Finite Group Databasebeta

← C22×#11758507 #16858933

Subgroups — C22×#11758507 #16858933

Order 3968.

2528 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 725 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×D124C211yesmaximal, characteristic, complemented
2C23×C22⋊D31C212yesmaximal, complemented
3C2×#11758507C2124yesmaximal, direct product factor
4C22×C4⋊Dic31C211yesmaximal, characteristic, complemented
5C22×(C2×D31)⋊C4C212yesmaximal, complemented
6C24×C4×C31C211yesmaximal, characteristic, complemented
7C25×C31C2211yescharacteristic
8C24×D31C2212yes—
9C23×C4×C31—28no—
10C23×C4×C31C2211yescharacteristic
11C23×C4×C31C2211yescharacteristic
12C23×C4×C31C22112yescomplemented
13C22×C22⋊D31—216no—
14C22×C22⋊D31C22124yescomplemented
15C23×Dic31C2212yes—
16C22×D124—28no—
17C22×D124C22112yescomplemented
18#11758507C22164yesdirect product factor
19C2×(C2×D31)⋊C4C22124yescomplemented
20C2×C4⋊Dic31C22112yescomplemented
21C24×C31—24no—
22C24×C31—24no—
23C24×C31D418yescomplemented
24C24×C31C2316yes—
25C24×C31C2311yescharacteristic
26C23×D31—28no—
27C23×D31—28no—
28C23×D31—28no—
29C23×D31C23112yes—
30C22×C4×C31—24no—
31C22×C4×C31—248no—
32C22×C4×C31D418yescomplemented
33C22×C4×C31C2316yes—
34C22×C4×C31C2316yes—
35C22×C4×C31C23116yescomplemented
36C2×C22⋊D31—296no—
37C2×C22⋊D31C23132yescomplemented
38C22×Dic31—28no—
39C22×Dic31C23112yes—
40C2×D124—248no—
41C2×D124C23116yescomplemented
42(C2×D31)⋊C4C23132yescomplemented
43C4⋊Dic31C23116yescomplemented
44C23×C31—232no—
45C23×C31—212no—
46C23×C31—212no—
47C23×C31C2×D414yes—
48C23×C31C2×D414yes—
49C23×C31C2×D4124yescomplemented
50C23×C31#4114yes—
51C23×C31C2414yes—
52C23×C31C2413yes—
53C22×D31—224no—
54C22×D31—224no—
55C22×D31—224no—
56C22×D31—432no—
57C22×D31C2418yes—
58C2×C4×C31—212no—
59C2×C4×C31—232no—
60C2×C4×C31C2×D4124yescomplemented
61C2×C4×C31C2414yes—
62C2×C4×C31C2414yes—
63C22⋊D31—264no—
64C2×Dic31—224no—
65C2×Dic31C2418yes—
66D124—232no—
67C22×#121—311nomaximal, Sylow
68C22×C31—248no—
69C22×C31—24no—
70C22×C31—24no—
71C22×C31#121116yescomplemented
72C22×C31C22×D416yes—
73C22×C31C22×D416yes—
74C22×C31C22×D418yescomplemented
75C22×C31C2×#4116yes—
76C22×C31C2511yescharacteristic
77C2×D31—28no—
78C2×D31—28no—
79C2×D31—28no—
80C2×D31—448no—
81C4×C31—24no—
82C4×C31C22×D418yescomplemented
83Dic31—28no—
84C23×D4—312no—
85C23×D4—311no—
86C24×C4D3111yescharacteristic, complemented
87C2×#121—3124no—
88C22×C4⋊C4—311no—
89C22×C22⋊C4—312no—
90C2×C31—28no—
91C2×C31C2×#121112yescomplemented
92C2×C31C23×D411yescharacteristic
93C2×C31C23×D411yescharacteristic
94C2×C31C22×#4111yescharacteristic
95D31—48no—
96C25—312no—
97C25C2×D3111yescharacteristic
98C22×D4—3124no—
99C22×D4—3112no—
100C22×D4—6216no—
101C22×D4—628no—
102C23×C4—28no—
103C23×C4—312no—
104C23×C4C2×D3111yescharacteristic
105C23×C4C2×D3111yescharacteristic
106C23×C4C2×D31112yescomplemented
107#121—3164no—
108C2×C4⋊C4—3112no—
