Finite Group Databasebeta

← C32×#4889 #13162325

Subgroups — C32×#4889 #13162325

Order 1728.

864 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#204C211yesmaximal, characteristic
2C32×#893C211yesmaximal, characteristic
3C32×#887C211yesmaximal, characteristic, complemented
4C33×#107C211yesmaximal, characteristic, complemented
5C32×#815C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7C32×#809C211yesmaximal, characteristic
8C32×#440—31nomaximal
9C3×#4889C314yesmaximal, direct product factor
10C2×C32×D12C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C32×#204C414yescomplemented
22C2×C32×SD16—31no—
23C32×C4×Q8—31no—
24C32×C4×D4—31no—
25C2×C3×#204C2×C314yes—
26C3×#893C2×C314yes—
27C3×#887C2×C314yescomplemented
28C32×#107—24no—
29C32×#107D311yescharacteristic, complemented
30C32×#107C2×C314yescomplemented
31C32×#103—31no—
32C32×#102—31no—
33C32×#97—31no—
34C3×#815C2×C314yescomplemented
35C3×#810C2×C314yescomplemented
36C3×#809C2×C314yes—
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C32×Dic3—21no—
42C2×C32×Dic3D411yescharacteristic
43C2×C32×Dic3D411yescharacteristic
44C32×D12—21no—
45C32×D12C2×C412yes—
46C32×C4×D3—42no—
47C32×Dic6—21no—
48C32×Dic6C2×C412yescomplemented
49C33×C8C2×C412yescomplemented
50C32×C3⋊C8—21no—
51C3×#440—34no—
52#4889C3211yescharacteristic, direct product factor
53C2×C32×Q8—31no—
54C2×C32×D4—31no—
55C22×C32×C4—61no—
56C2×C3×D12C22×C314yes—
57C2×C3×C4×D3—24no—
58C2×C3×Dic6C22×C314yes—
59C32×SD16—34no—
60C2×C32×C8—24no—
61C2×C32×C8—31no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C22×C314yes—
64C32×C4⋊C4—24no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4—61no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C4⋊C4C22×C314yes—
70C32×C4⋊C4C22×C314yes—
71C32×C22⋊C4—61no—
72C32×C42—31no—
73C32×C42—61no—
74C3×(C2×D3)⋊C4—24no—
75C3×Dic3⋊C4—24no—
76C3×C4×Dic3—24no—
77C3×C4×Dic3C22×C314yes—
78C2×C3×C3⋊C8C22×C314yes—
79C3×#204C3×C4116yescomplemented
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—41no—
85C33×C4—41no—
86C33×C4C22×C411yescharacteristic
87C33×C4#4111yescharacteristic
88C32×Dic3—22no—
89C32×Dic3—21no—
90C32×Dic3—21no—
91C2×C3×SD16—34no—
92C3×C4×Q8—34no—
93C3×C4×D4—34no—
94C2×#204C2×C3211yescharacteristic
95#893C2×C3211yescharacteristic
96#887C2×C3211yescharacteristic, complemented
97C3×#107—24no—
98C3×#107C3×D314yescomplemented
99C3×#107C2×C3211yescharacteristic, complemented
100C3×#103—34no—
101C3×#102—34no—
102C3×#97—34no—
103#815C2×C3211yescharacteristic, complemented
104#810C2×C3211yescharacteristic, complemented
105#809C2×C3211yescharacteristic
106C23×C32—61no—
107C22×C3×D3—24no—
108C32×Q8—32no—
109C32×Q8—61no—
110C32×D4—32no—
111C32×D4—61no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—31no—
118C2×C32×C4—31no—
119C2×C32×C4—44no—
120C2×C32×C4—44no—
121C2×C32×C4—61no—
122C2×C32×C4—122no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C314yes—
125C2×C3×Dic3—24no—
126C2×C3×Dic3C3×D414yes—
127C2×C3×Dic3C3×D414yes—
128C3×D12—24no—
129C3×D12C2×C3×C418yes—
130C3×C4×D3—48no—
131C3×Dic6—24no—
132C3×Dic6C2×C3×C418yescomplemented
133C32×C8—28no—
134C32×C8—61no—
135C32×C8C4×D312yescomplemented
136C32×C8C2×C3×C418yescomplemented
137C3×C3⋊C8—24no—
138#440—31noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C2×C33#13711yescharacteristic
142C32×D3—42no—
143C2×C3×Q8—34no—
144C2×C3×D4—34no—
145C22×C3×C4—64no—
146C2×D12C22×C3211yescharacteristic
147C2×C4×D3—21no—
148C2×Dic6C22×C3211yescharacteristic
149C3×SD16—316no—
150C2×C3×C8—24no—
151C2×C3×C8—34no—
152C2×C3×C8C2×C3×D314yes—
153C2×C3×C8C22×C3211yescharacteristic
154C3×C4⋊C4—24no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—64no—
157C3×C4⋊C4C2×C3×D314yes—
158C3×C4⋊C4C2×C3×D314yes—
159C3×C4⋊C4C22×C3211yescharacteristic
160C3×C4⋊C4C22×C3211yescharacteristic
161C3×C22⋊C4—64no—
162C3×C42—34no—
