Finite Group Databasebeta

← C3×#10591682 #13158697

Subgroups — C3×#10591682 #13158697

Order 1728.

622 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64160C211yesmaximal, characteristic
2C32×#951C211yesmaximal, characteristic
3C32×#948C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5C3×#63577C211yesmaximal, characteristic, complemented
6C3×#63561C211yesmaximal, characteristic, complemented
7C3×#63560C211yesmaximal, characteristic, complemented
8C3×#5204—31nomaximal
9C3×#5143—31nomaximal
10#10591682C311yesmaximal, characteristic, direct product factor
11C33×#41C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×#3756—21no—
15C3×#3754—21no—
16C3×#3748C2211yescharacteristic
17C32×#216—21no—
18C32×#214—21no—
19C32×#208—21no—
20C32×#208C2211yescharacteristic
21C32×#208C2211yescharacteristic
22C2×C32×C3⋊C8—21no—
23C2×C32×C3⋊C8C2211yescharacteristic
24C3×#3719—21no—
25C3×#3719—21no—
26C32×#137—31no—
27C32×#131—31no—
28#64160C2×C311yescharacteristic
29C3×#951—21no—
30C3×#951—31no—
31C3×#951—31no—
32C3×#951—61no—
33C3×#951D311yescharacteristic
34C3×#951C2×C311yescharacteristic
35C3×#948—21no—
36C3×#948D311yescharacteristic
37C3×#948C2×C311yescharacteristic
38C2×C3×#216—31no—
39C2×C3×Dic12—31no—
40C2×#3719C2×C311yescharacteristic, complemented
41C3×#836—31no—
42C3×#824—31no—
43C3×#824—31no—
44C3×#824—31no—
45C3×#822—31no—
46C3×#819—31no—
47#63577C2×C311yescharacteristic, complemented
48#63561C2×C311yescharacteristic, complemented
49#63560C2×C311yescharacteristic, complemented
50C33×Q8D411yescharacteristic
51C33×D4—21no—
52C33×D4D411yescharacteristic
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3—21no—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66C3×#473—91no—
67#5204—31no—
68#5143—31no—
69C32×#41—21no—
70C32×#41—21no—
71C32×#41—21no—
72C32×#41—41no—
73C32×#41C2×D311yescharacteristic
74C32×#41C2×D311yescharacteristic
75C32×#41C22×C311yescharacteristic
76C2×C32×Q8—31no—
77C2×#634C22×C311yescharacteristic
78C2×C3×Dic6—21no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C22×C311yescharacteristic
84C32×Q16—61no—
85C32×SD16—61no—
86C32×M16—31no—
87C32×M16—31no—
88C32×M16—61no—
89C2×C32×C8—31no—
90C2×C32×C8—61no—
91#3756—21no—
92#3754—21no—
93#3748C22×C311yescharacteristic
94C3×#216—22no—
95C3×#216—21no—
96C3×#216—21no—
97C3×#216—62no—
98C3×#216—61no—
99C3×#216—61no—
100C3×#216—61no—
101C3×#216—61no—
102C3×#214—22no—
103C3×#214—21no—
104C3×#214—21no—
105C3×#214—62no—
106C3×#214—61no—
107C3×#214—61no—
108C3×#208—22no—
109C3×#208—21no—
110C3×#208—21no—
111C3×#208—21no—
112C3×#208—21no—
113C3×#208—31no—
114C3×#208—31no—
115C3×#208—61no—
116C3×#208C2×D311yescharacteristic
117C3×#208C2×D311yescharacteristic
118C3×#208C22×C311yescharacteristic
119C3×#208C22×C311yescharacteristic
120C2×C3×C3⋊C8—22no—
121C2×C3×C3⋊C8—21no—
122C2×C3×C3⋊C8—21no—
123C2×C3×C3⋊C8—21no—
124C2×C3×C3⋊C8C2×D311yescharacteristic
125C2×C3×C3⋊C8C22×C311yescharacteristic
126C3×Dic12—61no—
127C3×Dic12—61no—
128#3719—21no—
129#3719—21no—
130C22×C33—21no—
131C22×C33#4111yescharacteristic
132C33×C4—21no—
133C33×C4C2×D411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C3×C3⋊Dic3—41no—
136C32×Dic3—41no—
137C3×#137—31no—
138C3×#137—31no—
139C3×#137—61no—
140C3×#137—91no—
141C2×C3×Q16—91no—
142C3×#131—31no—
143C3×#131—31no—
144C3×#131—61no—
145#951—31no—
146#951—31no—
147#951—61no—
148#951C3×D311yescharacteristic
149#948C3×D311yescharacteristic
150C2×#216—31no—
151C2×Dic12—31no—
152C3×#108—91no—
153C3×#101—91no—
154C3×#101—91no—
155#836—31no—
156#824—31no—
157#824—31no—
158#824—31no—
159#822—31no—
160#819—31no—
161C32×Q8—21no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—41no—
165C32×Q8—61no—
166C32×Q8—61no—
167C32×Q8D1211yescharacteristic
168C32×Q8#6711yescharacteristic
169C32×Q8C3×D411yescharacteristic
