Finite Group Databasebeta

← C3×#10596051 #13179331

Subgroups — C3×#10596051 #13179331

Order 1728.

1684 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C3×#64107C212yesmaximal, complemented
5C3×#64104C211yesmaximal, characteristic
6C3×#64103C212yesmaximal, complemented
7C3×#64099C211yesmaximal, characteristic, complemented
8C3×C4×C4⋊(C3⋊D3)C212yesmaximal, complemented
9C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
10C3×#64081C212yesmaximal, complemented
11C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
12#10596051C311yesmaximal, characteristic, direct product factor
13C3×#5580—34nomaximal
14C2×C33×Q8C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2212yes—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2212yes—
21C33×C4⋊C4C2211yescharacteristic
22C33×C4⋊C4C2212yes—
23C33×C42C2211yescharacteristic
24C33×C42C2212yes—
25C3×#3752C2212yes—
26C3×#3752C2212yes—
27C3×#3752C2212yes—
28C3×#3752C2212yes—
29C3×#3752C2212yes—
30C3×C4⋊(C3⋊Dic3)C2212yes—
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×#3750C2212yes—
33C3×#3750C2212yes—
34C3×C4×C3⋊Dic3C2211yescharacteristic
35C3×C4×C3⋊Dic3C2211yescharacteristic
36C3×C4×C3⋊Dic3C2211yescharacteristic
37C32×C4×Q8—24no—
38C32×C4×Q8D314yescomplemented
39C32×C4×Q8C2×C311yescharacteristic, complemented
40#64156C2×C311yescharacteristic, complemented
41#64155C2×C311yescharacteristic, complemented
42#64107C2×C312yescomplemented
43#64104C2×C311yescharacteristic
44#64103C2×C312yescomplemented
45#64099C2×C311yescharacteristic, complemented
46C4×C4⋊(C3⋊D3)C2×C312yescomplemented
47C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
48#64081C2×C312yescomplemented
49C42×C3⋊D3C2×C311yescharacteristic, complemented
50C3×#947—34no—
51C3×#946—34no—
52C3×#898—38no—
53C3×#895—34no—
54C3×#894—38no—
55C3×#890—34no—
56C3×C4×D12—38no—
57C3×C4×D12—34no—
58C3×#872—38no—
59C3×C42×D3—34no—
60C22×C3×C3⋊D3C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C33×Q8—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2312yes—
65C2×C33×C4C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C3×C4⋊(C3⋊D3)—24no—
73C3×C4⋊(C3⋊D3)—22no—
74C3×C4×C3⋊D3—22no—
75C3×C4×C3⋊D3—22no—
76C3×C4×C3⋊D3—22no—
77C3×C4×C3⋊D3—24no—
78C3×#529—91no—
79#5580—34no—
80C2×C32×Q8—24no—
81C2×C32×Q8C2×D314yes—
82C2×C32×Q8C22×C311yescharacteristic
83C2×C4⋊(C3⋊D3)C22×C312yes—
84C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C311yescharacteristic
86C2×C4×C3⋊D3C22×C311yescharacteristic
87C2×C4×C3⋊D3C22×C311yescharacteristic
88C2×C4×C3⋊D3C22×C312yes—
89C2×C3×D12—38no—
90C2×C3×D12—34no—
91C2×C3×C4×D3—34no—
92C2×C3×C4×D3—34no—
93C2×C3×C4×D3—34no—
94C2×C3×C4×D3—38no—
95C32×C4⋊C4—24no—
96C32×C4⋊C4—28no—
97C32×C4⋊C4C2×D314yes—
98C32×C4⋊C4C2×D318yes—
99C32×C4⋊C4C22×C311yescharacteristic
100C32×C4⋊C4C22×C312yes—
101C32×C42—24no—
102C32×C42—28no—
103C32×C42C2×D314yes—
104C32×C42C2×D318yes—
105C32×C42C22×C311yescharacteristic
106C32×C42C22×C312yes—
107#3752C22×C312yes—
108#3752C22×C312yes—
109#3752C22×C312yes—
