Finite Group Databasebeta

← C3×#10590538 #13157514

Subgroups — C3×#10590538 #13157514

Order 1728.

636 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64096C211yesmaximal, characteristic, complemented
2C3×#63863C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic
5C32×#810C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic
7C3×#63572C211yesmaximal, characteristic, complemented
8C3×#4798—31nomaximal
9C3×#4767—31nomaximal
10#10590538C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3719—22no—
23C3×#3716—22no—
24#64096C2×C311yescharacteristic, complemented
25C3×#947—31no—
26C2×C3×#216—31no—
27C3×#936—31no—
28C2×C3×#214—31no—
29C2×C3×Dic12—31no—
30C2×C3×#204—31no—
31C3×#887—31no—
32C3×#887—31no—
33C3×#887—61no—
34#63863C2×C311yescharacteristic
35C2×#3719C2×C311yescharacteristic, complemented
36C2×#3716C2×C311yescharacteristic
37C32×#103—31no—
38C32×#102—31no—
39C3×Dic3⋊C8—31no—
40C3×Dic3⋊C8—31no—
41C3×#810—21no—
42C3×#810D311yescharacteristic, complemented
43C3×#810C2×C311yescharacteristic, complemented
44C3×#809—21no—
45C3×#809D311yescharacteristic
46C3×#809C2×C311yescharacteristic
47#63572C2×C311yescharacteristic, complemented
48C22×C32×D3—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×#634—21no—
55C3×#634D412yes—
56C2×C32×Dic3—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×C3⋊C8—21no—
62C32×C3⋊C8—21no—
63C3×#464—91no—
64#4798—31no—
65#4767—31no—
66C2×C32×Q8—31no—
67C2×C32×D4—31no—
68C2×#634C22×C311yescharacteristic
69C2×C3×D12—21no—
70C2×C3×D12C2×D311yescharacteristic
71C2×C3×D12C22×C311yescharacteristic
72C2×C3×Dic6—21no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—31no—
75C2×C3×Dic6—61no—
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C22×C311yescharacteristic
78C2×C32×C8—31no—
79C2×C32×C8—31no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4—41no—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C22×C311yescharacteristic
87#3750—21no—
88C4×C3⋊Dic3—21no—
89C4×C3⋊Dic3C22×C311yescharacteristic
90C3×#217—61no—
91C3×#217—61no—
92C3×#216—62no—
93C3×#214—62no—
94C3×(C2×D3)⋊C4—61no—
95C3×(C2×D3)⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—62no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—61no—
103C3×C4×Dic3—61no—
104C3×C4×Dic3—62no—
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111C3×Dic12—62no—
112C3×#204—62no—
113(C2×D3)⋊Dic3—21no—
114(C2×D3)⋊Dic3—21no—
115#3719—22no—
116#3716—22no—
117C22×C33C2×D411yescharacteristic
118C2×C32×D3—41no—
119C2×C32×D3—41no—
120C2×C32×D3—41no—
121C33×C4—41no—
122C33×C4C2×D411yescharacteristic
123C33×C4#4111yescharacteristic
124C3×C3⋊Dic3—22no—
125C3×C3⋊Dic3—21no—
126C3×C3⋊Dic3—21no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×SD16—91no—
130C3×#124—91no—
131C3×C4×Q8—91no—
132#947—31no—
133C2×#216—31no—
134#936—31no—
135C2×#214—31no—
136C2×Dic12—31no—
137C2×#204—31no—
138#887—31no—
139#887—31no—
140#887—61no—
141C3×C4⋊C8—91no—
142C3×#103—31no—
143C3×#103—31no—
144C3×#103—61no—
145C3×#102—31no—
146C3×#102—31no—
147C3×#102—61no—
148Dic3⋊C8—31no—
149Dic3⋊C8—31no—
150#810C3×D311yescharacteristic, complemented
151#809C3×D311yescharacteristic
152C23×C32—61no—
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—41no—
156C32×Q8—61no—
157C32×Q8—61no—
158C32×D4—61no—
159C32×D4—61no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—42no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3C3×D411yescharacteristic
177C2×C3⋊Dic3C3×D411yescharacteristic
