Finite Group Databasebeta

← C22×#88790 #13176877

Subgroups — C22×#88790 #13176877

Order 1728.

2972 conjugacy classes of subgroups; only the 452 normal subgroups are shown.

normal subgroups only

This group has 2972 conjugacy classes of subgroups — too many to draw. Only its 452 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×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C22×C32×D12C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#88790C2112yesmaximal, direct product factor
5C22×#3753—31nomaximal
6C22×#3714—34nomaximal
7C23×C3⋊(C3⋊D3)—21no—
8C23×C32×D3—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C4⋊(C3⋊(C3⋊D3))—23no—
11C2×C4⋊(C3⋊(C3⋊D3))C2216yescomplemented
12C2×C32×D12—23no—
13C2×C32×D12C2216yescomplemented
14C2×C3×C3⋊(C3⋊C8)C2216yescomplemented
15#88790C22116yesdirect product factor
16C22×C32×D4—31no—
17C22×C4⋊(C3⋊D3)—31no—
18C22×C4⋊(C3⋊D3)—34no—
19C22×C4⋊(C3⋊D3)—64no—
20C22×C3×D12D314yescomplemented
21C2×#3753—312no—
22C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
23C22×C3×C3⋊C8—34no—
24C2×#3714—348no—
25C23×C33D411yescharacteristic
26C22×C3⋊(C3⋊D3)—26no—
27C22×C3⋊(C3⋊D3)—44no—
28C22×C32×D3—26no—
29C22×C32×D3—44no—
30C2×C33×C4D413yes—
31C2×C33×C4C2313yes—
32C4⋊(C3⋊(C3⋊D3))—26no—
33C4⋊(C3⋊(C3⋊D3))C2314yescomplemented
34C32×D12—26no—
35C32×D12C2314yescomplemented
36C3×C3⋊(C3⋊C8)C2314yescomplemented
37C22×#213—94no—
38C22×D24—91no—
39C24×C32—61no—
40C23×C3⋊D3—68no—
41C23×C3⋊D3—61no—
42C23×C3⋊D3—64no—
43C23×C3×D3—24no—
44C2×C32×D4—36no—
45C2×C32×D4—63no—
46C22×C32×C4—24no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D314yes—
49C2×C4⋊(C3⋊D3)—36no—
50C2×C4⋊(C3⋊D3)—324no—
51C2×C4⋊(C3⋊D3)—624no—
52C2×C4⋊(C3⋊D3)—624no—
53C2×C4⋊(C3⋊D3)—63no—
54C2×C4⋊(C3⋊D3)—612no—
55C2×C3×D12—212no—
56C2×C3×D12C2×D3124yescomplemented
57#3753—316no—
58C2×C3⋊(C3⋊C8)C2×D316yescomplemented
59C2×C3×C3⋊C8—324no—
60#3714—364no—
61C22×C33D814yescomplemented
62C22×C33C2×D413yes—
63C2×C3⋊(C3⋊D3)—24no—
64C2×C3⋊(C3⋊D3)—412no—
65C2×C32×D3—24no—
66C2×C32×D3—412no—
67C33×C4C2×D413yes—
68C33×C4C2411yescharacteristic
69C22×C3×D4—34no—
70C22×D12—94no—
71C22×D12—91no—
72C22×D12—184no—
73C22×D12C3⋊D311yescharacteristic, complemented
74C22×C3×C8—91no—
75C2×#213—948no—
76C22×C3⋊C8—34no—
77C2×D24—912no—
78C23×C32—24no—
79C23×C32—66no—
80C23×C32—124no—
81C23×C32D1211yescharacteristic
82C23×C32#6714yes—
83C22×C3⋊D3—648no—
84C22×C3⋊D3—66no—
85C22×C3⋊D3—624no—
86C22×C3⋊D3—1232no—
87C22×C3⋊D3—124no—
88C22×C3⋊D3—1216no—
89C22×C3×D3—224no—
90C22×C3×D3—416no—
91C32×D4—34no—
92C32×D4—66no—
93C2×C32×C4—212no—
94C2×C32×C4—212no—