110#3752C22×C312yes—
111#3752C22×C312yes—
112C4⋊(C3⋊Dic3)C22×C312yes—
113C4⋊(C3⋊Dic3)C22×C311yescharacteristic
114#3750C22×C312yes—
115#3750C22×C312yes—
116C4×C3⋊Dic3C22×C311yescharacteristic
117C4×C3⋊Dic3C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C311yescharacteristic
119C3×(C2×D3)⋊C4—38no—
120C3×(C2×D3)⋊C4—38no—
121C3×(C2×D3)⋊C4—38no—
122C3×(C2×D3)⋊C4—38no—
123C3×(C2×D3)⋊C4—38no—
124C3×C4⋊Dic3—38no—
125C3×C4⋊Dic3—34no—
126C3×Dic3⋊C4—38no—
127C3×Dic3⋊C4—38no—
128C3×C4×Dic3—34no—
129C3×C4×Dic3—34no—
130C3×C4×Dic3—34no—
131C22×C33C2411yescharacteristic
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—22no—
135C2×C3×C3⋊D3—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3#4112yes—
138C33×C4—22no—
139C33×C4—21no—
140C33×C4—22no—
141C33×C4#4112yes—
142C33×C4#4112yes—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—21no—
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3—21no—
147C3×#126—92no—
148C3×#125—91no—
149C3×#124—91no—
150C3×#123—92no—
151C3×C22⋊Q8—91no—
152C3×#121—91no—
153C3×C4×Q8—24no—
154C3×C4×Q8C3×D314yescomplemented
155C3×C4×Q8C3⋊D311yescharacteristic, complemented
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C3×#117—92no—
159C2×C3×C42—91no—
160#947—34no—
161#946—34no—
162#898—38no—
163#895—34no—
164#894—38no—
165#890—34no—
166C4×D12—38no—
167C4×D12—34no—
168#872—38no—
169C42×D3—34no—
170C22×C3⋊D3C23×C312yes—
171C22×C3⋊D3C23×C311yescharacteristic
172C22×C3×D3—34no—
173C22×C3×D3—38no—
174C32×Q8—216no—
175C32×Q8—28no—
176C32×Q8—22no—
177C2×C32×C4—24no—
178C2×C32×C4—28no—
179C2×C32×C4—24no—
180C2×C32×C4—24no—
181C2×C32×C4—28no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D318yes—
184C2×C32×C4C22×D318yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C23×C311yescharacteristic
188C2×C32×C4C23×C312yes—
189C2×C32×C4C23×C312yes—
190C2×C32×C4C23×C311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C2×C3⋊Dic3C23×C311yescharacteristic
193C2×C3⋊Dic3C23×C311yescharacteristic
194C2×C3⋊Dic3C23×C311yescharacteristic
195C2×C3⋊Dic3C23×C312yes—
196C4⋊(C3⋊D3)—24no—
197C4⋊(C3⋊D3)—22no—
198C4×C3⋊D3—22no—
199C4×C3⋊D3—22no—
200C4×C3⋊D3—22no—
201C4×C3⋊D3—24no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3—34no—
205C2×C3×Dic3—38no—
206C3×D12—616no—
207C3×D12—68no—
208C3×C4×D3—68no—
209C3×C4×D3—68no—
210C3×C4×D3—68no—
211C3×C4×D3—616no—
212#529—91noSylow
213C2×C33C2×#4111yescharacteristic
214C2×C33C2×#4111yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C3×C3⋊D3—22no—
217C3×C3⋊D3—42no—
218C2×C3×Q8—24no—
219C2×C3×Q8C2×C3×D314yes—
220C2×C3×Q8C2×C3⋊D311yescharacteristic
221C2×C3×D4—92no—
222C2×C3×D4—91no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4—92no—
227C2×D12—38no—
228C2×D12—34no—
229C2×C4×D3—34no—
230C2×C4×D3—34no—
231C2×C4×D3—34no—
232C2×C4×D3—38no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4C2×C3×D314yes—
240C3×C4⋊C4C2×C3×D318yes—
241C3×C4⋊C4C2×C3⋊D311yescharacteristic
242C3×C4⋊C4C2×C3⋊D312yes—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C42—24no—