178#634—21no—
179#634C3×D412yes—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C3×D12—22no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—41no—
198C3×Dic6—22no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—32no—
204C3×Dic6—32no—
205C3×Dic6—41no—
206C3×Dic6—62no—
207C3×Dic6—62no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C32×C8—61no—
211C32×C8—61no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—41no—
217C3×C3⋊C8—41no—
218#464—91noSylow
219C2×C33#12111yescharacteristic
220C2×C33#13511yescharacteristic
221C2×C33#13711yescharacteristic
222C32×D3—81no—
223C2×C3×Q8—31no—
224C2×C3×Q8—31no—
225C2×C3×Q8—61no—
226C2×C3×Q8—91no—
227C2×C3×D4—31no—
228C2×C3×D4—31no—
229C2×C3×D4—61no—
230C2×D12C2×C3×D311yescharacteristic
231C2×Dic6—31no—
232C2×Dic6—31no—
233C2×Dic6—61no—
234C2×Dic6C2×C3×D311yescharacteristic
235C3×Q16—182no—
236C3×SD16—182no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—61no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—181no—
248C3×C4⋊C4D3211yescharacteristic
249C3×C4⋊C4C2×C3×D311yescharacteristic
250C3×C4⋊C4C2×C3×D311yescharacteristic
251C3×C22⋊C4—181no—
252C3×C22⋊C4—181no—
253C3×C42—91no—
254C3×C42—181no—
255#217—61no—
256#217—61no—
257#216—62no—
258#214—62no—
259(C2×D3)⋊C4—61no—
260(C2×D3)⋊C4—61no—
261Dic3⋊C4—61no—
262Dic3⋊C4—61no—
263Dic3⋊C4—62no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—61no—
267C4×Dic3—61no—
268C4×Dic3—61no—
269C4×Dic3—62no—
270C2×C3⋊C8C2×C3×D311yescharacteristic
271C2×C3⋊C8C2×C3×D311yescharacteristic
272Dic12—62no—
273#204—62no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—41no—
278C22×C32—121no—
279C22×C32—121no—
280C22×C32—121no—
281C22×C32D3×D411yescharacteristic
282C22×C32D3×D411yescharacteristic
283C22×C32C2×C3×D411yescharacteristic
284C2×C3×D3—42no—
285C2×C3×D3—41no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C2×C3×D3—41no—
291C2×C3×D3—81no—
292C2×C3×D3—81no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4—42no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—81no—
306C32×C4—81no—
307C32×C4—82no—
308C32×C4—121no—
309C32×C4#23511yescharacteristic
310C32×C4#23511yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4D3×D411yescharacteristic
313C32×C4C2×C3×D411yescharacteristic
314C32×C4C3×#4111yescharacteristic
315C3⋊Dic3—22no—
316C3⋊Dic3—21no—
317C3⋊Dic3—21no—
318C3×Dic3—42no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—64no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—121no—
330C2×Q16—91no—
331C2×SD16—91no—
332#124—91no—
333C4×Q8—91no—
334C4⋊C8—91no—
335#103—31no—
336#102—31no—
337C33#46411yescharacteristic, complemented, Sylow
338C23×C3—61no—
339C23×C3—61no—
340C23×C3—121no—
341C22×D3—21no—
342C3×Q8—62no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—92no—
348C3×Q8—121no—
349C3×Q8—181no—
350C3×D4—62no—
351C3×D4—61no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—121no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—121no—
372C2×C3×C4—181no—
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C22×C3×D311yescharacteristic
375C2×C3×C4C22×C3×D311yescharacteristic
376C2×Dic3—21no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—62no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386D12—21no—
387D12—21no—
388Dic6—21no—
389Dic6—21no—
390Dic6—32no—
391Dic6—32no—
392Dic6—62no—
393Dic6—62no—
394Dic6—61no—
395Dic6—61no—
396C3×C8—61no—
397C3×C8—61no—
398C3×C8—61no—
399C3×C8—61no—
400C3×C8—121no—
