Finite Group Databasebeta

← C22×#10620215 #72220272

Subgroups — C22×#10620215 #72220272

Order 2592.

8538 conjugacy classes of subgroups; only the 763 normal subgroups are shown.

normal subgroups only

This group has 8538 conjugacy classes of subgroups — too many to draw. Only its 763 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×C3⋊D32C213yesmaximal, complemented
2C22×C3×#1113C214yesmaximal, complemented
3C2×#10620215C2124yesmaximal, direct product factor
4C22×D33—34nomaximal
5C22×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
6C22×C32×C3⋊D3C2216yescomplemented
7C2×C3⋊D32C22136yescomplemented
8C2×C3×#1113C22148yescomplemented
9#10620215C22164yesdirect product factor
10C22×#1113D314yescomplemented
11C22×D3×C3⋊D3—312no—
12C22×D3×C3⋊D3—66no—
13C22×C3×D32—312no—
14C2×D33—396no—
15C22×C34C2311yescharacteristic, complemented
16C2×C3⋊(C3⋊(C3⋊D3))C2316yescomplemented
17C2×C32×C3⋊D3C23136yescomplemented
18C3⋊D32C23148yescomplemented
19C3×#1113C23164yescomplemented
20C23×D32—96no—
21C22×C3⋊(C3⋊D3)—34no—
22C22×C3⋊(C3⋊D3)—66no—
23C22×C3⋊(C3⋊D3)—124no—
24C22×C3⋊(C3⋊D3)—122no—
25C22×C3×C3⋊D3—26no—
26C22×C3×C3⋊D3C2×D3112yescomplemented
27C22×C32×D3—312no—
28C22×C32×D3—66no—
29C2×#1113C2×D3148yescomplemented
30C2×D3×C3⋊D3—3144no—
31C2×D3×C3⋊D3—672no—
32C2×C3×D32—3144no—
33D33—3256no—
34C2×C34C2413yescomplemented
35C3⋊(C3⋊(C3⋊D3))C2414yescomplemented
36C32×C3⋊D3C24124yescomplemented
37C23×C3⋊D3—96no—
38C23×C3×D3—912no—
39C22×D32—312no—
40C22×D32—9144no—
41C22×D32—912no—
42C22×D32—1812no—
43C22×D32—186no—
44C22×C33—26no—
45C22×C33—42no—
46C22×C33—44no—
47C22×C33C22×D314yescomplemented
48C2×C3⋊(C3⋊D3)—324no—
49C2×C3⋊(C3⋊D3)—636no—
50C2×C3⋊(C3⋊D3)—1224no—
51C2×C3⋊(C3⋊D3)—1212no—
52C2×C3×C3⋊D3—236no—
53C2×C3×C3⋊D3C22×D3172yescomplemented
54C2×C32×D3—372no—
55C2×C32×D3—636no—
56#1113C22×D3164yescomplemented
57D3×C3⋊D3—3192no—
58D3×C3⋊D3—696no—
59C3×D32—3192no—
60C24×D3—274no—
61C34C2511yescharacteristic, complemented, Sylow
62C23×C32—96no—
63C22×C3⋊D3—96no—
64C22×C3⋊D3—972no—
65C22×C3⋊D3—1812no—
66C22×C3⋊D3—186no—
67C22×C3⋊D3—364no—
68C22×C3⋊D3—364no—
69C22×C3⋊D3—3612no—
70C22×C3⋊D3—362no—
71C22×C3⋊D3D3216yescomplemented
72C22×C3×D3—324no—
73C22×C3×D3—612no—
74C22×C3×D3—612no—
75C22×C3×D3—612no—
76C22×C3×D3—9144no—
77C2×D32—3144no—
78C2×D32—9384no—
79C2×D32—9144no—
80C2×D32—18144no—
81C2×D32—1872no—
82C2×C33—218no—
83C2×C33—46no—
84C2×C33—412no—
85C2×C33C23×D3112yescomplemented
86C3⋊(C3⋊D3)—316no—
87C3⋊(C3⋊D3)—624no—
88C3⋊(C3⋊D3)—1216no—
89C3⋊(C3⋊D3)—128no—
90C3×C3⋊D3—224no—
91C3×C3⋊D3C23×D3148yescomplemented
92C32×D3—348no—
93C32×D3—624no—
94C24×C3—274no—
95C23×D3—912no—
96C23×D3—2712no—
97C23×D3—2796no—
98C23×D3—546no—
99C22×C32—26no—
100C22×C32—212no—
101C22×C32—44no—
102C22×C32—412no—
103C22×C32—42no—
104C22×C32—44no—
105C22×C32—936no—
106C22×C32C2×D3216yescomplemented
107C2×C3⋊D3—936no—
108C2×C3⋊D3—996no—
109C2×C3⋊D3—1872no—
110C2×C3⋊D3—1836no—
111C2×C3⋊D3—3624no—
112C2×C3⋊D3—3624no—
113C2×C3⋊D3—3672no—
114C2×C3⋊D3—3612no—
115C2×C3⋊D3C2×D32136yescomplemented
116C2×C3×D3—3144no—
117C2×C3×D3—672no—
