Finite Group Databasebeta

← C3×#5968⋊C3 #13181453

Subgroups — C3×#5968⋊C3 #13181453

Order 1728.

1048 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C3×D3×#41⋊C3C211yesmaximal, characteristic, complemented
3C3×C3⋊#994C211yesmaximal, characteristic, complemented
4C3×#1007⋊C3C212yesmaximal, complemented
5C2×C3×C3⋊(#41⋊C3)C212yesmaximal, complemented
6C3×#5968C311yesmaximal, characteristic, complemented
7C3×#5945—31nomaximal
8#5968⋊C3C313yesmaximal, direct product factor
9C32×#237—41nomaximal
10C32×#41⋊C3C2211yescharacteristic, complemented
11C2×C32×SL(2,𝔽3)C2212yescomplemented
12C3×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
13C3×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
14C3×C3⋊(#41⋊C3)C2212yescomplemented
15C32×2−1+4C2×C311yescharacteristic, complemented
16C3×#1009—32no—
17C3×#1009—31no—
18C3×#1009C2×C311yescharacteristic, complemented
19C3×D3×#41—31no—
20C3×D3×#41—32no—
21C3×D3×#41C2×C311yescharacteristic, complemented
22C3×#1007—31no—
23C3×#1007C2×C312yescomplemented
24C2×C3×#239—31no—
25C2×C3×#239C2×C312yescomplemented
26C3×2−1+4⋊C3—21no—
27C3×2−1+4⋊C3—23no—
28C3×2−1+4⋊C3—32no—
29C3×2−1+4⋊C3D311yescharacteristic, complemented
30C3×2−1+4⋊C3C2×C313yescomplemented
31C3×#994—31no—
32C2×C3×#41⋊C3—31no—
33C2×C3×#41⋊C3—32no—
34D3×#41⋊C3C2×C313yescomplemented
35C3⋊#994C2×C313yescomplemented
36#1007⋊C3C2×C316yescomplemented
37C2×C3⋊(#41⋊C3)C2×C316yescomplemented
38C33×D4—41no—
39C32×#67—42no—
40C2×C32×Dic3—42no—
41C32×C4×D3—41no—
42C32×Dic6—41no—
43C32×SL(2,𝔽3)C2311yescharacteristic
44C3×#585—31no—
45#5968C3211yescharacteristic, complemented
46#5945—33no—
47C32×#41—31no—
48C32×#41—32no—
49C32×#41—121no—
50C32×#41C22×C311yescharacteristic
51C2×C32×Q8—31no—
52C2×C32×Q8C22×C312yes—
53C3×#239—32no—
54C3×#239—32no—
55C3×#239—31no—
56C3×#239—32no—
57C3×#239—32no—
58C3×#239C22×C311yescharacteristic
59C3×#239C22×C312yes—
60C3×D3×Q8—31no—
61C3×D3×Q8—32no—
62C3×D3×Q8C22×C311yescharacteristic
63C3×#237—32no—
64C3×#237—31no—
65C3×#237—43no—
66C3×#237A411yescharacteristic
67C3×D3×D4—34no—
68C3×D3×D4—31no—
69C3×D3×D4—31no—
70C3×D3×D4—32no—
71C3×D3×D4—32no—
72C3×#235—34no—
73C3×#235—32no—
74C3×#235—32no—
75C3×#235—31no—
76C3×#235—31no—
77C2×C3×D12—32no—
78C2×C3×D12—32no—
79C2×C3×D12—31no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—31no—
83C3×#41⋊C3—21no—
84C3×#41⋊C3—23no—
85C3×#41⋊C3—31no—
86C3×#41⋊C3—32no—
87C3×#41⋊C3C2×D311yescharacteristic, complemented
88C3×#41⋊C3C22×C313yescomplemented
89C2×C3×SL(2,𝔽3)—22no—
90C2×C3×SL(2,𝔽3)—26no—
91C2×C3×SL(2,𝔽3)—31no—
