4—838no—
159C2×D4—1668no—
160C2×D4—16632no—
161C2×D4—16616no—
162C2×D4—16616no—
163C2×D4—1664no—
164C2×D4—1664no—
165C2×D4—1664no—
166C22×C4—24no—
167C22×C4—834no—
168C22×C4C22⋊D8314yescomplemented
169C22×C4C22×D8312yes—
170C22×C4C22×D8311yescharacteristic
171C22×C4C22×D8311yescharacteristic
172C4⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—838no—
175C22⋊C4—16616no—
176C42—1664no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—22no—
182C23—24no—
183C23—3328no—
184C23—33232no—
185C23—3328no—
186C23—838no—
187C23—16616no—
188C23—1668no—
189C23—1664no—
190C23—1664no—
191C23—1668no—
192C23D4×D8314yes—
193C23C2×C22⋊D8312yes—
194C23C23×D8311yescharacteristic
195D4—48no—
196D4—44no—
197D4—33232no—
198D4—16616no—
199D4—16616no—
200C2×C4—22no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—838no—
205C2×C4—1664no—
206C2×C4—1664no—
207C2×C4D4×D8314yes—
208C2×C4C2×C22⋊D8312yes—
209C2×C4C2×C22⋊D8312yes—
210C22—28no—
211C22—22no—
212C22—22no—
213C22—48no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—33232no—
218C22—3328no—
219C22—3324no—
220C22—3324no—
221C22—16616no—
222C22—1668no—
223C22#1458178814yescomplemented
224C22C2×D4×D8312yes—
225C22C2×D4×D8312yes—
226C22C22×C22⋊D8311yescharacteristic
227C22#1458183812yes—
228C4—22no—
229C4—42no—
230C4—1668no—
231C4#1458176014yescomplemented
232C2—24no—
233C2—82no—
234C2—3328no—
235C2C2×#1458176011yescharacteristic
236C2C2×#1458178811yescharacteristic
237C2#1912826511yescharacteristic
238C2#1912833514yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2433
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×D332C2semidirect
2. C2×#14581788C2semidirect
3. C2×#14581760C2semidirect
4. #19127610C2semidirect
5. #19127602C2semidirect
6. C4×C22⋊Dic83C2semidirect
7. C2×#121×C83C2semidirect
13. C22×C22⋊D83C22semidirect
17. C2×(C2×D83)⋊C4C22semidirect
18. C2×C4×Dic83C22semidirect
19. C83⋊#95C22semidirect
21. C22×D4×C83C22semidirect
24. C2×C4⋊C4×C83C22semidirect
25. C2×C22⋊C4×C83C22semidirect
36. C2×D332D4semidirect
41. C22⋊Dic83D4semidirect
51. C22×C4×C83D4semidirect
87. C2×Dic83C2×D4semidirect
97. C22×C83#127semidirect
110. C4×C83#120semidirect
120. C83#2433semidirect
123. C2×#121D83semidirect
135. C22×D4C2×D83semidirect
140. C2×C4⋊C4C2×D83semidirect
143. C2×C22⋊C4C2×D83semidirect
168. C22×C4C22⋊D83semidirect
223. C22#14581788semidirect
231. C4#14581760semidirect

Maximal subgroups

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

subgroupindex
C23×D3322
C2×#145817882
C2×#145817602
#191276102
#191276022
C4×C22⋊Dic832
C2×#121×C832
#243383