Finite Group Databasebeta

← C32×#5039 #13162473

Subgroups — C32×#5039 #13162473

Order 1728.

1156 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#121C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3C32×#930C211yesmaximal, characteristic, complemented
4C2×C32×#208C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7C32×#808C211yesmaximal, characteristic, complemented
8C32×#469—31nomaximal
9C3×#5039C314yesmaximal, direct product factor
10C2×C33×D4—21no—
11C2×C33×D4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C2×C32×#67—21no—
14C2×C32×D12C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C33×C22⋊C4—21no—
17C32×#213—22no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×#208—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23C2×C32×D8—31no—
24C2×C32×M16—31no—
25C32×#121—24no—
26C32×#121—31no—
27C32×#121D311yescharacteristic, complemented
28C32×#121C2×C314yescomplemented
29C2×C3×#213C2×C314yescomplemented
30C3×#930C2×C314yescomplemented
31C2×C3×#208C2×C314yescomplemented
32C32×#106—31no—
33C32×#102—31no—
34C32×#102—31no—
35C3×#832C2×C314yescomplemented
36C3×#809C2×C314yescomplemented
37C3×#808C2×C314yescomplemented
38C23×C33—21no—
39C23×C33D411yescharacteristic
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C33×D4—42no—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C32×#67—42no—
50C2×C32×Dic3—21no—
51C32×D12—21no—
52C32×D12—21no—
53C32×C3⋊C8—21no—
54C32×C3⋊C8D412yescomplemented
55C3×#469—34no—
56#5039C3211yescharacteristic, direct product factor
57C2×C32×D4—28no—
58C2×C32×D4—24no—
59C2×C32×D4—21no—
60C2×C32×D4—24no—
61C2×C32×D4—31no—
62C2×C32×D4—61no—
63C2×C32×D4C2×D311yescharacteristic
64C2×C32×D4C22×C314yes—
65C22×C32×C4—24no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C314yes—
68C2×C3×#67—24no—
69C2×C3×D12C22×C314yes—
70C32×D8—62no—
71C32×M16—62no—
72C2×C32×C8—31no—
73C2×C32×C8—31no—
74C32×C4⋊C4—24no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C4⋊C4C22×C314yes—
78C32×C22⋊C4—28no—
79C32×C22⋊C4—21no—
80C32×C22⋊C4—24no—
81C32×C22⋊C4—61no—
82C3×#213—28no—
83C3×(C2×D3)⋊C4—24no—
84C3×C4⋊Dic3C22×C314yes—
85C3×#208—28no—
86C2×C3×C3⋊C8C22×C314yes—
87C2×C3×C3⋊C8C22×C314yes—
88C22×C33—21no—
89C22×C33—21no—
90C22×C33—21no—
91C22×C33—41no—
92C22×C33—41no—
93C22×C33—41no—
94C22×C33C2×D411yescharacteristic
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C33×C4—21no—
99C33×C4—41no—
100C33×C4C2×D411yescharacteristic
101C33×C4#4111yescharacteristic
102C32×Dic3—41no—
103C2×C3×D8—34no—
104C2×C3×M16—34no—
105C3×#121—24no—
106C3×#121—34no—
107C3×#121C3×D314yescomplemented
108C3×#121C2×C3211yescharacteristic, complemented
109C2×#213C2×C3211yescharacteristic, complemented
110#930C2×C3211yescharacteristic, complemented
111C2×#208C2×C3211yescharacteristic, complemented
112C3×#106—34no—
113C3×#102—34no—
114C3×#102—34no—
115#832C2×C3211yescharacteristic, complemented
116#809C2×C3211yescharacteristic, complemented
117#808C2×C3211yescharacteristic, complemented
118C23×C32—28no—
119C23×C32—21no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—61no—
123C23×C32#6711yescharacteristic
124C23×C32C3×D414yes—
125C22×C3×D3—24no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—24no—
129C32×D4—24no—
130C32×D4—44no—
131C32×D4—44no—
132C32×D4—416no—
133C32×D4—42no—
134C32×D4—48no—
135C32×D4—61no—
136C32×D4—61no—
137C32×D4—122no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—28no—
141C2×C32×C4—21no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—24no—
