Finite Group Databasebeta

← C32×#5766 #13179143

Subgroups — C32×#5766 #13179143

Order 1728.

1728 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1007C212yesmaximal, complemented
2C2×C33×Q16C211yesmaximal, characteristic, complemented
3C32×#942C212yesmaximal, complemented
4C32×#919C212yesmaximal, complemented
5C32×#918C214yesmaximal, complemented
6C32×D3×Q16C212yesmaximal, complemented
7C32×#904C211yesmaximal, characteristic, complemented
8C32×#901C211yesmaximal, characteristic, complemented
9C32×#578—31nomaximal
10C3×#5766C314yesmaximal, direct product factor
11C2×C33×Q8C2212yes—
12C32×#239—22no—
13C32×#239C2214yes—
14C32×D3×Q8—22no—
15C32×D3×Q8C2214yes—
16C32×#235—24no—
17C32×#235C2211yescharacteristic
18C32×#235C2212yes—
19C33×Q16C2212yes—
20C33×Q16C2212yes—
21C2×C33×C8C2211yescharacteristic
22C32×#216C2214yes—
23C32×#215C2214yes—
24C32×#208C2211yescharacteristic
25C32×Dic12C2211yescharacteristic
26C32×D24C2211yescharacteristic
27C32×#204C2212yes—
28C32×ΓL(1,𝔽25)C2212yes—
29C32×D3×C8C2212yes—
30C32×2−1+4—32no—
31C3×#1007C2×C318yescomplemented
32C32×#137—32no—
33C32×#137—34no—
34C32×#135—32no—
35C32×#135—31no—
36C2×C32×Q16—24no—
37C2×C32×Q16—32no—
38C2×C32×Q16D311yescharacteristic, complemented
39C2×C32×Q16C2×C314yescomplemented
40C32×#131—31no—
41C3×#942C2×C318yescomplemented
42C3×#919C2×C318yescomplemented
43C3×#918C2×C3116yescomplemented
44C3×D3×Q16C2×C318yescomplemented
45C3×#904C2×C314yescomplemented
46C3×#901C2×C314yescomplemented
47C33×Q8—22no—
48C33×Q8C2314yes—
49C2×C33×C4—22no—
50C2×C33×C4C2311yescharacteristic
51C32×#67—22no—
52C32×#67D412yes—
53C32×D12—24no—
54C32×D12D411yescharacteristic
55C32×D12C2312yes—
56C32×C4×D3—24no—
57C32×C4×D3—22no—
58C32×C4×D3—24no—
59C32×C4×D3C2312yes—
60C32×Dic6—24no—
61C32×Dic6D411yescharacteristic
62C32×Dic6C2312yes—
63C33×C8C2311yescharacteristic
64C33×C8C2311yescharacteristic
65C32×C3⋊C8C2312yes—
66C3×#578—34no—
67#5766C3211yescharacteristic, direct product factor
68C32×#41—31no—
69C32×#41—34no—
70C32×#41—32no—
71C32×#41—64no—
72C32×#41—62no—
73C2×C32×Q8—28no—
74C2×C32×Q8—34no—
75C2×C32×Q8—62no—
76C2×C32×Q8C2×D312yes—
77C2×C32×Q8C22×C318yes—
78C3×#239—28no—
79C3×#239C22×C3116yes—
80C3×D3×Q8—28no—
81C3×D3×Q8C22×C3116yes—
82C3×#235—216no—
83C3×#235C22×C314yes—
84C3×#235C22×C318yes—
85C32×Q16—28no—
86C32×Q16—28no—
87C32×Q16—34no—
88C32×Q16—31no—
89C32×Q16C2×D312yes—
90C32×Q16C2×D312yes—
91C32×Q16C22×C318yes—
92C32×Q16C22×C318yes—
93C32×SD16—34no—
94C32×SD16—32no—
95C32×D8—31no—
96C32×M16—31no—
97C32×M16—32no—
98C2×C32×C8—24no—
99C2×C32×C8—32no—
100C2×C32×C8C2×D311yescharacteristic
101C2×C32×C8C22×C314yes—
102C3×#216C22×C3116yes—
103C3×#215C22×C3116yes—
104C3×#208C22×C314yes—
105C3×Dic12C22×C314yes—
106C3×D24C22×C314yes—
107C3×#204C22×C318yes—
108C3×ΓL(1,𝔽25)C22×C318yes—
109C3×D3×C8C22×C318yes—
110C22×C33C2×D411yescharacteristic
111C2×C32×D3—22no—
112C2×C32×D3C2×D412yes—
113C33×C4—24no—
114C33×C4C2×D411yescharacteristic
115C33×C4C2411yescharacteristic
116C32×Dic3—22no—
