Finite Group Databasebeta

← C3×#5964⋊C3 #13181449

Subgroups — C3×#5964⋊C3 #13181449

Order 1728.

1051 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×#1010⋊C3C211yesmaximal, characteristic, complemented
3C3×D3×#41⋊C3C212yesmaximal, complemented
4C3×C3⋊#994C211yesmaximal, characteristic, complemented
5C3×#1007⋊C3C212yesmaximal, complemented
6C3×#5964C311yesmaximal, characteristic, complemented
7C3×#5945—31nomaximal
8#5964⋊C3C313yesmaximal, direct product factor
9C32×#235—41nomaximal
10C32×#41⋊C3C2211yescharacteristic, complemented
11C32×#41⋊C3C2211yescharacteristic, complemented
12C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
13C3×D3×SL(2,𝔽3)C2212yescomplemented
14C3×C3⋊(#41⋊C3)C2212yescomplemented
15C2×C32×#41C2×C311yescharacteristic, complemented
16C3×#1010—31no—
17C3×#1010C2×C311yescharacteristic, complemented
18C3×#1009—31no—
19C3×#1009C2×C311yescharacteristic, complemented
20C3×D3×#41—32no—
21C3×D3×#41C2×C312yescomplemented
22C3×#1007C2×C312yescomplemented
23C3×#1004—32no—
24C2×C3×#235—31no—
25C2×C3×#235—31no—
26C3×2−1+4⋊C3—31no—
27C3×2−1+4⋊C3—32no—
28C3×#994—31no—
29C2×C3×#41⋊C3—21no—
30C2×C3×#41⋊C3—23no—
31C2×C3×#41⋊C3—32no—
32C2×C3×#41⋊C3D311yescharacteristic, complemented
33C2×C3×#41⋊C3C2×C313yescomplemented
34#1010⋊C3C2×C313yescomplemented
35D3×#41⋊C3C2×C316yescomplemented
36C3⋊#994C2×C313yescomplemented
37#1007⋊C3C2×C316yescomplemented
38C2×C33×C4—41no—
39C32×#67—42no—
40C32×D12—41no—
41C32×C4×D3—42no—
42C32×Dic6—41no—
43C32×SL(2,𝔽3)C2311yescharacteristic
44C3×#585—31no—
45#5964C3211yescharacteristic, complemented
46#5945—33no—
47C32×#41—31no—
48C32×#41—31no—
49C32×#41—121no—
50C32×#41C22×C311yescharacteristic
51C32×#41C22×C311yescharacteristic
52C2×C32×Q8C22×C311yescharacteristic
53C2×C32×D4—31no—
54C22×C32×C4—31no—
55C2×C3×#67—32no—
56C2×C3×#67—32no—
57C3×#239—32no—
58C3×#239C22×C312yes—
59C3×D3×Q8—32no—
60C3×D3×Q8C22×C312yes—
61C3×#237—34no—
62C3×#237—32no—
63C3×#237—32no—
64C3×D3×D4—34no—
65C3×D3×D4—32no—
66C3×D3×D4—32no—
67C3×#235—32no—
68C3×#235—31no—
69C3×#235—32no—
70C3×#235—32no—
71C3×#235—31no—
72C3×#235—32no—
73C3×#235—32no—
74C3×#235—31no—
75C3×#235—43no—
76C3×#235A411yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—32no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C3×#41⋊C3—21no—
84C3×#41⋊C3—21no—
85C3×#41⋊C3—23no—
86C3×#41⋊C3—23no—
87C3×#41⋊C3—32no—
88C3×#41⋊C3C2×D311yescharacteristic, complemented
89C3×#41⋊C3C2×D311yescharacteristic, complemented
90C3×#41⋊C3C22×C313yescomplemented
91C3×#41⋊C3C22×C313yescomplemented
92C2×C3×SL(2,𝔽3)—21no—
93C2×C3×SL(2,𝔽3)—23no—
94C2×C3×SL(2,𝔽3)—32no—
95C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
