Finite Group Databasebeta

← C32×#5059 #13162492

Subgroups — C32×#5059 #13162492

Order 1728.

1084 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#124C211yesmaximal, characteristic, complemented
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5C2×C32×#213C211yesmaximal, characteristic
6C32×#875C211yesmaximal, characteristic, complemented
7C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
8C32×#495—31nomaximal
9C3×#5059C314yesmaximal, direct product factor
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C22⋊C4—22no—
15C33×C42C2211yescharacteristic
16C32×#216—22no—
17C32×#215—22no—
18C32×#214—22no—
19C32×#213—22no—
20C32×(C2×D3)⋊C4—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23C2×C32×Q16—31no—
24C2×C32×SD16—31no—
25C2×C32×SD16—31no—
26C2×C32×D8—31no—
27C32×#124—24no—
28C32×#124—31no—
29C32×#124D311yescharacteristic, complemented
30C32×#124C2×C314yescomplemented
31C2×C3×#216C2×C314yescomplemented
32C2×C3×#215C2×C314yescomplemented
33C2×C3×#214C2×C314yescomplemented
34C2×C3×#213C2×C314yes—
35C3×#875C2×C314yescomplemented
36C32×C4×C8—31no—
37C3×C4×C3⋊C8C2×C314yescomplemented
38C23×C33—21no—
39C22×C32×D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4—21no—
45C2×C33×C4D412yes—
46C2×C33×C4C2311yescharacteristic
47C2×C32×Dic3—21no—
48C32×D12—21no—
49C32×D12—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C32×C3⋊C8D412yes—
53C32×C3⋊C8D412yes—
54C3×#495—34no—
55#5059C3211yescharacteristic, direct product factor
56C2×C32×Q8—24no—
57C2×C32×Q8—31no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C314yes—
60C2×C32×D4—24no—
61C2×C32×D4—31no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C22×C314yes—
64C2×C3×D12C22×C314yes—
65C2×C3×Dic6C22×C314yes—
66C32×Q16—62no—
67C32×SD16—62no—
68C32×SD16—62no—
69C32×D8—62no—
70C2×C32×C8—31no—
71C2×C32×C8—31no—
72C32×C22⋊C4—216no—
73C32×C22⋊C4—22no—
74C32×C22⋊C4—28no—
75C32×C22⋊C4—62no—
76C32×C42—24no—
77C32×C42C2×D311yescharacteristic
78C32×C42C22×C314yes—
79C3×#216—28no—
80C3×#215—28no—
81C3×#214—28no—
82C3×#213—28no—
83C3×(C2×D3)⋊C4—28no—
84C2×C3×C3⋊C8C22×C314yes—
85C2×C3×C3⋊C8C22×C314yes—
86C22×C33—41no—
87C22×C33—42no—
88C22×C33C2×D411yescharacteristic
89C2×C32×D3—41no—
90C2×C32×D3—42no—
91C33×C4—22no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4C2×D411yescharacteristic
95C32×Dic3—41no—
96C2×C3×Q16—34no—
97C2×C3×SD16—34no—
98C2×C3×SD16—34no—
99C2×C3×D8—34no—
100C3×#124—24no—
101C3×#124—34no—
102C3×#124C3×D314yescomplemented
103C3×#124C2×C3211yescharacteristic, complemented
104C2×#216C2×C3211yescharacteristic, complemented
105C2×#215C2×C3211yescharacteristic, complemented
106C2×#214C2×C3211yescharacteristic, complemented
107C2×#213C2×C3211yescharacteristic
108#875C2×C3211yescharacteristic, complemented
109C3×C4×C8—34no—
110C4×C3⋊C8C2×C3211yescharacteristic, complemented
111C23×C32—28no—
112C23×C32—21no—
113C23×C32—24no—
114C23×C32—61no—
115C22×C3×D3—24no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—24no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8—44no—
122C32×Q8—61no—
123C32×Q8—61no—
124C32×D4—21no—
125C32×D4—21no—
126C32×D4—24no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4—44no—
130C32×D4—61no—
131C32×D4—61no—
132C2×C32×C4—28no—
133C2×C32×C4—21no—
134C2×C32×C4—24no—
135C2×C32×C4—28no—
136C2×C32×C4—24no—
137C2×C32×C4—61no—
138C2×C32×C4#6712yes—
139C2×C32×C4C3×D418yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C23×C314yes—
142C2×C3×Dic3—24no—
143C3×D12—24no—
144C3×D12—24no—
145C3×Dic6—24no—
146C3×Dic6—24no—
147C32×C8—32no—
148C32×C8—32no—
149C3×C3⋊C8C3×D418yes—
150C3×C3⋊C8C3×D418yes—
151#495—31noSylow
152C2×C33—81no—
153C2×C33#12711yescharacteristic
154C2×C33#13512yes—
155C32×D3—81no—
156C2×C3×Q8—24no—
157C2×C3×Q8—34no—
158C2×C3×Q8C2×C3×D314yes—
159C2×C3×Q8C22×C3211yescharacteristic
160C2×C3×D4—24no—
161C2×C3×D4—34no—
162C2×C3×D4C2×C3×D314yes—
163C2×C3×D4C22×C3211yescharacteristic
164C2×D12C22×C3211yescharacteristic
165C2×Dic6C22×C3211yescharacteristic
166C3×Q16—68no—
167C3×SD16—68no—
