Finite Group Databasebeta

← C3×#10596052 #13179332

Subgroups — C3×#10596052 #13179332

Order 1728.

1564 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic, complemented
2C3×#64155C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C3×#64103C212yesmaximal
5C3×#64101C211yesmaximal, characteristic, complemented
6C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
7C3×#64098C212yesmaximal, complemented
8C3×#64085C212yesmaximal, complemented
9C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
10C3×#64081C212yesmaximal, complemented
11C3×#64079C211yesmaximal, characteristic, complemented
12#10596052C311yesmaximal, characteristic, direct product factor
13C3×#5581—34nomaximal
14C2×C33×Q8C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C33×C4⋊C4C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C33×C42C2211yescharacteristic
22C33×C42C2212yes—
23C3×#3752C2211yescharacteristic
24C3×#3752C2212yes—
25C3×#3752C2211yescharacteristic
26C3×#3752C2212yes—
27C3×#3752C2212yes—
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C3×#3750C2212yes—
32C3×#3750C2212yes—
33C3×#3750C2212yes—
34C3×#3750C2212yes—
35C3×#3750C2212yes—
36C3×C4×C3⋊Dic3C2212yes—
37C32×C4×Q8—24no—
38C32×C4×Q8D314yescomplemented
39C32×C4×Q8C2×C311yescharacteristic, complemented
40#64155C2×C311yescharacteristic, complemented
41#64155C2×C311yescharacteristic, complemented
42#64103C2×C312yes—
43#64101C2×C311yescharacteristic, complemented
44C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
45#64098C2×C312yescomplemented
46#64085C2×C312yescomplemented
47C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
48#64081C2×C312yescomplemented
49#64079C2×C311yescharacteristic, complemented
50C3×#946—34no—
51C3×#946—34no—
52C3×#894—38no—
53C3×#892—34no—
54C3×D3×C4⋊C4—34no—
55C3×#889—38no—
56C3×#876—38no—
57C3×C4×D12—34no—
58C3×#872—38no—
59C3×#870—34no—
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C33×Q8—22no—
63C2×C33×C4C2312yes—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4⋊(C3⋊D3)—22no—
73C3×C4×C3⋊D3—24no—
74C3×C4×C3⋊D3—22no—
75C3×C4×C3⋊D3—22no—
76C3×#552—91no—
77#5581—34no—
78C2×C32×Q8—24no—
79C2×C32×Q8C2×D314yes—
80C2×C32×Q8C22×C311yescharacteristic
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×C4×C3⋊D3C22×C311yescharacteristic
84C2×C4×C3⋊D3C22×C311yescharacteristic
85C2×C3×D12—34no—
86C2×C3×C4×D3—38no—
87C2×C3×C4×D3—34no—
88C2×C3×C4×D3—34no—
89C32×C4⋊C4—28no—
90C32×C4⋊C4—24no—
91C32×C4⋊C4C2×D318yes—
92C32×C4⋊C4C2×D314yes—
93C32×C4⋊C4C22×C312yes—
94C32×C4⋊C4C22×C311yescharacteristic
95C32×C42—24no—
96C32×C42—28no—
97C32×C42C2×D314yes—
98C32×C42C2×D318yes—
99C32×C42C22×C311yescharacteristic
100C32×C42C22×C312yes—
101#3752C22×C311yescharacteristic
102#3752C22×C312yes—
103#3752C22×C311yescharacteristic
104#3752C22×C312yes—
105#3752C22×C312yes—
106C4⋊(C3⋊Dic3)C22×C311yescharacteristic