96C2×C3×SL(2,𝔽3)C22×C313yescomplemented
97D3×SL(2,𝔽3)C22×C316yescomplemented
98C3⋊(#41⋊C3)C22×C316yescomplemented
99C22×C33—41no—
100C2×C32×D3—42no—
101C33×C4—41no—
102C33×C4—41no—
103C32×Dic3—42no—
104C3×2−1+4—31no—
105C3×2−1+4—32no—
106C3×2−1+4—91no—
107C3×H(5,𝔽2)—31no—
108C3×H(5,𝔽2)—92no—
109C3×H(5,𝔽2)—91no—
110C2×C3×#41—21no—
111C2×C3×#41—32no—
112C2×C3×#41—91no—
113C2×C3×#41—91no—
114C2×C3×#41—92no—
115C2×C3×#41C3×D311yescharacteristic, complemented
116C2×C3×#41C2×C3211yescharacteristic, complemented
117#1010—31no—
118#1010C2×C3211yescharacteristic, complemented
119#1009—31no—
120#1009C2×C3211yescharacteristic, complemented
121D3×#41—32no—
122D3×#41C2×C3212yescomplemented
123#1007C2×C3212yescomplemented
124#1004—32no—
125C2×#235—31no—
126C2×#235—31no—
1272−1+4⋊C3—33no—
1282−1+4⋊C3—36no—
129#994—33no—
130C2×#41⋊C3—23no—
131C2×#41⋊C3—36no—
132C2×#41⋊C3C3×D313yescomplemented
133C23×C32—31no—
134C22×C3×D3—32no—
135C22×C3×D3—32no—
136C32×Q8—31no—
137C32×Q8—121no—
138C32×Q8C23×C311yescharacteristic
139C32×D4—31no—
140C32×D4—31no—
141C32×D4—32no—
142C32×D4—121no—
143C32×D4—122no—
144C2×C32×C4—31no—
145C2×C32×C4—31no—
146C2×C32×C4—31no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—41no—
150C2×C32×C4—43no—
151C2×C32×C4—83no—
152C2×C32×C4—81no—
153C2×C32×C4—122no—
154C2×C32×C4C2×A411yescharacteristic
155C3×#67—34no—
156C3×#67—32no—
157C3×#67—32no—
158C3×#67—32no—
159C3×#67—46no—
160C3×#67C2×A412yes—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C3×D12—32no—
164C3×D12—32no—
165C3×D12—31no—
166C3×D12—43no—
167C3×D12C2×A411yescharacteristic
168C3×C4×D3—34no—
169C3×C4×D3—32no—
170C3×C4×D3—32no—
171C3×C4×D3—32no—
172C3×C4×D3—46no—
173C3×C4×D3C2×A412yes—
174C3×Dic6—31no—
175C3×Dic6—32no—
176C3×Dic6—32no—
177C3×Dic6—43no—
178C3×Dic6C2×A411yescharacteristic
179C3×SL(2,𝔽3)—21no—
180C3×SL(2,𝔽3)—23no—
181C3×SL(2,𝔽3)C22×D311yescharacteristic
182C3×SL(2,𝔽3)C23×C313yes—
183#585—31noSylow
184C2×C33—41no—
185C2×C33—81no—
186C32×D3—82no—
187C3×#41—21no—
188C3×#41—21no—
189C3×#41—31no—
190C3×#41—31no—
191C3×#41—31no—
192C3×#41—31no—
193C3×#41—31no—
194C3×#41—32no—
195C3×#41—61no—
196C3×#41—61no—
197C3×#41—92no—
198C3×#41—92no—
199C3×#41—91no—
200C3×#41—91no—
201C3×#41—92no—
202C3×#41—92no—
203C3×#41—92no—
204C3×#41—94no—
205C3×#41—92no—
206C3×#41—92no—
207C3×#41—91no—
208C3×#41—92no—
209C3×#41—123no—
210C3×#41C2×C3×D311yescharacteristic
211C3×#41C2×C3×D311yescharacteristic
212C3×#41C22×C3211yescharacteristic
213C3×#41C22×C3211yescharacteristic
214C2×C3×Q8—21no—
215C2×C3×Q8—32no—
216C2×C3×Q8—91no—
217C2×C3×Q8—91no—
218C2×C3×Q8—92no—
219C2×C3×Q8C2×C3×D311yescharacteristic
