Finite Group Databasebeta

← C32×#5729 #13179106

Subgroups — C32×#5729 #13179106

Order 1728.

1508 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C32×#946C212yesmaximal
3C32×#944C212yesmaximal, complemented
4C32×#896C212yesmaximal
5C32×#892C211yesmaximal, characteristic
6C32×D3×C4⋊C4C211yesmaximal, characteristic
7C32×#890C211yesmaximal, characteristic, complemented
8C32×#889C212yesmaximal, complemented
9C32×#888C211yesmaximal, characteristic, complemented
10C32×#872C211yesmaximal, characteristic
11C32×#870C211yesmaximal, characteristic, complemented
12C32×#544—31nomaximal
13C3×#5729C314yesmaximal, direct product factor
14C2×C33×Q8C2212yes—
15C2×C32×C4×D3C2212yes—
16C2×C32×C4×D3C2211yescharacteristic
17C2×C32×Dic6C2212yes—
18C33×C4⋊C4C2211yescharacteristic
19C33×C4⋊C4C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C33×C42C2211yescharacteristic
22C32×(C2×D3)⋊C4C2211yescharacteristic
23C32×(C2×D3)⋊C4C2212yes—
24C32×(C2×D3)⋊C4C2211yescharacteristic
25C32×C4⋊Dic3C2212yes—
26C32×C4⋊Dic3C2211yescharacteristic
27C32×C4⋊Dic3C2211yescharacteristic
28C32×Dic3⋊C4C2211yescharacteristic
29C32×Dic3⋊C4C2212yes—
30C32×Dic3⋊C4C2212yes—
31C32×Dic3⋊C4C2212yes—
32C32×Dic3⋊C4C2212yes—
33C32×Dic3⋊C4C2212yes—
34C32×Dic3⋊C4C2211yescharacteristic
35C32×C4×Dic3C2212yes—
36C32×C4×Dic3C2211yescharacteristic
37C32×C4⋊Q8—24no—
38C32×C4⋊Q8—32no—
39C32×C4⋊Q8—31no—
40C32×C4⋊Q8D311yescharacteristic, complemented
41C32×C4⋊Q8C2×C314yescomplemented
42C32×#125—32no—
43C32×#125—31no—
44C32×#125—31no—
45C32×C22⋊Q8—32no—
46C32×C22⋊Q8—32no—
47C32×#117—31no—
48C32×#117—31no—
49C2×C32×C4⋊C4—31no—
50C3×#946C2×C318yes—
51C3×#944C2×C318yescomplemented
52C3×#896C2×C318yes—
53C3×#892C2×C314yes—
54C3×D3×C4⋊C4C2×C314yes—
55C3×#890C2×C314yescomplemented
56C3×#889C2×C318yescomplemented
57C3×#888C2×C314yescomplemented
58C3×#872C2×C314yes—
59C3×#870C2×C314yescomplemented
60C22×C32×D3C2311yescharacteristic
61C33×Q8—42no—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2312yes—
65C2×C33×C4C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C32×Dic3C2311yescharacteristic
68C2×C32×Dic3C2312yes—
69C2×C32×Dic3C2312yes—
70C2×C32×Dic3C2311yescharacteristic
71C2×C32×Dic3C2311yescharacteristic
72C32×C4×D3—42no—
73C32×C4×D3Q814yescomplemented
74C32×Dic6—42no—
75C3×#544—34no—
76#5729C3211yescharacteristic, direct product factor
77C2×C32×Q8—28no—
78C2×C32×Q8—32no—
79C2×C32×Q8C2×D312yes—
80C2×C32×Q8C22×C318yes—
81C22×C32×C4—31no—
82C22×C32×C4—32no—
83C2×C3×C4×D3C22×C318yes—
84C2×C3×C4×D3C22×C314yes—
85C2×C3×Dic6C22×C318yes—
86C32×C4⋊C4—24no—
87C32×C4⋊C4—24no—
88C32×C4⋊C4—28no—
89C32×C4⋊C4—32no—
90C32×C4⋊C4—31no—
91C32×C4⋊C4—31no—
92C32×C4⋊C4—31no—
93C32×C4⋊C4—31no—
94C32×C4⋊C4—32no—
95C32×C4⋊C4—32no—
96C32×C4⋊C4—32no—
97C32×C4⋊C4—32no—
98C32×C4⋊C4—32no—
99C32×C4⋊C4C2×D311yescharacteristic
100C32×C4⋊C4C2×D311yescharacteristic
101C32×C4⋊C4C2×D312yes—
102C32×C4⋊C4C22×C314yes—
103C32×C4⋊C4C22×C318yes—
104C32×C4⋊C4C22×C314yes—
