Finite Group Databasebeta

← C32×#5602 #13178981

Subgroups — C32×#5602 #13178981

Order 1728.

1684 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#121C211yesmaximal, characteristic
2C32×#935C212yesmaximal, complemented
3C32×#935C212yesmaximal, complemented
4C32×D4×Dic3C212yesmaximal, complemented
5C32×#924C211yesmaximal, characteristic, complemented
6C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C32×#889C212yesmaximal, complemented
8C32×#879C212yesmaximal, complemented
9C32×#878C212yesmaximal, complemented
10C32×#539—31nomaximal
11C3×#5602C314yesmaximal, direct product factor
12C2×C33×D4C2212yes—
13C2×C33×D4C2211yescharacteristic
14C22×C33×C4C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C22×C32×Dic3C2212yes—
18C2×C32×Dic6C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C33×C22⋊C4C2212yes—
21C32×#217C2212yes—
22C32×#217C2212yes—
23C32×#217C2212yes—
24C32×#217C2212yes—
25C32×C4⋊Dic3C2211yescharacteristic
26C32×C4⋊Dic3C2212yes—
27C32×Dic3⋊C4C2212yes—
28C32×Dic3⋊C4C2212yes—
29C32×Dic3⋊C4C2212yes—
30C32×Dic3⋊C4C2212yes—
31C32×Dic3⋊C4C2212yes—
32C32×C4×Dic3C2212yes—
33C32×#126—32no—
34C32×#125—32no—
35C32×#123—32no—
36C32×#123—32no—
37C32×C22⋊Q8—31no—
38C32×C22⋊Q8—32no—
39C32×#121—24no—
40C32×#121D311yescharacteristic
41C32×#121C2×C314yes—
42C32×C4×D4—32no—
43C2×C32×C4⋊C4—31no—
44C3×#935C2×C318yescomplemented
45C3×#935C2×C318yescomplemented
46C3×D4×Dic3C2×C318yescomplemented
47C3×#924C2×C314yescomplemented
48C2×C3×Dic3⋊C4C2×C314yescomplemented
49C3×#889C2×C318yescomplemented
50C3×#879C2×C318yescomplemented
51C3×#878C2×C318yescomplemented
52C23×C33C2312yes—
53C23×C33C2311yescharacteristic
54C33×D4—24no—
55C33×D4—41no—
56C2×C33×C4—41no—
57C2×C33×C4C2311yescharacteristic
58C2×C33×C4C2312yes—
59C2×C33×C4C2311yescharacteristic
60C2×C32×Dic3—22no—
61C2×C32×Dic3—42no—
62C2×C32×Dic3—41no—
63C2×C32×Dic3C2312yes—
64C2×C32×Dic3C2312yes—
65C2×C32×Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67C32×Dic6—41no—
68C3×#539—34no—
69#5602C3211yescharacteristic, direct product factor
70C2×C32×Q8—31no—
71C2×C32×D4—24no—
72C2×C32×D4—28no—
73C2×C32×D4C2×D312yes—
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C318yes—
76C2×C32×D4C22×C314yes—
77C22×C32×C4—24no—
78C22×C32×C4—31no—
79C22×C32×C4—31no—
80C22×C32×C4—32no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C22×C314yes—
83C22×C3×Dic3C22×C314yes—
84C22×C3×Dic3C22×C318yes—
85C22×C3×Dic3C22×C314yes—
86C2×C3×Dic6C22×C314yes—
87C32×C4⋊C4—24no—
88C32×C4⋊C4—32no—
89C32×C4⋊C4—32no—
90C32×C4⋊C4—32no—
91C32×C4⋊C4—32no—
92C32×C4⋊C4—32no—
93C32×C4⋊C4—31no—
94C32×C4⋊C4—32no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C22×C314yes—
97C32×C22⋊C4—28no—
98C32×C22⋊C4—32no—
99C32×C22⋊C4—32no—
100C32×C22⋊C4—32no—
101C32×C22⋊C4—32no—
102C32×C22⋊C4C2×D312yes—
103C32×C22⋊C4C22×C318yes—
104C32×C42—32no—
105C3×#217C22×C318yes—
106C3×#217C22×C318yes—
107C3×#217C22×C318yes—
108C3×#217C22×C318yes—
109C3×C4⋊Dic3C22×C318yes—
110C3×C4⋊Dic3C22×C314yes—
111C3×Dic3⋊C4C22×C318yes—
112C3×Dic3⋊C4C22×C318yes—
113C3×Dic3⋊C4C22×C318yes—
114C3×Dic3⋊C4C22×C318yes—
115C3×Dic3⋊C4C22×C318yes—
116C3×C4×Dic3C22×C318yes—
117C22×C33—21no—
118C22×C33—21no—
119C22×C33—22no—
120C22×C33—22no—
121C22×C33—22no—
122C22×C33#4112yes—
123C22×C33C2411yescharacteristic
124C33×C4—22no—
125C33×C4—21no—
126C33×C4—21no—
127C32×Dic3—22no—
128C32×Dic3—22no—
129C32×Dic3—22no—
130C32×Dic3—22no—
131C3×#126—38no—
132C3×#125—38no—
133C3×#123—38no—
134C3×#123—38no—