117C32×Dic3C2×D412yes—
118C3×2−1+4—38no—
119#1007C2×C3212yescomplemented
120C3×#137—38no—
121C3×#137—316no—
122C3×#135—38no—
123C3×#135—34no—
124C2×C3×Q16—24no—
125C2×C3×Q16—38no—
126C2×C3×Q16C3×D314yescomplemented
127C2×C3×Q16C2×C3211yescharacteristic, complemented
128C3×#131—34no—
129#942C2×C3212yescomplemented
130#919C2×C3212yescomplemented
131#918C2×C3214yescomplemented
132D3×Q16C2×C3212yescomplemented
133#904C2×C3211yescharacteristic, complemented
134#901C2×C3211yescharacteristic, complemented
135C32×Q8—22no—
136C32×Q8—216no—
137C32×Q8—28no—
138C32×Q8—31no—
139C32×Q8—32no—
140C32×Q8—48no—
141C32×Q8—64no—
142C32×Q8C22×D314yes—
143C32×Q8C23×C3116yes—
144C32×D4—32no—
145C32×D4—31no—
146C32×D4—32no—
147C32×D4—64no—
148C32×D4—62no—
149C2×C32×C4—22no—
150C2×C32×C4—24no—
151C2×C32×C4—28no—
152C2×C32×C4—32no—
153C2×C32×C4—48no—
154C2×C32×C4—64no—
155C2×C32×C4—64no—
156C2×C32×C4—62no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C23×C314yes—
159C3×#67—28no—
160C3×#67C3×D418yes—
161C3×D12—216no—
162C3×D12C3×D414yes—
163C3×D12C23×C318yes—
164C3×C4×D3—216no—
165C3×C4×D3—28no—
166C3×C4×D3—216no—
167C3×C4×D3C23×C318yes—
168C3×Dic6—216no—
169C3×Dic6C3×D414yes—
170C3×Dic6C23×C318yes—
171C32×C8—24no—
172C32×C8—24no—
173C32×C8—32no—
174C32×C8C22×D311yescharacteristic
175C32×C8C22×D311yescharacteristic
176C32×C8C23×C314yes—
177C32×C8C23×C314yes—
178C3×C3⋊C8C23×C318yes—
179#578—31noSylow
180C2×C33—21no—
181C2×C33C22×D411yescharacteristic
182C32×D3—22no—
183C32×D3—42no—
184C3×#41—34no—
185C3×#41—316no—
186C3×#41—38no—
187C3×#41—616no—
188C3×#41—68no—
189C2×C3×Q8—28no—
190C2×C3×Q8—316no—
191C2×C3×Q8—68no—
192C2×C3×Q8C2×C3×D318yes—
193C2×C3×Q8C22×C3212yes—
194#239—22no—
195#239C22×C3214yes—
196D3×Q8—22no—
197D3×Q8C22×C3214yes—
198#235—24no—
199#235C22×C3211yescharacteristic
200#235C22×C3212yes—
201C3×Q16—28no—
202C3×Q16—28no—
203C3×Q16—316no—
204C3×Q16—34no—
205C3×Q16C2×C3×D318yes—
206C3×Q16C2×C3×D318yes—
207C3×Q16C22×C3212yes—
208C3×Q16C22×C3212yes—
209C3×SD16—316no—
210C3×SD16—38no—
211C3×D8—34no—
212C3×M16—34no—
213C3×M16—38no—
214C2×C3×C8—24no—
215C2×C3×C8—38no—
216C2×C3×C8C2×C3×D314yes—
217C2×C3×C8C22×C3211yescharacteristic
218#216C22×C3214yes—
219#215C22×C3214yes—
220#208C22×C3211yescharacteristic
221Dic12C22×C3211yescharacteristic
222D24C22×C3211yescharacteristic
223#204C22×C3212yes—
224ΓL(1,𝔽25)C22×C3212yes—
225D3×C8C22×C3212yes—
226C22×C32—24no—
227C22×C32—32no—
228C22×C32—62no—
229C22×C32D3×D411yescharacteristic
230C22×C32C2×C3×D414yes—
231C2×C3×D3—28no—
232C2×C3×D3C2×C3×D418yes—
233C32×C4—24no—
234C32×C4—24no—
235C32×C4—24no—
236C32×C4—216no—
237C32×C4—32no—
238C32×C4—416no—
239C32×C4—62no—
240C32×C4D3×D411yescharacteristic
241C32×C4C2×C3×D414yes—
242C32×C4C23×D311yescharacteristic
243C32×C4C24×C314yes—
244C3×Dic3—28no—
245C3×Dic3C2×C3×D418yes—
2462−1+4—32no—
247#137—32no—
248#137—34no—
249#135—32no—
250#135—31no—
251C2×Q16—32no—
252C2×Q16C32×D311yescharacteristic, complemented
253#131—31no—
254C33#57811yescharacteristic, complemented, Sylow