105C32×C22⋊C4—32no—
106C32×C22⋊C4—31no—
107C32×C22⋊C4—31no—
108C32×C42—24no—
109C32×C42—32no—
110C32×C42—31no—
111C32×C42C2×D311yescharacteristic
112C32×C42C22×C314yes—
113C3×(C2×D3)⋊C4C22×C318yes—
114C3×(C2×D3)⋊C4C22×C314yes—
115C3×(C2×D3)⋊C4C22×C314yes—
116C3×C4⋊Dic3C22×C314yes—
117C3×C4⋊Dic3C22×C318yes—
118C3×C4⋊Dic3C22×C314yes—
119C3×Dic3⋊C4C22×C318yes—
120C3×Dic3⋊C4C22×C314yes—
121C3×Dic3⋊C4C22×C318yes—
122C3×Dic3⋊C4C22×C318yes—
123C3×Dic3⋊C4C22×C318yes—
124C3×Dic3⋊C4C22×C314yes—
125C3×Dic3⋊C4C22×C318yes—
126C3×C4×Dic3C22×C314yes—
127C3×C4×Dic3C22×C318yes—
128C22×C33C2411yescharacteristic
129C2×C32×D3—21no—
130C2×C32×D3—21no—
131C2×C32×D3C2×Q812yes—
132C33×C4—21no—
133C33×C4—22no—
134C33×C4—22no—
135C33×C4—21no—
136C33×C4C2×Q812yes—
137C32×Dic3—21no—
138C32×Dic3—22no—
139C32×Dic3—22no—
140C32×Dic3—21no—
141C32×Dic3C2×Q812yes—
142C3×C4⋊Q8—24no—
143C3×C4⋊Q8—38no—
144C3×C4⋊Q8—34no—
145C3×C4⋊Q8C3×D314yescomplemented
146C3×C4⋊Q8C2×C3211yescharacteristic, complemented
147C3×#125—38no—
148C3×#125—34no—
149C3×#125—34no—
150C3×C22⋊Q8—38no—
151C3×C22⋊Q8—38no—
152C3×#117—34no—
153C3×#117—34no—
154C2×C3×C4⋊C4—34no—
155#946C2×C3212yes—
156#944C2×C3212yescomplemented
157#896C2×C3212yes—
158#892C2×C3211yescharacteristic
159D3×C4⋊C4C2×C3211yescharacteristic
160#890C2×C3211yescharacteristic, complemented
161#889C2×C3212yescomplemented
162#888C2×C3211yescharacteristic, complemented
163#872C2×C3211yescharacteristic
164#870C2×C3211yescharacteristic, complemented
165C23×C32—31no—
166C22×C3×D3C23×C314yes—
167C32×Q8—42no—
168C32×Q8—48no—
169C32×Q8—416no—
170C32×Q8—122no—
171C2×C32×C4—24no—
172C2×C32×C4—24no—
173C2×C32×C4—24no—
174C2×C32×C4—28no—
175C2×C32×C4—28no—
176C2×C32×C4—31no—
177C2×C32×C4—34no—
178C2×C32×C4—31no—
179C2×C32×C4—32no—
180C2×C32×C4—31no—
181C2×C32×C4—32no—
182C2×C32×C4—122no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D312yes—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C23×C318yes—
189C2×C32×C4C23×C314yes—
190C2×C32×C4C23×C314yes—
191C2×C32×C4C23×C318yes—
192C2×C32×C4C23×C314yes—
193C2×C3×Dic3C23×C314yes—
194C2×C3×Dic3C23×C318yes—
195C2×C3×Dic3C23×C314yes—
196C2×C3×Dic3C23×C318yes—
197C2×C3×Dic3C23×C314yes—
198C3×C4×D3—48no—
199C3×C4×D3C3×Q8116yescomplemented
200C3×Dic6—48no—
201#544—31noSylow
202C2×C33C22×Q811yescharacteristic
203C2×C33H(5,𝔽2)11yescharacteristic
204C2×C332−1+411yescharacteristic
205C32×D3—22no—
206C2×C3×Q8—28no—
207C2×C3×Q8—38no—
208C2×C3×Q8C2×C3×D318yes—
209C2×C3×Q8C22×C3212yes—
210C22×C3×C4—34no—
211C22×C3×C4—38no—
212C2×C4×D3C22×C3212yes—
213C2×C4×D3C22×C3211yescharacteristic
214C2×Dic6C22×C3212yes—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—38no—
219C3×C4⋊C4—34no—
220C3×C4⋊C4—34no—
221C3×C4⋊C4—38no—
222C3×C4⋊C4—38no—
223C3×C4⋊C4—38no—
224C3×C4⋊C4—38no—
225C3×C4⋊C4—34no—
226C3×C4⋊C4—34no—
227C3×C4⋊C4—38no—
228C3×C4⋊C4C2×C3×D314yes—
229C3×C4⋊C4C2×C3×D314yes—
230C3×C4⋊C4C2×C3×D318yes—
231C3×C4⋊C4C22×C3211yescharacteristic