92C2×C3×SL(2,𝔽3)C2×D312yescomplemented
93C2×C3×SL(2,𝔽3)C22×C316yescomplemented
94D3×SL(2,𝔽3)C22×C313yescomplemented
95C3⋊(#41⋊C3)C22×C313yescomplemented
96C3⋊(#41⋊C3)C22×C316yescomplemented
97C22×C33—42no—
98C2×C32×D3—41no—
99C33×C4—41no—
100C32×Dic3—42no—
101C32×Dic3—41no—
102C3×2−1+4—21no—
103C3×2−1+4—32no—
104C3×2−1+4—91no—
105C3×2−1+4C3×D311yescharacteristic, complemented
106C3×2−1+4C2×C3211yescharacteristic, complemented
107C3×H(5,𝔽2)—31no—
108C3×H(5,𝔽2)—91no—
109C3×H(5,𝔽2)—92no—
110C2×C3×#41—31no—
111C2×C3×#41—32no—
112C2×C3×#41—91no—
113C2×C3×#41—91no—
114C2×C3×#41—92no—
115#1009—31no—
116#1009—32no—
117#1009C2×C3211yescharacteristic, complemented
118D3×#41—31no—
119D3×#41—32no—
120D3×#41C2×C3211yescharacteristic, complemented
121#1007—31no—
122#1007C2×C3212yescomplemented
123C2×#239—31no—
124C2×#239C2×C3212yescomplemented
1252−1+4⋊C3—23no—
1262−1+4⋊C3—36no—
1272−1+4⋊C3C3×D313yescomplemented
128#994—33no—
129C2×#41⋊C3—33no—
130C2×#41⋊C3—36no—
131C22×C3×D3—32no—
132C22×C3×D3—32no—
133C22×C3×D3—31no—
134C32×Q8—32no—
135C32×Q8—31no—
136C32×Q8—121no—
137C32×Q8C23×C311yescharacteristic
138C32×D4—31no—
139C32×D4—32no—
140C32×D4—41no—
141C32×D4—43no—
142C32×D4—83no—
143C32×D4—81no—
144C32×D4—122no—
145C32×D4C2×A411yescharacteristic
146C2×C32×C4—32no—
147C2×C32×C4—32no—
148C2×C32×C4—31no—
149C2×C32×C4—121no—
150C2×C32×C4—122no—
151C3×#67—32no—
152C3×#67—32no—
153C3×#67—32no—
154C3×#67—46no—
155C3×#67C2×A412yes—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—46no—
158C2×C3×Dic3C2×A412yes—
159C3×D12—31no—
160C3×D12—32no—
161C3×D12—34no—
162C3×D12—32no—
163C3×D12—31no—
164C3×C4×D3—34no—
165C3×C4×D3—31no—
166C3×C4×D3—31no—
167C3×C4×D3—32no—
168C3×C4×D3—31no—
169C3×C4×D3—32no—
170C3×C4×D3—32no—
171C3×C4×D3—43no—
172C3×C4×D3C2×A411yescharacteristic
173C3×Dic6—32no—
174C3×Dic6—31no—
175C3×Dic6—43no—
176C3×Dic6C2×A411yescharacteristic
177C3×SL(2,𝔽3)—21no—
178C3×SL(2,𝔽3)—23no—
179C3×SL(2,𝔽3)C22×D311yescharacteristic
180C3×SL(2,𝔽3)C23×C313yes—
181#585—31noSylow
182C2×C33—41no—
183C2×C33—82no—
184C32×D3—81no—
185C3×#41—21no—
186C3×#41—31no—
187C3×#41—31no—
188C3×#41—31no—
189C3×#41—32no—
190C3×#41—32no—
191C3×#41—31no—
192C3×#41—32no—
193C3×#41—61no—
194C3×#41—62no—
195C3×#41—92no—
196C3×#41—92no—
197C3×#41—91no—
198C3×#41—91no—
199C3×#41—94no—
200C3×#41—92no—
201C3×#41—91no—
202C3×#41—92no—
203C3×#41—92no—
204C3×#41—91no—
205C3×#41—92no—
206C3×#41—92no—
207C3×#41—123no—
208C3×#41C2×C3×D311yescharacteristic
209C3×#41C22×C3211yescharacteristic
210C2×C3×Q8—22no—
211C2×C3×Q8—31no—
212C2×C3×Q8—31no—
