Finite Group Databasebeta

← C32×#5041 #13162474

Subgroups — C32×#5041 #13162474

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#121C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C32×#924C211yesmaximal, characteristic, complemented
4C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7C32×#808C211yesmaximal, characteristic, complemented
8C32×#465—31nomaximal
9C3×#5041C314yesmaximal, direct product factor
10C2×C33×D4—21no—
11C2×C33×D4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C33×C22⋊C4—21no—
16C32×#217—21no—
17C32×#214—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C2×C32×C3⋊C8—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23C2×C32×SD16—31no—
24C22×C32×C8—31no—
25C32×C22⋊Q8—31no—
26C32×#121—24no—
27C32×#121D311yescharacteristic
28C32×#121C2×C314yes—
29C2×C3×#214C2×C314yescomplemented
30C3×#924C2×C314yescomplemented
31C22×C3×C3⋊C8C2×C314yescomplemented
32C32×#106—31no—
33C32×#103—31no—
34C32×#102—31no—
35C3×#832C2×C314yescomplemented
36C3×#810C2×C314yescomplemented
37C3×#808C2×C314yescomplemented
38C23×C33—21no—
39C23×C33D411yescharacteristic
40C33×D4—21no—
41C33×D4—21no—
42C33×D4—42no—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C32×C3⋊C8—21no—
53C32×C3⋊C8D412yescomplemented
54C3×#465—34no—
55#5041C3211yescharacteristic, direct product factor
56C2×C32×Q8—31no—
57C2×C32×D4—28no—
58C2×C32×D4—24no—
59C2×C32×D4—21no—
60C2×C32×D4—24no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C314yes—
63C22×C32×C4—24no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C22×C314yes—
66C2×C3×Dic6C22×C314yes—
67C32×SD16—62no—
68C2×C32×C8—31no—
69C2×C32×C8—31no—
70C2×C32×C8—62no—
71C32×C4⋊C4—24no—
72C32×C4⋊C4—31no—
73C32×C4⋊C4—61no—
74C32×C4⋊C4C2×D311yescharacteristic
75C32×C4⋊C4C22×C314yes—
76C32×C22⋊C4—28no—
77C32×C22⋊C4—21no—
78C32×C22⋊C4—24no—
79C32×C22⋊C4—61no—
80C3×#217—24no—
81C3×#214—28no—
82C3×C4⋊Dic3C22×C314yes—
83C3×Dic3⋊C4—24no—
84C2×C3×C3⋊C8—28no—
85C2×C3×C3⋊C8C22×C314yes—
86C2×C3×C3⋊C8C22×C314yes—
87C22×C33—21no—
88C22×C33—21no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33SD1612yescomplemented
93C22×C33C2×D411yescharacteristic
94C33×C4—21no—
95C33×C4—41no—
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C32×Dic3—41no—
99C32×Dic3—41no—
100C2×C3×SD16—34no—
101C22×C3×C8—34no—
102C3×C22⋊Q8—34no—
103C3×#121—24no—
104C3×#121C3×D314yes—
105C3×#121C2×C3211yescharacteristic
106C2×#214C2×C3211yescharacteristic, complemented
107#924C2×C3211yescharacteristic, complemented
108C22×C3⋊C8C2×C3211yescharacteristic, complemented
109C3×#106—34no—
110C3×#103—34no—
111C3×#102—34no—
112#832C2×C3211yescharacteristic, complemented
113#810C2×C3211yescharacteristic, complemented
114#808C2×C3211yescharacteristic, complemented
115C23×C32—28no—
116C23×C32—21no—
117C23×C32—24no—
118C23×C32—24no—
119C23×C32#6711yescharacteristic
120C23×C32C3×D414yes—
121C32×Q8—61no—
122C32×Q8—61no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—24no—
126C32×D4—24no—
127C32×D4—44no—
128C32×D4—44no—
129C32×D4—416no—
130C32×D4—42no—
131C32×D4—48no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—28no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—44no—
142C2×C32×C4—44no—
143C2×C32×C4—61no—
144C2×C32×C4—61no—
145C2×C32×C4#6711yescharacteristic
146C2×C32×C4C3×D414yes—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C23×C314yes—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—24no—
151C3×Dic6—24no—
152C3×Dic6—24no—
153C32×C8—32no—
154C32×C8—61no—
155C3×C3⋊C8—24no—
156C3×C3⋊C8C3×D418yescomplemented
157#465—31noSylow
158C2×C33—22no—
159C2×C33—81no—
160C2×C33#12111yescharacteristic
161C2×C33C2×SD1611yescharacteristic
162C2×C33#13511yescharacteristic
163C2×C3×Q8—34no—
164C2×C3×D4—28no—
165C2×C3×D4—24no—
166C2×C3×D4—21no—
167C2×C3×D4—24no—
168C2×C3×D4C2×C3×D314yes—
169C2×C3×D4C22×C3211yescharacteristic
170C22×C3×C4—24no—
171C22×C3×C4C2×C3×D314yes—
172C22×C3×C4C22×C3211yescharacteristic
173C2×Dic6C22×C3211yescharacteristic
174C3×SD16—68no—
175C2×C3×C8—34no—
176C2×C3×C8—34no—
177C2×C3×C8—68no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—34no—
180C3×C4⋊C4—64no—
181C3×C4⋊C4C2×C3×D314yes—
182C3×C4⋊C4C22×C3211yescharacteristic
183C3×C22⋊C4—28no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—64no—
187#217—21no—
188#214—22no—
189C4⋊Dic3C22×C3211yescharacteristic
190Dic3⋊C4—21no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8C22×C3211yescharacteristic
