Finite Group Databasebeta

← C32×#4804 #13162239

Subgroups — C32×#4804 #13162239

Order 1728.

1116 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic
2C2×C32×D3×C8C211yesmaximal, characteristic, complemented
3C32×#892C211yesmaximal, characteristic, complemented
4C33×#103C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7C32×#809C211yesmaximal, characteristic, complemented
8C32×#416—31nomaximal
9C3×#4804C314yesmaximal, direct product factor
10C2×C33×Q8C2211yescharacteristic
11C32×#239—22no—
12C32×#239C414yescomplemented
13C2×C32×D12—21no—
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3—21no—
16C2×C32×C4×D3C2211yescharacteristic
17C2×C33×C8C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21C32×C4×Dic3—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23C32×D3×C8—22no—
24C2×C32×#41—31no—
25C2×C3×#239C2×C314yes—
26C22×C32×C8—31no—
27C32×#117—31no—
28C2×C3×D3×C8C2×C314yescomplemented
29C3×#892C2×C314yescomplemented
30C32×#103—24no—
31C32×#103—31no—
32C32×#103D311yescharacteristic, complemented
33C32×#103C2×C314yescomplemented
34C32×#102—31no—
35C32×#102—31no—
36C3×#835C2×C314yescomplemented
37C3×#821C2×C314yescomplemented
38C3×#809C2×C314yescomplemented
39C22×C32×D3—21no—
40C22×C32×D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8C2×C412yes—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C32×D12—24no—
49C32×D12—21no—
50C32×D12C2×C412yes—
51C32×C4×D3—24no—
52C32×C4×D3C2×C412yes—
53C32×C4×D3D412yes—
54C33×C8—21no—
55C32×C3⋊C8—21no—
56C3×#416—34no—
57#4804C3211yescharacteristic, direct product factor
58C32×#41—34no—
59C32×#41—62no—
60C2×C32×Q8—24no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C22×C314yes—
63C2×C32×D4—31no—
64C2×C32×D4—61no—
65C22×C32×C4—31no—
66C22×C32×C4—61no—
67C3×#239—28no—
68C3×#239C3×C4116yescomplemented
69C2×C3×D12—24no—
70C2×C3×D12C22×C314yes—
71C2×C3×C4×D3—24no—
72C2×C3×C4×D3C22×C314yes—
73C2×C32×C8—24no—
74C2×C32×C8—31no—
75C2×C32×C8—62no—
76C2×C32×C8C2×D311yescharacteristic
77C2×C32×C8C22×C314yes—
78C32×C4⋊C4—24no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4C2×D311yescharacteristic
81C32×C4⋊C4C22×C314yes—
82C32×C22⋊C4—61no—
83C32×C42—61no—
84C3×(C2×D3)⋊C4—24no—
85C3×C4⋊Dic3C22×C314yes—
86C3×C4×Dic3—24no—
87C2×C3×C3⋊C8C22×C314yes—
88C3×D3×C8—28no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3—22no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3C22⋊C412yes—
96C33×C4—22no—
97C33×C4—41no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C32×Dic3—41no—
101C32×Dic3C22⋊C412yes—
102C2×C3×#41—34no—
103C2×#239C2×C3211yescharacteristic
104C22×C3×C8—34no—
105C3×#117—34no—
106C2×D3×C8C2×C3211yescharacteristic, complemented
107#892C2×C3211yescharacteristic, complemented
108C3×#103—24no—
109C3×#103—34no—
110C3×#103C3×D314yescomplemented
111C3×#103C2×C3211yescharacteristic, complemented
112C3×#102—34no—
113C3×#102—34no—
114#835C2×C3211yescharacteristic, complemented
115#821C2×C3211yescharacteristic, complemented
116#809C2×C3211yescharacteristic, complemented
117C23×C32—31no—
118C23×C32—61no—
119C22×C3×D3—24no—
120C22×C3×D3C3×D414yes—
121C32×Q8—21no—
122C32×Q8—28no—
123C32×Q8—24no—
124C32×Q8—44no—
125C32×Q8C4×D312yes—
126C32×Q8C2×C3×C418yes—
127C32×D4—32no—
128C32×D4—64no—
129C32×D4—61no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—31no—
136C2×C32×C4—32no—
137C2×C32×C4—32no—
138C2×C32×C4—44no—
139C2×C32×C4—44no—
140C2×C32×C4—64no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C23×C314yes—
144C2×C3×Dic3—24no—
145C2×C3×Dic3C3×D414yes—
146C3×D12—216no—
147C3×D12—24no—
148C3×D12C2×C3×C418yes—
149C3×C4×D3—216no—
150C3×C4×D3C2×C3×C418yes—
151C3×C4×D3C3×D418yes—
152C32×C8—21no—
153C32×C8—24no—
154C32×C8—44no—
155C32×C8—61no—
156C3×C3⋊C8—24no—
157#416—31noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33#13511yescharacteristic
161C32×D3—22no—
162C32×D3—42no—
163C3×#41—316no—
164C3×#41—68no—
165C2×C3×Q8—24no—
166C2×C3×Q8C2×C3×D314yes—
167C2×C3×Q8C22×C3211yescharacteristic
168C2×C3×D4—34no—
169C2×C3×D4—64no—
170C22×C3×C4—34no—
171C22×C3×C4—64no—
172#239—22no—
173#239C32×C414yescomplemented
174C2×D12—21no—
175C2×D12C22×C3211yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3C22×C3211yescharacteristic
178C2×C3×C8—24no—
179C2×C3×C8—34no—
180C2×C3×C8—68no—
181C2×C3×C8C2×C3×D314yes—
182C2×C3×C8C22×C3211yescharacteristic
183C3×C4⋊C4—24no—
184C3×C4⋊C4—34no—
185C3×C4⋊C4C2×C3×D314yes—