213C2×C3×Q8—31no—
214C2×C3×Q8—61no—
215C2×C3×Q8—91no—
216C2×C3×Q8—92no—
217C2×C3×Q8C2×C3×D312yes—
218C2×C3×Q8C22×C3212yes—
219C2×C3×D4—92no—
220C2×C3×D4—92no—
221C2×C3×D4—91no—
222C2×C3×D4—92no—
223C2×C3×D4—91no—
224C2×C3×D4—94no—
225C2×C3×D4—92no—
226C2×C3×D4—91no—
227C22×C3×C4—92no—
228C22×C3×C4—91no—
229C22×C3×C4—92no—
230#239—32no—
231#239—32no—
232#239—31no—
233#239—32no—
234#239—32no—
235#239C22×C3211yescharacteristic
236#239C22×C3212yes—
237D3×Q8—31no—
238D3×Q8—32no—
239D3×Q8C22×C3211yescharacteristic
240#237—32no—
241#237—31no—
242#237C3×A411yescharacteristic
243D3×D4—31no—
244D3×D4—34no—
245D3×D4—31no—
246D3×D4—32no—
247D3×D4—32no—
248#235—34no—
249#235—32no—
250#235—32no—
251#235—31no—
252#235—31no—
253C2×D12—32no—
254C2×D12—32no—
255C2×D12—31no—
256C2×C4×D3—32no—
257C2×C4×D3—32no—
258C2×C4×D3—31no—
259#41⋊C3—23no—
260#41⋊C3—33no—
261#41⋊C3—36no—
262#41⋊C3C2×C3×D313yescomplemented
263C2×SL(2,𝔽3)—26no—
264C2×SL(2,𝔽3)—33no—
265C2×SL(2,𝔽3)C2×C3×D316yescomplemented
266C22×C32—31no—
267C22×C32—42no—
268C22×C32—46no—
269C22×C32—86no—
270C22×C32—82no—
271C22×C32—121no—
272C22×C32C22×A412yes—
273C2×C3×D3—32no—
274C2×C3×D3—31no—
275C2×C3×D3—43no—
276C2×C3×D3—121no—
277C2×C3×D3—122no—
278C2×C3×D3—122no—
279C2×C3×D3C22×A411yescharacteristic
280C32×C4—32no—
281C32×C4—31no—
282C32×C4—41no—
283C32×C4—43no—
284C32×C4—83no—
285C32×C4—81no—
286C32×C4—121no—
287C32×C4—122no—
288C32×C4C22×A411yescharacteristic
289C3×Dic3—31no—
290C3×Dic3—46no—
291C3×Dic3—43no—
292C3×Dic3C22×A412yes—
293C3×Dic3C22×A411yescharacteristic
2942−1+4—32no—
2952−1+4—91no—
2962−1+4C32×D311yescharacteristic, complemented
297H(5,𝔽2)—31no—
298H(5,𝔽2)—91no—
299H(5,𝔽2)—92no—
300C2×#41—31no—
301C2×#41—32no—
302C2×#41—91no—
303C2×#41—91no—
304C2×#41—92no—
305C33—41noSylow
306C23×C3—91no—
307C23×C3—92no—
308C23×C3—92no—
309C22×D3—32no—
310C22×D3—32no—
311C22×D3—31no—
312C3×Q8—21no—
313C3×Q8—32no—
314C3×Q8—32no—
315C3×Q8—31no—
316C3×Q8—31no—
317C3×Q8—31no—
318C3×Q8—61no—
319C3×Q8—62no—
320C3×Q8—92no—
321C3×Q8—91no—
322C3×Q8—123no—
323C3×Q8C22×C3×D311yescharacteristic
324C3×Q8C23×C3211yescharacteristic
325C3×D4—21no—
326C3×D4—31no—
327C3×D4—31no—
328C3×D4—32no—
329C3×D4—32no—
330C3×D4—32no—
331C3×D4—43no—
332C3×D4—62no—
333C3×D4—61no—
334C3×D4—83no—
335C3×D4—92no—
336C3×D4—92no—
337C3×D4—91no—
338C3×D4—92no—
339C3×D4—94no—
340C3×D4—92no—
341C3×D4—92no—
342C3×D4—91no—
343C3×D4—126no—
344C3×D4D3×A411yescharacteristic
345C3×D4C2×C3×A411yescharacteristic
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—32no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—91no—
