Finite Group Databasebeta

← C3×#10591611 #13158653

Subgroups — C3×#10591611 #13158653

Order 1728.

730 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64153C211yesmaximal, characteristic, complemented
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic
4C3×#63863C211yesmaximal, characteristic
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3716C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8C3×#5190—31nomaximal
9C3×#5167—31nomaximal
10#10591611C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3750—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#216—22no—
18C32×#215—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3719—22no—
23C3×#3716—22no—
24C2×C32×Q16—31no—
25C2×C32×SD16—31no—
26#64153C2×C311yescharacteristic, complemented
27C3×#947—31no—
28C3×#944—31no—
29C3×#944—31no—
30C3×#944—61no—
31C2×C3×#216—21no—
32C2×C3×#216—31no—
33C2×C3×#216D311yescharacteristic, complemented
34C2×C3×#216C2×C311yescharacteristic, complemented
35C2×C3×#215—21no—
36C2×C3×#215D311yescharacteristic
37C2×C3×#215C2×C311yescharacteristic
38C3×#936—31no—
39C2×C3×#214—31no—
40C2×C3×Dic12—31no—
41C2×C3×#204—31no—
42#63863C2×C311yescharacteristic
43C2×#3719C2×C311yescharacteristic, complemented
44C2×#3716C2×C311yescharacteristic
45C3×#815—31no—
46C3×#815—31no—
47#63555C2×C311yescharacteristic, complemented
48C22×C32×D3—21no—
49C33×Q8—21no—
50C33×Q8—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D412yes—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C32×C3⋊C8D412yes—
63C32×C3⋊C8D412yes—
64C3×#497—91no—
65#5190—31no—
66#5167—31no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—31no—
71C2×C32×Q8—41no—
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C22×C311yescharacteristic
75C2×C32×D4—31no—
76C2×#634C22×C311yescharacteristic
77C2×C3×D12—21no—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C22×C311yescharacteristic
80C2×C3×Dic6—21no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×Q16—62no—
87C32×SD16—62no—
88C2×C32×C8—31no—
89C2×C32×C8—31no—
90#3750—22no—
91C4×C3⋊Dic3C22×C311yescharacteristic
92C3×#217—62no—
93C3×#216—24no—
94C3×#216—22no—
95C3×#216—22no—
96C3×#216—62no—
97C3×#215—24no—
98C3×#215—22no—
99C3×#215—22no—
100C3×#214—62no—
101C3×(C2×D3)⋊C4—62no—
102C3×Dic3⋊C4—62no—
103C3×Dic3⋊C4—62no—
104C3×Dic3⋊C4—64no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C3×Dic12—62no—
115C3×#204—62no—
116(C2×D3)⋊Dic3—22no—
117#3719—22no—
118#3716—22no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—41no—
121C2×C32×D3—42no—
122C33×C4—41no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2×D411yescharacteristic
125C3×C3⋊Dic3—22no—
126C3×C3⋊Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—31no—
129C2×C3×Q16—31no—
130C2×C3×Q16—61no—
131C2×C3×Q16—91no—
132C2×C3×SD16—31no—
133C2×C3×SD16—31no—
134C2×C3×SD16—61no—
135C2×C3×SD16—91no—
136C3×C4⋊Q8—91no—
137C3×#124—91no—
138#947—31no—
139#944—31no—
140#944—31no—
141#944—61no—
142C2×#216—31no—
143C2×#216C3×D311yescharacteristic, complemented
144C2×#215C3×D311yescharacteristic
145#936—31no—
146C2×#214—31no—
147C2×Dic12—31no—
148C2×#204—31no—
149C3×#97—91no—
150#815—31no—
151#815—31no—
152C23×C32—61no—
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—41no—
156C32×Q8—21no—
157C32×Q8—21no—
158C32×Q8—22no—
159C32×Q8—22no—
160C32×Q8—21no—
161C32×Q8—21no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—41no—
165C32×Q8—41no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—42no—
169C32×Q8—42no—
170C32×Q8—61no—
171C32×Q8—61no—
172C32×D4—61no—