186C3×C4⋊C4C22×C3211yescharacteristic
187C3×C22⋊C4—64no—
188C3×C42—64no—
189(C2×D3)⋊C4—21no—
190C4⋊Dic3C22×C3211yescharacteristic
191C4×Dic3—21no—
192C2×C3⋊C8C22×C3211yescharacteristic
193D3×C8—22no—
194C22×C32—24no—
195C22×C32—32no—
196C22×C32—61no—
197C22×C32—61no—
198C22×C32—62no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32D3×D411yescharacteristic
202C22×C32C2×C3×D414yes—
203C2×C3×D3—24no—
204C2×C3×D3—24no—
205C2×C3×D3—28no—
206C2×C3×D3—44no—
207C2×C3×D3—44no—
208C2×C3×D3C3×C22⋊C418yes—
209C32×C4—22no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—28no—
213C32×C4—32no—
214C32×C4—48no—
215C32×C4—41no—
216C32×C4—44no—
217C32×C4—84no—
218C32×C4—121no—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4D3×D411yescharacteristic
221C32×C4C22×C3×C414yes—
222C32×C4C2×C3×D414yes—
223C3×Dic3—44no—
224C3×Dic3C3×C22⋊C418yes—
225C2×#41—31no—
226C22×C8—31no—
227#117—31no—
228#103—31no—
229#103C32×D311yescharacteristic, complemented
230#102—31no—
231#102—31no—
232C33#41611yescharacteristic, complemented, Sylow
233C23×C3—34no—
234C23×C3—64no—
235C22×D3—21no—
236C22×D3C32×D411yescharacteristic
237C3×Q8—24no—
238C3×Q8—21no—
239C3×Q8—28no—
240C3×Q8—44no—
241C3×Q8C3×C4×D318yes—
242C3×Q8C2×C32×C412yes—
243C3×D4—38no—
244C3×D4—616no—
245C3×D4—64no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—24no—
251C2×C3×C4—34no—
252C2×C3×C4—38no—
253C2×C3×C4—38no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—616no—
257C2×C3×C4—64no—
258C2×C3×C4C22×C3×D314yes—
259C2×C3×C4C23×C3211yescharacteristic
260C2×Dic3—21no—
261C2×Dic3C32×D411yescharacteristic
262D12—24no—
263D12—21no—
264D12C2×C32×C412yes—
265C4×D3—24no—
266C4×D3C2×C32×C412yes—
267C4×D3C32×D412yes—
268C3×C8—24no—
269C3×C8—21no—
270C3×C8—44no—
271C3×C8—64no—
272C3⋊C8—21no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—62no—
277C2×C32—122no—
278C2×C32D3×C22⋊C411yescharacteristic
279C2×C32#91611yescharacteristic
280C2×C32#91911yescharacteristic
281C2×C32C2×C3×C22⋊C414yes—
282C2×C32C3×#13514yes—
283C2×C32C3×#13514yes—
284C3×D3—28no—
285C3×D3—48no—
286#41—34no—
287#41—62no—
288C2×Q8C2×C32×D311yescharacteristic
289C2×D4—31no—
290C2×D4—61no—
291C22×C4—31no—
292C22×C4—61no—
293C2×C8—31no—
294C2×C8—62no—
295C2×C8C2×C32×D311yescharacteristic
296C4⋊C4—31no—
297C4⋊C4C2×C32×D311yescharacteristic
298C22⋊C4—61no—
299C42—61no—
300C22×C3—24no—
301C22×C3—38no—
302C22×C3—64no—
303C22×C3—64no—
304C22×C3—68no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3C3×D3×D414yes—
308C22×C3C2×C32×D411yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—22no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3C32×C22⋊C412yes—
315C3×C4—28no—
316C3×C4—22no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—38no—
320C3×C4—48no—
321C3×C4—44no—
322C3×C4—41no—
323C3×C4—84no—
324C3×C4—124no—
325C3×C4C2×C3×C4×D314yes—
326C3×C4C3×D3×D414yes—
327C3×C4C22×C32×C411yescharacteristic
328C3×C4C2×C32×D411yescharacteristic
329Dic3—41no—
330Dic3C32×C22⋊C412yes—
331C32—24no—
332C32#480411yescharacteristic, direct product factor
333C32C3×#41614yescomplemented
334C23—31no—
335C23—61no—
336Q8—21no—
337Q8C32×C4×D312yes—
338D4—32no—
339D4—64no—
340D4—61no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—31no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—64no—
347C2×C4—61no—
348C2×C4C22×C32×D311yescharacteristic
349C8—21no—
350C8—61no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—24no—
354C2×C3—68no—
355C2×C3—128no—
356C2×C3C3×D3×C22⋊C414yes—
357C2×C3C3×#91614yes—
358C2×C3C3×#91914yes—
359C2×C3C2×C32×C22⋊C411yescharacteristic
360C2×C3C32×#13511yescharacteristic
361C2×C3C32×#13511yescharacteristic
362D3—22no—
363D3—42no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—62no—
368C22—121no—
369C22—121no—
370C22C32×D3×D411yescharacteristic
371C4—22no—
372C4—32no—
373C4—41no—
374C4—121no—
375C4C2×C32×C4×D311yescharacteristic
376C4C32×D3×D411yescharacteristic
377C3—24no—
378C3C3×#480414yesdirect product factor
379C3C32×#41611yescharacteristic, complemented
380C2—62no—
381C2—122no—
382C2C32×D3×C22⋊C411yescharacteristic
383C2C32×#91611yescharacteristic
384C2C32×#91911yescharacteristic
subgroupG/H
centerC22×C32D3×D4
commutatorC3×C4C22×C32×C4
FrattiniC2×C4C22×C32×D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×C32×#239C2
cosocleC2×C3×C4C23×C32
hypercenterC32×#103D3
hypocenterC3C32×#416
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×C4 ◁ C32×#103