255C3×Q8—28no—
256C3×Q8—22no—
257C3×Q8—216no—
258C3×Q8—34no—
259C3×Q8—38no—
260C3×Q8—48no—
261C3×Q8—616no—
262C3×Q8C22×C3×D3116yes—
263C3×Q8C23×C3214yes—
264C3×D4—38no—
265C3×D4—38no—
266C3×D4—34no—
267C3×D4—616no—
268C3×D4—68no—
269C2×C3×C4—28no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—38no—
273C2×C3×C4—48no—
274C2×C3×C4—616no—
275C2×C3×C4—616no—
276C2×C3×C4—68no—
277C2×C3×C4C22×C3×D314yes—
278C2×C3×C4C23×C3211yescharacteristic
279#67—22no—
280#67C32×D412yes—
281D12—24no—
282D12C32×D411yescharacteristic
283D12C23×C3212yes—
284C4×D3—24no—
285C4×D3—22no—
286C4×D3—24no—
287C4×D3C23×C3212yes—
288Dic6—24no—
289Dic6C32×D411yescharacteristic
290Dic6C23×C3212yes—
291C3×C8—24no—
292C3×C8—24no—
293C3×C8—38no—
294C3×C8C22×C3×D314yes—
295C3×C8C22×C3×D314yes—
296C3×C8C23×C3211yescharacteristic
297C3×C8C23×C3211yescharacteristic
298C3⋊C8C23×C3212yes—
299C2×C32—28no—
300C2×C32—24no—
301C2×C32—21no—
302C2×C32—24no—
303C2×C32—62no—
304C2×C32—122no—
305C2×C32C2×D3×D411yescharacteristic
306C2×C32C22×C3×D414yes—
307C3×D3—28no—
308C3×D3—48no—
309#41—31no—
310#41—34no—
311#41—32no—
312#41—64no—
313#41—62no—
314C2×Q8—34no—
315C2×Q8—62no—
316C2×Q8C2×C32×D312yes—
317Q16—34no—
318Q16—31no—
319Q16C2×C32×D312yes—
320Q16C2×C32×D312yes—
321SD16—34no—
322SD16—32no—
323D8—31no—
324M16—31no—
325M16—32no—
326C2×C8—32no—
327C2×C8C2×C32×D311yescharacteristic
328C22×C3—24no—
329C22×C3—38no—
330C22×C3—68no—
331C22×C3C3×D3×D414yes—
332C22×C3C2×C32×D411yescharacteristic
333C2×D3—22no—
334C2×D3C2×C32×D412yes—
335C3×C4—216no—
336C3×C4—24no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—38no—
340C3×C4—416no—
341C3×C4—68no—
342C3×C4C3×D3×D414yes—
343C3×C4C2×C32×D411yescharacteristic
344C3×C4C23×C3×D314yes—
345C3×C4C24×C3211yescharacteristic
346Dic3—22no—
347Dic3C2×C32×D412yes—
348C32—24no—
349C32#576611yescharacteristic, direct product factor
350C32C3×#57814yescomplemented
351Q8—22no—
352Q8—31no—
353Q8—32no—
354Q8—64no—
355Q8C22×C32×D314yes—
356D4—32no—
357D4—32no—
358D4—31no—
359D4—64no—
360D4—62no—
361C2×C4—22no—
362C2×C4—32no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4—62no—
366C2×C4C22×C32×D311yescharacteristic
367C8—32no—
368C8C22×C32×D311yescharacteristic
369C8C22×C32×D311yescharacteristic
370C2×C3—28no—
371C2×C3—24no—
372C2×C3—21no—
373C2×C3—24no—
374C2×C3—68no—
375C2×C3—128no—
376C2×C3C2×C3×D3×D414yes—
377C2×C3C22×C32×D411yescharacteristic
378D3—22no—
379D3—42no—
380C22—32no—
381C22—62no—
382C22C32×D3×D411yescharacteristic
383C4—24no—
384C4—32no—
385C4—62no—
386C4C32×D3×D411yescharacteristic
387C4C23×C32×D311yescharacteristic
388C3—24no—
389C3C3×#576614yesdirect product factor
390C3C32×#57811yescharacteristic, complemented
391C2—21no—
392C2—62no—
393C2—122no—
394C2C2×C32×D3×D411yescharacteristic
subgroupG/H
centerC2×C32C2×D3×D4
commutator, cosocleC3×C4C24×C32
FrattiniC4C23×C32×D3
FittingC2×C33×Q16C2
socleC2×C33C22×D4
hypercenterC2×C32×Q16D3
hypocenterC3C32×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C2×C32×Q16

