Finite Group Databasebeta

← #16849024

Subgroups — #16849024

Order 3968.

1564 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1564 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×C22⋊D31C212yesmaximal, complemented
2C2×#11758459C214yesmaximal, complemented
3#14520958C214yesmaximal, complemented
4#14520926C214yesmaximal, complemented
5C31×#379C211yesmaximal, characteristic, complemented
6C25×C31C2211yescharacteristic
7C24×D31C2212yescomplemented
8C22×C22⋊D31—28no—
9C22×C22⋊D31—216no—
10C22×C22⋊D31C2214yes—
11C22×C22⋊D31C2214yescomplemented
12C23×Dic31C2212yescomplemented
13C22×D124C2214yescomplemented
14C2×C22⋊C4×C31C2214yescomplemented
15C2×C22⋊C4×C31C2212yescomplemented
16C2×(C2×D31)⋊C4C2218yescomplemented
17#11758459—216no—
18C31⋊#95C2214yescomplemented
19C24×C31—28no—
20C24×C31—42no—
21C24×C31D414yescomplemented
22C24×C31C2311yescharacteristic
23C24×C31C2312yes—
24C23×D31—24no—
25C23×D31—28no—
26C23×D31—28no—
27C23×D31—44no—
28C23×D31C2314yes—
29C22×C4×C31C2314yes—
30C2×C22⋊D31—264no—
31C2×C22⋊D31—28no—
32C2×C22⋊D31—28no—
33C2×C22⋊D31—28no—
34C2×C22⋊D31—28no—
35C2×C22⋊D31—28no—
36C2×C22⋊D31D4116yescomplemented
37C22×Dic31—24no—
38C22×Dic31—28no—
39C22×Dic31C2314yes—
40C2×D124—216no—
41C2×D124—44no—
42C22⋊C4×C31—28no—
43C22⋊C4×C31—42no—
44(C2×D31)⋊C4—216no—
45C23×C31—24no—
46C23×C31—24no—
47C23×C31—216no—
48C23×C31—24no—
49C23×C31—22no—
50C23×C31—44no—
51C23×C31—416no—
52C23×C31C2×D412yes—
53C23×C31C2×D414yes—
54C23×C31C2×D418yes—
55C23×C31C2411yescharacteristic
56C22×D31—216no—
57C22×D31—28no—
58C22×D31—24no—
59C22×D31—24no—
60C22×D31—28no—
61C22×D31—432no—
62C22×D31—48no—
63C22×D31—48no—
64C22×D31C2×D418yescomplemented
65C2×C4×C31—28no—
66C2×C4×C31—24no—
67C22⋊D31—232no—
68C22⋊D31—232no—
69C2×Dic31—216no—
70C2×Dic31—24no—
71C2×Dic31—24no—
72C2×Dic31C2×D418yescomplemented
73D124—416no—
74#2403—311nomaximal, Sylow
75C22×C31—24no—
76C22×C31—24no—
77C22×C31—216no—
78C22×C31—24no—
79C22×C31—22no—
80C22×C31—416no—
81C22×C31—44no—
82C22×C31#12018yescomplemented
83C22×C31C22×D411yescharacteristic
84C22×C31C22×D414yes—
85C22×C31H(5,𝔽2)12yes—
86C2×D31—28no—
87C2×D31—216no—
88C2×D31—432no—
89C2×D31—44no—
90C2×D31—44no—
91C2×D31—48no—
92C4×C31—44no—
93Dic31—28no—
94C23×D4—312no—
95C2×#120—314no—
96#392—314no—
97#386—314no—
98#379D3111yescharacteristic, complemented
99C2×C31—28no—
100C2×C31—42no—
101C2×C31C2×#12012yes—
102C2×C31#53411yescharacteristic
103C2×C31D4214yes—
104D31—48no—
105C25—312no—
106C25C2×D3111yescharacteristic
107C22×D4—314no—
108C22×D4—314no—
109C22×D4—314no—
110C22×D4—6216no—
111C22×D4—628no—
112C23×C4—312no—
113#120—6216no—
114C2×C22⋊C4—318no—
115C2×C22⋊C4C2×D3114yescomplemented
116C2×C22⋊C4C2×D3112yescomplemented
117#95—314no—
118C31#240311yescharacteristic, complemented, Sylow
119C24—28no—
120C24—42no—
121C24—314no—
122C24—628no—
123C24—624no—
124C24—628no—
125C24—1244no—
126C24D12414yescomplemented
127C24C22×D3111yescharacteristic
128C24C22×D3112yes—
129C2×D4—3116no—
130C2×D4—6216no—
131C2×D4—628no—
132C2×D4—628no—
133C2×D4—6264no—
134C2×D4—628no—
135C2×D4—628no—
136C2×D4—628no—
137C2×D4—1244no—
138C22×C4—314no—
139C22×C4—628no—
140C22×C4—624no—
141C22×C4C22×D3114yes—
142C22⋊C4—28no—
143C22⋊C4—42no—
144C22⋊C4—6216no—
145C23—24no—
146C23—24no—
147C23—216no—
148C23—24no—
149C23—22no—
150C23—44no—
151C23—416no—
152C23—318no—
153C23—6216no—
154C23—624no—
155C23—628no—
156C23—624no—
157C23—628no—
158C23—12432no—
159C23—1248no—
160C23—1248no—
161C23C2×D12412yes—
162C23C2×D12414yes—
163C23D4×D3118yes—
164C23C23×D3111yescharacteristic
165D4—6232no—
166D4—6232no—
167D4—12416no—
168C2×C4—28no—
169C2×C4—24no—
170C2×C4—318no—
171C2×C4—6216no—
172C2×C4—624no—
173C2×C4—624no—
174C22—24no—
175C22—216no—
176C22—24no—
177C22—22no—
178C22—24no—
179C22—416no—
180C22—44no—
181C22—6216no—
182C22—628no—
183C22—12432no—
184C22—1244no—
185C22—1248no—
186C22—1244no—
187C22#1175845918yescomplemented
188C22C22×D12411yescharacteristic
189C22C2×D4×D3114yes—
190C22#1175856112yes—
191C4—44no—
192C4—628no—
193C2—28no—
194C2—42no—
195C2—1248no—
196C2C2×#1175845912yes—
197C2#1452161111yescharacteristic
198C2#1452168114yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#379C2
hypercenter#379D31
hypocenterC31#2403
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D31C2semidirect
2. C2×#11758459C2semidirect
3. #14520958C2semidirect
4. #14520926C2semidirect
5. C31×#379C2semidirect
7. C24×D31C22semidirect
11. C22×C22⋊D31C22semidirect
12. C23×Dic31C22semidirect
13. C22×D124C22semidirect
14. C2×C22⋊C4×C31C22semidirect
15. C2×C22⋊C4×C31C22semidirect
16. C2×(C2×D31)⋊C4C22semidirect
18. C31⋊#95C22semidirect
21. C24×C31D4semidirect
36. C2×C22⋊D31D4semidirect
64. C22×D31C2×D4semidirect
72. C2×Dic31C2×D4semidirect
82. C22×C31#120semidirect
98. #379D31semidirect
115. C2×C22⋊C4C2×D31semidirect
116. C2×C22⋊C4C2×D31semidirect
118. C31#2403semidirect
126. C24D124semidirect
187. C22#11758459semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D312
C2×#117584592
#145209582
#145209262
C31×#3792
#240331