95C2×C32×C4D1213yes—
96C2×C32×C4#67112yes—
97C2×C32×C4C22×D3112yes—
98C2×C32×C4C22×D313yes—
99C4⋊(C3⋊D3)—34no—
100C4⋊(C3⋊D3)—316no—
101C4⋊(C3⋊D3)—616no—
102C4⋊(C3⋊D3)—648no—
103C4⋊(C3⋊D3)—66no—
104C4⋊(C3⋊D3)—624no—
105C3×D12—224no—
106C3×D12C22×D3116yescomplemented
107C3⋊(C3⋊C8)C22×D314yescomplemented
108C3×C3⋊C8—316no—
109C22×D8—271noSylow
110C2×C33C2×D816yescomplemented
111C2×C33C22×D411yescharacteristic
112C3⋊(C3⋊D3)—44no—
113C32×D3—44no—
114C24×C3—64no—
115C23×D3—21no—
116C23×D3—188no—
117C23×D3—184no—
118C23×D3—181no—
119C2×C3×D4—324no—
120C2×C3×D4—612no—
121C22×C3×C4—24no—
122C22×C3×C4D3214yes—
123C22×C3×C4C2×C3⋊D311yescharacteristic
124C2×D12—23no—
125C2×D12—924no—
126C2×D12—96no—
127C2×D12—1824no—
128C2×D12—1824no—
129C2×D12—1812no—
130C2×D12—183no—
131C2×D12C2×C3⋊D316yescomplemented
132C2×C3×C8—96no—
133#213—964no—
134C2×C3⋊C8—324no—
135D24—916no—
136C22×C32—216no—
137C22×C32—212no—
138C22×C32—64no—
139C22×C32—1212no—
140C22×C32D2414yescomplemented
141C22×C32#213116yescomplemented
142C22×C32C2×D1213yes—
143C22×C32C2×#67112yes—
144C2×C3⋊D3—632no—
145C2×C3⋊D3—64no—
146C2×C3⋊D3—616no—
147C2×C3⋊D3—1296no—
148C2×C3⋊D3—1212no—
149C2×C3⋊D3—1248no—
150C2×C3×D3—216no—
151C2×C3×D3—448no—
152C32×C4—24no—
153C32×C4—212no—
154C32×C4C2×D1213yes—
155C32×C4C2×#67112yes—
156C32×C4C23×D314yes—
157C32×C4C23×D311yescharacteristic
158C22×D4—31no—
159C22×D4—271no—
160C2×D8—2712no—
161C22×C8—91no—
162C33C22×D811yescharacteristic, complemented, Sylow
163C23×C3—24no—
164C23×C3—624no—
165C23×C3—1216no—
166C23×C3#62614yes—
167C23×C3#63811yescharacteristic
168C22×D3—26no—
169C22×D3—44no—
170C22×D3—1848no—
171C22×D3—1824no—
172C22×D3—186no—
173C22×D3—3632no—
174C22×D3—3616no—
175C22×D3—364no—
176C3×D4—316no—
177C3×D4—624no—
178C2×C3×C4—212no—
179C2×C3×C4—212no—
180C2×C3×C4#626112yes—
181C2×C3×C4#63813yes—
182C2×C3×C4C2×D32112yes—
183C2×C3×C4C22×C3⋊D313yes—
184D12—26no—
185D12—916no—
186D12—94no—
187D12—1816no—
188D12—1848no—
189D12—1824no—
190D12—186no—
191D12C22×C3⋊D314yescomplemented
192C3×C8—94no—
193C3⋊C8—316no—
194C2×C32—24no—
195C2×C32—224no—
196C2×C32—124no—
197C2×C32C2×D2416yescomplemented
198C2×C32C2×#213124yescomplemented
199C2×C32C22×D1211yescharacteristic
200C2×C32C22×#6714yes—
201C3⋊D3—1232no—
202C3⋊D3—124no—
203C3⋊D3—1216no—
204C3×D3—416no—
205C24—61no—
206C24—541no—
207C2×D4—36no—
208C2×D4—63no—
209C2×D4—276no—
210C2×D4—543no—
211C22×C4D3×C3⋊D311yescharacteristic
212D8—2716no—
213C2×C8—96no—
214C22×C3—212no—
215C22×C3—216no—
216C22×C3—616no—
217C22×C3—1248no—
218C22×C3#3714116yescomplemented
