Finite Group Databasebeta

← #24907902

Subgroups — #24907902

Order 10624.

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

normal subgroups only

This group has 412 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
1C22×#13050574C211yesmaximal, characteristic, complemented
2C2×#14581713C212yesmaximal, complemented
3#19127561C211yesmaximal, characteristic, complemented
4#19127463C212yesmaximal, complemented
5C2×#121×C83C211yesmaximal, characteristic
6C2×#13050574—24no—
7C2×#13050574—22no—
8C22×C83⋊C8C2212yes—
9C2×C4⋊Dic83C2212yes—
10C2×C22⋊Dic83—21no—
11#14581713—24no—
12C83⋊#95—21no—
13C22×D4×C83—21no—
14C22×D4×C83C2211yescharacteristic
15C23×C4×C83C2211yescharacteristic
16#121×C83—22no—
17#121×C83C414yescomplemented
18C2×C4⋊C4×C83C2211yescharacteristic
19C2×C22⋊C4×C83—21no—
20C22×Dic83—22no—
21#13050574—28no—
22C2×C83⋊C8—22no—
23C2×C83⋊C8—22no—
24C2×C83⋊C8D414yescomplemented
25C4⋊Dic83—22no—
26C4⋊Dic83—22no—
27C22⋊Dic83—42no—
28C24×C83—21no—
29C24×C83D411yescharacteristic
30C2×D4×C83—24no—
31C2×D4×C83—22no—
32C2×D4×C83—21no—
33C2×D4×C83—22no—
34C2×D4×C83—44no—
35C2×D4×C83C2×C412yes—
36C22×C4×C83—22no—
37C22×C4×C83—22no—
38C22×C4×C83—21no—
39C22×C4×C83C2×C412yes—
40C22×C4×C83D411yescharacteristic
41C22×C4×C83D412yes—
42C22×C4×C83C2311yescharacteristic
43C4⋊C4×C83—21no—
44C4⋊C4×C83C2×C412yes—
45C22⋊C4×C83—24no—
46C23×C83—22no—
47C23×C83—22no—
48C23×C83—21no—
49C23×C83—21no—
50C23×C83—22no—
51C23×C83—42no—
52C23×C83—42no—
53C23×C83—41no—
54C23×C83—41no—
55C23×C83C22⋊C412yes—
56C23×C83C2×D411yescharacteristic
57D4×C83—24no—
58D4×C83—48no—
59D4×C83—41no—
60D4×C83—41no—
61C2×C4×C83—22no—
62C2×C4×C83—24no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—42no—
67C2×C4×C83C22⋊C412yes—
68C2×C4×C83C22×C411yescharacteristic
69C2×C4×C83C2×D412yes—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83#4112yes—
72C2×Dic83—42no—
73C2×Dic83—42no—
74C2×Dic83—42no—
75C83⋊C8—24no—
76C22×C83—22no—
77C22×C83—22no—
78C22×C83—22no—
79C22×C83—42no—
80C22×C83—42no—
81C22×C83—42no—
82C22×C83—42no—
83C22×C83—42no—
84C22×C83—82no—
85C22×C83—82no—
86C22×C83C2×C22⋊C411yescharacteristic
87C22×C83#12112yes—
88C22×C83AGL(1,ℤ/8)11yescharacteristic
89C22×C83AGL(1,ℤ/8)12yes—
90C22×C83#13711yescharacteristic
91C4×C83—22no—
92C4×C83—42no—
93C4×C83C4×D412yes—
94C4×C83#12011yescharacteristic
95C4×C83#12311yescharacteristic
96Dic83—82no—
97C2×C83—42no—
98C2×C83—82no—
99C2×C83#38611yescharacteristic
100C2×C83#41711yescharacteristic
101C2×C83#41811yescharacteristic
102C2×C83#46812yes—
103C2×C83#46912yes—
104#1895—831nomaximal, Sylow
105C83#189511yescharacteristic, complemented, Sylow
106C22×M16—831no—
107C2×#121D8311yescharacteristic
108C2×#102—832no—
109#380—831no—
110#340—832no—
111C22×D4—21no—
112C22×D4C2×D8311yescharacteristic
113C23×C4C2×D8311yescharacteristic
114C2×M16—1664no—
115C2×M16—1662no—
116C22×C8—832no—
117#121—22no—
118#121Dic8314yescomplemented
119C2×C4⋊C4—832no—
120C2×C4⋊C4C2×D8311yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—1661no—
123#102—1664no—
124#95—1661no—
125C24—21no—
126C24C22⋊D8311yescharacteristic
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4C2×Dic8312yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—1662no—
137C22×C4C2×Dic8312yes—
138C22×C4C22⋊D8311yescharacteristic
139C22×C4C22⋊D8312yes—
140C22×C4C22×D8311yescharacteristic
141M16—1668no—
142C2×C8—834no—
143C2×C8—1662no—
144C2×C8—1662no—
145C4⋊C4—21no—
146C4⋊C4—1662no—
147C4⋊C4—1662no—
148C4⋊C4C2×Dic8312yes—
149C22⋊C4—24no—
150C22⋊C4—3322no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—42no—
157C23—41no—
158C23—42no—
159C23—41no—
160C23C22⋊Dic8312yes—
161C23C2×C22⋊D8311yescharacteristic
162D4—24no—
163D4—48no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—3322no—
173C2×C4—3322no—
174C2×C4—3322no—
175C2×C4C22⋊Dic8312yes—
176C2×C4C22×Dic8311yescharacteristic
177C2×C4D4×D8312yes—
178C2×C4C2×C22⋊D8311yescharacteristic
179C2×C4#1305058912yes—
180C8—1664no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22C2×C22⋊Dic8311yescharacteristic
192C22#1458176412yes—
193C22#1458181711yescharacteristic
194C22#1458181812yes—
195C22#1458182211yescharacteristic
196C4—22no—
197C4—42no—
198C4—6642no—
199C4D4×Dic8312yes—
200C4#1458176011yescharacteristic
201C4#1458177311yescharacteristic
202C2—42no—
203C2—82no—
204C2#1912758111yescharacteristic
205C2#1912770211yescharacteristic
206C2#1912770611yescharacteristic
207C2#1912794512yes—
208C2#1912795212yes—
subgroupG/H
centerC23C2×C22⋊D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting, prime radicalC2×#121×C83C2
socleC23×C83C2×D4
cosocleC22×C4×C83C23
hypercenterC2×#121D83
hypocenterC83#1895
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13050574C2semidirect
2. C2×#14581713C2semidirect
3. #19127561C2semidirect
4. #19127463C2semidirect
17. #121×C83C4semidirect
24. C2×C83⋊C8D4semidirect
105. C83#1895semidirect
118. #121Dic83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505742
C2×#145817132
#191275612
#191274632
C2×#121×C832
#189583