Finite Group Databasebeta

← #24915534

Subgroups — #24915534

Order 10624.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581785C212yesmaximal, complemented
2#19127635C212yesmaximal, complemented
3#19127631C211yesmaximal, characteristic, complemented
4#19127624C212yesmaximal, complemented
5#19127624C212yesmaximal, complemented
6#19127624C212yesmaximal, complemented
7#19127615C212yesmaximal
8#19127570C211yesmaximal, characteristic, complemented
9#394×C83C211yesmaximal, characteristic
10#14581785—28no—
11C2×C4⋊Dic83C2212yes—
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2212yes—
27C22×D4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2212yes—
31#95×C83C2211yescharacteristic
32C22×Dic83C2312yes—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C4⋊Dic83—24no—
37C4⋊Dic83—41no—
38C4⋊Dic83—41no—
39Dic83⋊C4—24no—
40Dic83⋊C4—24no—
41Dic83⋊C4—42no—
42Dic83⋊C4—42no—
43Dic83⋊C4—42no—
44C22⋊Dic83—24no—
45C22⋊Dic83—24no—
46C22⋊Dic83—42no—
47C22⋊Dic83—42no—
48C4×Dic83D414yescomplemented
49C24×C83C2312yes—
50C2×D4×C83—41no—
51C2×D4×C83—42no—
52C22×C4×C83C2312yes—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C4⋊C4×C83—41no—
57C22⋊C4×C83—24no—
58C22⋊C4×C83—42no—
59C23×C83—22no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—22no—
63C23×C83—22no—
64C23×C83—22no—
65C23×C83—22no—
66C23×C83C2411yescharacteristic
67D4×C83—44no—
68C2×C4×C83—21no—
69C2×C4×C83—21no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—21no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83C2×D412yes—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83C2×D414yes—
90C22×C83—42no—
91C22×C83—42no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83—42no—
97C22×C83C22×D411yescharacteristic
98C22×C83H(5,𝔽2)11yescharacteristic
99C22×C83H(5,𝔽2)12yes—
100C22×C832−1+412yes—
101C22×C832−1+411yescharacteristic
102C4×C83—22no—
103C4×C83—42no—
104C4×C83—41no—
105C4×C83—41no—
106Dic83—24no—
107Dic83—42no—
108Dic83—42no—
109Dic83—42no—
110C2×C83—82no—
111C2×C83#53711yescharacteristic
112C2×C83#53712yes—
113C2×C83#56011yescharacteristic
114C2×C83#56311yescharacteristic
115C2×C83#56312yes—
116#2823—831nomaximal, Sylow
117C83#282311yescharacteristic, complemented, Sylow
118C2×#126—832no—
119#398—831no—
120#398—832no—
121#396—832no—
122#396—832no—
123#396—832no—
124#394—832no—
125#394D8311yescharacteristic
126#384—831no—
127C22×D4C2×D8311yescharacteristic
128#126—1668no—
129C2×C4⋊C4—832no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4—831no—
132C2×C4⋊C4—832no—
133C2×C4⋊C4—831no—
134C2×C4⋊C4—832no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4C2×D8311yescharacteristic
138C2×C22⋊C4—832no—
139C2×C22⋊C4—832no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4—832no—
142C2×C22⋊C4C2×D8312yes—
143C2×C22⋊C4C2×D8312yes—
144C2×C42—831no—
145#95—832no—
146#95—831no—
147#95—832no—
148#95C2×D8311yescharacteristic
149C24C22×D8312yes—
150C2×D4—41no—
151C2×D4—42no—
152C22×C4—832no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4—832no—
156C22×C4C22×D8311yescharacteristic
157C22×C4C22×D8312yes—
158C22×C4C22×D8311yescharacteristic
159C22×C4C22×D8311yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—3322no—
162C4⋊C4—3322no—
163C4⋊C4—3322no—
164C4⋊C4—3321no—
165C4⋊C4—3321no—
166C4⋊C4—1664no—
167C4⋊C4—1664no—
168C4⋊C4—1664no—
169C22⋊C4—24no—
170C22⋊C4—42no—
171C22⋊C4—3322no—
172C22⋊C4—3322no—
173C22⋊C4—1664no—
174C22⋊C4—1664no—
175C42—834no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23C23×D8311yescharacteristic
184D4—44no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—834no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4D4×D8312yes—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×D4×D8311yescharacteristic
215C22#1458183711yescharacteristic
216C22#1458183712yes—
217C22#1458183912yes—
218C22#1458183911yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—3322no—
224C4—3322no—
225C4—3322no—
226C4—1664no—
227C2—82no—
228C2#1912827811yescharacteristic
229C2#1912828012yes—
230C2#1912844011yescharacteristic
231C2#1912845111yescharacteristic
232C2#1912845412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2823
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581785C2semidirect
2. #19127635C2semidirect
3. #19127631C2semidirect
4. #19127624C2semidirect
5. #19127624C2semidirect
6. #19127624C2semidirect
8. #19127570C2semidirect
48. C4×Dic83D4semidirect
117. C83#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#145817852
#191276352
#191276312
#191276242
#191276152
#191275702
#394×C832
#282383