401C3×C8—121no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—41no—
416C2×C32—241no—
417C2×C32#88411yescharacteristic
418C2×C32#88411yescharacteristic
419C2×C32#91211yescharacteristic
420C2×C32#91511yescharacteristic
421C2×C32#91611yescharacteristic
422C2×C32#91811yescharacteristic
423C2×C32C3×#12111yescharacteristic
424C2×C32C3×#13511yescharacteristic
425C2×C32C3×#13711yescharacteristic
426C3×D3—82no—
427C3×D3—81no—
428C3×D3—81no—
429C2×Q8—31no—
430C2×Q8—91no—
431C2×D4—31no—
432Q16—182no—
433SD16—182no—
434C2×C8—31no—
435C2×C8—31no—
436C4⋊C4—181no—
437C4⋊C4C3×D3211yescharacteristic
438C22⋊C4—181no—
439C22⋊C4—181no—
440C42—91no—
441C42—181no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—41no—
446C22×C3—122no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3—121no—
452C22×C3—121no—
453C22×C3—241no—
454C22×C3—241no—
455C22×C3#381111yescharacteristic
456C22×C3C3×D3×D411yescharacteristic
457C22×C3C3×D3×D411yescharacteristic
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—82no—
476C3×C4—122no—
477C3×C4—121no—
478C3×C4—121no—
479C3×C4—182no—
480C3×C4—181no—
481C3×C4—181no—
482C3×C4#379811yescharacteristic
483C3×C4#380211yescharacteristic
484C3×C4C3×#23511yescharacteristic
485C3×C4C3×#23511yescharacteristic
486C3×C4C3×D3×D411yescharacteristic
487C3×C4C3×D3×D411yescharacteristic
488Dic3—41no—
489Dic3—64no—
490Dic3—62no—
491Dic3—62no—
492Dic3—62no—
493Dic3—61no—
494Dic3—61no—
495Dic3—61no—
496Dic3—61no—
497Dic3—121no—
498C32—21no—
499C32—21no—
500C32—21no—
501C32—41no—
502C32#476711yescharacteristic, complemented
503C32#479811yescharacteristic, complemented
504C32C3×#46411yescharacteristic, complemented
505C23—61no—
506Q8—61no—
507Q8—61no—
508Q8—92no—
509Q8—181no—
510D4—61no—
511D4—61no—
512C2×C4—21no—
513C2×C4—61no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—181no—
517C2×C4C2×C3×D3211yescharacteristic
518C8—61no—
519C8—61no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—242no—
533C2×C3—241no—
534C2×C3—241no—
535C2×C3#6391211yescharacteristic
536C2×C3#6393611yescharacteristic
537C2×C3#6395111yescharacteristic
538C2×C3C3×#88411yescharacteristic
539C2×C3C3×#88411yescharacteristic
540C2×C3C3×#91211yescharacteristic
541C2×C3C3×#91511yescharacteristic
542C2×C3C3×#91611yescharacteristic
543C2×C3C3×#91811yescharacteristic
544D3—81no—
545C22—121no—
546C22—121no—
547C22—121no—
548C22C3×#381111yescharacteristic
549C4—41no—
550C4—121no—
551C4—181no—
552C4—182no—
553C4—181no—
554C4C3×#379811yescharacteristic
555C4C3×#380211yescharacteristic
556C3—21no—
557C3—21no—
558C3—21no—
559C3—41no—
560C3#1059053811yescharacteristic, direct product factor
561C3C3×#476711yescharacteristic, complemented
562C3C3×#479811yescharacteristic, complemented
563C2—241no—
564C2C3×#6391211yescharacteristic
565C2C3×#6393611yescharacteristic
566C2C3×#6395111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64096C2semidirect
3. C2×C3×#3719C2semidirect
5. C32×#810C2semidirect
7. C3×#63572C2semidirect
10. #10590538C3direct (N × H)
24. #64096C2×C3semidirect
35. C2×#3719C2×C3semidirect
42. C3×#810D3semidirect
43. C3×#810C2×C3semidirect
47. #63572C2×C3semidirect
150. #810C3×D3semidirect
337. C33#464semidirect
502. C32#4767semidirect
503. C32#4798semidirect
504. C32C3×#464semidirect
560. C3#10590538direct (N × H)
561. C3C3×#4767semidirect
562. C3C3×#4798semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#638632
C2×C3×#37192
C2×C3×#37162
C32×#8102
C32×#8092
C3×#635722
C3×#47983
C3×#47673
#105905383