170C32×D4—21no—
171C32×D4—22no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—42no—
181C32×D4D1211yescharacteristic
182C32×D4#6711yescharacteristic
183C32×D4C3×D411yescharacteristic
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—41no—
192C2×C32×C4—41no—
193C2×C32×C4—41no—
194C2×C32×C4—42no—
195C2×C32×C4—61no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199C2×C3⋊Dic3—21no—
200#634—21no—
201#634—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—41no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—121no—
208C3×Dic6—22no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—41no—
214C3×Dic6—62no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—121no—
220C32×C8—31no—
221C32×C8—31no—
222C32×C8—61no—
223C32×C8—61no—
224C3⋊(C3⋊C8)—21no—
225C3×C3⋊C8—22no—
226C3×C3⋊C8—22no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—21no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—21no—
231C3×C3⋊C8—21no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—62no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—61no—
236C3×C3⋊C8#6711yescharacteristic
237C3×C3⋊C8#6711yescharacteristic
238C3×C3⋊C8C3×D411yescharacteristic
239C3×C3⋊C8C3×D411yescharacteristic
240#473—91noSylow
241C2×C33—21no—
242C2×C33—41no—
243C2×C33#12111yescharacteristic
244C3×#41—21no—
245C3×#41—21no—
246C3×#41—21no—
247C3×#41—41no—
248C3×#41D3211yescharacteristic
249C3×#41C2×C3×D311yescharacteristic
250C3×#41C2×C3×D311yescharacteristic
251C2×C3×Q8—31no—
252C2×C3×Q8—31no—
253C2×C3×Q8—61no—
254C2×C3×Q8—91no—
255C2×Dic6—31no—
256C2×Dic6—31no—
257C2×Dic6—61no—
258C2×Dic6C2×C3×D311yescharacteristic
259C3×Q16—62no—
260C3×Q16—61no—
261C3×Q16—61no—
262C3×Q16—181no—
263C3×Q16—181no—
264C3×Q16—181no—
265C3×SD16—62no—
266C3×SD16—61no—
267C3×SD16—61no—
268C3×SD16—181no—
269C3×M16—31no—
270C3×M16—31no—
271C3×M16—31no—
272C3×M16—31no—
273C3×M16—61no—
274C3×M16—61no—
275C3×M16—61no—
276C3×M16—61no—
277C3×M16—62no—
278C3×M16—91no—
279C2×C3×C8—31no—
280C2×C3×C8—31no—
281C2×C3×C8—61no—
282C2×C3×C8—61no—
283C2×C3×C8—61no—
284C2×C3×C8—62no—
285#216—21no—
286#216—62no—
287#216—61no—
288#216—61no—
289#216—61no—
290#216—61no—
291#214—21no—
292#214—62no—
293#214—61no—
294#214—61no—
295#208—21no—
296#208—31no—
297#208—31no—
298#208—61no—
299#208C2×C3×D311yescharacteristic
300#208C2×C3×D311yescharacteristic
301C2×C3⋊C8—21no—
302C2×C3⋊C8C2×C3×D311yescharacteristic
303Dic12—61no—
304Dic12—61no—
305C22×C32—21no—
306C22×C32—22no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—41no—
313C22×C32—41no—
314C22×C32—41no—
315C22×C32—42no—
316C22×C32#23511yescharacteristic
317C22×C32#23711yescharacteristic
318C22×C32C3×#4111yescharacteristic
319C32×C4—21no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—41no—
330C32×C4—41no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—42no—
334C32×C4—121no—
335C32×C4C2×D1211yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C32×C4C2×C3×D411yescharacteristic
340C32×C4C2×C3×D411yescharacteristic
341C3⋊Dic3—41no—
342C3×Dic3—42no—
343C3×Dic3—41no—
344C3×Dic3—41no—
345C3×Dic3—122no—
346C3×Dic3—121no—
347C3×Dic3—121no—
348#137—31no—
349#137—91no—
350C2×Q16—91no—
351#131—31no—
352#108—91no—
353#101—91no—
354#101—91no—
355C33#47311yescharacteristic, complemented, Sylow
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—41no—
360C3×Q8—62no—
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—121no—
366C3×Q8—181no—
367C3×Q8—181no—
368C3×Q8#62611yescharacteristic
369C3×Q8C3×D1211yescharacteristic
370C3×Q8C3×#6711yescharacteristic
371C3×D4—22no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—42no—
381C3×D4—41no—
382C3×D4#62611yescharacteristic
383C3×D4C3×D1211yescharacteristic
384C3×D4C3×#6711yescharacteristic
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—121no—