358C2×C3×C4—92no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—94no—
364C2×C3×C4—91no—
365C2×C3×C4—123no—
366C2×C3×C4—126no—
367#67—32no—
368#67—32no—
369#67—32no—
370#67C2×C3×A412yes—
371C2×Dic3—31no—
372C2×Dic3C2×C3×A412yes—
373D12—31no—
374D12—32no—
375D12—34no—
376D12—32no—
377D12—31no—
378C4×D3—34no—
379C4×D3—31no—
380C4×D3—31no—
381C4×D3—32no—
382C4×D3—31no—
383C4×D3—32no—
384C4×D3—32no—
385C4×D3C2×C3×A411yescharacteristic
386Dic6—32no—
387Dic6—31no—
388Dic6C2×C3×A411yescharacteristic
389SL(2,𝔽3)—23no—
390SL(2,𝔽3)C22×C3×D313yes—
391C2×C32—22no—
392C2×C32—41no—
393C2×C32—43no—
394C2×C32—61no—
395C2×C32—83no—
396C2×C32—82no—
397C2×C32—86no—
398C2×C32—81no—
399C2×C32—162no—
400C2×C32—166no—
401C2×C32—241no—
402C2×C32C23×A411yescharacteristic
403C3×D3—21no—
404C3×D3—61no—
405C3×D3—62no—
406C3×D3—83no—
407#41—31no—
408#41—31no—
409#41—32no—
410#41—31no—
411#41—32no—
412#41—92no—
413#41—92no—
414#41—91no—
415#41—91no—
416#41—94no—
417#41—92no—
418#41—91no—
419#41—92no—
420#41—92no—
421#41—91no—
422#41—92no—
423#41—92no—
424#41C2×C32×D311yescharacteristic
425C2×Q8—31no—
426C2×Q8—31no—
427C2×Q8—91no—
428C2×Q8—92no—
429C2×Q8C2×C32×D312yes—
430C2×D4—92no—
431C2×D4—92no—
432C2×D4—91no—
433C2×D4—92no—
434C2×D4—91no—
435C2×D4—94no—
436C2×D4—92no—
437C2×D4—91no—
438C22×C4—92no—
439C22×C4—91no—
440C22×C4—92no—
441C22×C3—22no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—46no—
446C22×C3—61no—
447C22×C3—86no—
448C22×C3—91no—
449C22×C3—92no—
450C22×C3—123no—
451C22×C3—362no—
452C22×C3—361no—
453C22×C3—362no—
454C22×C3C2×D3×A412yes—
455C22×C3C22×C3×A412yes—
456C2×D3—32no—
457C2×D3—31no—
458C2×D3—121no—
459C2×D3—122no—
460C2×D3—122no—
461C2×D3C22×C3×A411yescharacteristic
462C3×C4—21no—
463C3×C4—31no—
464C3×C4—32no—
465C3×C4—31no—
466C3×C4—32no—
467C3×C4—31no—
468C3×C4—32no—
469C3×C4—43no—
470C3×C4—61no—
471C3×C4—62no—
472C3×C4—83no—
473C3×C4—91no—
474C3×C4—123no—
475C3×C4—126no—
476C3×C4C2×D3×A411yescharacteristic
477C3×C4C22×C3×A411yescharacteristic
478Dic3—31no—
479Dic3C22×C3×A412yes—
480Dic3C22×C3×A411yescharacteristic
481C32—41no—
482C32—43no—
483C32—83no—
484C32—81no—
485C32#594511yescharacteristic, complemented
486C23—91no—
487C23—92no—
488C23—92no—
489Q8—32no—
490Q8—31no—
491Q8—31no—
492Q8—92no—
493Q8—91no—
494Q8C22×C32×D311yescharacteristic
495D4—31no—
496D4—32no—
497D4—32no—
498D4—92no—
499D4—92no—
500D4—91no—
501D4—92no—
502D4—94no—
503D4—92no—
504D4—92no—
505D4—91no—
506D4C3×D3×A411yescharacteristic
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—91no—