232C3×C4⋊C4C22×C3212yes—
233C3×C4⋊C4C22×C3211yescharacteristic
234C3×C22⋊C4—38no—
235C3×C22⋊C4—34no—
236C3×C22⋊C4—34no—
237C3×C42—24no—
238C3×C42—38no—
239C3×C42—34no—
240C3×C42C2×C3×D314yes—
241C3×C42C22×C3211yescharacteristic
242(C2×D3)⋊C4C22×C3211yescharacteristic
243(C2×D3)⋊C4C22×C3211yescharacteristic
244(C2×D3)⋊C4C22×C3212yes—
245C4⋊Dic3C22×C3211yescharacteristic
246C4⋊Dic3C22×C3211yescharacteristic
247C4⋊Dic3C22×C3212yes—
248Dic3⋊C4C22×C3212yes—
249Dic3⋊C4C22×C3212yes—
250Dic3⋊C4C22×C3211yescharacteristic
251Dic3⋊C4C22×C3211yescharacteristic
252Dic3⋊C4C22×C3212yes—
253Dic3⋊C4C22×C3212yes—
254Dic3⋊C4C22×C3212yes—
255C4×Dic3C22×C3211yescharacteristic
256C4×Dic3C22×C3212yes—
257C22×C32—24no—
258C22×C32—32no—
259C22×C32—61no—
260C22×C32—61no—
261C22×C32C23×D311yescharacteristic
262C22×C32C24×C314yes—
263C2×C3×D3—24no—
264C2×C3×D3—24no—
265C2×C3×D3C2×C3×Q818yes—
266C32×C4—28no—
267C32×C4—28no—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—24no—
273C32×C4—24no—
274C32×C4—28no—
275C32×C4—28no—
276C32×C4—28no—
277C32×C4—32no—
278C32×C4—48no—
279C32×C4—48no—
280C32×C4—62no—
281C32×C4—61no—
282C32×C4—61no—
283C32×C4—62no—
284C32×C4D3×Q812yes—
285C32×C4C2×C3×Q818yes—
286C3×Dic3—24no—
287C3×Dic3—28no—
288C3×Dic3—24no—
289C3×Dic3—28no—
290C3×Dic3C2×C3×Q818yes—
291C4⋊Q8—32no—
292C4⋊Q8—31no—
293C4⋊Q8C32×D311yescharacteristic, complemented
294#125—32no—
295#125—31no—
296#125—31no—
297C22⋊Q8—32no—
298C22⋊Q8—32no—
299#117—31no—
300#117—31no—
301C2×C4⋊C4—31no—
302C33#54411yescharacteristic, complemented, Sylow
303C23×C3—34no—
304C22×D3C23×C3211yescharacteristic
305C3×Q8—48no—
306C3×Q8—42no—
307C3×Q8—416no—
308C3×Q8—128no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—28no—
313C2×C3×C4—28no—
314C2×C3×C4—34no—
315C2×C3×C4—316no—
316C2×C3×C4—34no—
317C2×C3×C4—38no—
318C2×C3×C4—34no—
319C2×C3×C4—38no—
320C2×C3×C4—128no—
321C2×C3×C4C22×C3×D318yes—
322C2×C3×C4C22×C3×D314yes—
323C2×C3×C4C22×C3×D314yes—
324C2×C3×C4C22×C3×D318yes—
325C2×C3×C4C22×C3×D314yes—
326C2×C3×C4C23×C3212yes—
327C2×C3×C4C23×C3211yescharacteristic
328C2×C3×C4C23×C3211yescharacteristic
329C2×C3×C4C23×C3212yes—
330C2×C3×C4C23×C3211yescharacteristic
331C2×Dic3C23×C3211yescharacteristic
332C2×Dic3C23×C3212yes—
333C2×Dic3C23×C3211yescharacteristic
334C2×Dic3C23×C3212yes—
335C2×Dic3C23×C3211yescharacteristic
336C4×D3—42no—
337C4×D3C32×Q814yescomplemented
338Dic6—42no—
339C2×C32—24no—
340C2×C32—24no—
341C2×C32—24no—
342C2×C32—62no—
343C2×C32#100411yescharacteristic
344C2×C32C2×D3×Q811yescharacteristic
345C2×C32#100711yescharacteristic
346C2×C32C22×C3×Q814yes—
347C2×C32C3×H(5,𝔽2)14yes—
348C2×C32C3×2−1+414yes—
349C3×D3—28no—
350C2×Q8—32no—
351C2×Q8C2×C32×D312yes—
352C22×C4—31no—
353C22×C4—32no—
354C4⋊C4—32no—
355C4⋊C4—31no—
356C4⋊C4—31no—
357C4⋊C4—31no—
358C4⋊C4—31no—
359C4⋊C4—32no—
360C4⋊C4—32no—
361C4⋊C4—32no—
362C4⋊C4—32no—
363C4⋊C4—32no—
364C4⋊C4C2×C32×D311yescharacteristic
365C4⋊C4C2×C32×D311yescharacteristic
366C4⋊C4C2×C32×D312yes—