109C2×C22⋊C4—3124no—
110C31C22×#12111yescharacteristic, complemented, Sylow
111C24—24no—
112C24—24no—
113C24—3112no—
114C24—628no—
115C24—628no—
116C24—628no—
117C24D12418yescomplemented
118C24C22×D3116yes—
119C24C22×D3111yescharacteristic
120C2×D4—3132no—
121C2×D4—3116no—
122C2×D4—6296no—
123C2×D4—6248no—
124C22×C4—24no—
125C22×C4—248no—
126C22×C4—3112no—
127C22×C4—628no—
128C22×C4C22⋊D3118yescomplemented
129C22×C4C22×D3116yes—
130C22×C4C22×D3116yes—
131C22×C4C22×D31116yescomplemented
132C4⋊C4—3116no—
133C22⋊C4—3132no—
134C23—232no—
135C23—212no—
136C23—212no—
137C23—318no—
138C23—6224no—
139C23—6224no—
140C23—6224no—
141C23—12432no—
142C23C2×D12414yes—
143C23C2×D124124yescomplemented
144C23#9249514yes—
145C23C2×C22⋊D3114yes—
146C23C23×D3114yes—
147C23C23×D3113yes—
148D4—6264no—
149D4—6232no—
150C2×C4—212no—
151C2×C4—232no—
152C2×C4—318no—
153C2×C4—6224no—
154C2×C4C2×C22⋊D31124yescomplemented
155C2×C4C23×D3114yes—
156C2×C4C23×D3114yes—
157C22—24no—
158C22—24no—
159C22—248no—
160C22—628no—
161C22—628no—
162C22—628no—
163C22—12448no—
164C22#11758507116yesdirect product factor
165C22C22×D12416yes—
166C22C22×D12418yescomplemented
167C22C2×#9249516yes—
168C22C22×C22⋊D3116yes—
169C22C24×D3111yescharacteristic
170C4—24no—
171C4—628no—
172C4C22×C22⋊D3118yescomplemented
173C2—28no—
174C2—1248no—
175C2C2×#11758507112yesdirect product factor
176C2C23×C22⋊D3111yescharacteristic
177C2C23×D12411yescharacteristic
178C2C22×#9249511yescharacteristic
subgroupG/H
centerC24C22×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC24×C4×C31C2
socleC24×C31C23
hypercenterC24×C4D31
hypocenterC31C22×#121
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C24 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C23×D124C2semidirect
2. C23×C22⋊D31C2semidirect
3. C2×#11758507C2direct (N × H)
4. C22×C4⋊Dic31C2semidirect
5. C22×(C2×D31)⋊C4C2semidirect
6. C24×C4×C31C2semidirect
12. C23×C4×C31C22semidirect
14. C22×C22⋊D31C22semidirect
17. C22×D124C22semidirect
18. #11758507C22direct (N × H)
19. C2×(C2×D31)⋊C4C22semidirect
20. C2×C4⋊Dic31C22semidirect
23. C24×C31D4semidirect
32. C22×C4×C31D4semidirect
35. C22×C4×C31C23semidirect
37. C2×C22⋊D31C23semidirect
41. C2×D124C23semidirect
42. (C2×D31)⋊C4C23semidirect
43. C4⋊Dic31C23semidirect
49. C23×C31C2×D4semidirect
60. C2×C4×C31C2×D4semidirect
71. C22×C31#121semidirect
74. C22×C31C22×D4semidirect
82. C4×C31C22×D4semidirect
86. C24×C4D31semidirect
91. C2×C31C2×#121semidirect
106. C23×C4C2×D31semidirect
110. C31C22×#121semidirect
117. C24D124semidirect
128. C22×C4C22⋊D31semidirect
131. C22×C4C22×D31semidirect
143. C23C2×D124semidirect
154. C2×C4C2×C22⋊D31semidirect
164. C22#11758507direct (N × H)
166. C22C22×D124semidirect
172. C4C22×C22⋊D31semidirect
175. C2C2×#11758507direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D1242
C23×C22⋊D312
C2×#117585072
C22×C4⋊Dic312
C22×(C2×D31)⋊C42
C24×C4×C312
C22×#12131