193C2×C3⋊C8C22×C3211yescharacteristic
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—28no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—44no—
201C22×C32—44no—
202C22×C32—48no—
203C22×C32—48no—
204C22×C32—48no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—44no—
209C22×C32—44no—
210C22×C32—44no—
211C22×C32#21412yescomplemented
212C22×C32C3×SD1618yescomplemented
213C22×C32C2×#6711yescharacteristic
214C22×C32C2×C3×D414yes—
215C32×C4—21no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—24no—
219C32×C4—44no—
220C32×C4—48no—
221C32×C4—41no—
222C32×C4—44no—
223C32×C4—121no—
224C32×C4—121no—
225C32×C4D3×D411yescharacteristic
226C32×C4#23711yescharacteristic
227C32×C4C2×C3×D414yes—
228C32×C4C3×#4114yes—
229C3×Dic3—44no—
230C3×Dic3—44no—
231C2×SD16—31no—
232C22×C8—31no—
233C22⋊Q8—31no—
234#121C32×D311yescharacteristic
235#106—31no—
236#103—31no—
237#102—31no—
238C33#46511yescharacteristic, complemented, Sylow
239C23×C3—28no—
240C23×C3—24no—
241C23×C3—21no—
242C23×C3—24no—
243C23×C3C3×#6714yes—
244C23×C3C32×D411yescharacteristic
245C3×Q8—64no—
246C3×Q8—64no—
247C3×D4—24no—
248C3×D4—24no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—44no—
252C3×D4—44no—
253C3×D4—416no—
254C3×D4—48no—
255C3×D4—42no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—28no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—44no—
266C2×C3×C4—44no—
267C2×C3×C4—64no—
268C2×C3×C4—64no—
269C2×C3×C4C3×#6714yes—
270C2×C3×C4C32×D411yescharacteristic
271C2×C3×C4C22×C3×D314yes—
272C2×C3×C4C23×C3211yescharacteristic
273C2×Dic3—21no—
274C2×Dic3—21no—
275Dic6—21no—
276Dic6—21no—
277C3×C8—38no—
278C3×C8—64no—
279C3⋊C8—21no—
280C3⋊C8C32×D412yescomplemented
281C2×C32—22no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—28no—
286C2×C32—48no—
287C2×C32—88no—
288C2×C32—81no—
289C2×C32—84no—
290C2×C32C2×#21411yescharacteristic
291C2×C32#93911yescharacteristic
292C2×C32#95011yescharacteristic
293C2×C32C3×#12114yes—
294C2×C32C2×C3×SD1614yes—
295C2×C32C3×#13514yes—
296C2×Q8—31no—
297C2×D4—21no—
298C2×D4C2×C32×D311yescharacteristic
299C22×C4C2×C32×D311yescharacteristic
300SD16—62no—
301C2×C8—31no—
302C2×C8—31no—
303C2×C8—62no—
304C4⋊C4—31no—
305C4⋊C4—61no—
306C4⋊C4C2×C32×D311yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—61no—
309C22×C3—24no—
310C22×C3—24no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—24no—
314C22×C3—28no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3—48no—
318C22×C3—48no—
319C22×C3—48no—
320C22×C3—44no—
321C22×C3—44no—
322C22×C3—44no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3C3×#21418yescomplemented
327C22×C3C32×SD1612yescomplemented
328C22×C3C2×C3×#6714yes—
329C22×C3C2×C32×D411yescharacteristic
330C3×C4—24no—
331C3×C4—21no—
332C3×C4—24no—
333C3×C4—24no—
334C3×C4—44no—
335C3×C4—48no—
336C3×C4—44no—
337C3×C4—41no—
338C3×C4—124no—
339C3×C4—124no—
340C3×C4C3×D3×D414yes—
341C3×C4C3×#23714yes—
342C3×C4C2×C32×D411yescharacteristic
343C3×C4C32×#4111yescharacteristic
344Dic3—41no—
345Dic3—41no—
346C32—24no—
347C32#504111yescharacteristic, direct product factor
348C32C3×#46514yescomplemented
349C23—21no—
350C23C32×#6711yescharacteristic
351Q8—61no—
352Q8—61no—
353D4—21no—
354D4—21no—
355D4—42no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4C32×#6711yescharacteristic
362C2×C4C22×C32×D311yescharacteristic
363C8—32no—
364C8—61no—
365C2×C3—22no—
366C2×C3—28no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—24no—
370C2×C3—48no—
371C2×C3—88no—
372C2×C3—81no—
373C2×C3—84no—
374C2×C3C2×C3×#21414yes—
375C2×C3C3×#93914yes—
376C2×C3C3×#95014yes—
377C2×C3C32×#12111yescharacteristic
378C2×C3C2×C32×SD1611yescharacteristic
379C2×C3C32×#13511yescharacteristic
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22C32×#21412yescomplemented
386C22C2×C32×#6711yescharacteristic
387C4—21no—
388C4—41no—
389C4—121no—
390C4—121no—
391C4C32×D3×D411yescharacteristic
392C4C32×#23711yescharacteristic
393C3—24no—
394C3C3×#504114yesdirect product factor
395C3C32×#46511yescharacteristic, complemented
396C2—22no—
397C2—81no—
398C2C2×C32×#21411yescharacteristic
399C2C32×#93911yescharacteristic
400C2C32×#95011yescharacteristic
subgroupG/H
centerC22×C32C2×#67
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
Fitting, prime radicalC33×#121C2
socleC22×C33C2×D4
hypercenterC32×#121D3
hypocenterC3C32×#465
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C22×C32×C4 ◁ C32×#121

