Finite Group Databasebeta

← C22×#11758472 #16858934

Subgroups — C22×#11758472 #16858934

Order 3968.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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×D124C212yesmaximal, complemented
2C23×D124C211yesmaximal, characteristic, complemented
3C23×C4×D31C211yesmaximal, characteristic, complemented
4C2×#11758472C2124yesmaximal, direct product factor
5C22×(C2×D31)⋊C4C212yesmaximal, complemented
6C22×C4⋊C4×C31C211yesmaximal, characteristic, complemented
7C24×D31C2212yes—
8C24×D31C2211yescharacteristic
9C23×C4×C31C2212yes—
10C23×C4×C31C2211yescharacteristic
11C23×Dic31C2211yescharacteristic
12C22×D124—216no—
13C22×D124—28no—
14C22×D124C22124yescomplemented
15C22×D124C22112yescomplemented
16C22×C4×D31—28no—
17C22×C4×D31C22112yescomplemented
18C2×C4⋊C4×C31C22112yescomplemented
19C2×(C2×D31)⋊C4C22124yescomplemented
20#11758472C22164yesdirect product factor
21C24×C31C2311yescharacteristic
22C23×D31—24no—
23C23×D31—24no—
24C23×D31—28no—
25C23×D31—28no—
26C23×D31—28no—
27C23×D31D418yescomplemented
28C23×D31C23112yes—
29C23×D31C2316yes—
30C22×C4×C31—28no—
31C22×C4×C31D418yescomplemented
32C22×C4×C31C23112yes—
33C22×C4×C31C2316yes—
34C22×Dic31—24no—
35C22×Dic31C2316yes—
36C2×D124—296no—
37C2×D124—248no—
38C2×D124C23132yescomplemented
39C2×D124C23116yescomplemented
40C2×C4×D31—248no—
41C2×C4×D31C23116yescomplemented
42C4⋊C4×C31C23116yescomplemented
43(C2×D31)⋊C4C23132yescomplemented
44C23×C31C2×D414yes—
45C23×C31C2×D414yes—
46C23×C31#4114yes—
47C23×C31C2413yes—
48C22×D31—232no—
49C22×D31—212no—
50C22×D31—212no—
51C22×D31—224no—
52C22×D31—224no—
53C22×D31—224no—
54C22×D31—432no—
55C22×D31C2×D4124yescomplemented
56C22×D31C2418yes—
57C22×D31C2414yes—
58C2×C4×C31—224no—
59C2×C4×C31C2×D4124yescomplemented
60C2×C4×C31C2418yes—
61C2×C4×C31C2414yes—
62C2×Dic31—212no—
63C2×Dic31C2414yes—
64D124—264no—
65D124—232no—
66C4×D31—232no—
67C22×#121—311nomaximal, Sylow
68C22×C31#121116yescomplemented
69C22×C31C22×D416yes—
70C22×C31C22×D416yes—
71C22×C31C2×#4116yes—
72C22×C31C2511yescharacteristic
73C2×D31—248no—
74C2×D31—24no—
75C2×D31—24no—
76C2×D31—28no—
77C2×D31—28no—
78C2×D31—28no—
79C2×D31—448no—
80C2×D31C22×D418yescomplemented
81C4×C31—28no—
82C4×C31C22×D418yescomplemented
83Dic31—24no—
84C23×D4—312no—
85C23×D4—311no—
86C24×C4—311no—
87C2×#121—3124no—
88C22×C4⋊C4D3111yescharacteristic, complemented
89C22×C22⋊C4—312no—
90C2×C31C2×#121112yescomplemented
91C2×C31C23×D411yescharacteristic
92C2×C31C23×D411yescharacteristic
93C2×C31C22×#4111yescharacteristic
94D31—28no—
95D31—48no—
96C25—311no—
97C25—312no—
98C22×D4—3124no—
99C22×D4—3112no—
100C22×D4—6216no—
101C22×D4—628no—
102C23×C4—311no—
103C23×C4—3112no—
104C23×C4—628no—
105C23×C4C2×D3112yes—