220C2×C3×Q8C22×C3211yescharacteristic
221C2×C3×D4—31no—
222C2×C3×D4—31no—
223C2×C3×D4—61no—
224C2×C3×D4—91no—
225C2×C3×D4—91no—
226C2×C3×D4—92no—
227C2×C3×D4—92no—
228C2×C3×D4—94no—
229C2×C3×D4—92no—
230C2×C3×D4—92no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—61no—
234C22×C3×C4—92no—
235C22×C3×C4—92no—
236C2×#67—32no—
237C2×#67—32no—
238#239—32no—
239#239C22×C3212yes—
240D3×Q8—32no—
241D3×Q8C22×C3212yes—
242#237—34no—
243#237—32no—
244#237—32no—
245D3×D4—32no—
246D3×D4—34no—
247D3×D4—32no—
248#235—32no—
249#235—31no—
250#235—32no—
251#235—32no—
252#235—31no—
253#235—32no—
254#235—32no—
255#235—31no—
256#235C3×A411yescharacteristic
257C2×D12—31no—
258C2×D12—31no—
259C2×C4×D3—32no—
260C2×C4×D3—32no—
261C2×Dic6—31no—
262C2×Dic6—31no—
263#41⋊C3—23no—
264#41⋊C3—23no—
265#41⋊C3—36no—
266#41⋊C3C2×C3×D313yescomplemented
267#41⋊C3C2×C3×D313yescomplemented
268C2×SL(2,𝔽3)—23no—
269C2×SL(2,𝔽3)—36no—
270C2×SL(2,𝔽3)C2×C3×D313yescomplemented
271C22×C32—31no—
272C22×C32—31no—
273C22×C32—41no—
274C22×C32—43no—
275C22×C32—83no—
276C22×C32—81no—
277C22×C32—122no—
278C22×C32—121no—
279C22×C32C22×A411yescharacteristic
280C2×C3×D3—32no—
281C2×C3×D3—46no—
282C2×C3×D3—122no—
283C2×C3×D3—122no—
284C2×C3×D3C22×A412yes—
285C32×C4—31no—
286C32×C4—31no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—43no—
290C32×C4—43no—
291C32×C4—83no—
292C32×C4—83no—
293C32×C4—81no—
294C32×C4—81no—
295C32×C4—122no—
296C32×C4C22×A411yescharacteristic
297C32×C4C22×A411yescharacteristic
298C3×Dic3—32no—
299C3×Dic3—46no—
300C3×Dic3C22×A412yes—
3012−1+4—31no—
3022−1+4—32no—
3032−1+4—91no—
304H(5,𝔽2)—31no—
305H(5,𝔽2)—92no—
306H(5,𝔽2)—91no—
307C2×#41—32no—
308C2×#41—91no—
309C2×#41—91no—
310C2×#41—92no—
311C2×#41C32×D311yescharacteristic, complemented
312C33—41noSylow
313C23×C3—31no—
314C23×C3—31no—
315C23×C3—61no—
316C23×C3—92no—
317C23×C3—92no—
318C22×D3—32no—
319C22×D3—32no—
320C3×Q8—21no—
321C3×Q8—31no—
322C3×Q8—31no—
323C3×Q8—31no—
324C3×Q8—61no—
325C3×Q8—92no—
326C3×Q8—92no—
327C3×Q8—91no—
328C3×Q8—123no—
329C3×Q8C22×C3×D311yescharacteristic
330C3×Q8C23×C3211yescharacteristic
331C3×D4—31no—
332C3×D4—31no—
333C3×D4—31no—
334C3×D4—31no—
335C3×D4—32no—
336C3×D4—32no—
337C3×D4—32no—
338C3×D4—31no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—62no—
342C3×D4—92no—
343C3×D4—92no—
344C3×D4—92no—
345C3×D4—92no—
346C3×D4—94no—
347C3×D4—92no—
348C3×D4—91no—
349C3×D4—123no—
350C3×D4—126no—
351C2×C3×C4—21no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—43no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—83no—
370C2×C3×C4—92no—
371C2×C3×C4—92no—
372C2×C3×C4—92no—
373C2×C3×C4—94no—
374C2×C3×C4—92no—
375C2×C3×C4—92no—
