Finite Group Databasebeta

← C32×#5735 #13179112

Subgroups — C32×#5735 #13179112

Order 1728.

1448 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C32×#947C211yesmaximal, characteristic
3C32×#946C212yesmaximal, complemented
4C32×#944C211yesmaximal, characteristic, complemented
5C32×#898C212yesmaximal, complemented
6C32×#896C212yesmaximal, complemented
7C32×#894C212yesmaximal
8C32×#889C212yesmaximal, complemented
9C32×#876C212yesmaximal, complemented
10C32×#563—31nomaximal
11C3×#5735C314yesmaximal, direct product factor
12C2×C33×Q8C2211yescharacteristic
13C2×C33×Q8C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3C2212yes—
16C2×C32×Dic6C2211yescharacteristic
17C33×C4⋊C4C2212yes—
18C33×C4⋊C4C2212yes—
19C33×C42C2211yescharacteristic
20C32×(C2×D3)⋊C4C2212yes—
21C32×(C2×D3)⋊C4C2212yes—
22C32×(C2×D3)⋊C4C2212yes—
23C32×(C2×D3)⋊C4C2212yes—
24C32×(C2×D3)⋊C4C2212yes—
25C32×C4⋊Dic3C2212yes—
26C32×Dic3⋊C4C2212yes—
27C32×Dic3⋊C4C2212yes—
28C32×Dic3⋊C4C2212yes—
29C32×Dic3⋊C4C2212yes—
30C32×Dic3⋊C4C2212yes—
31C32×C4×Dic3C2211yescharacteristic
32C32×C4×Dic3C2211yescharacteristic
33C32×C4⋊Q8—24no—
34C32×C4⋊Q8—31no—
35C32×C4⋊Q8D311yescharacteristic, complemented
36C32×C4⋊Q8C2×C314yescomplemented
37C32×#126—32no—
38C32×#126—32no—
39C32×#125—32no—
40C32×#124—31no—
41C32×#123—32no—
42C32×C22⋊Q8—32no—
43C32×C22⋊Q8—32no—
44C3×#947C2×C314yes—
45C3×#946C2×C318yescomplemented
46C3×#944C2×C314yescomplemented
47C3×#898C2×C318yescomplemented
48C3×#896C2×C318yescomplemented
49C3×#894C2×C318yes—
50C3×#889C2×C318yescomplemented
51C3×#876C2×C318yescomplemented
52C22×C32×D3C2312yes—
53C33×Q8—41no—
54C33×Q8—41no—
55C2×C33×C4C2312yes—
56C2×C33×C4C2311yescharacteristic
57C2×C33×C4C2311yescharacteristic
58C2×C33×C4C2312yes—
59C2×C33×C4C2311yescharacteristic
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2312yes—
62C2×C32×Dic3C2312yes—
63C32×D12—41no—
64C32×C4×D3—42no—
65C32×Dic6—41no—
66C3×#563—34no—
67#5735C3211yescharacteristic, direct product factor
68C2×C32×Q8—24no—
69C2×C32×Q8—24no—
70C2×C32×Q8—31no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C314yes—
74C2×C32×Q8C22×C314yes—
75C2×C32×D4—31no—
76C22×C32×C4—32no—
77C2×C3×D12C22×C314yes—
78C2×C3×C4×D3C22×C318yes—
79C2×C3×Dic6C22×C314yes—
80C32×C4⋊C4—28no—
81C32×C4⋊C4—28no—
82C32×C4⋊C4—32no—
83C32×C4⋊C4—32no—
84C32×C4⋊C4—32no—
85C32×C4⋊C4—32no—
86C32×C4⋊C4—32no—
87C32×C4⋊C4—32no—
88C32×C4⋊C4C2×D312yes—
89C32×C4⋊C4C2×D312yes—
90C32×C4⋊C4C22×C318yes—
91C32×C4⋊C4C22×C318yes—
92C32×C22⋊C4—32no—
93C32×C22⋊C4—32no—
94C32×C22⋊C4—32no—
95C32×C22⋊C4—32no—
