Finite Group Databasebeta

← C22×#88810 #13178510

Subgroups — C22×#88810 #13178510

Order 1728.

1954 conjugacy classes of subgroups; only the 311 normal subgroups are shown.

normal subgroups only

This group has 1954 conjugacy classes of subgroups — too many to draw. Only its 311 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×C4⋊(C3⋊D3)C212yesmaximal, complemented
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C2×#88810C2112yesmaximal, direct product factor
4C22×#3714—32nomaximal
5C22×#3713—31nomaximal
6C23×C3×C3⋊D3—22no—
7C22×C33×C4C2211yescharacteristic
8C2×C3×C4⋊(C3⋊D3)—26no—
9C2×C3×C4⋊(C3⋊D3)C22112yescomplemented
10C2×C3×C3⋊(C3⋊C8)C2216yescomplemented
11#88810C22116yesdirect product factor
12C22×C4⋊(C3⋊D3)D312yescomplemented
13C22×C3×D12—32no—
14C22×C3×D12—32no—
15C22×C3×D12—62no—
16C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
17C22×C3×C3⋊C8—32no—
18C22×C3×C3⋊C8—61no—
19C2×#3714—324no—
20C2×#3713—312no—
21C23×C33D411yescharacteristic
22C22×C3×C3⋊D3—212no—
23C22×C3×C3⋊D3—48no—
24C2×C33×C4D413yes—
25C2×C33×C4C2313yes—
26C3×C4⋊(C3⋊D3)—212no—
27C3×C4⋊(C3⋊D3)C2318yescomplemented
28C3×C3⋊(C3⋊C8)C2314yescomplemented
29C22×#213—92no—
30C22×D24—91no—
31C23×C3⋊D3—22no—
32C23×C3×D3—64no—
33C23×C3×D3—62no—
34C23×C3×D3—62no—
35C22×C32×C4—21no—
36C22×C32×C4—22no—
37C22×C32×C4—41no—
38C22×C32×C4C2×D311yescharacteristic
39C22×C32×C4C2×D312yes—
40C2×C4⋊(C3⋊D3)—26no—
41C2×C4⋊(C3⋊D3)C2×D3112yescomplemented
42C2×C3×D12—312no—
43C2×C3×D12—312no—
44C2×C3×D12—612no—
45C2×C3×D12—612no—
46C2×C3×D12—66no—
47C2×C3×D12—66no—
48C2×C3⋊(C3⋊C8)C2×D316yescomplemented
49C2×C3×C3⋊C8—312no—
50C2×C3×C3⋊C8—66no—
51#3714—332no—
52#3713—316no—
53C22×C33D814yescomplemented
54C22×C33C2×D413yes—
55C2×C3×C3⋊D3—28no—
56C2×C3×C3⋊D3—424no—
57C33×C4C2×D413yes—
58C33×C4C2411yescharacteristic
59C22×C3×D4—92no—
60C22×D12—32no—
61C22×D12—32no—
62C22×D12—62no—
63C22×C3×C8—91no—
64C2×#213—924no—
65C22×C3⋊C8—32no—
66C22×C3⋊C8—61no—
67C2×D24—912no—
68C23×C32—21no—
69C23×C32—22no—
70C23×C32—41no—
71C23×C32D1211yescharacteristic
72C23×C32#6712yes—
73C22×C3⋊D3—212no—
74C22×C3⋊D3—48no—
75C22×C3×D3—624no—
76C22×C3×D3—612no—
77C22×C3×D3—612no—
78C22×C3×D3—1216no—
79C22×C3×D3—128no—
80C22×C3×D3—128no—
81C2×C32×C4—23no—
82C2×C32×C4—26no—
83C2×C32×C4—23no—
84C2×C32×C4—26no—
85C2×C32×C4—43no—
86C2×C32×C4—43no—
87C2×C32×C4D1213yes—
88C2×C32×C4#6716yes—
89C2×C32×C4C22×D313yes—
90C2×C32×C4C22×D316yes—
91C4⋊(C3⋊D3)—212no—
92C4⋊(C3⋊D3)C22×D318yescomplemented