163C3×C42—64no—
164(C2×D3)⋊C4—21no—
165Dic3⋊C4—21no—
166C4×Dic3—21no—
167C4×Dic3C22×C3211yescharacteristic
168C2×C3⋊C8C22×C3211yescharacteristic
169#204C32×C414yescomplemented
170C22×C32—24no—
171C22×C32—62no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32D3×D411yescharacteristic
175C22×C32C2×C3×D414yes—
176C2×C3×D3—28no—
177C2×C3×D3—44no—
178C2×C3×D3—44no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—41no—
182C32×C4—41no—
183C32×C4—44no—
184C32×C4—44no—
185C32×C4—62no—
186C32×C4—61no—
187C32×C4—61no—
188C32×C4—84no—
189C32×C4—84no—
190C32×C4C2×C4×D311yescharacteristic
191C32×C4#23911yescharacteristic
192C32×C4C22×C3×C414yes—
193C32×C4C3×#4114yes—
194C3×Dic3—28no—
195C3×Dic3—24no—
196C3×Dic3—24no—
197C2×SD16—31no—
198C4×Q8—31no—
199C4×D4—31no—
200#107C32×D311yescharacteristic, complemented
201#103—31no—
202#102—31no—
203#97—31no—
204C33#44011yescharacteristic, complemented, Sylow
205C23×C3—64no—
206C22×D3—21no—
207C3×Q8—38no—
208C3×Q8—64no—
209C3×D4—38no—
210C3×D4—64no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—34no—
217C2×C3×C4—34no—
218C2×C3×C4—44no—
219C2×C3×C4—44no—
220C2×C3×C4—64no—
221C2×C3×C4—128no—
222C2×C3×C4C22×C3×D314yes—
223C2×C3×C4C23×C3211yescharacteristic
224C2×Dic3—21no—
225C2×Dic3C32×D411yescharacteristic
226C2×Dic3C32×D411yescharacteristic
227D12—21no—
228D12C2×C32×C412yes—
229C4×D3—42no—
230Dic6—21no—
231Dic6C2×C32×C412yescomplemented
232C3×C8—28no—
233C3×C8—64no—
234C3×C8C3×C4×D318yescomplemented
235C3×C8C2×C32×C412yescomplemented
236C3⋊C8—21no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—122no—
241C2×C32#89311yescharacteristic
242C2×C32#91111yescharacteristic
243C2×C32#91811yescharacteristic
244C2×C32C3×C4×D414yes—
245C2×C32C3×AGL(1,ℤ/8)14yes—
246C2×C32C3×#13714yes—
247C3×D3—48no—
248C2×Q8—31no—
249C2×D4—31no—
250C22×C4—61no—
251SD16—34no—
252C2×C8—31no—
253C2×C8C2×C32×D311yescharacteristic
254C4⋊C4—61no—
255C4⋊C4C2×C32×D311yescharacteristic
256C4⋊C4C2×C32×D311yescharacteristic
257C22⋊C4—61no—
258C42—31no—
259C42—61no—
260C22×C3—24no—
261C22×C3—68no—
262C22×C3—124no—
263C22×C3—124no—
264C22×C3C3×D3×D414yes—
265C22×C3C2×C32×D411yescharacteristic
266C2×D3—22no—
267C2×D3—41no—
268C2×D3—41no—
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—44no—
272C3×C4—44no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—68no—
276C3×C4—64no—
277C3×C4—64no—
278C3×C4—84no—
279C3×C4—84no—
280C3×C4C2×C3×C4×D314yes—
281C3×C4C3×#23914yes—
282C3×C4C22×C32×C411yescharacteristic
283C3×C4C32×#4111yescharacteristic
284Dic3—22no—
285Dic3—21no—
286Dic3—21no—
287C32—24no—
288C32#488911yescharacteristic, direct product factor
289C32C3×#44014yescomplemented
290C23—61no—
291Q8—32no—
292Q8—61no—
293D4—32no—
294D4—61no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—61no—
300C2×C4—122no—
301C2×C4C22×C32×D311yescharacteristic
302C8—61no—
303C8C32×C4×D312yescomplemented
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—128no—
308C2×C3C3×#89314yes—
309C2×C3C3×#91114yes—
310C2×C3C3×#91814yes—
311C2×C3C32×C4×D411yescharacteristic
312C2×C3C32×AGL(1,ℤ/8)11yescharacteristic
313C2×C3C32×#13711yescharacteristic
314D3—42no—
315C22—62no—
316C22—121no—
317C22—121no—
318C22C32×D3×D411yescharacteristic
319C4—41no—
320C4—41no—
321C4—61no—
322C4—61no—
323C4—62no—
324C4C2×C32×C4×D311yescharacteristic
325C4C32×#23911yescharacteristic
326C3—24no—
327C3C3×#488914yesdirect product factor
328C3C32×#44011yescharacteristic, complemented
329C2—122no—
330C2C32×#89311yescharacteristic
331C2C32×#91111yescharacteristic
332C2C32×#91811yescharacteristic
subgroupG/H
centerC22×C32D3×D4
commutatorC3×C4C22×C32×C4
FrattiniC2×C4C22×C32×D3
FittingC33×#107C2
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C3×C4C23×C32
hypercenterC32×#107D3
hypocenterC3C32×#440
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×C4 ◁ C32×#107