367C22⋊C4—32no—
368C22⋊C4—31no—
369C22⋊C4—31no—
370C42—32no—
371C42—31no—
372C42C2×C32×D311yescharacteristic
373C22×C3—24no—
374C22×C3—38no—
375C22×C3—64no—
376C22×C3—64no—
377C22×C3C23×C3×D314yes—
378C22×C3C24×C3211yescharacteristic
379C2×D3—21no—
380C2×D3—21no—
381C2×D3C2×C32×Q812yes—
382C3×C4—28no—
383C3×C4—28no—
384C3×C4—28no—
385C3×C4—24no—
386C3×C4—28no—
387C3×C4—24no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—28no—
393C3×C4—38no—
394C3×C4—48no—
395C3×C4—48no—
396C3×C4—68no—
397C3×C4—64no—
398C3×C4—64no—
399C3×C4—68no—
400C3×C4C3×D3×Q818yes—
401C3×C4C2×C32×Q812yes—
402Dic3—21no—
403Dic3—22no—
404Dic3—21no—
405Dic3—22no—
406Dic3C2×C32×Q812yes—
407C32—24no—
408C32#572911yescharacteristic, direct product factor
409C32C3×#54414yescomplemented
410C23—31no—
411Q8—42no—
412Q8—122no—
413C2×C4—31no—
414C2×C4—34no—
415C2×C4—31no—
416C2×C4—32no—
417C2×C4—31no—
418C2×C4—32no—
419C2×C4—122no—
420C2×C4C22×C32×D312yes—
421C2×C4C22×C32×D311yescharacteristic
422C2×C4C22×C32×D311yescharacteristic
423C2×C4C22×C32×D312yes—
424C2×C4C22×C32×D311yescharacteristic
425C2×C3—24no—
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—68no—
429C2×C3C3×#100414yes—
430C2×C3C2×C3×D3×Q814yes—
431C2×C3C3×#100714yes—
432C2×C3C22×C32×Q811yescharacteristic
433C2×C3C32×H(5,𝔽2)11yescharacteristic
434C2×C3C32×2−1+411yescharacteristic
435D3—22no—
436C22—32no—
437C22—61no—
438C22—61no—
439C22C23×C32×D311yescharacteristic
440C4—22no—
441C4—21no—
442C4—22no—
443C4—21no—
444C4—32no—
445C4—62no—
446C4—61no—
447C4—61no—
448C4—62no—
449C4C32×D3×Q812yes—
450C3—24no—
451C3C3×#572914yesdirect product factor
452C3C32×#54411yescharacteristic, complemented
453C2—62no—
454C2C32×#100411yescharacteristic
455C2C2×C32×D3×Q811yescharacteristic
456C2C32×#100711yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC33×C4⋊Q8C2
socleC22×C33C24
prime radicalC22×C32×D3C23
hypercenterC32×C4⋊Q8D3
hypocenterC3C32×#544
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×#944C2semidirect
7. C32×#890C2semidirect
8. C32×#889C2semidirect
9. C32×#888C2semidirect
11. C32×#870C2semidirect
13. C3×#5729C3direct (N × H)
40. C32×C4⋊Q8D3semidirect
41. C32×C4⋊Q8C2×C3semidirect
51. C3×#944C2×C3semidirect
55. C3×#890C2×C3semidirect
56. C3×#889C2×C3semidirect
57. C3×#888C2×C3semidirect
59. C3×#870C2×C3semidirect
73. C32×C4×D3Q8semidirect
76. #5729C32direct (N × H)
145. C3×C4⋊Q8C3×D3semidirect
146. C3×C4⋊Q8C2×C32semidirect
156. #944C2×C32semidirect
160. #890C2×C32semidirect
161. #889C2×C32semidirect
162. #888C2×C32semidirect
164. #870C2×C32semidirect
199. C3×C4×D3C3×Q8semidirect
293. C4⋊Q8C32×D3semidirect
302. C33#544semidirect
337. C4×D3C32×Q8semidirect
408. C32#5729direct (N × H)
409. C32C3×#544semidirect
451. C3C3×#5729direct (N × H)
452. C3C32×#544semidirect

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C32×#9462
C32×#9442
C32×#8962
C32×#8922
C32×D3×C4⋊C42
C32×#8902
C32×#8892
C32×#8882
C32×#8722
C32×#8702
C32×#5443
C3×#57293