Complements

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

NH ≅ G/Ntype
1. C32×#1007C2semidirect
2. C2×C33×Q16C2semidirect
3. C32×#942C2semidirect
4. C32×#919C2semidirect
5. C32×#918C2semidirect
6. C32×D3×Q16C2semidirect
7. C32×#904C2semidirect
8. C32×#901C2semidirect
10. C3×#5766C3direct (N × H)
31. C3×#1007C2×C3semidirect
38. C2×C32×Q16D3semidirect
39. C2×C32×Q16C2×C3semidirect
41. C3×#942C2×C3semidirect
42. C3×#919C2×C3semidirect
43. C3×#918C2×C3semidirect
44. C3×D3×Q16C2×C3semidirect
45. C3×#904C2×C3semidirect
46. C3×#901C2×C3semidirect
67. #5766C32direct (N × H)
119. #1007C2×C32semidirect
126. C2×C3×Q16C3×D3semidirect
127. C2×C3×Q16C2×C32semidirect
129. #942C2×C32semidirect
130. #919C2×C32semidirect
131. #918C2×C32semidirect
132. D3×Q16C2×C32semidirect
133. #904C2×C32semidirect
134. #901C2×C32semidirect
252. C2×Q16C32×D3semidirect
254. C33#578semidirect
349. C32#5766direct (N × H)
350. C32C3×#578semidirect
389. C3C3×#5766direct (N × H)
390. C3C32×#578semidirect

Maximal subgroups

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

subgroupindex
C32×#10072
C2×C33×Q162
C32×#9422
C32×#9192
C32×#9182
C32×D3×Q162
C32×#9042
C32×#9012
C32×#5783
C3×#57663