135C3×C22⋊Q8—34no—
136C3×C22⋊Q8—38no—
137C3×#121—24no—
138C3×#121C3×D314yes—
139C3×#121C2×C3211yescharacteristic
140C3×C4×D4—38no—
141C2×C3×C4⋊C4—34no—
142#935C2×C3212yescomplemented
143#935C2×C3212yescomplemented
144D4×Dic3C2×C3212yescomplemented
145#924C2×C3211yescharacteristic, complemented
146C2×Dic3⋊C4C2×C3211yescharacteristic, complemented
147#889C2×C3212yescomplemented
148#879C2×C3212yescomplemented
149#878C2×C3212yescomplemented
150C23×C32—24no—
151C23×C32—28no—
152C23×C32C22×D312yes—
153C23×C32C22×D311yescharacteristic
154C23×C32C23×C318yes—
155C23×C32C23×C314yes—
156C32×Q8—121no—
157C32×D4—24no—
158C32×D4—216no—
159C32×D4—416no—
160C32×D4—41no—
161C32×D4—44no—
162C32×D4—48no—
163C2×C32×C4—24no—
164C2×C32×C4—24no—
165C2×C32×C4—28no—
166C2×C32×C4—32no—
167C2×C32×C4—32no—
168C2×C32×C4—32no—
169C2×C32×C4—32no—
170C2×C32×C4—41no—
171C2×C32×C4—44no—
172C2×C32×C4—48no—
173C2×C32×C4—62no—
174C2×C32×C4—121no—
175C2×C32×C4—122no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C23×C318yes—
180C2×C32×C4C23×C314yes—
181C2×C32×C4C23×C314yes—
182C2×C3×Dic3—28no—
183C2×C3×Dic3—48no—
184C2×C3×Dic3—44no—
185C2×C3×Dic3C23×C318yes—
186C2×C3×Dic3C23×C318yes—
187C2×C3×Dic3C23×C318yes—
188C2×C3×Dic3C23×C318yes—
189C3×Dic6—44no—
190#539—31noSylow
191C2×C33—22no—
192C2×C33—42no—
193C2×C33C2×#4111yescharacteristic
194C2×C33H(5,𝔽2)11yescharacteristic
195C2×C332−1+411yescharacteristic
196C2×C3×Q8—34no—
197C2×C3×D4—28no—
198C2×C3×D4—24no—
199C2×C3×D4C2×C3×D318yes—
200C2×C3×D4C2×C3×D314yes—
201C2×C3×D4C22×C3212yes—
202C2×C3×D4C22×C3211yescharacteristic
203C22×C3×C4—24no—
204C22×C3×C4—34no—
205C22×C3×C4—34no—
206C22×C3×C4—38no—
207C22×C3×C4C2×C3×D314yes—
208C22×C3×C4C22×C3211yescharacteristic
209C22×Dic3C22×C3211yescharacteristic
210C22×Dic3C22×C3212yes—
211C22×Dic3C22×C3211yescharacteristic
212C2×Dic6C22×C3211yescharacteristic
213C3×C4⋊C4—24no—
214C3×C4⋊C4—38no—
215C3×C4⋊C4—38no—
216C3×C4⋊C4—38no—
217C3×C4⋊C4—38no—
218C3×C4⋊C4—38no—
219C3×C4⋊C4—38no—
220C3×C4⋊C4—34no—
221C3×C4⋊C4C2×C3×D314yes—
222C3×C4⋊C4C22×C3211yescharacteristic
223C3×C22⋊C4—28no—
224C3×C22⋊C4—38no—
225C3×C22⋊C4—38no—
226C3×C22⋊C4—38no—
227C3×C22⋊C4—38no—
228C3×C22⋊C4C2×C3×D318yes—
229C3×C22⋊C4C22×C3212yes—
230C3×C42—38no—
231#217C22×C3212yes—
232#217C22×C3212yes—
233#217C22×C3212yes—
234#217C22×C3212yes—
235C4⋊Dic3C22×C3212yes—
236C4⋊Dic3C22×C3211yescharacteristic
237Dic3⋊C4C22×C3212yes—
238Dic3⋊C4C22×C3212yes—
239Dic3⋊C4C22×C3212yes—
240Dic3⋊C4C22×C3212yes—
241Dic3⋊C4C22×C3212yes—
242C4×Dic3C22×C3212yes—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—24no—
249C22×C32—28no—
250C22×C32—24no—
251C22×C32—24no—
252C22×C32—28no—
253C22×C32—28no—
254C22×C32—28no—
255C22×C32—44no—
256C22×C32—44no—
257C22×C32—48no—
258C22×C32—48no—
259C22×C32—48no—
260C22×C32#23712yes—
261C22×C32C3×#4118yes—
262C22×C32C23×D311yescharacteristic
263C22×C32C24×C314yes—
264C32×C4—28no—
265C32×C4—22no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—24no—
269C32×C4—28no—
270C32×C4—24no—
271C32×C4—44no—
272C32×C4—48no—
273C32×C4—62no—
274C32×C4—62no—
275C32×C4—62no—
276C32×C4—62no—
277C3×Dic3—28no—
278C3×Dic3—28no—
279C3×Dic3—28no—
280C3×Dic3—28no—
281#126—32no—
282#125—32no—
283#123—32no—
284#123—32no—
285C22⋊Q8—31no—
286C22⋊Q8—32no—
287#121C32×D311yescharacteristic
288C4×D4—32no—
289C2×C4⋊C4—31no—
290C33#53911yescharacteristic, complemented, Sylow
291C23×C3—24no—
292C23×C3—28no—
293C23×C3C22×C3×D318yes—
294C23×C3C22×C3×D314yes—
295C23×C3C23×C3212yes—