Complements

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

NH ≅ G/Ntype
2. C2×C32×D3×C8C2semidirect
3. C32×#892C2semidirect
4. C33×#103C2semidirect
5. C32×#835C2semidirect
6. C32×#821C2semidirect
7. C32×#809C2semidirect
9. C3×#4804C3direct (N × H)
12. C32×#239C4semidirect
28. C2×C3×D3×C8C2×C3semidirect
29. C3×#892C2×C3semidirect
32. C32×#103D3semidirect
33. C32×#103C2×C3semidirect
36. C3×#835C2×C3semidirect
37. C3×#821C2×C3semidirect
38. C3×#809C2×C3semidirect
57. #4804C32direct (N × H)
68. C3×#239C3×C4semidirect
106. C2×D3×C8C2×C32semidirect
107. #892C2×C32semidirect
110. C3×#103C3×D3semidirect
111. C3×#103C2×C32semidirect
114. #835C2×C32semidirect
115. #821C2×C32semidirect
116. #809C2×C32semidirect
173. #239C32×C4semidirect
229. #103C32×D3semidirect
232. C33#416semidirect
332. C32#4804direct (N × H)
333. C32C3×#416semidirect
378. C3C3×#4804direct (N × H)
379. C3C32×#416semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×C32×D3×C82
C32×#8922
C33×#1032
C32×#8352
C32×#8212
C32×#8092
C32×#4163
C3×#48043