Finite Group Databasebeta

← C3×#10591474 #13158530

Subgroups — C3×#10591474 #13158530

Order 1728.

794 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64145C211yesmaximal, characteristic, complemented
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic
4C3×#63863C211yesmaximal, characteristic, complemented
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3716C211yesmaximal, characteristic, complemented
7C3×#63553C211yesmaximal, characteristic, complemented
8C3×#5168—31nomaximal
9C3×#5155—31nomaximal
10#10591474C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3757—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#214—22no—
18C32×#213—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3719—22no—
23C3×#3716—22no—
24C2×C32×SD16—31no—
25C2×C32×D8—31no—
26#64145C2×C311yescharacteristic, complemented
27C3×#947—31no—
28C2×C3×#216—31no—
29C3×#936—31no—
30C3×#936—31no—
31C3×#936—31no—
32C3×#936—61no—
33C2×C3×#214—21no—
34C2×C3×#214—31no—
35C2×C3×#214D311yescharacteristic, complemented
36C2×C3×#214C2×C311yescharacteristic, complemented
37C2×C3×#213—21no—
38C2×C3×#213D311yescharacteristic
39C2×C3×#213C2×C311yescharacteristic
40C2×C3×Dic12—31no—
41C2×C3×#204—31no—
42#63863C2×C311yescharacteristic, complemented
43C2×#3719C2×C311yescharacteristic, complemented
44C2×#3716C2×C311yescharacteristic, complemented
45C3×C8×Dic3—31no—
46C3×C8×Dic3—31no—
47#63553C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C32×D3—21no—
50C33×D4—21no—
51C33×D4—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×#495—91no—
65#5168—31no—
66#5155—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—31no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
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×SD16—62no—
87C32×D8—62no—
88C2×C32×C8—31no—
89C2×C32×C8—31no—
90#3757—22no—
91C4×C3⋊Dic3C22×C311yescharacteristic
92C3×#217—62no—
93C3×#217—62no—
94C3×#217—62no—
95C3×#217—64no—
96C3×#216—62no—
97C3×#214—24no—
98C3×#214—22no—
99C3×#214—22no—
100C3×#214—62no—
101C3×#213—24no—
102C3×#213—22no—
103C3×#213—22no—
104C3×(C2×D3)⋊C4—62no—
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×C33—41no—
120C22×C33—42no—
121C22×C33C2×D411yescharacteristic
122C2×C32×D3—41no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3—41no—
128C32×Dic3—41no—
129C2×C3×Q16—91no—
130C2×C3×SD16—31no—
131C2×C3×SD16—31no—
132C2×C3×SD16—61no—
133C2×C3×SD16—91no—
134C2×C3×D8—31no—
135C2×C3×D8—31no—
136C2×C3×D8—61no—
137C3×#124—91no—
138C3×#124—91no—
139#947—31no—
140C2×#216—31no—
141#936—31no—
142#936—31no—
143#936—31no—
144#936—61no—
145C2×#214—31no—
146C2×#214C3×D311yescharacteristic, complemented
147C2×#213C3×D311yescharacteristic
148C2×Dic12—31no—
149C2×#204—31no—
150C3×C4×C8—91no—
151C8×Dic3—31no—
152C8×Dic3—31no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—41no—
158C23×C32—41no—
159C23×C32—42no—
160C23×C32—61no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—41no—
164C32×Q8—61no—
165C32×Q8—61no—
166C32×D4—21no—
167C32×D4—21no—
168C32×D4—22no—
169C32×D4—22no—
170C32×D4—21no—
171C32×D4—21no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—41no—
175C32×D4—41no—
176C32×D4—41no—
177C32×D4—41no—
178C32×D4—42no—
179C32×D4—42no—
180C32×D4—61no—
181C32×D4—61no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4—61no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C23×C311yescharacteristic
190C2×C3⋊Dic3—21no—
191C2×C3⋊Dic3C3×D412yes—
192#634—21no—
193#634—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—41no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C3×D12—22no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—41no—