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic
108C4⋊(C3⋊Dic3)C22×C311yescharacteristic
109#3750C22×C312yes—
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113#3750C22×C312yes—
114C4×C3⋊Dic3C22×C312yes—
115C3×(C2×D3)⋊C4—38no—
116C3×(C2×D3)⋊C4—38no—
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4—34no—
119C3×(C2×D3)⋊C4—34no—
120C3×C4⋊Dic3—34no—
121C3×C4⋊Dic3—34no—
122C3×C4⋊Dic3—34no—
123C3×Dic3⋊C4—38no—
124C3×Dic3⋊C4—38no—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4—38no—
127C3×Dic3⋊C4—38no—
128C3×C4×Dic3—38no—
129C22×C33C2411yescharacteristic
130C2×C3×C3⋊D3—21no—
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3#4112yes—
136C33×C4—22no—
137C33×C4—22no—
138C33×C4—21no—
139C33×C4—21no—
140C33×C4#4112yes—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—22no—
145C3×#126—92no—
146C3×#125—92no—
147C3×#125—91no—
148C3×#123—92no—
149C3×C22⋊Q8—91no—
150C3×C22⋊Q8—91no—
151C3×C4×Q8—24no—
152C3×C4×Q8C3×D314yescomplemented
153C3×C4×Q8C3⋊D311yescharacteristic, complemented
154C3×C4×D4—91no—
155C3×#117—91no—
156C3×#117—92no—
157C2×C3×C4⋊C4—91no—
158#946—34no—
159#946—34no—
160#894—38no—
161#892—34no—
162D3×C4⋊C4—34no—
163#889—38no—
164#876—38no—
165C4×D12—34no—
166#872—38no—
167#870—34no—
168C22×C3⋊D3C23×C311yescharacteristic
169C22×C3⋊D3C23×C311yescharacteristic
170C22×C3×D3—34no—
171C22×C3×D3—34no—
172C32×Q8—216no—
173C32×Q8—28no—
174C32×Q8—22no—
175C2×C32×C4—24no—
176C2×C32×C4—28no—
177C2×C32×C4—24no—
178C2×C32×C4—28no—
179C2×C32×C4—24no—
180C2×C32×C4C22×D318yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D318yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C23×C312yes—
186C2×C32×C4C23×C311yescharacteristic
187C2×C32×C4C23×C311yescharacteristic
188C2×C32×C4C23×C312yes—
189C2×C32×C4C23×C311yescharacteristic
190C2×C3⋊Dic3C23×C311yescharacteristic
191C2×C3⋊Dic3C23×C312yes—
192C2×C3⋊Dic3C23×C312yes—
193C2×C3⋊Dic3C23×C311yescharacteristic
194C4⋊(C3⋊D3)—22no—
195C4×C3⋊D3—24no—
196C4×C3⋊D3—22no—
197C4×C3⋊D3—22no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—38no—
202C3×D12—68no—
203C3×C4×D3—616no—
204C3×C4×D3—68no—
205C3×C4×D3—68no—
206#552—91noSylow
207C2×C33C2×#4111yescharacteristic
208C2×C33C2×#4111yescharacteristic
209C2×C332−1+411yescharacteristic
210C3×C3⋊D3—22no—
211C3×C3⋊D3—41no—
212C2×C3×Q8—24no—
213C2×C3×Q8C2×C3×D314yes—
214C2×C3×Q8C2×C3⋊D311yescharacteristic
215C2×C3×D4—91no—
216C22×C3×C4—92no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C2×D12—34no—
220C2×C4×D3—38no—
221C2×C4×D3—34no—
222C2×C4×D3—34no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—28no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—92no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4C2×C3×D318yes—
234C3×C4⋊C4C2×C3×D314yes—
235C3×C4⋊C4C2×C3⋊D311yescharacteristic
236C3×C4⋊C4C2×C3⋊D312yes—
237C3×C22⋊C4—91no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C42—28no—