93C3×D12—38no—
94C3×D12—38no—
95C3×D12—68no—
96C3×D12—624no—
97C3×D12—612no—
98C3×D12—612no—
99C3⋊(C3⋊C8)C22×D314yescomplemented
100C3×C3⋊C8—38no—
101C3×C3⋊C8—64no—
102C22×D8—271noSylow
103C2×C33C2×D816yescomplemented
104C2×C33C22×D411yescharacteristic
105C3×C3⋊D3—48no—
106C24×C3—182no—
107C23×D3—64no—
108C23×D3—62no—
109C23×D3—62no—
110C2×C3×D4—912no—
111C2×C3×D4—186no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—41no—
115C22×C3×C4D3211yescharacteristic
116C22×C3×C4D3212yes—
117C2×D12—312no—
118C2×D12—312no—
119C2×D12—612no—
120C2×D12—612no—
121C2×D12—66no—
122C2×D12—66no—
123C2×C3×C8—96no—
124#213—932no—
125C2×C3⋊C8—312no—
126C2×C3⋊C8—66no—
127D24—916no—
128C22×C32—24no—
129C22×C32—28no—
130C22×C32—23no—
131C22×C32—26no—
132C22×C32—44no—
133C22×C32—43no—
134C22×C32D2414yescomplemented
135C22×C32#21318yescomplemented
136C22×C32C2×D1213yes—
137C22×C32C2×#6716yes—
138C2×C3⋊D3—28no—
139C2×C3⋊D3—424no—
140C2×C3×D3—616no—
141C2×C3×D3—68no—
142C2×C3×D3—68no—
143C2×C3×D3—1248no—
144C2×C3×D3—1224no—
145C2×C3×D3—1224no—
146C32×C4—21no—
147C32×C4—22no—
148C32×C4—23no—
149C32×C4—26no—
150C32×C4—43no—
151C32×C4—41no—
152C32×C4C2×D1213yes—
153C32×C4C2×#6716yes—
154C32×C4C23×D311yescharacteristic
155C32×C4C23×D312yes—
156C22×D4—92no—
157C2×D8—2712no—
158C22×C8—91no—
159C33C22×D811yescharacteristic, complemented, Sylow
160C23×C3—21no—
161C23×C3—22no—
162C23×C3—41no—
163C23×C3—1812no—
164C23×C3—368no—
165C23×C3#62511yescharacteristic
166C23×C3#62612yes—
167C22×D3—624no—
168C22×D3—612no—
169C22×D3—612no—
170C22×D3—1216no—
171C22×D3—128no—
172C22×D3—128no—
173C3×D4—98no—
174C3×D4—1812no—
175C2×C3×C4—23no—
176C2×C3×C4—26no—
177C2×C3×C4—23no—
178C2×C3×C4—26no—
179C2×C3×C4—43no—
180C2×C3×C4—43no—
181C2×C3×C4#62513yes—
182C2×C3×C4#62616yes—
183C2×C3×C4C2×D3213yes—
184C2×C3×C4C2×D3216yes—
185D12—38no—
186D12—38no—
187D12—68no—
188D12—624no—
189D12—612no—
190D12—612no—
191C3×C8—94no—
192C3⋊C8—38no—
193C3⋊C8—64no—
194C2×C32—21no—
195C2×C32—22no—
196C2×C32—26no—
197C2×C32—212no—
198C2×C32—46no—
199C2×C32—41no—
200C2×C32C2×D2416yescomplemented
201C2×C32C2×#213112yescomplemented
202C2×C32C22×D1211yescharacteristic
203C2×C32C22×#6712yes—
204C3⋊D3—48no—
205C3×D3—1216no—
206C3×D3—128no—
207C3×D3—128no—
208C24—182no—
209C2×D4—912no—
210C2×D4—186no—
211C22×C4#111311yescharacteristic
212D8—2716no—
213C2×C8—96no—
214C22×C3—26no—
215C22×C3—23no—
216C22×C3—28no—
217C22×C3—24no—
218C22×C3—43no—
219C22×C3—44no—
220C22×C3—188no—
221C22×C3—3624no—