145C2×C32×C4—24no—
146C2×C32×C4—24no—
147C2×C32×C4—44no—
148C2×C32×C4—44no—
149C2×C32×C4—61no—
150C2×C32×C4#6711yescharacteristic
151C2×C32×C4C3×D414yes—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C314yes—
154C3×#67—48no—
155C2×C3×Dic3—24no—
156C3×D12—24no—
157C3×D12—24no—
158C32×C8—32no—
159C32×C8—61no—
160C3×C3⋊C8—24no—
161C3×C3⋊C8C3×D418yescomplemented
162#469—31noSylow
163C2×C33—41no—
164C2×C33—81no—
165C2×C33#12111yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C2×C33AGL(1,ℤ/8)11yescharacteristic
168C32×D3—81no—
169C2×C3×D4—28no—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—24no—
173C2×C3×D4—34no—
174C2×C3×D4—64no—
175C2×C3×D4C2×C3×D314yes—
176C2×C3×D4C22×C3211yescharacteristic
177C22×C3×C4—24no—
178C22×C3×C4C2×C3×D314yes—
179C22×C3×C4C22×C3211yescharacteristic
180C2×#67—21no—
181C2×D12C22×C3211yescharacteristic
182C3×D8—68no—
183C3×M16—68no—
184C2×C3×C8—34no—
185C2×C3×C8—34no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—34no—
188C3×C4⋊C4C2×C3×D314yes—
189C3×C4⋊C4C22×C3211yescharacteristic
190C3×C22⋊C4—28no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—64no—
194#213—22no—
195(C2×D3)⋊C4—21no—
196C4⋊Dic3C22×C3211yescharacteristic
197#208—22no—
198C2×C3⋊C8C22×C3211yescharacteristic
199C2×C3⋊C8C22×C3211yescharacteristic
200C22×C32—28no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—24no—
205C22×C32—24no—
206C22×C32—24no—
207C22×C32—24no—
208C22×C32—44no—
209C22×C32—44no—
210C22×C32—48no—
211C22×C32—48no—
212C22×C32—48no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—44no—
217C22×C32—44no—
218C22×C32—44no—
219C22×C32—121no—
220C22×C32—121no—
221C22×C32—121no—
222C22×C32C2×#6711yescharacteristic
223C22×C32C2×C3×D414yes—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C2×C3×D3—44no—
227C32×C4—21no—
228C32×C4—24no—
229C32×C4—24no—
230C32×C4—24no—
231C32×C4—44no—
232C32×C4—48no—
233C32×C4—41no—
234C32×C4—44no—
235C32×C4—121no—
236C32×C4D3×D411yescharacteristic
237C32×C4#23711yescharacteristic
238C32×C4C2×C3×D414yes—
239C32×C4C3×#4114yes—
240C3×Dic3—44no—
241C2×D8—31no—
242C2×M16—31no—
243#121—31no—
244#121C32×D311yescharacteristic, complemented
245#106—31no—
246#102—31no—
247#102—31no—
248C33#46911yescharacteristic, complemented, Sylow
249C23×C3—28no—
250C23×C3—24no—
251C23×C3—21no—
252C23×C3—24no—
253C23×C3—64no—
254C23×C3C3×#6714yes—
255C23×C3C32×D411yescharacteristic
256C22×D3—21no—
257C3×D4—24no—
258C3×D4—24no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—44no—
262C3×D4—44no—
263C3×D4—416no—
264C3×D4—48no—
265C3×D4—42no—
266C3×D4—64no—
267C3×D4—64no—
268C3×D4—128no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—28no—
274C2×C3×C4—24no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—64no—
281C2×C3×C4C3×#6714yes—
282C2×C3×C4C32×D411yescharacteristic
283C2×C3×C4C22×C3×D314yes—
284C2×C3×C4C23×C3211yescharacteristic
285#67—42no—
286C2×Dic3—21no—
287D12—21no—
288D12—21no—
289C3×C8—38no—
290C3×C8—64no—
291C3⋊C8—21no—
292C3⋊C8C32×D412yescomplemented
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—48no—
297C2×C32—41no—
298C2×C32—44no—
299C2×C32—88no—
300C2×C32—81no—
301C2×C32—84no—
302C2×C32—241no—
303C2×C32#93211yescharacteristic
304C2×C32#93911yescharacteristic
305C2×C32#94911yescharacteristic
306C2×C32C3×#12114yes—
307C2×C32C3×AGL(1,ℤ/8)14yes—
308C2×C32C3×AGL(1,ℤ/8)14yes—
309C3×D3—84no—
310C2×D4—21no—
311C2×D4—31no—
312C2×D4—61no—
313C2×D4C2×C32×D311yescharacteristic