399C2×C3×C4—181no—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403C2×Dic3—21no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—121no—
407Dic6—21no—
408Dic6—21no—
409Dic6—62no—
410Dic6—61no—
411Dic6—61no—
412Dic6—61no—
413Dic6—61no—
414Dic6—121no—
415C3×C8—31no—
416C3×C8—31no—
417C3×C8—31no—
418C3×C8—31no—
419C3×C8—61no—
420C3×C8—61no—
421C3×C8—62no—
422C3×C8—62no—
423C3×C8—61no—
424C3×C8—61no—
425C3×C8—61no—
426C3×C8—61no—
427C3×C8—181no—
428C3⋊C8—21no—
429C3⋊C8—21no—
430C3⋊C8—62no—
431C3⋊C8—61no—
432C3⋊C8—61no—
433C3⋊C8C3×#6711yescharacteristic
434C3⋊C8C3×#6711yescharacteristic
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—41no—
442C2×C32—41no—
443C2×C32—41no—
444C2×C32—41no—
445C2×C32—41no—
446C2×C32—42no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—81no—
450C2×C32—81no—
451C2×C32—81no—
452C2×C32—82no—
453C2×C32#93011yescharacteristic
454C2×C32#93911yescharacteristic
455C2×C32C3×#12111yescharacteristic
456#41C3×D3211yescharacteristic
457C2×Q8—31no—
458C2×Q8—91no—
459Q16—61no—
460Q16—181no—
461Q16—181no—
462Q16—181no—
463SD16—61no—
464SD16—181no—
465M16—31no—
466M16—31no—
467M16—61no—
468M16—91no—
469C2×C8—31no—
470C2×C8—61no—
471C22×C3—22no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—42no—
481C22×C3—41no—
482C22×C3#380411yescharacteristic
483C22×C3C3×#23511yescharacteristic
484C22×C3C3×#23711yescharacteristic
485C3×C4—22no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—42no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—122no—
501C3×C4—121no—
502C3×C4—121no—
503C3×C4—361no—
504C3×C4C2×#62611yescharacteristic
505C3×C4D3×#6711yescharacteristic
506C3×C4C2×C3×D1211yescharacteristic
507C3×C4C3×D3×D411yescharacteristic
508C3×C4C2×C3×#6711yescharacteristic
509C3×C4C2×C3×#6711yescharacteristic
510Dic3—41no—
511Dic3—122no—
512Dic3—121no—
513Dic3—121no—
514C32—21no—
515C32—21no—
516C32—21no—
517C32—41no—
518C32#514311yescharacteristic, complemented
519C32#520411yescharacteristic, complemented
520C32C3×#47311yescharacteristic, complemented
521Q8—61no—
522Q8—61no—
523Q8—181no—
524Q8—181no—
525Q8C3×#62611yescharacteristic
526D4—21no—
527D4C3×#62611yescharacteristic
528C2×C4—21no—
529C2×C4—61no—
530C2×C4—181no—
531C2×C4C2×C3×D3211yescharacteristic
532C8—31no—
533C8—31no—
534C8—61no—
535C8—61no—
536C8—181no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—42no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—81no—
552C2×C3—81no—
553C2×C3—81no—
554C2×C3—82no—
555C2×C3#6397811yescharacteristic
556C2×C3C3×#93011yescharacteristic
557C2×C3C3×#93911yescharacteristic
558C22—21no—
559C22C3×#380411yescharacteristic
560C4—21no—
561C4—121no—
562C4—361no—
563C4C2×C3×#62611yescharacteristic
564C4C3×D3×#6711yescharacteristic
565C3—21no—
566C3—21no—
567C3—21no—
568C3—41no—
569C3#1059168211yescharacteristic, direct product factor
570C3C3×#514311yescharacteristic, complemented
571C3C3×#520411yescharacteristic, complemented
572C2—21no—
573C2—41no—
574C2C3×#6397811yescharacteristic
subgroupG/H
centerC2×C3#63978
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC33×#41C22
socleC2×C33#121
hypercenterC3×#41D32
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
4. C2×C3×#3719C2semidirect
5. C3×#63577C2semidirect
6. C3×#63561C2semidirect
7. C3×#63560C2semidirect
10. #10591682C3direct (N × H)
40. C2×#3719C2×C3semidirect
47. #63577C2×C3semidirect
48. #63561C2×C3semidirect
49. #63560C2×C3semidirect
355. C33#473semidirect
518. C32#5143semidirect
519. C32#5204semidirect
520. C32C3×#473semidirect
569. C3#10591682direct (N × H)
570. C3C3×#5143semidirect
571. C3C3×#5204semidirect

Maximal subgroups

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

subgroupindex
C3×#641602
C32×#9512
C32×#9482
C2×C3×#37192
C3×#635772
C3×#635612
C3×#635602
C3×#52043
C3×#51433
#105916823