96C32×C22⋊C4—32no—
97C32×C42—24no—
98C32×C42—31no—
99C32×C42—31no—
100C32×C42C2×D311yescharacteristic
101C32×C42C22×C314yes—
102C3×(C2×D3)⋊C4C22×C318yes—
103C3×(C2×D3)⋊C4C22×C318yes—
104C3×(C2×D3)⋊C4C22×C318yes—
105C3×(C2×D3)⋊C4C22×C318yes—
106C3×(C2×D3)⋊C4C22×C318yes—
107C3×C4⋊Dic3C22×C318yes—
108C3×Dic3⋊C4C22×C318yes—
109C3×Dic3⋊C4C22×C318yes—
110C3×Dic3⋊C4C22×C318yes—
111C3×Dic3⋊C4C22×C318yes—
112C3×Dic3⋊C4C22×C318yes—
113C3×C4×Dic3C22×C314yes—
114C3×C4×Dic3C22×C314yes—
115C22×C33C2411yescharacteristic
116C2×C32×D3—22no—
117C2×C32×D3—24no—
118C33×C4—21no—
119C33×C4—21no—
120C33×C4—22no—
121C33×C4—21no—
122C33×C4—22no—
123C32×Dic3—22no—
124C32×Dic3—22no—
125C32×Dic3—22no—
126C3×C4⋊Q8—24no—
127C3×C4⋊Q8—34no—
128C3×C4⋊Q8C3×D314yescomplemented
129C3×C4⋊Q8C2×C3211yescharacteristic, complemented
130C3×#126—38no—
131C3×#126—38no—
132C3×#125—38no—
133C3×#124—34no—
134C3×#123—38no—
135C3×C22⋊Q8—38no—
136C3×C22⋊Q8—38no—
137#947C2×C3211yescharacteristic
138#946C2×C3212yescomplemented
139#944C2×C3211yescharacteristic, complemented
140#898C2×C3212yescomplemented
141#896C2×C3212yescomplemented
142#894C2×C3212yes—
143#889C2×C3212yescomplemented
144#876C2×C3212yescomplemented
145C23×C32—32no—
146C22×C3×D3C23×C318yes—
147C32×Q8—41no—
148C32×Q8—41no—
149C32×Q8—44no—
150C32×Q8—44no—
151C32×Q8—48no—
152C32×Q8—48no—
153C32×Q8—121no—
154C32×D4—121no—
155C2×C32×C4—28no—
156C2×C32×C4—24no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—28no—
160C2×C32×C4—32no—
161C2×C32×C4—32no—
162C2×C32×C4—32no—
163C2×C32×C4—122no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D312yes—
169C2×C32×C4C23×C314yes—
170C2×C32×C4C23×C314yes—
171C2×C32×C4C23×C318yes—
172C2×C32×C4C23×C314yes—
173C2×C32×C4C23×C318yes—
174C2×C3×Dic3C23×C318yes—
175C2×C3×Dic3C23×C318yes—
176C2×C3×Dic3C23×C318yes—
177C3×D12—44no—
178C3×C4×D3—48no—
179C3×Dic6—44no—
180#563—31noSylow
181C2×C33H(5,𝔽2)11yescharacteristic
182C2×C332−1+412yes—
183C32×D3—42no—
184C2×C3×Q8—24no—
185C2×C3×Q8—24no—
186C2×C3×Q8—34no—
187C2×C3×Q8C2×C3×D314yes—
188C2×C3×Q8C2×C3×D314yes—
189C2×C3×Q8C22×C3211yescharacteristic
190C2×C3×Q8C22×C3211yescharacteristic
191C2×C3×D4—34no—
192C22×C3×C4—38no—
193C2×D12C22×C3211yescharacteristic
194C2×C4×D3C22×C3212yes—
195C2×Dic6C22×C3211yescharacteristic
196C3×C4⋊C4—28no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—38no—
199C3×C4⋊C4—38no—
200C3×C4⋊C4—38no—
201C3×C4⋊C4—38no—
202C3×C4⋊C4—38no—
203C3×C4⋊C4—38no—
204C3×C4⋊C4C2×C3×D318yes—
205C3×C4⋊C4C2×C3×D318yes—
206C3×C4⋊C4C22×C3212yes—
207C3×C4⋊C4C22×C3212yes—