314C22×C4C2×C32×D311yescharacteristic
315D8—62no—
316M16—62no—
317C2×C8—31no—
318C2×C8—31no—
319C4⋊C4—31no—
320C4⋊C4C2×C32×D311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—61no—
323C22×C3—28no—
324C22×C3—24no—
325C22×C3—24no—
326C22×C3—24no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—24no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—48no—
334C22×C3—48no—
335C22×C3—48no—
336C22×C3—44no—
337C22×C3—44no—
338C22×C3—44no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—124no—
343C22×C3—124no—
344C22×C3—124no—
345C22×C3C2×C3×#6714yes—
346C22×C3C2×C32×D411yescharacteristic
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—41no—
350C3×C4—24no—
351C3×C4—21no—
352C3×C4—24no—
353C3×C4—24no—
354C3×C4—44no—
355C3×C4—48no—
356C3×C4—44no—
357C3×C4—41no—
358C3×C4—124no—
359C3×C4C3×D3×D414yes—
360C3×C4C3×#23714yes—
361C3×C4C2×C32×D411yescharacteristic
362C3×C4C32×#4111yescharacteristic
363Dic3—41no—
364C32—24no—
365C32#503911yescharacteristic, direct product factor
366C32C3×#46914yescomplemented
367C23—21no—
368C23—61no—
369C23C32×#6711yescharacteristic
370D4—21no—
371D4—21no—
372D4—42no—
373D4—61no—
374D4—61no—
375D4—122no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—61no—
380C2×C4C32×#6711yescharacteristic
381C2×C4C22×C32×D311yescharacteristic
382C8—32no—
383C8—61no—
384C2×C3—24no—
385C2×C3—24no—
386C2×C3—24no—
387C2×C3—48no—
388C2×C3—41no—
389C2×C3—44no—
390C2×C3—88no—
391C2×C3—81no—
392C2×C3—84no—
393C2×C3—244no—
394C2×C3C3×#93214yes—
395C2×C3C3×#93914yes—
396C2×C3C3×#94914yes—
397C2×C3C32×#12111yescharacteristic
398C2×C3C32×AGL(1,ℤ/8)11yescharacteristic
399C2×C3C32×AGL(1,ℤ/8)11yescharacteristic
400D3—81no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—121no—
408C22—121no—
409C22—121no—
410C22C2×C32×#6711yescharacteristic
411C4—21no—
412C4—41no—
413C4—121no—
414C4C32×D3×D411yescharacteristic
415C4C32×#23711yescharacteristic
416C3—24no—
417C3C3×#503914yesdirect product factor
418C3C32×#46911yescharacteristic, complemented
419C2—41no—
420C2—81no—
421C2—241no—
422C2C32×#93211yescharacteristic
423C2C32×#93911yescharacteristic
424C2C32×#94911yescharacteristic
subgroupG/H
centerC22×C32C2×#67
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
FittingC33×#121C2
socleC22×C33C2×D4
hypercenterC32×#121D3
hypocenterC3C32×#469
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
1. C33×#121C2semidirect
2. C2×C32×#213C2semidirect
3. C32×#930C2semidirect
4. C2×C32×#208C2semidirect
5. C32×#832C2semidirect
6. C32×#809C2semidirect
7. C32×#808C2semidirect
9. C3×#5039C3direct (N × H)
27. C32×#121D3semidirect
28. C32×#121C2×C3semidirect
29. C2×C3×#213C2×C3semidirect
30. C3×#930C2×C3semidirect
31. C2×C3×#208C2×C3semidirect
35. C3×#832C2×C3semidirect
36. C3×#809C2×C3semidirect
37. C3×#808C2×C3semidirect
54. C32×C3⋊C8D4semidirect
56. #5039C32direct (N × H)
107. C3×#121C3×D3semidirect
108. C3×#121C2×C32semidirect
109. C2×#213C2×C32semidirect
110. #930C2×C32semidirect
111. C2×#208C2×C32semidirect
115. #832C2×C32semidirect
116. #809C2×C32semidirect
117. #808C2×C32semidirect
161. C3×C3⋊C8C3×D4semidirect
244. #121C32×D3semidirect
248. C33#469semidirect
292. C3⋊C8C32×D4semidirect
365. C32#5039direct (N × H)
366. C32C3×#469semidirect
417. C3C3×#5039direct (N × H)
418. C3C32×#469semidirect

Maximal subgroups

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

subgroupindex
C33×#1212
C2×C32×#2132
C32×#9302
C2×C32×#2082
C32×#8322
C32×#8092
C32×#8082
C32×#4693
C3×#50393