249C3×C42—28no—
250C3×C42—91no—
251C3×C42—91no—
252C3×C42—91no—
253C3×C42C2×C3×D314yes—
254C3×C42C2×C3×D318yes—
255C3×C42C2×C3⋊D311yescharacteristic
256C3×C42C2×C3⋊D312yes—
257(C2×D3)⋊C4—38no—
258(C2×D3)⋊C4—38no—
259(C2×D3)⋊C4—38no—
260(C2×D3)⋊C4—38no—
261(C2×D3)⋊C4—38no—
262C4⋊Dic3—38no—
263C4⋊Dic3—34no—
264Dic3⋊C4—38no—
265Dic3⋊C4—38no—
266C4×Dic3—34no—
267C4×Dic3—34no—
268C4×Dic3—34no—
269C22×C32—24no—
270C22×C32C23×D314yes—
271C22×C32C24×C311yescharacteristic
272C2×C3⋊D3—21no—
273C2×C3⋊D3—21no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3—22no—
276C2×C3⋊D3—22no—
277C2×C3⋊D3C3×#4112yes—
278C2×C3×D3—38no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C2×C3×D3—68no—
282C2×C3×D3—68no—
283C2×C3×D3—68no—
284C32×C4—216no—
285C32×C4—28no—
286C32×C4—21no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—28no—
290C32×C4—28no—
291C32×C4—24no—
292C32×C4—28no—
293C32×C4—44no—
294C32×C4—48no—
295C32×C4#23518yes—
296C32×C4#23918yes—
297C32×C4C3×#4112yes—
298C32×C4C3×#4112yes—
299C3⋊Dic3—21no—
300C3⋊Dic3—21no—
301C3⋊Dic3—22no—
302C3⋊Dic3—21no—
303C3×Dic3—64no—
304C3×Dic3—64no—
305C3×Dic3—68no—
306C3×Dic3—64no—
307#126—92no—
308#125—91no—
309#124—91no—
310#123—92no—
311C22⋊Q8—91no—
312#121—91no—
313C4×Q8C3×C3⋊D311yescharacteristic, complemented
314C4×D4—91no—
315C4×D4—92no—
316#117—92no—
317C2×C42—91no—
318C33#52911yescharacteristic, complemented, Sylow
319C23×C3—91no—
320C23×C3—92no—
321C22×D3—34no—
322C22×D3—38no—
323C3×Q8—216no—
324C3×Q8—28no—
325C3×Q8—22no—
326C3×D4—184no—
327C3×D4—182no—
328C2×C3×C4—24no—
329C2×C3×C4—28no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—28no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—92no—
337C2×C3×C4—182no—
338C2×C3×C4—182no—
339C2×C3×C4—182no—
340C2×C3×C4—184no—
341C2×C3×C4C22×C3×D314yes—
342C2×C3×C4C22×C3×D318yes—
343C2×C3×C4C22×C3×D318yes—
344C2×C3×C4C22×C3×D314yes—
345C2×C3×C4C22×C3×D314yes—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347C2×C3×C4C22×C3⋊D312yes—
348C2×C3×C4C22×C3⋊D312yes—
349C2×C3×C4C22×C3⋊D311yescharacteristic
350C2×C3×C4C22×C3⋊D311yescharacteristic
351C2×Dic3—34no—
352C2×Dic3—34no—
353C2×Dic3—34no—
354C2×Dic3—38no—
355D12—616no—
356D12—68no—
357C4×D3—68no—
358C4×D3—68no—
359C4×D3—68no—
360C4×D3—616no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—24no—
364C2×C32C2×#23514yes—
365C2×C32C2×#23914yes—
366C2×C32D3×#4114yes—
367C2×C32C2×C3×#4111yescharacteristic
368C2×C32C2×C3×#4111yescharacteristic
369C2×C32C2×C3×#4111yescharacteristic
370C3⋊D3—22no—
371C3⋊D3—42no—
372C3×D3—68no—
373C3×D3—128no—
374C2×Q8C2×C3×C3⋊D311yescharacteristic
375C2×D4—92no—
376C2×D4—91no—
377C22×C4—91no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—92no—
381C4⋊C4—92no—
382C4⋊C4—92no—
383C4⋊C4—91no—
384C4⋊C4—92no—
385C4⋊C4C2×C3×C3⋊D311yescharacteristic
386C4⋊C4C2×C3×C3⋊D312yes—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C22⋊C4—92no—
391C22⋊C4—92no—
392C42—91no—
393C42—91no—
394C42—91no—
395C42C2×C3×C3⋊D311yescharacteristic