118C2×C3×D3—672no—
119C2×C3×D3—672no—
120C2×C3×D3—9192no—
121D32—3192no—
122D32—9192no—
123D32—18192no—
124D32—1896no—
125C25—811noSylow
126C33—26no—
127C33—42no—
128C33—44no—
129C33C24×D314yescomplemented
130C23×C3—912no—
131C23×C3—186no—
132C23×C3—2748no—
133C22×D3—312no—
134C22×D3—66no—
135C22×D3—9144no—
136C22×D3—27144no—
137C22×D3—27256no—
138C22×D3—274no—
139C22×D3—5472no—
140C22×D3—546no—
141C22×D3—1084no—
142C22×D3—1082no—
143C2×C32—218no—
144C2×C32—236no—
145C2×C32—412no—
146C2×C32—436no—
147C2×C32—46no—
148C2×C32—412no—
149C2×C32—924no—
150C2×C32C22×D32118yescomplemented
151C3⋊D3—924no—
152C3⋊D3—1848no—
153C3⋊D3—1824no—
154C3⋊D3—3616no—
155C3⋊D3—3616no—
156C3⋊D3—3648no—
157C3⋊D3—368no—
158C3⋊D3C22×D32124yescomplemented
159C3×D3—396no—
160C3×D3—648no—
161C3×D3—648no—
162C3×D3—648no—
163C24—274no—
164C24—813no—
165C24—8124no—
166C22×C3—26no—
167C22×C3—42no—
168C22×C3—44no—
169C22×C3—972no—
170C22×C3—1836no—
171C22×C3—2764no—
172C22×C3D3314yescomplemented
173C2×D3—372no—
174C2×D3—636no—
175C2×D3—9192no—
176C2×D3—27192no—
177C2×D3—2724no—
178C2×D3—5496no—
179C2×D3—5436no—
180C2×D3—10824no—
181C2×D3—10812no—
182C32—26no—
183C32—212no—
184C32—44no—
185C32—412no—
186C32—42no—
187C32—44no—
188C32C23×D3216yescomplemented
189C23—96no—
190C23—2748no—
191C23—8136no—
192C23—8164no—
193C23—811no—
194C2×C3—218no—
195C2×C3—46no—
196C2×C3—412no—
197C2×C3—948no—
198C2×C3—1824no—
199C2×C3C2×D33112yescomplemented
200D3—348no—
201D3—624no—
202D3—2716no—
203D3—5424no—
204D3—10816no—
205D3—1088no—
206C22—936no—
207C22—2764no—
208C22—8148no—
209C22—816no—
210C22#1062021511yescharacteristic, direct product factor
211C3—26no—
212C3—42no—
213C3—44no—
214C3C22×D3314yescomplemented
215C2—924no—
216C2—814no—
217C2C2×#1062021513yesdirect product factor
subgroupG/H
center, hypercenterC22#10620215
commutator, cosocle, hypocenterC34C25
FrattiniC1G
Fitting, socleC22×C34C23
LCSG ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊D32C2semidirect
2. C22×C3×#1113C2semidirect
3. C2×#10620215C2direct (N × H)
5. C22×C3⋊(C3⋊(C3⋊D3))C22semidirect
6. C22×C32×C3⋊D3C22semidirect
7. C2×C3⋊D32C22semidirect
8. C2×C3×#1113C22semidirect
9. #10620215C22direct (N × H)
10. C22×#1113D3semidirect
15. C22×C34C23semidirect
16. C2×C3⋊(C3⋊(C3⋊D3))C23semidirect
17. C2×C32×C3⋊D3C23semidirect
18. C3⋊D32C23semidirect
19. C3×#1113C23semidirect
26. C22×C3×C3⋊D3C2×D3semidirect
29. C2×#1113C2×D3semidirect
34. C2×C34C24semidirect
35. C3⋊(C3⋊(C3⋊D3))C24semidirect
36. C32×C3⋊D3C24semidirect
47. C22×C33C22×D3semidirect
53. C2×C3×C3⋊D3C22×D3semidirect
56. #1113C22×D3semidirect
61. C34C25semidirect
71. C22×C3⋊D3D32semidirect
85. C2×C33C23×D3semidirect
91. C3×C3⋊D3C23×D3semidirect
106. C22×C32C2×D32semidirect
115. C2×C3⋊D3C2×D32semidirect
129. C33C24×D3semidirect
150. C2×C32C22×D32semidirect
158. C3⋊D3C22×D32semidirect
172. C22×C3D33semidirect
188. C32C23×D32semidirect
199. C2×C3C2×D33semidirect
210. C22#10620215direct (N × H)
214. C3C22×D33semidirect
217. C2C2×#10620215direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊D322
C22×C3×#11132
C2×#106202152
C22×D333