168C3×SD16—68no—
169C3×D8—68no—
170C2×C3×C8—34no—
171C2×C3×C8—34no—
172C3×C22⋊C4—216no—
173C3×C22⋊C4—28no—
174C3×C22⋊C4—22no—
175C3×C22⋊C4—68no—
176C3×C42—24no—
177C3×C42C2×C3×D314yes—
178C3×C42C22×C3211yescharacteristic
179#216—22no—
180#215—22no—
181#214—22no—
182#213—22no—
183(C2×D3)⋊C4—22no—
184C2×C3⋊C8C22×C3211yescharacteristic
185C2×C3⋊C8C22×C3211yescharacteristic
186C22×C32—24no—
187C22×C32—48no—
188C22×C32—416no—
189C22×C32—41no—
190C22×C32—42no—
191C22×C32—44no—
192C22×C32—48no—
193C22×C32—121no—
194C22×C32—122no—
195C22×C32C2×#6711yescharacteristic
196C22×C32C2×C3×D414yes—
197C2×C3×D3—44no—
198C2×C3×D3—48no—
199C32×C4—22no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—28no—
203C32×C4—48no—
204C32×C4—48no—
205C32×C4—41no—
206C32×C4—44no—
207C32×C4—121no—
208C32×C4D3×D411yescharacteristic
209C32×C4D3×D411yescharacteristic
210C32×C4C2×C3×D414yes—
211C32×C4C2×C3×D414yes—
212C3×Dic3—44no—
213C2×Q16—31no—
214C2×SD16—31no—
215C2×SD16—31no—
216C2×D8—31no—
217#124—31no—
218#124C32×D311yescharacteristic, complemented
219C4×C8—31no—
220C33#49511yescharacteristic, complemented, Sylow
221C23×C3—28no—
222C23×C3—24no—
223C23×C3—21no—
224C23×C3—64no—
225C22×D3—21no—
226C3×Q8—24no—
227C3×Q8—24no—
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—44no—
231C3×Q8—44no—
232C3×Q8—64no—
233C3×Q8—64no—
234C3×D4—24no—
235C3×D4—24no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—44no—
239C3×D4—44no—
240C3×D4—64no—
241C3×D4—64no—
242C2×C3×C4—28no—
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—28no—
246C2×C3×C4—24no—
247C2×C3×C4—64no—
248C2×C3×C4C3×#6718yes—
249C2×C3×C4C32×D412yes—
250C2×C3×C4C22×C3×D314yes—
251C2×C3×C4C23×C3211yescharacteristic
252C2×Dic3—21no—
253D12—21no—
254D12—21no—
255Dic6—21no—
256Dic6—21no—
257C3×C8—38no—
258C3×C8—38no—
259C3⋊C8C32×D412yes—
260C3⋊C8C32×D412yes—
261C2×C32—24no—
262C2×C32—28no—
263C2×C32—88no—
264C2×C32—81no—
265C2×C32—84no—
266C2×C32—241no—
267C2×C32#94011yescharacteristic
268C2×C32#95012yes—
269C2×C32C3×#12714yes—
270C2×C32C3×#13518yes—
271C3×D3—84no—
272C2×Q8—31no—
273C2×Q8C2×C32×D311yescharacteristic
274C2×D4—31no—
275C2×D4C2×C32×D311yescharacteristic
276Q16—62no—
277SD16—62no—
278SD16—62no—
279D8—62no—
280C2×C8—31no—
281C2×C8—31no—
282C22⋊C4—22no—
283C22⋊C4—62no—
284C42C2×C32×D311yescharacteristic
285C22×C3—24no—
286C22×C3—48no—
287C22×C3—416no—
288C22×C3—44no—
289C22×C3—48no—
290C22×C3—41no—
291C22×C3—42no—
292C22×C3—124no—
293C22×C3—128no—
294C22×C3C2×C3×#6714yes—
295C22×C3C2×C32×D411yescharacteristic
296C2×D3—41no—
297C2×D3—42no—
298C3×C4—28no—
299C3×C4—22no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—48no—
303C3×C4—48no—
304C3×C4—44no—
305C3×C4—41no—
306C3×C4—124no—
307C3×C4C3×D3×D414yes—
308C3×C4C3×D3×D414yes—
309C3×C4C2×C32×D411yescharacteristic
310C3×C4C2×C32×D411yescharacteristic
311Dic3—41no—
312C32—24no—
313C32#505911yescharacteristic, direct product factor
314C32C3×#49514yescomplemented
315C23—21no—
316C23—61no—
317Q8—21no—
318Q8—21no—
319Q8—61no—
320Q8—61no—
321D4—21no—
322D4—21no—
323D4—61no—
324D4—61no—
325C2×C4—21no—
326C2×C4—61no—
327C2×C4C32×#6712yes—
328C2×C4C22×C32×D311yescharacteristic
329C8—32no—
330C8—32no—
331C2×C3—24no—
332C2×C3—28no—
333C2×C3—88no—
334C2×C3—81no—
335C2×C3—84no—
336C2×C3—244no—
337C2×C3C3×#94014yes—
338C2×C3C3×#95018yes—
339C2×C3C32×#12711yescharacteristic
340C2×C3C32×#13512yes—
341D3—81no—
342C22—41no—
343C22—42no—
344C22—121no—
345C22—122no—
346C22C2×C32×#6711yescharacteristic
347C4—22no—
348C4—41no—
349C4—121no—
350C4C32×D3×D411yescharacteristic
351C4C32×D3×D411yescharacteristic
352C3—24no—
353C3C3×#505914yesdirect product factor
354C3C32×#49511yescharacteristic, complemented
355C2—81no—
356C2—241no—
357C2C32×#94011yescharacteristic
358C2C32×#95012yes—
subgroupG/H
centerC22×C32C2×#67
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
FittingC33×#124C2
socleC22×C33C2×D4
prime radicalC2×C32×#213C2
hypercenterC32×#124D3
hypocenterC3C32×#495
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×C42 ◁ C32×#124