208C3×C22⋊C4—38no—
209C3×C22⋊C4—38no—
210C3×C22⋊C4—38no—
211C3×C22⋊C4—38no—
212C3×C22⋊C4—38no—
213C3×C42—24no—
214C3×C42—34no—
215C3×C42—34no—
216C3×C42C2×C3×D314yes—
217C3×C42C22×C3211yescharacteristic
218(C2×D3)⋊C4C22×C3212yes—
219(C2×D3)⋊C4C22×C3212yes—
220(C2×D3)⋊C4C22×C3212yes—
221(C2×D3)⋊C4C22×C3212yes—
222(C2×D3)⋊C4C22×C3212yes—
223C4⋊Dic3C22×C3212yes—
224Dic3⋊C4C22×C3212yes—
225Dic3⋊C4C22×C3212yes—
226Dic3⋊C4C22×C3212yes—
227Dic3⋊C4C22×C3212yes—
228Dic3⋊C4C22×C3212yes—
229C4×Dic3C22×C3211yescharacteristic
230C4×Dic3C22×C3211yescharacteristic
231C22×C32—24no—
232C22×C32—62no—
233C22×C32—64no—
234C22×C32C23×D311yescharacteristic
235C22×C32C24×C314yes—
236C2×C3×D3—28no—
237C2×C3×D3—216no—
238C32×C4—216no—
239C32×C4—28no—
240C32×C4—22no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—22no—
245C32×C4—24no—
246C32×C4—28no—
247C32×C4—24no—
248C32×C4—28no—
249C32×C4—24no—
250C32×C4—48no—
251C32×C4—44no—
252C32×C4—44no—
253C32×C4—62no—
254C32×C4—62no—
255C32×C4—62no—
256C3×Dic3—28no—
257C3×Dic3—28no—
258C3×Dic3—28no—
259C4⋊Q8—31no—
260C4⋊Q8C32×D311yescharacteristic, complemented
261#126—32no—
262#126—32no—
263#125—32no—
264#124—31no—
265#123—32no—
266C22⋊Q8—32no—
267C22⋊Q8—32no—
268C33#56311yescharacteristic, complemented, Sylow
269C23×C3—38no—
270C22×D3C23×C3212yes—
271C3×Q8—44no—
272C3×Q8—44no—
273C3×Q8—41no—
274C3×Q8—41no—
275C3×Q8—48no—
276C3×Q8—48no—
277C3×Q8—124no—
278C3×D4—124no—
279C2×C3×C4—28no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—28no—
284C2×C3×C4—38no—
285C2×C3×C4—38no—
286C2×C3×C4—38no—
287C2×C3×C4—128no—
288C2×C3×C4C22×C3×D314yes—
289C2×C3×C4C22×C3×D314yes—
290C2×C3×C4C22×C3×D318yes—
291C2×C3×C4C22×C3×D314yes—
292C2×C3×C4C22×C3×D318yes—
293C2×C3×C4C23×C3211yescharacteristic
294C2×C3×C4C23×C3211yescharacteristic
295C2×C3×C4C23×C3212yes—
296C2×C3×C4C23×C3211yescharacteristic
297C2×C3×C4C23×C3212yes—
298C2×Dic3C23×C3212yes—
299C2×Dic3C23×C3212yes—
300C2×Dic3C23×C3212yes—
301D12—41no—
302C4×D3—42no—
303Dic6—41no—
304C2×C32—24no—
305C2×C32—28no—
306C2×C32—122no—
307C2×C32#100411yescharacteristic
308C2×C32#100712yes—
309C2×C32C3×H(5,𝔽2)14yes—
310C2×C32C3×2−1+418yes—
311C3×D3—48no—
312C2×Q8—31no—
313C2×Q8C2×C32×D311yescharacteristic
314C2×Q8C2×C32×D311yescharacteristic
315C2×D4—31no—
316C22×C4—32no—
317C4⋊C4—32no—
318C4⋊C4—32no—
319C4⋊C4—32no—
320C4⋊C4—32no—
321C4⋊C4—32no—
322C4⋊C4—32no—
323C4⋊C4C2×C32×D312yes—
324C4⋊C4C2×C32×D312yes—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—32no—
330C42—31no—
331C42—31no—
332C42C2×C32×D311yescharacteristic