219C22×C3#375314yescomplemented
220C22×C3C2×#626112yes—
221C22×C3C2×#63813yes—
222C2×D3—24no—
223C2×D3—412no—
224C2×D3—1832no—
225C2×D3—1816no—
226C2×D3—184no—
227C2×D3—3696no—
228C2×D3—3648no—
229C2×D3—3612no—
230C3×C4—24no—
231C3×C4—212no—
232C3×C4C2×#626112yes—
233C3×C4C2×#63813yes—
234C3×C4C22×D3214yes—
235C3×C4C23×C3⋊D311yescharacteristic
236C32—24no—
237C32C22×D2411yescharacteristic, complemented
238C32C22×#21314yescomplemented
239C23—66no—
240C23—124no—
241C23—546no—
242C23—1084no—
243C23#629111yescharacteristic
244D4—34no—
245D4—66no—
246D4—274no—
247D4—546no—
248C2×C4#629113yes—
249C2×C4C2×D3×C3⋊D313yes—
250C8—94no—
251C2×C3—24no—
252C2×C3—224no—
253C2×C3—1216no—
254C2×C3C2×#3714124yescomplemented
255C2×C3C2×#375316yescomplemented
256C2×C3C22×#62614yes—
257C2×C3C22×#63811yescharacteristic
258D3—44no—
259D3—3632no—
260D3—364no—
261D3—3616no—
262C22—64no—
263C22—1212no—
264C22—544no—
265C22—10812no—
266C22#8879014yesdirect product factor
267C22C2×#629113yes—
268C4C2×#629113yes—
269C4C22×D3×C3⋊D311yescharacteristic
270C3—24no—
271C3C22×#371414yescomplemented
272C3C22×#375311yescharacteristic, complemented
273C2—124no—
274C2—1084no—
275C2C2×#8879016yesdirect product factor
276C2C22×#629111yescharacteristic
subgroupG/H
centerC23#6291
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C22×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊(C3⋊(C3⋊D3))C2semidirect
2. C22×C32×D12C2semidirect
3. C22×C3×C3⋊(C3⋊C8)C2semidirect
4. C2×#88790C2direct (N × H)
11. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
13. C2×C32×D12C22semidirect
14. C2×C3×C3⋊(C3⋊C8)C22semidirect
15. #88790C22direct (N × H)
20. C22×C3×D12D3semidirect
22. C22×C3⋊(C3⋊C8)D3semidirect
33. C4⋊(C3⋊(C3⋊D3))C23semidirect
35. C32×D12C23semidirect
36. C3×C3⋊(C3⋊C8)C23semidirect
56. C2×C3×D12C2×D3semidirect
58. C2×C3⋊(C3⋊C8)C2×D3semidirect
61. C22×C33D8semidirect
73. C22×D12C3⋊D3semidirect
106. C3×D12C22×D3semidirect
107. C3⋊(C3⋊C8)C22×D3semidirect
110. C2×C33C2×D8semidirect
131. C2×D12C2×C3⋊D3semidirect
140. C22×C32D24semidirect
141. C22×C32#213semidirect
162. C33C22×D8semidirect
191. D12C22×C3⋊D3semidirect
197. C2×C32C2×D24semidirect
198. C2×C32C2×#213semidirect
218. C22×C3#3714semidirect
219. C22×C3#3753semidirect
237. C32C22×D24semidirect
238. C32C22×#213semidirect
254. C2×C3C2×#3714semidirect
255. C2×C3C2×#3753semidirect
266. C22#88790direct (N × H)
271. C3C22×#3714semidirect
272. C3C22×#3753semidirect
275. C2C2×#88790direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊(C3⋊(C3⋊D3))2
C22×C32×D122
C22×C3×C3⋊(C3⋊C8)2
C2×#887902
C22×#37533
C22×#37143