Finite Group Databasebeta

← #24907877

Subgroups — #24907877

Order 10624.

386 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#19128176C211yesmaximal, characteristic
2C2×#14581707C211yesmaximal, characteristic
3#19127564C212yesmaximal, complemented
4#19127467C212yesmaximal, complemented
5C2×#123×C83C211yesmaximal, characteristic
6D4×Dic83—24no—
7#14581751—21no—
8C2×C22⋊Dic83—21no—
9C2×C22⋊Dic83C2212yes—
10C2×C22⋊Dic83C2212yes—
11#14581707—22no—
12C83⋊#95—24no—
13C22×D4×C83C2211yescharacteristic
14C23×C4×C83C2211yescharacteristic
15#123×C83—22no—
16#123×C83C414yescomplemented
17C2×C4⋊C4×C83—21no—
18C2×C22⋊C4×C83—21no—
19C2×C22⋊C4×C83C2211yescharacteristic
20C22×Dic83—22no—
21C22×Dic83—22no—
22C22×Dic83D414yes—
23C4⋊Dic83—22no—
24C22⋊Dic83—22no—
25C22⋊Dic83—22no—
26C22⋊Dic83—22no—
27C22⋊Dic83—22no—
28C22⋊Dic83—22no—
29C4×Dic83—22no—
30C24×C83D411yescharacteristic
31C24×C83C2311yescharacteristic
32C2×D4×C83—22no—
33C2×D4×C83—22no—
34C2×D4×C83C2×C412yes—
35C2×D4×C83D412yes—
36C22×C4×C83—21no—
37C22×C4×C83—21no—
38C22×C4×C83—21no—
39C22×C4×C83—22no—
40C22×C4×C83—22no—
41C22×C4×C83C2×C412yescomplemented
42C22×C4×C83D411yescharacteristic
43C4⋊C4×C83—24no—
44C22⋊C4×C83—24no—
45C22⋊C4×C83—21no—
46C22⋊C4×C83C2×C412yescomplemented
47C23×C83—21no—
48C23×C83—21no—
49C23×C83—42no—
50C23×C83—42no—
51C23×C83C22⋊C412yes—
52C23×C83C22×C411yescharacteristic
53C23×C83C2×D412yes—
54C23×C83C2×D411yescharacteristic
55C23×C83C2×D411yescharacteristic
56C23×C83#4112yes—
57D4×C83—24no—
58D4×C83—24no—
59C2×C4×C83—21no—
60C2×C4×C83—21no—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—24no—
64C2×C4×C83—42no—
65C2×C4×C83—41no—
66C2×C4×C83—42no—
67C2×C4×C83—41no—
68C2×C4×C83C22⋊C412yes—
69C2×Dic83—24no—
70C2×Dic83—44no—
71C2×Dic83—44no—
72C2×Dic83—42no—
73C2×Dic83—82no—
74C22×C83—24no—
75C22×C83—22no—
76C22×C83—22no—
77C22×C83—41no—
78C22×C83—41no—
79C22×C83—42no—
80C22×C83—42no—
81C22×C83—42no—
82C22×C83—81no—
83C22×C83C2×C22⋊C411yescharacteristic
84C22×C83C4×D412yes—
85C22×C83#12011yescharacteristic
86C22×C83#12112yes—
87C22×C83#12311yescharacteristic
88C4×C83—22no—
89C4×C83—42no—
90C4×C83—42no—
91Dic83—44no—
92Dic83—82no—
93C2×C83—22no—
94C2×C83—21no—
95C2×C83—21no—
96C2×C83—22no—
97C2×C83—42no—
98C2×C83#38611yescharacteristic
99C2×C83#45812yes—
100#1890—831nomaximal, Sylow
101C83#189011yescharacteristic, complemented, Sylow
102C2×#123D8311yescharacteristic
103#518—831no—
104C2×#99—831no—
105#381—832no—
106#342—832no—
107C22×D4C2×D8311yescharacteristic
108C23×C4C2×D8311yescharacteristic
109#123—22no—
110#123Dic8314yescomplemented
111C4×D4—1664no—
112#117—1661no—
113C2×C4⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—832no—
116C2×C22⋊C4—832no—
117C2×C22⋊C4—1661no—
118C2×C22⋊C4C2×D8311yescharacteristic
119#99—1662no—
120#95—1664no—
121C24C22⋊D8311yescharacteristic
122C24C22×D8311yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C2×Dic8312yes—
126C2×D4C22⋊D8312yes—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—834no—
133C22×C4—1662no—
134C22×C4—1662no—
135C22×C4C2×Dic8312yescomplemented
136C22×C4C22⋊D8311yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—1662no—
139C22⋊C4—24no—
140C22⋊C4—21no—
141C22⋊C4—1662no—
142C22⋊C4—1662no—
143C22⋊C4—1662no—
144C22⋊C4—1662no—
145C22⋊C4—1662no—
146C22⋊C4C2×Dic8312yescomplemented
147C42—1662no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23C22⋊Dic8312yes—
153C23C22×Dic8311yescharacteristic
154C23D4×D8312yes—
155C23C2×C22⋊D8311yescharacteristic
156C23C2×C22⋊D8311yescharacteristic
157C23#1305058912yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—3324no—
170C2×C4—3324no—
171C2×C4—3322no—
172C2×C4—6642no—
173C2×C4—1664no—
174C2×C4C22⋊Dic8312yes—
175C22—22no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×C22⋊Dic8311yescharacteristic
185C22D4×Dic8312yes—
186C22#1458176011yescharacteristic
187C22#1458176412yes—
188C22#1458177311yescharacteristic
189C4—22no—
190C4—42no—
191C4—42no—
192C4—3324no—
193C4—6642no—
194C2—21no—
195C2—22no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#1912758111yescharacteristic
200C2#1912789012yes—
subgroupG/H
centerC22#14581760
commutatorC23×C83C22×C4
FrattiniC24C22×D83
FittingC2×#123×C83C2
socleC22×C83#120
prime radicalC22×D4×C83C22
cosocleC24×C83C23
hypercenterC2×#123D83
hypocenterC83#1890
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #19127564C2semidirect
4. #19127467C2semidirect
16. #123×C83C4semidirect
41. C22×C4×C83C2×C4semidirect
46. C22⋊C4×C83C2×C4semidirect
101. C83#1890semidirect
110. #123Dic83semidirect
135. C22×C4C2×Dic83semidirect
146. C22⋊C4C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#191281762
C2×#145817072
#191275642
#191274672
C2×#123×C832
#189083