513C2×C4—92no—
514C2×C4—92no—
515C2×C4—92no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—94no—
519C2×C4—91no—
520C2×C3—22no—
521C2×C3—22no—
522C2×C3—21no—
523C2×C3—42no—
524C2×C3—43no—
525C2×C3—61no—
526C2×C3—61no—
527C2×C3—61no—
528C2×C3—86no—
529C2×C3—83no—
530C2×C3—121no—
531C2×C3—166no—
532C2×C3—181no—
533C2×C3—182no—
534C2×C3—243no—
535C2×C3C22×D3×A411yescharacteristic
536C2×C3C23×C3×A411yescharacteristic
537D3—21no—
538D3—61no—
539D3—62no—
540C22—31no—
541C22—31no—
542C22—91no—
543C22—92no—
544C22—362no—
545C22—361no—
546C22—362no—
547C22C2×C3×D3×A412yes—
548C4—31no—
549C4—32no—
550C4—31no—
551C4—32no—
552C4—91no—
553C4C2×C3×D3×A411yescharacteristic
554C3—21no—
555C3—43no—
556C3—83no—
557C3#5968⋊C311yescharacteristic, direct product factor
558C3C3×#594511yescharacteristic, complemented
559C2—22no—
560C2—61no—
561C2—61no—
562C2—181no—
563C2—182no—
564C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC2×C3C22×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC32×2−1+4C2×C3
socleC2×C32C23×A4
hypercenterC3×D4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. C3×D3×#41⋊C3C2semidirect
3. C3×C3⋊#994C2semidirect
4. C3×#1007⋊C3C2semidirect
5. C2×C3×C3⋊(#41⋊C3)C2semidirect
6. C3×#5968C3semidirect
8. #5968⋊C3C3direct (N × H)
10. C32×#41⋊C3C22semidirect
11. C2×C32×SL(2,𝔽3)C22semidirect
12. C3×D3×SL(2,𝔽3)C22semidirect
13. C3×C3⋊(#41⋊C3)C22semidirect
14. C3×C3⋊(#41⋊C3)C22semidirect
15. C32×2−1+4C2×C3semidirect
18. C3×#1009C2×C3semidirect
21. C3×D3×#41C2×C3semidirect
23. C3×#1007C2×C3semidirect
25. C2×C3×#239C2×C3semidirect
29. C3×2−1+4⋊C3D3semidirect
30. C3×2−1+4⋊C3C2×C3semidirect
34. D3×#41⋊C3C2×C3semidirect
35. C3⋊#994C2×C3semidirect
36. #1007⋊C3C2×C3semidirect
37. C2×C3⋊(#41⋊C3)C2×C3semidirect
45. #5968C32semidirect
87. C3×#41⋊C3C2×D3semidirect
88. C3×#41⋊C3C22×C3semidirect
92. C2×C3×SL(2,𝔽3)C2×D3semidirect
93. C2×C3×SL(2,𝔽3)C22×C3semidirect
94. D3×SL(2,𝔽3)C22×C3semidirect
95. C3⋊(#41⋊C3)C22×C3semidirect
96. C3⋊(#41⋊C3)C22×C3semidirect
105. C3×2−1+4C3×D3semidirect
106. C3×2−1+4C2×C32semidirect
117. #1009C2×C32semidirect
120. D3×#41C2×C32semidirect
122. #1007C2×C32semidirect
124. C2×#239C2×C32semidirect
127. 2−1+4⋊C3C3×D3semidirect
262. #41⋊C3C2×C3×D3semidirect
265. C2×SL(2,𝔽3)C2×C3×D3semidirect
296. 2−1+4C32×D3semidirect
485. C32#5945semidirect
557. C3#5968⋊C3direct (N × H)
558. C3C3×#5945semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
C3×D3×#41⋊C32
C3×C3⋊#9942
C3×#1007⋊C32
C2×C3×C3⋊(#41⋊C3)2
C3×#59683
C3×#59453
#5968⋊C33
C32×#2374