173C32×D4—61no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—41no—
182C2×C32×C4—41no—
183C2×C32×C4—41no—
184C2×C32×C4—42no—
185C2×C32×C4—61no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C23×C311yescharacteristic
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3C3×D412yes—
191#634—21no—
192#634—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C3×D12—22no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—41no—
208C3×Dic6—22no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—41no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—121no—
219C3×Dic6—121no—
220C32×C8—32no—
221C32×C8—32no—
222C3×C3⋊C8—22no—
223C3×C3⋊C8—22no—
224C3×C3⋊C8#6712yes—
225C3×C3⋊C8#6712yes—
226C3×C3⋊C8C3×D412yes—
227C3×C3⋊C8C3×D412yes—
228#497—91noSylow
229C2×C33#12711yescharacteristic
230C2×C33#13712yes—
231C32×D3—81no—
232C2×C3×Q8—21no—
233C2×C3×Q8—21no—
234C2×C3×Q8—21no—
235C2×C3×Q8—31no—
236C2×C3×Q8—31no—
237C2×C3×Q8—41no—
238C2×C3×Q8—61no—
239C2×C3×Q8—91no—
240C2×C3×Q8D3211yescharacteristic
241C2×C3×Q8C2×C3×D311yescharacteristic
242C2×C3×Q8C2×C3×D311yescharacteristic
243C2×C3×D4—31no—
244C2×C3×D4—31no—
245C2×C3×D4—61no—
246C2×D12C2×C3×D311yescharacteristic
247C2×Dic6—31no—
248C2×Dic6—31no—
249C2×Dic6—61no—
250C2×Dic6C2×C3×D311yescharacteristic
251C3×Q16—62no—
252C3×Q16—62no—
253C3×Q16—64no—
254C3×Q16—182no—
255C3×SD16—62no—
256C3×SD16—62no—
257C3×SD16—64no—
258C3×SD16—182no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—61no—
264C2×C3×C8—61no—
265C3×C4⋊C4—182no—
266C3×C22⋊C4—182no—
267C3×C42—91no—
268#217—62no—
269#216—22no—
270#216—62no—
271#215—22no—
272#214—62no—
273(C2×D3)⋊C4—62no—
274Dic3⋊C4—62no—
275Dic3⋊C4—62no—
276Dic3⋊C4—64no—
277C4×Dic3—31no—
278C4×Dic3—31no—
279C4×Dic3—61no—
280C2×C3⋊C8C2×C3×D311yescharacteristic
281C2×C3⋊C8C2×C3×D311yescharacteristic
282Dic12—62no—
283#204—62no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32—121no—
289C22×C32—122no—
290C22×C32D3×D411yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32C2×C3×D411yescharacteristic
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—42no—
298C2×C3×D3—82no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—42no—
308C32×C4—42no—
309C32×C4—42no—
310C32×C4—44no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4—121no—
315C32×C4D3×D411yescharacteristic
316C32×C4D3×D411yescharacteristic
317C32×C4C2×#6711yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C32×C4C2×C3×D411yescharacteristic
320C32×C4C2×C3×D411yescharacteristic
321C3⋊Dic3—22no—
322C3⋊Dic3—41no—
323C3×Dic3—42no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—64no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—122no—
330C3×Dic3—121no—
331C3×Dic3—121no—
332C2×Q16—31no—
333C2×Q16—91no—
334C2×SD16—31no—
335C2×SD16—91no—
336C4⋊Q8—91no—
337#124—91no—
338#97—91no—
339C33#49711yescharacteristic, complemented, Sylow
340C23×C3—61no—
341C23×C3—61no—
342C23×C3—121no—
343C22×D3—21no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—21no—
347C3×Q8—21no—
348C3×Q8—21no—
349C3×Q8—21no—
350C3×Q8—21no—
351C3×Q8—21no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—42no—
357C3×Q8—42no—
358C3×Q8—62no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—121no—
364C3×Q8—181no—
365C3×Q8—181no—
366C3×D4—62no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—121no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—92no—
386C2×C3×C4—121no—
387C2×C3×C4—181no—
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×C3×C4C22×C3×D311yescharacteristic