222C22×C3#371314yescomplemented
223C22×C3#371418yescomplemented
224C22×C3C2×#62513yes—
225C22×C3C2×#62616yes—
226C2×D3—616no—
227C2×D3—68no—
228C2×D3—68no—
229C2×D3—1248no—
230C2×D3—1224no—
231C2×D3—1224no—
232C3×C4—21no—
233C3×C4—22no—
234C3×C4—26no—
235C3×C4—23no—
236C3×C4—41no—
237C3×C4—43no—
238C3×C4C2×#62513yes—
239C3×C4C2×#62616yes—
240C3×C4C22×D3211yescharacteristic
241C3×C4C22×D3212yes—
242C32—21no—
243C32—22no—
244C32—41no—
245C32C22×D2411yescharacteristic, complemented
246C32C22×#21312yescomplemented
247C23—1812no—
248C23—368no—
249C23#629511yescharacteristic
250D4—98no—
251D4—1812no—
252C2×C4#629513yes—
253C2×C4C2×#111313yes—
254C8—94no—
255C2×C3—22no—
256C2×C3—21no—
257C2×C3—212no—
258C2×C3—26no—
259C2×C3—41no—
260C2×C3—46no—
261C2×C3—368no—
262C2×C3C2×#371316yescomplemented
263C2×C3C2×#3714112yescomplemented
264C2×C3C22×#62511yescharacteristic
265C2×C3C22×#62612yes—
266D3—1216no—
267D3—128no—
268D3—128no—
269C22—188no—
270C22—3624no—
271C22#8881014yesdirect product factor
272C22C2×#629513yes—
273C4C2×#629513yes—
274C4C22×#111311yescharacteristic
275C3—22no—
276C3—21no—
277C3—41no—
278C3C22×#371311yescharacteristic, complemented
279C3C22×#371412yescomplemented
280C2—368no—
281C2C2×#8881016yesdirect product factor
282C2C22×#629511yescharacteristic
subgroupG/H
centerC23#6295
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC22×C33×C4C22
socleC23×C33D4
hypercenterC22×C4#1113
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×C3×C4⋊(C3⋊D3)C2semidirect
2. C22×C3×C3⋊(C3⋊C8)C2semidirect
3. C2×#88810C2direct (N × H)
9. C2×C3×C4⋊(C3⋊D3)C22semidirect
10. C2×C3×C3⋊(C3⋊C8)C22semidirect
11. #88810C22direct (N × H)
12. C22×C4⋊(C3⋊D3)D3semidirect
16. C22×C3⋊(C3⋊C8)D3semidirect
27. C3×C4⋊(C3⋊D3)C23semidirect
28. C3×C3⋊(C3⋊C8)C23semidirect
41. C2×C4⋊(C3⋊D3)C2×D3semidirect
48. C2×C3⋊(C3⋊C8)C2×D3semidirect
53. C22×C33D8semidirect
92. C4⋊(C3⋊D3)C22×D3semidirect
99. C3⋊(C3⋊C8)C22×D3semidirect
103. C2×C33C2×D8semidirect
134. C22×C32D24semidirect
135. C22×C32#213semidirect
159. C33C22×D8semidirect
200. C2×C32C2×D24semidirect
201. C2×C32C2×#213semidirect
222. C22×C3#3713semidirect
223. C22×C3#3714semidirect
245. C32C22×D24semidirect
246. C32C22×#213semidirect
262. C2×C3C2×#3713semidirect
263. C2×C3C2×#3714semidirect
271. C22#88810direct (N × H)
278. C3C22×#3713semidirect
279. C3C22×#3714semidirect
281. C2C2×#88810direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C22×C3×C3⋊(C3⋊C8)2
C2×#888102
C22×#37143
C22×#37133