396C42C2×C3×C3⋊D312yes—
397C22×C3—24no—
398C22×C3—92no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3C23×C3×D314yes—
405C22×C3C23×C3⋊D311yescharacteristic
406C2×D3—38no—
407C2×D3—64no—
408C2×D3—64no—
409C2×D3—68no—
410C2×D3—68no—
411C2×D3—68no—
412C3×C4—216no—
413C3×C4—28no—
414C3×C4—24no—
415C3×C4—28no—
416C3×C4—22no—
417C3×C4—21no—
418C3×C4—22no—
419C3×C4—28no—
420C3×C4—28no—
421C3×C4—44no—
422C3×C4—48no—
423C3×C4—181no—
424C3×C4—181no—
425C3×C4—182no—
426C3×C4—181no—
427C3×C4C3×#23518yes—
428C3×C4C3×#23918yes—
429C3×C4#382812yes—
430C3×C4#383212yes—
431Dic3—64no—
432Dic3—64no—
433Dic3—68no—
434Dic3—64no—
435C32—24no—
436C32#558014yescomplemented
437C32C3×#52911yescharacteristic, complemented
438C23—91no—
439C23—92no—
440Q8—22no—
441D4—184no—
442D4—182no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—92no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—184no—
451C2×C4C22×C3×C3⋊D311yescharacteristic
452C2×C4C22×C3×C3⋊D312yes—
453C2×C4C22×C3×C3⋊D312yes—
454C2×C4C22×C3×C3⋊D311yescharacteristic
455C2×C4C22×C3×C3⋊D311yescharacteristic
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—182no—
460C2×C3—362no—
461C2×C3C2×C3×#23514yes—
462C2×C3C2×C3×#23914yes—
463C2×C3C3×D3×#4114yes—
464C2×C3C2×#382811yescharacteristic
465C2×C3C2×#383211yescharacteristic
466C2×C3#41×C3⋊D311yescharacteristic
467D3—68no—
468D3—128no—
469C22—92no—
470C22—181no—
471C22—181no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22C23×C3×C3⋊D311yescharacteristic
476C4—22no—
477C4—21no—
478C4—22no—
479C4—181no—
480C4—181no—
481C4—182no—
482C4—181no—
483C4C3×#382812yes—
484C4C3×#383212yes—
485C3—24no—
486C3C3×#558014yescomplemented
487C3#1059605111yescharacteristic, direct product factor
488C2—182no—
489C2—362no—
490C2C2×C3×#382811yescharacteristic
491C2C2×C3×#383211yescharacteristic
492C2C3×#41×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×Q8C2
socleC22×C33C24
prime radicalC3×#64104C2
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4×Q8C2semidirect
2. C3×#64156C2semidirect
3. C3×#64155C2semidirect
4. C3×#64107C2semidirect
6. C3×#64103C2semidirect
7. C3×#64099C2semidirect
8. C3×C4×C4⋊(C3⋊D3)C2semidirect
9. C3×C4×C4⋊(C3⋊D3)C2semidirect
10. C3×#64081C2semidirect
11. C3×C42×C3⋊D3C2semidirect
12. #10596051C3direct (N × H)
38. C32×C4×Q8D3semidirect
39. C32×C4×Q8C2×C3semidirect
40. #64156C2×C3semidirect
41. #64155C2×C3semidirect
42. #64107C2×C3semidirect
44. #64103C2×C3semidirect
45. #64099C2×C3semidirect
46. C4×C4⋊(C3⋊D3)C2×C3semidirect
47. C4×C4⋊(C3⋊D3)C2×C3semidirect
48. #64081C2×C3semidirect
49. C42×C3⋊D3C2×C3semidirect
154. C3×C4×Q8C3×D3semidirect
155. C3×C4×Q8C3⋊D3semidirect
313. C4×Q8C3×C3⋊D3semidirect
318. C33#529semidirect
436. C32#5580semidirect
437. C32C3×#529semidirect
486. C3C3×#5580semidirect
487. C3#10596051direct (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×#641562
C3×#641552
C3×#641072
C3×#641042
C3×#641032
C3×#640992
C3×C4×C4⋊(C3⋊D3)2
C3×#640812
C3×C42×C3⋊D32
#105960513
C3×#55803