Complements

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

NH ≅ G/Ntype
3. C32×#887C2semidirect
4. C33×#107C2semidirect
5. C32×#815C2semidirect
6. C32×#810C2semidirect
9. C3×#4889C3direct (N × H)
21. C32×#204C4semidirect
27. C3×#887C2×C3semidirect
29. C32×#107D3semidirect
30. C32×#107C2×C3semidirect
34. C3×#815C2×C3semidirect
35. C3×#810C2×C3semidirect
48. C32×Dic6C2×C4semidirect
49. C33×C8C2×C4semidirect
52. #4889C32direct (N × H)
79. C3×#204C3×C4semidirect
96. #887C2×C32semidirect
98. C3×#107C3×D3semidirect
99. C3×#107C2×C32semidirect
103. #815C2×C32semidirect
104. #810C2×C32semidirect
132. C3×Dic6C2×C3×C4semidirect
135. C32×C8C4×D3semidirect
136. C32×C8C2×C3×C4semidirect
169. #204C32×C4semidirect
200. #107C32×D3semidirect
204. C33#440semidirect
231. Dic6C2×C32×C4semidirect
234. C3×C8C3×C4×D3semidirect
235. C3×C8C2×C32×C4semidirect
288. C32#4889direct (N × H)
289. C32C3×#440semidirect
303. C8C32×C4×D3semidirect
327. C3C3×#4889direct (N × H)
328. C3C32×#440semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2042
C32×#8932
C32×#8872
C33×#1072
C32×#8152
C32×#8102
C32×#8092
C32×#4403
C3×#48893