243C3×C42—24no—
244C3×C42—92no—
245C3×C42C2×C3×D314yes—
246C3×C42C2×C3×D318yes—
247C3×C42C2×C3⋊D312yes—
248C3×C42C2×C3⋊D311yescharacteristic
249(C2×D3)⋊C4—34no—
250(C2×D3)⋊C4—38no—
251(C2×D3)⋊C4—34no—
252(C2×D3)⋊C4—38no—
253(C2×D3)⋊C4—38no—
254C4⋊Dic3—34no—
255C4⋊Dic3—34no—
256C4⋊Dic3—34no—
257Dic3⋊C4—38no—
258Dic3⋊C4—38no—
259Dic3⋊C4—38no—
260Dic3⋊C4—38no—
261Dic3⋊C4—38no—
262C4×Dic3—38no—
263C22×C32—24no—
264C22×C32C23×D314yes—
265C22×C32C24×C311yescharacteristic
266C2×C3⋊D3—21no—
267C2×C3⋊D3—21no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3⋊D3—21no—
271C2×C3⋊D3C3×#4112yes—
272C2×C3×D3—38no—
273C2×C3×D3—64no—
274C2×C3×D3—64no—
275C2×C3×D3—64no—
276C2×C3×D3—64no—
277C2×C3×D3—64no—
278C32×C4—28no—
279C32×C4—216no—
280C32×C4—28no—
281C32×C4—24no—
282C32×C4—22no—
283C32×C4—21no—
284C32×C4—22no—
285C32×C4—21no—
286C32×C4—28no—
287C32×C4—28no—
288C32×C4—24no—
289C32×C4—48no—
290C32×C4—44no—
291C32×C4#23518yes—
292C32×C4C3×#4112yes—
293C3⋊Dic3—21no—
294C3⋊Dic3—21no—
295C3⋊Dic3—22no—
296C3⋊Dic3—22no—
297C3×Dic3—64no—
298C3×Dic3—64no—
299C3×Dic3—68no—
300C3×Dic3—68no—
301#126—92no—
302#125—92no—
303#125—91no—
304#123—92no—
305C22⋊Q8—91no—
306C22⋊Q8—91no—
307C4×Q8C3×C3⋊D311yescharacteristic, complemented
308C4×D4—91no—
309#117—91no—
310#117—92no—
311C2×C4⋊C4—91no—
312C33#55211yescharacteristic, complemented, Sylow
313C23×C3—91no—
314C23×C3—91no—
315C22×D3—34no—
316C22×D3—34no—
317C3×Q8—216no—
318C3×Q8—28no—
319C3×Q8—22no—
320C3×D4—182no—
321C2×C3×C4—24no—
322C2×C3×C4—28no—
323C2×C3×C4—24no—
324C2×C3×C4—28no—
325C2×C3×C4—24no—
326C2×C3×C4—91no—
327C2×C3×C4—92no—
328C2×C3×C4—91no—
329C2×C3×C4—92no—
330C2×C3×C4—184no—
331C2×C3×C4—182no—
332C2×C3×C4—182no—
333C2×C3×C4C22×C3×D318yes—
334C2×C3×C4C22×C3×D314yes—
335C2×C3×C4C22×C3×D314yes—
336C2×C3×C4C22×C3×D318yes—
337C2×C3×C4C22×C3×D314yes—
338C2×C3×C4C22×C3⋊D312yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340C2×C3×C4C22×C3⋊D312yes—
341C2×C3×C4C22×C3⋊D311yescharacteristic
342C2×C3×C4C22×C3⋊D311yescharacteristic
343C2×Dic3—34no—
344C2×Dic3—38no—
345C2×Dic3—34no—
346C2×Dic3—38no—
347D12—68no—
348C4×D3—616no—
349C4×D3—68no—
350C4×D3—68no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32C2×#23514yes—
355C2×C32#100714yes—
356C2×C32D3×#4114yes—
357C2×C32C2×C3×#4111yescharacteristic
358C2×C32C2×C3×#4111yescharacteristic
359C2×C32C3×2−1+411yescharacteristic
360C3⋊D3—22no—
361C3⋊D3—41no—
362C3×D3—68no—
363C3×D3—124no—
364C2×Q8C2×C3×C3⋊D311yescharacteristic
365C2×D4—91no—
366C22×C4—92no—
367C22×C4—91no—
368C22×C4—91no—
369C4⋊C4—91no—
370C4⋊C4—92no—
371C4⋊C4—92no—
372C4⋊C4—91no—
373C4⋊C4—92no—
374C4⋊C4—91no—
375C4⋊C4—92no—
376C4⋊C4—92no—
377C4⋊C4C2×C3×C3⋊D311yescharacteristic
378C4⋊C4C2×C3×C3⋊D312yes—
379C22⋊C4—91no—
380C22⋊C4—92no—
381C22⋊C4—92no—
382C22⋊C4—91no—
383C22⋊C4—92no—
384C42—92no—
385C42C2×C3×C3⋊D312yes—