296C23×C3C23×C3211yescharacteristic
297C3×Q8—124no—
298C3×D4—216no—
299C3×D4—24no—
300C3×D4—416no—
301C3×D4—44no—
302C3×D4—41no—
303C3×D4—48no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—28no—
307C2×C3×C4—38no—
308C2×C3×C4—38no—
309C2×C3×C4—38no—
310C2×C3×C4—38no—
311C2×C3×C4—44no—
312C2×C3×C4—41no—
313C2×C3×C4—48no—
314C2×C3×C4—68no—
315C2×C3×C4—124no—
316C2×C3×C4—128no—
317C2×C3×C4C22×C3×D318yes—
318C2×C3×C4C22×C3×D314yes—
319C2×C3×C4C22×C3×D314yes—
320C2×C3×C4C23×C3212yes—
321C2×C3×C4C23×C3211yescharacteristic
322C2×C3×C4C23×C3211yescharacteristic
323C2×Dic3—22no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3C23×C3212yes—
327C2×Dic3C23×C3212yes—
328C2×Dic3C23×C3212yes—
329C2×Dic3C23×C3212yes—
330Dic6—41no—
331C2×C32—22no—
332C2×C32—24no—
333C2×C32—24no—
334C2×C32—24no—
335C2×C32—28no—
336C2×C32—48no—
337C2×C32—42no—
338C2×C32—48no—
339C2×C32—88no—
340C2×C32C2×#23711yescharacteristic
341C2×C32#100411yescharacteristic
342C2×C32#101011yescharacteristic
343C2×C32C2×C3×#4114yes—
344C2×C32C3×H(5,𝔽2)14yes—
345C2×C32C3×2−1+414yes—
346C2×Q8—31no—
347C2×D4C2×C32×D312yes—
348C2×D4C2×C32×D311yescharacteristic
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4C2×C32×D311yescharacteristic
353C4⋊C4—32no—
354C4⋊C4—32no—
355C4⋊C4—32no—
356C4⋊C4—32no—
357C4⋊C4—32no—
358C4⋊C4—31no—
359C4⋊C4—32no—
360C4⋊C4C2×C32×D311yescharacteristic
361C22⋊C4—32no—
362C22⋊C4—32no—
363C22⋊C4—32no—
364C22⋊C4—32no—
365C22⋊C4C2×C32×D312yes—
366C42—32no—
367C22×C3—24no—
368C22×C3—24no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—28no—
372C22×C3—28no—
373C22×C3—28no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—28no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—48no—
382C22×C3—48no—
383C22×C3—48no—
384C22×C3C3×#23718yes—
385C22×C3C32×#4112yes—
386C22×C3C23×C3×D314yes—
387C22×C3C24×C3211yescharacteristic
388C3×C4—28no—
389C3×C4—28no—
390C3×C4—24no—
391C3×C4—24no—
392C3×C4—22no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—44no—
396C3×C4—48no—
397C3×C4—68no—
398C3×C4—68no—
399C3×C4—68no—
400C3×C4—68no—
401Dic3—22no—
402Dic3—22no—
403Dic3—22no—
404Dic3—22no—
405C32—24no—
406C32#560211yescharacteristic, direct product factor
407C32C3×#53914yescomplemented
408C23C22×C32×D312yes—
409C23C22×C32×D311yescharacteristic
410Q8—121no—
411D4—24no—
412D4—41no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—41no—
418C2×C4—62no—
419C2×C4—121no—
420C2×C4—122no—
421C2×C4C22×C32×D312yes—
422C2×C4C22×C32×D311yescharacteristic
423C2×C4C22×C32×D311yescharacteristic
424C2×C3—28no—
425C2×C3—22no—
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—24no—
429C2×C3—48no—
430C2×C3—48no—
431C2×C3—42no—
432C2×C3—88no—
433C2×C3C2×C3×#23714yes—
434C2×C3C3×#100414yes—
435C2×C3C3×#101014yes—
436C2×C3C2×C32×#4111yescharacteristic
437C2×C3C32×H(5,𝔽2)11yescharacteristic
438C2×C3C32×2−1+411yescharacteristic
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22C32×#23712yes—
445C22C23×C32×D311yescharacteristic
446C4—22no—
447C4—21no—
448C4—21no—
449C4—62no—
450C4—62no—
451C4—62no—
452C4—62no—
453C3—24no—
454C3C3×#560214yesdirect product factor
455C3C32×#53911yescharacteristic, complemented
456C2—22no—
457C2—42no—
458C2C2×C32×#23711yescharacteristic
459C2C32×#100411yescharacteristic
460C2C32×#101011yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
Fitting, prime radicalC33×#121C2
socleC22×C33C24
hypercenterC32×#121D3
hypocenterC3C32×#539
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×#121