106C23×C4C2×D3111yescharacteristic
107#121—3164no—
108C2×C4⋊C4C2×D31112yescomplemented
109C2×C22⋊C4—3124no—
110C31C22×#12111yescharacteristic, complemented, Sylow
111C24—318no—
112C24—316no—
113C24—3112no—
114C24—624no—
115C24—624no—
116C24—628no—
117C24—628no—
118C24—628no—
119C24C22×D3111yescharacteristic
120C2×D4—3132no—
121C2×D4—3116no—
122C2×D4—6296no—
123C2×D4—6248no—
124C22×C4—28no—
125C22×C4—316no—
126C22×C4—3116no—
127C22×C4—624no—
128C22×C4—6248no—
129C22×C4D12418yescomplemented
130C22×C4C22×D31112yes—
131C22×C4C22×D3116yes—
132C4⋊C4C22×D31116yescomplemented
133C22⋊C4—3132no—
134C23—3124no—
135C23—314no—
136C23—318no—
137C23—6232no—
138C23—6212no—
139C23—6212no—
140C23—6224no—
141C23—6224no—
142C23—6224no—
143C23—12432no—
144C23C2×D12414yes—
145C23D4×D3114yes—
146C23#9249914yes—
147C23C23×D3113yes—
148D4—6264no—
149D4—6232no—
150C2×C4—224no—
151C2×C4—314no—
152C2×C4—6212no—
153C2×C4—6232no—
154C2×C4C2×D124124yescomplemented
155C2×C4C23×D3118yes—
156C2×C4C23×D3114yes—
157C22—318no—
158C22—624no—
159C22—624no—
160C22—6248no—
161C22—628no—
162C22—628no—
163C22—628no—
164C22—12448no—
165C22#11758472116yesdirect product factor
166C22C22×D12416yes—
167C22C2×D4×D3116yes—
168C22C2×#9249916yes—
169C22C24×D3111yescharacteristic
170C4—28no—
171C4—624no—
172C4C22×D12418yescomplemented
173C2—628no—
174C2—1248no—
175C2C2×#11758472112yesdirect product factor
176C2C22×D4×D3111yescharacteristic
177C2C23×D12411yescharacteristic
178C2C22×#9249911yescharacteristic
subgroupG/H
centerC24C22×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC22×C4⋊C4×C31C2
socleC24×C31C23
hypercenterC22×C4⋊C4D31
hypocenterC31C22×#121
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D124C2semidirect
2. C23×D124C2semidirect
3. C23×C4×D31C2semidirect
4. C2×#11758472C2direct (N × H)
5. C22×(C2×D31)⋊C4C2semidirect
6. C22×C4⋊C4×C31C2semidirect
14. C22×D124C22semidirect
15. C22×D124C22semidirect
17. C22×C4×D31C22semidirect
18. C2×C4⋊C4×C31C22semidirect
19. C2×(C2×D31)⋊C4C22semidirect
20. #11758472C22direct (N × H)
27. C23×D31D4semidirect
31. C22×C4×C31D4semidirect
38. C2×D124C23semidirect
39. C2×D124C23semidirect
41. C2×C4×D31C23semidirect
42. C4⋊C4×C31C23semidirect
43. (C2×D31)⋊C4C23semidirect
55. C22×D31C2×D4semidirect
59. C2×C4×C31C2×D4semidirect
68. C22×C31#121semidirect
80. C2×D31C22×D4semidirect
82. C4×C31C22×D4semidirect
88. C22×C4⋊C4D31semidirect
90. C2×C31C2×#121semidirect
108. C2×C4⋊C4C2×D31semidirect
110. C31C22×#121semidirect
129. C22×C4D124semidirect
132. C4⋊C4C22×D31semidirect
154. C2×C4C2×D124semidirect
165. C22#11758472direct (N × H)
172. C4C22×D124semidirect
175. C2C2×#11758472direct (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×C4×D312
C2×#117584722
C22×(C2×D31)⋊C42
C22×C4⋊C4×C312
C22×#12131