Complements

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

NH ≅ G/Ntype
1. C33×#124C2semidirect
2. C2×C32×#216C2semidirect
3. C2×C32×#215C2semidirect
4. C2×C32×#214C2semidirect
6. C32×#875C2semidirect
7. C32×C4×C3⋊C8C2semidirect
9. C3×#5059C3direct (N × H)
29. C32×#124D3semidirect
30. C32×#124C2×C3semidirect
31. C2×C3×#216C2×C3semidirect
32. C2×C3×#215C2×C3semidirect
33. C2×C3×#214C2×C3semidirect
35. C3×#875C2×C3semidirect
37. C3×C4×C3⋊C8C2×C3semidirect
55. #5059C32direct (N × H)
102. C3×#124C3×D3semidirect
103. C3×#124C2×C32semidirect
104. C2×#216C2×C32semidirect
105. C2×#215C2×C32semidirect
106. C2×#214C2×C32semidirect
108. #875C2×C32semidirect
110. C4×C3⋊C8C2×C32semidirect
218. #124C32×D3semidirect
220. C33#495semidirect
313. C32#5059direct (N × H)
314. C32C3×#495semidirect
353. C3C3×#5059direct (N × H)
354. C3C32×#495semidirect

Maximal subgroups

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

subgroupindex
C33×#1242
C2×C32×#2162
C2×C32×#2152
C2×C32×#2142
C2×C32×#2132
C32×#8752
C32×C4×C3⋊C82
C32×#4953
C3×#50593