333C22×C3—24no—
334C22×C3—68no—
335C22×C3—616no—
336C22×C3C23×C3×D314yes—
337C22×C3C24×C3211yescharacteristic
338C2×D3—22no—
339C2×D3—24no—
340C3×C4—28no—
341C3×C4—216no—
342C3×C4—28no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—24no—
346C3×C4—28no—
347C3×C4—21no—
348C3×C4—22no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—22no—
352C3×C4—48no—
353C3×C4—44no—
354C3×C4—44no—
355C3×C4—68no—
356C3×C4—68no—
357C3×C4—68no—
358Dic3—22no—
359Dic3—22no—
360Dic3—22no—
361C32—24no—
362C32#573511yescharacteristic, direct product factor
363C32C3×#56314yescomplemented
364C23—32no—
365Q8—41no—
366Q8—41no—
367Q8—121no—
368D4—121no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—122no—
373C2×C4C22×C32×D311yescharacteristic
374C2×C4C22×C32×D311yescharacteristic
375C2×C4C22×C32×D312yes—
376C2×C4C22×C32×D311yescharacteristic
377C2×C4C22×C32×D312yes—
378C2×C3—24no—
379C2×C3—28no—
380C2×C3—128no—
381C2×C3C3×#100414yes—
382C2×C3C3×#100718yes—
383C2×C3C32×H(5,𝔽2)11yescharacteristic
384C2×C3C32×2−1+412yes—
385D3—42no—
386C22—62no—
387C22—64no—
388C22C23×C32×D311yescharacteristic
389C4—22no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—62no—
395C4—62no—
396C4—62no—
397C3—24no—
398C3C3×#573514yesdirect product factor
399C3C32×#56311yescharacteristic, complemented
400C2—122no—
401C2C32×#100411yescharacteristic
402C2C32×#100712yes—
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC33×C4⋊Q8C2
socleC22×C33C24
prime radicalC2×C32×D12C22
hypercenterC32×C4⋊Q8D3
hypocenterC3C32×#563
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4⋊Q8C2semidirect
3. C32×#946C2semidirect
4. C32×#944C2semidirect
5. C32×#898C2semidirect
6. C32×#896C2semidirect
8. C32×#889C2semidirect
9. C32×#876C2semidirect
11. C3×#5735C3direct (N × H)
35. C32×C4⋊Q8D3semidirect
36. C32×C4⋊Q8C2×C3semidirect
45. C3×#946C2×C3semidirect
46. C3×#944C2×C3semidirect
47. C3×#898C2×C3semidirect
48. C3×#896C2×C3semidirect
50. C3×#889C2×C3semidirect
51. C3×#876C2×C3semidirect
67. #5735C32direct (N × H)
128. C3×C4⋊Q8C3×D3semidirect
129. C3×C4⋊Q8C2×C32semidirect
138. #946C2×C32semidirect
139. #944C2×C32semidirect
140. #898C2×C32semidirect
141. #896C2×C32semidirect
143. #889C2×C32semidirect
144. #876C2×C32semidirect
260. C4⋊Q8C32×D3semidirect
268. C33#563semidirect
362. C32#5735direct (N × H)
363. C32C3×#563semidirect
398. C3C3×#5735direct (N × H)
399. C3C32×#563semidirect

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C32×#9472
C32×#9462
C32×#9442
C32×#8982
C32×#8962
C32×#8942
C32×#8892
C32×#8762
C32×#5633
C3×#57353