209C3×Dic6—22no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—41no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—121no—
220C3×Dic6—121no—
221C32×C8—32no—
222C32×C8—32no—
223C3×C3⋊C8—22no—
224C3×C3⋊C8—22no—
225C3×C3⋊C8#6712yes—
226C3×C3⋊C8#6712yes—
227C3×C3⋊C8C3×D412yes—
228C3×C3⋊C8C3×D412yes—
229#495—91noSylow
230C2×C33—81no—
231C2×C33#12711yescharacteristic
232C2×C33#13512yes—
233C32×D3—81no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×Q8—91no—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—31no—
242C2×C3×D4—31no—
243C2×C3×D4—41no—
244C2×C3×D4—61no—
245C2×C3×D4D3211yescharacteristic
246C2×C3×D4C2×C3×D311yescharacteristic
247C2×C3×D4C2×C3×D311yescharacteristic
248C2×D12C2×C3×D311yescharacteristic
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C2×Dic6C2×C3×D311yescharacteristic
253C3×Q16—182no—
254C3×SD16—62no—
255C3×SD16—62no—
256C3×SD16—64no—
257C3×SD16—182no—
258C3×D8—62no—
259C3×D8—62no—
260C3×D8—64no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—61no—
266C2×C3×C8—61no—
267C3×C22⋊C4—182no—
268C3×C22⋊C4—182no—
269C3×C42—91no—
270#217—62no—
271#217—62no—
272#217—62no—
273#217—64no—
274#216—62no—
275#214—22no—
276#214—62no—
277#213—22no—
278(C2×D3)⋊C4—62no—
279C4×Dic3—31no—
280C4×Dic3—31no—
281C4×Dic3—61no—
282C2×C3⋊C8C2×C3×D311yescharacteristic
283C2×C3⋊C8C2×C3×D311yescharacteristic
284Dic12—62no—
285#204—62no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—42no—
291C22×C32—42no—
292C22×C32—42no—
293C22×C32—44no—
294C22×C32—41no—
295C22×C32—41no—
296C22×C32—41no—
297C22×C32—42no—
298C22×C32—44no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32—82no—
302C22×C32—82no—
303C22×C32—84no—
304C22×C32—121no—
305C22×C32—122no—
306C22×C32D3×D411yescharacteristic
307C22×C32D3×D411yescharacteristic
308C22×C32C2×C3×D411yescharacteristic
309C2×C3×D3—42no—
310C2×C3×D3—42no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—42no—
314C2×C3×D3—82no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—121no—
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4C2×#6711yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C32×C4C2×C3×D411yescharacteristic
329C32×C4C2×C3×D411yescharacteristic
330C3⋊Dic3—22no—
331C3⋊Dic3—41no—
332C3×Dic3—42no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337C3×Dic3—122no—
338C3×Dic3—122no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C2×Q16—91no—
342C2×SD16—31no—
343C2×SD16—91no—
344C2×D8—31no—
345#124—91no—
346#124—91no—
347C4×C8—91no—
348C33#49511yescharacteristic, complemented, Sylow
349C23×C3—22no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—42no—
356C23×C3—61no—
357C23×C3—61no—
358C23×C3—121no—
359C22×D3—21no—
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×D4—22no—
369C3×D4—22no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—21no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—42no—
381C3×D4—42no—
382C3×D4—62no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—61no—
386C3×D4—61no—
387C3×D4—121no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—92no—
395C2×C3×C4—121no—
396C2×C3×C4—181no—
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×Dic3—21no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—62no—
404C2×Dic3—62no—
405C2×Dic3—61no—
406C2×Dic3—61no—
407D12—21no—
408D12—21no—
409Dic6—21no—
410Dic6—21no—
411Dic6—61no—
412Dic6—61no—
413Dic6—61no—
414Dic6—61no—
415Dic6—121no—
416Dic6—121no—
417C3×C8—32no—
418C3×C8—32no—
419C3×C8—32no—
420C3×C8—32no—