391C2×Dic3—21no—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398D12—21no—
399D12—21no—
400Dic6—21no—
401Dic6—21no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—61no—
406Dic6—121no—
407Dic6—121no—
408C3×C8—32no—
409C3×C8—32no—
410C3×C8—32no—
411C3×C8—32no—
412C3×C8—62no—
413C3×C8—62no—
414C3⋊C8C3×#6712yes—
415C3⋊C8C3×#6712yes—
416C2×C32—22no—
417C2×C32—22no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—41no—
423C2×C32—42no—
424C2×C32—241no—
425C2×C32#91512yes—
426C2×C32#91812yes—
427C2×C32#94011yescharacteristic
428C2×C32#94011yescharacteristic
429C2×C32C3×#12711yescharacteristic
430C2×C32C3×#13712yes—
431C3×D3—82no—
432C3×D3—81no—
433C3×D3—81no—
434C2×Q8—31no—
435C2×Q8—91no—
436C2×Q8C3×D3211yescharacteristic
437C2×D4—31no—
438Q16—62no—
439Q16—182no—
440SD16—62no—
441SD16—182no—
442C2×C8—31no—
443C2×C8—31no—
444C4⋊C4—182no—
445C22⋊C4—182no—
446C42—91no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—122no—
452C22×C3—122no—
453C22×C3—121no—
454C22×C3—122no—
455C22×C3—121no—
456C22×C3—242no—
457C22×C3#381111yescharacteristic
458C22×C3C3×D3×D411yescharacteristic
459C22×C3C3×D3×D411yescharacteristic
460C2×D3—42no—
461C2×D3—41no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—44no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—122no—
478C3×C4—121no—
479C3×C4—121no—
480C3×C4—182no—
481C3×C4—361no—
482C3×C4D3×#6711yescharacteristic
483C3×C4D3×#6711yescharacteristic
484C3×C4C3×D3×D411yescharacteristic
485C3×C4C3×D3×D411yescharacteristic
486C3×C4C2×C3×#6711yescharacteristic
487C3×C4C2×C3×#6711yescharacteristic
488Dic3—41no—
489Dic3—64no—
490Dic3—62no—
491Dic3—62no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#516711yescharacteristic, complemented
500C32#519011yescharacteristic, complemented
501C32C3×#49711yescharacteristic, complemented
502C23—61no—
503Q8—21no—
504Q8—21no—
505Q8—61no—
506Q8—61no—
507Q8—181no—
508Q8—181no—
509D4—61no—
510D4—61no—
511C2×C4—21no—
512C2×C4—61no—
513C2×C4—92no—
514C2×C4—181no—
515C2×C4C2×C3×D3211yescharacteristic
516C8—32no—
517C8—32no—
518C2×C3—22no—
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—41no—
526C2×C3—242no—
527C2×C3—241no—
528C2×C3—241no—
529C2×C3#6395112yes—
530C2×C3#6397911yescharacteristic
531C2×C3C3×#91512yes—
532C2×C3C3×#91812yes—
533C2×C3C3×#94011yescharacteristic
534C2×C3C3×#94011yescharacteristic
535D3—81no—
536C22—122no—
537C22—121no—
538C22C3×#381111yescharacteristic
539C4—41no—
540C4—121no—
541C4—182no—
542C4—361no—
543C4C3×D3×#6711yescharacteristic
544C4C3×D3×#6711yescharacteristic
545C3—21no—
546C3—21no—
547C3—21no—
548C3—41no—
549C3#1059161111yescharacteristic, direct product factor
550C3C3×#516711yescharacteristic, complemented
551C3C3×#519011yescharacteristic, complemented
552C2—241no—
553C2C3×#6395112yes—
554C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C3×Q8D32
hypocenterC32C3×#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64153C2semidirect
2. C2×C32×#216C2semidirect
5. C2×C3×#3719C2semidirect
7. C3×#63555C2semidirect
10. #10591611C3direct (N × H)
26. #64153C2×C3semidirect
33. C2×C3×#216D3semidirect
34. C2×C3×#216C2×C3semidirect
43. C2×#3719C2×C3semidirect
47. #63555C2×C3semidirect
143. C2×#216C3×D3semidirect
339. C33#497semidirect
499. C32#5167semidirect
500. C32#5190semidirect
501. C32C3×#497semidirect
549. C3#10591611direct (N × H)
550. C3C3×#5167semidirect
551. C3C3×#5190semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C32×#2162
C2×C32×#2152
C3×#638632
C2×C3×#37192
C2×C3×#37162
C3×#635552
C3×#51903
C3×#51673
#105916113