386C42C2×C3×C3⋊D311yescharacteristic
387C22×C3—24no—
388C22×C3—92no—
389C22×C3—181no—
390C22×C3—181no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3C23×C3×D314yes—
395C22×C3C23×C3⋊D311yescharacteristic
396C2×D3—38no—
397C2×D3—64no—
398C2×D3—64no—
399C2×D3—64no—
400C2×D3—64no—
401C2×D3—64no—
402C3×C4—216no—
403C3×C4—28no—
404C3×C4—28no—
405C3×C4—24no—
406C3×C4—28no—
407C3×C4—24no—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—28no—
413C3×C4—48no—
414C3×C4—44no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4—182no—
418C3×C4—182no—
419C3×C4C3×#23518yes—
420C3×C4#382812yes—
421Dic3—64no—
422Dic3—64no—
423Dic3—68no—
424Dic3—68no—
425C32—24no—
426C32#558114yescomplemented
427C32C3×#55211yescharacteristic, complemented
428C23—91no—
429C23—91no—
430Q8—22no—
431D4—182no—
432C2×C4—91no—
433C2×C4—92no—
434C2×C4—91no—
435C2×C4—92no—
436C2×C4—184no—
437C2×C4—182no—
438C2×C4—182no—
439C2×C4C22×C3×C3⋊D312yes—
440C2×C4C22×C3×C3⋊D311yescharacteristic
441C2×C4C22×C3×C3⋊D312yes—
442C2×C4C22×C3×C3⋊D311yescharacteristic
443C2×C4C22×C3×C3⋊D311yescharacteristic
444C2×C3—24no—
445C2×C3—24no—
446C2×C3—24no—
447C2×C3—182no—
448C2×C3—361no—
449C2×C3C2×C3×#23514yes—
450C2×C3C3×#100714yes—
451C2×C3C3×D3×#4114yes—
452C2×C3C2×#382811yescharacteristic
453C2×C3#6436411yescharacteristic
454C2×C3#41×C3⋊D311yescharacteristic
455D3—68no—
456D3—124no—
457C22—92no—
458C22—181no—
459C22—181no—
460C22—181no—
461C22—181no—
462C22—181no—
463C22C23×C3×C3⋊D311yescharacteristic
464C4—22no—
465C4—21no—
466C4—22no—
467C4—21no—
468C4—181no—
469C4—181no—
470C4—182no—
471C4—182no—
472C4C3×#382812yes—
473C3—24no—
474C3C3×#558114yescomplemented
475C3#1059605211yescharacteristic, direct product factor
476C2—182no—
477C2—361no—
478C2C2×C3×#382811yescharacteristic
479C2C3×#6436411yescharacteristic
480C2C3×#41×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×Q8C2
socleC22×C33C24
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4×Q8C2semidirect
2. C3×#64155C2semidirect
3. C3×#64155C2semidirect
5. C3×#64101C2semidirect
6. C3×C4⋊C4×C3⋊D3C2semidirect
7. C3×#64098C2semidirect
8. C3×#64085C2semidirect
10. C3×#64081C2semidirect
11. C3×#64079C2semidirect
12. #10596052C3direct (N × H)
38. C32×C4×Q8D3semidirect
39. C32×C4×Q8C2×C3semidirect
40. #64155C2×C3semidirect
41. #64155C2×C3semidirect
43. #64101C2×C3semidirect
44. C4⋊C4×C3⋊D3C2×C3semidirect
45. #64098C2×C3semidirect
46. #64085C2×C3semidirect
48. #64081C2×C3semidirect
49. #64079C2×C3semidirect
152. C3×C4×Q8C3×D3semidirect
153. C3×C4×Q8C3⋊D3semidirect
307. C4×Q8C3×C3⋊D3semidirect
312. C33#552semidirect
426. C32#5581semidirect
427. C32C3×#552semidirect
474. C3C3×#5581semidirect
475. C3#10596052direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×Q82
C3×#641552
C3×#641032
C3×#641012
C3×C4⋊C4×C3⋊D32
C3×#640982
C3×#640852
C3×C4×C4⋊(C3⋊D3)2
C3×#640812
C3×#640792
#105960523
C3×#55813