376C2×C3×C4—126no—
377C2×C3×C4D3×A411yescharacteristic
378C2×C3×C4C2×C3×A411yescharacteristic
379#67—34no—
380#67—32no—
381#67—32no—
382#67—32no—
383#67C2×C3×A412yes—
384C2×Dic3—32no—
385C2×Dic3—32no—
386D12—32no—
387D12—32no—
388D12—31no—
389D12C2×C3×A411yescharacteristic
390C4×D3—34no—
391C4×D3—32no—
392C4×D3—32no—
393C4×D3—32no—
394C4×D3C2×C3×A412yes—
395Dic6—31no—
396Dic6—32no—
397Dic6—32no—
398Dic6C2×C3×A411yescharacteristic
399SL(2,𝔽3)—23no—
400SL(2,𝔽3)C22×C3×D313yes—
401C2×C32—21no—
402C2×C32—41no—
403C2×C32—43no—
404C2×C32—61no—
405C2×C32—61no—
406C2×C32—83no—
407C2×C32—81no—
408C2×C32—83no—
409C2×C32—86no—
410C2×C32—82no—
411C2×C32—81no—
412C2×C32—242no—
413C2×C32C23×A411yescharacteristic
414C3×D3—22no—
415C3×D3—62no—
416C3×D3—86no—
417#41—31no—
418#41—31no—
419#41—31no—
420#41—32no—
421#41—92no—
422#41—92no—
423#41—91no—
424#41—91no—
425#41—92no—
426#41—92no—
427#41—92no—
428#41—94no—
429#41—92no—
430#41—92no—
431#41—91no—
432#41—92no—
433#41C2×C32×D311yescharacteristic
434#41C2×C32×D311yescharacteristic
435C2×Q8—32no—
436C2×Q8—91no—
437C2×Q8—91no—
438C2×Q8—92no—
439C2×Q8C2×C32×D311yescharacteristic
440C2×D4—31no—
441C2×D4—91no—
442C2×D4—91no—
443C2×D4—92no—
444C2×D4—92no—
445C2×D4—94no—
446C2×D4—92no—
447C2×D4—92no—
448C22×C4—31no—
449C22×C4—92no—
450C22×C4—92no—
451C22×C3—21no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—32no—
455C22×C3—31no—
456C22×C3—31no—
457C22×C3—43no—
458C22×C3—61no—
459C22×C3—61no—
460C22×C3—83no—
461C22×C3—92no—
462C22×C3—126no—
463C22×C3—121no—
464C22×C3—121no—
465C22×C3—122no—
466C22×C3—362no—
467C22×C3—362no—
468C22×C3C2×D3×A411yescharacteristic
469C22×C3C22×C3×A411yescharacteristic
470C2×D3—32no—
471C2×D3—122no—
472C2×D3—122no—
473C2×D3C22×C3×A412yes—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—31no—
477C3×C4—31no—
478C3×C4—32no—
479C3×C4—31no—
480C3×C4—31no—
481C3×C4—43no—
482C3×C4—43no—
483C3×C4—61no—
484C3×C4—61no—
485C3×C4—83no—
486C3×C4—83no—
487C3×C4—92no—
488C3×C4—126no—
489C3×C4C2×D3×A411yescharacteristic
490C3×C4C2×D3×A411yescharacteristic
491C3×C4C22×C3×A411yescharacteristic
492C3×C4C22×C3×A411yescharacteristic
493Dic3—32no—
494Dic3C22×C3×A412yes—
495C32—41no—
496C32—43no—
497C32—83no—
498C32—81no—
499C32#594511yescharacteristic, complemented
500C23—31no—
501C23—92no—
502C23—92no—
503Q8—31no—
504Q8—31no—
505Q8—92no—
506Q8—92no—
507Q8—91no—
508Q8C22×C32×D311yescharacteristic
509D4—31no—
510D4—31no—
511D4—32no—
512D4—32no—
513D4—31no—
514D4—92no—
515D4—92no—
516D4—92no—
517D4—92no—
518D4—94no—
519D4—92no—
520D4—91no—
521C2×C4—31no—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—31no—
525C2×C4—31no—