Complements

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

NH ≅ G/Ntype
2. C2×C32×#214C2semidirect
3. C32×#924C2semidirect
4. C22×C32×C3⋊C8C2semidirect
5. C32×#832C2semidirect
6. C32×#810C2semidirect
7. C32×#808C2semidirect
9. C3×#5041C3direct (N × H)
29. C2×C3×#214C2×C3semidirect
30. C3×#924C2×C3semidirect
31. C22×C3×C3⋊C8C2×C3semidirect
35. C3×#832C2×C3semidirect
36. C3×#810C2×C3semidirect
37. C3×#808C2×C3semidirect
53. C32×C3⋊C8D4semidirect
55. #5041C32direct (N × H)
92. C22×C33SD16semidirect
106. C2×#214C2×C32semidirect
107. #924C2×C32semidirect
108. C22×C3⋊C8C2×C32semidirect
112. #832C2×C32semidirect
113. #810C2×C32semidirect
114. #808C2×C32semidirect
156. C3×C3⋊C8C3×D4semidirect
211. C22×C32#214semidirect
212. C22×C32C3×SD16semidirect
238. C33#465semidirect
280. C3⋊C8C32×D4semidirect
326. C22×C3C3×#214semidirect
327. C22×C3C32×SD16semidirect
347. C32#5041direct (N × H)
348. C32C3×#465semidirect
385. C22C32×#214semidirect
394. C3C3×#5041direct (N × H)
395. C3C32×#465semidirect

Maximal subgroups

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

subgroupindex
C33×#1212
C2×C32×#2142
C32×#9242
C22×C32×C3⋊C82
C32×#8322
C32×#8102
C32×#8082
C32×#4653
C3×#50413