Complements

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

NH ≅ G/Ntype
2. C32×#935C2semidirect
3. C32×#935C2semidirect
4. C32×D4×Dic3C2semidirect
5. C32×#924C2semidirect
6. C2×C32×Dic3⋊C4C2semidirect
7. C32×#889C2semidirect
8. C32×#879C2semidirect
9. C32×#878C2semidirect
11. C3×#5602C3direct (N × H)
44. C3×#935C2×C3semidirect
45. C3×#935C2×C3semidirect
46. C3×D4×Dic3C2×C3semidirect
47. C3×#924C2×C3semidirect
48. C2×C3×Dic3⋊C4C2×C3semidirect
49. C3×#889C2×C3semidirect
50. C3×#879C2×C3semidirect
51. C3×#878C2×C3semidirect
69. #5602C32direct (N × H)
142. #935C2×C32semidirect
143. #935C2×C32semidirect
144. D4×Dic3C2×C32semidirect
145. #924C2×C32semidirect
146. C2×Dic3⋊C4C2×C32semidirect
147. #889C2×C32semidirect
148. #879C2×C32semidirect
149. #878C2×C32semidirect
290. C33#539semidirect
406. C32#5602direct (N × H)
407. C32C3×#539semidirect
454. C3C3×#5602direct (N × H)
455. C3C32×#539semidirect

Maximal subgroups

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

subgroupindex
C33×#1212
C32×#9352
C32×D4×Dic32
C32×#9242
C2×C32×Dic3⋊C42
C32×#8892
C32×#8792
C32×#8782
C32×#5393
C3×#56023