526C2×C4—32no—
527C2×C4—92no—
528C2×C4—92no—
529C2×C4—92no—
530C2×C4—94no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4C3×D3×A411yescharacteristic
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—43no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—62no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—83no—
545C2×C3—86no—
546C2×C3—83no—
547C2×C3—121no—
548C2×C3—121no—
549C2×C3—182no—
550C2×C3—246no—
551C2×C3C22×D3×A411yescharacteristic
552C2×C3C23×C3×A411yescharacteristic
553D3—22no—
554D3—62no—
555C22—31no—
556C22—31no—
557C22—32no—
558C22—92no—
559C22—121no—
560C22—362no—
561C22—362no—
562C22C2×C3×D3×A411yescharacteristic
563C4—31no—
564C4—31no—
565C4—32no—
566C4—92no—
567C4C2×C3×D3×A411yescharacteristic
568C4C2×C3×D3×A411yescharacteristic
569C3—21no—
570C3—43no—
571C3—83no—
572C3#5964⋊C311yescharacteristic, direct product factor
573C3C3×#594511yescharacteristic, complemented
574C2—21no—
575C2—61no—
576C2—61no—
577C2—62no—
578C2—182no—
579C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC3×C4C2×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC2×C32×#41C2×C3
socleC2×C32C23×A4
hypercenterC2×C3×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C3×#1010⋊C3C2semidirect
3. C3×D3×#41⋊C3C2semidirect
4. C3×C3⋊#994C2semidirect
5. C3×#1007⋊C3C2semidirect
6. C3×#5964C3semidirect
8. #5964⋊C3C3direct (N × H)
10. C32×#41⋊C3C22semidirect
11. C32×#41⋊C3C22semidirect
12. C2×C32×SL(2,𝔽3)C22semidirect
13. C3×D3×SL(2,𝔽3)C22semidirect
14. C3×C3⋊(#41⋊C3)C22semidirect
15. C2×C32×#41C2×C3semidirect
17. C3×#1010C2×C3semidirect
19. C3×#1009C2×C3semidirect
21. C3×D3×#41C2×C3semidirect
22. C3×#1007C2×C3semidirect
32. C2×C3×#41⋊C3D3semidirect
33. C2×C3×#41⋊C3C2×C3semidirect
34. #1010⋊C3C2×C3semidirect
35. D3×#41⋊C3C2×C3semidirect
36. C3⋊#994C2×C3semidirect
37. #1007⋊C3C2×C3semidirect
45. #5964C32semidirect
88. C3×#41⋊C3C2×D3semidirect
89. C3×#41⋊C3C2×D3semidirect
90. C3×#41⋊C3C22×C3semidirect
91. C3×#41⋊C3C22×C3semidirect
95. C2×C3×SL(2,𝔽3)C2×D3semidirect
96. C2×C3×SL(2,𝔽3)C22×C3semidirect
97. D3×SL(2,𝔽3)C22×C3semidirect
98. C3⋊(#41⋊C3)C22×C3semidirect
115. C2×C3×#41C3×D3semidirect
116. C2×C3×#41C2×C32semidirect
118. #1010C2×C32semidirect
120. #1009C2×C32semidirect
122. D3×#41C2×C32semidirect
123. #1007C2×C32semidirect
132. C2×#41⋊C3C3×D3semidirect
266. #41⋊C3C2×C3×D3semidirect
267. #41⋊C3C2×C3×D3semidirect
270. C2×SL(2,𝔽3)C2×C3×D3semidirect
311. C2×#41C32×D3semidirect
499. C32#5945semidirect
572. C3#5964⋊C3direct (N × H)
573. C3C3×#5945semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×#1010⋊C32
C3×D3×#41⋊C32
C3×C3⋊#9942
C3×#1007⋊C32
C3×#59643
C3×#59453
#5964⋊C33
C32×#2354