421C3×C8—62no—
422C3×C8—62no—
423C3⋊C8C3×#6712yes—
424C3⋊C8C3×#6712yes—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32—82no—
434C2×C32—82no—
435C2×C32—82no—
436C2×C32—84no—
437C2×C32—81no—
438C2×C32—81no—
439C2×C32—81no—
440C2×C32—241no—
441C2×C32#91212yes—
442C2×C32#91612yes—
443C2×C32#94011yescharacteristic
444C2×C32#94011yescharacteristic
445C2×C32C3×#12711yescharacteristic
446C2×C32C3×#13512yes—
447C3×D3—82no—
448C3×D3—81no—
449C3×D3—81no—
450C2×Q8—31no—
451C2×Q8—91no—
452C2×D4—31no—
453C2×D4C3×D3211yescharacteristic
454Q16—182no—
455SD16—62no—
456SD16—182no—
457D8—62no—
458C2×C8—31no—
459C2×C8—31no—
460C22⋊C4—182no—
461C22⋊C4—182no—
462C42—91no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—84no—
481C22×C3—122no—
482C22×C3—122no—
483C22×C3—121no—
484C22×C3—122no—
485C22×C3—121no—
486C22×C3—242no—
487C22×C3#381111yescharacteristic
488C22×C3C3×D3×D411yescharacteristic
489C22×C3C3×D3×D411yescharacteristic
490C2×D3—42no—
491C2×D3—41no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—122no—
501C3×C4—121no—
502C3×C4—121no—
503C3×C4—182no—
504C3×C4—361no—
505C3×C4D3×#6711yescharacteristic
506C3×C4D3×#6711yescharacteristic
507C3×C4C3×D3×D411yescharacteristic
508C3×C4C3×D3×D411yescharacteristic
509C3×C4C2×C3×#6711yescharacteristic
510C3×C4C2×C3×#6711yescharacteristic
511Dic3—41no—
512Dic3—62no—
513Dic3—62no—
514Dic3—122no—
515Dic3—122no—
516Dic3—121no—
517Dic3—121no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#515511yescharacteristic, complemented
523C32#516811yescharacteristic, complemented
524C32C3×#49511yescharacteristic, complemented
525C23—21no—
526C23—61no—
527Q8—61no—
528Q8—61no—
529Q8—181no—
530Q8—181no—
531D4—21no—
532D4—21no—
533D4—61no—
534D4—61no—
535C2×C4—61no—
536C2×C4—92no—
537C2×C4—181no—
538C2×C4C2×C3×D3211yescharacteristic
539C8—32no—
540C8—32no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—41no—
548C2×C3—42no—
549C2×C3—82no—
550C2×C3—82no—
551C2×C3—82no—
552C2×C3—84no—
553C2×C3—81no—
554C2×C3—81no—
555C2×C3—81no—
556C2×C3—242no—
557C2×C3—241no—
558C2×C3—241no—
559C2×C3#6393612yes—
560C2×C3#6397911yescharacteristic
561C2×C3C3×#91212yes—
562C2×C3C3×#91612yes—
563C2×C3C3×#94011yescharacteristic
564C2×C3C3×#94011yescharacteristic
565D3—81no—
566C22—41no—
567C22—42no—
568C22—121no—
569C22—122no—
570C22C3×#381111yescharacteristic
571C4—121no—
572C4—182no—
573C4—361no—
574C4C3×D3×#6711yescharacteristic
575C4C3×D3×#6711yescharacteristic
576C3—21no—
577C3—21no—
578C3—21no—
579C3—41no—
580C3#1059147411yescharacteristic, direct product factor
581C3C3×#515511yescharacteristic, complemented
582C3C3×#516811yescharacteristic, complemented
583C2—81no—
584C2—241no—
585C2C3×#6393612yes—
586C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#213C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64145C2semidirect
2. C2×C32×#214C2semidirect
4. C3×#63863C2semidirect
5. C2×C3×#3719C2semidirect
6. C2×C3×#3716C2semidirect
7. C3×#63553C2semidirect
10. #10591474C3direct (N × H)
26. #64145C2×C3semidirect
35. C2×C3×#214D3semidirect
36. C2×C3×#214C2×C3semidirect
42. #63863C2×C3semidirect
43. C2×#3719C2×C3semidirect
44. C2×#3716C2×C3semidirect
47. #63553C2×C3semidirect
146. C2×#214C3×D3semidirect
348. C33#495semidirect
522. C32#5155semidirect
523. C32#5168semidirect
524. C32C3×#495semidirect
580. C3#10591474direct (N × H)
581. C3C3×#5155semidirect
582. C3C3×#5168semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C2×C32×#2142
C2×C32×#2132
C3×#638632
C2×C3×#37192
C2×C3×#37162
C3×#635532
C3×#51683
C3×#51553
#105914743