Finite Group Databasebeta

← C3×#10590460 #13157599

Subgroups — C3×#10590460 #13157599

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64096C211yesmaximal, characteristic, complemented
2C3×#63898C211yesmaximal, characteristic
3C2×C3×#3716C211yesmaximal, characteristic
4C32×#809C211yesmaximal, characteristic
5C32×#808C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63553C211yesmaximal, characteristic, complemented
8C3×#4852—31nomaximal
9C3×#4758—31nomaximal
10#10590460C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3716C414yescomplemented
23#64096C2×C311yescharacteristic, complemented
24C3×D4×Dic3—31no—
25C2×C3×#214—31no—
26C2×C3×#204—31no—
27C3×#893—31no—
28C3×#887—31no—
29C3×#887—31no—
30C3×#887—61no—
31#63898C2×C311yescharacteristic
32C2×#3716C2×C311yescharacteristic
33C32×#106—31no—
34C32×#102—31no—
35C3×#816—31no—
36C3×C8×Dic3—31no—
37C3×C8×Dic3—31no—
38C3×#810—31no—
39C3×#809—21no—
40C3×#809D311yescharacteristic
41C3×#809C2×C311yescharacteristic
42C3×#808—21no—
43C3×#808D311yescharacteristic, complemented
44C3×#808C2×C311yescharacteristic, complemented
45#63567C2×C311yescharacteristic, complemented
46#63553C2×C311yescharacteristic, complemented
47C22×C32×D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yes—
58C3×D3×Dic3—42no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×C4×SD16—91no—
62#4852—31no—
63#4758—31no—
64C2×C32×D4—31no—
65C2×#634C22×C311yescharacteristic
66C22×C3×Dic3—61no—
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C22×C311yescharacteristic
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×D3×Dic3—21no—
75C2×C32×C8—31no—
76C2×C32×C8—31no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—31no—
81C32×C4⋊C4—41no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85#3750—21no—
86C4×C3⋊Dic3—21no—
87C4×C3⋊Dic3C22×C311yescharacteristic
88C3×#217—61no—
89C3×#214—34no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—62no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—62no—
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C3×#204—34no—
110(C2×D3)⋊Dic3—21no—
111#3716C3×C414yescomplemented
112C22×C33C2×D411yescharacteristic
113C2×C32×D3—22no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C33×C4—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3SD1612yescomplemented
122C32×Dic3—41no—
123C2×C3×SD16—91no—
124C3×C4×Q8—91no—
125C3×C4×D4—91no—
126D4×Dic3—31no—
127C2×#214—31no—
128C2×#204—31no—
129#893—31no—
130#887—31no—
131#887—31no—
132#887—61no—
133C3×#106—31no—
134C3×#106—31no—
135C3×#106—61no—
136C3×#103—91no—
137C3×#102—31no—
138C3×#102—31no—
139C3×#102—61no—
140C3×C4×C8—91no—
141#816—31no—
142C8×Dic3—31no—
143C8×Dic3—31no—
144#810—31no—
145#809C3×D311yescharacteristic
146#808C3×D311yescharacteristic, complemented
147C23×C32—61no—
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—41no—
151C32×D4—32no—
152C32×D4—61no—
153C2×C32×C4—21no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4—41no—
162C2×C32×C4—41no—
163C2×C32×C4—42no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C23×C311yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3C3×D411yescharacteristic
170C2×C3⋊Dic3C3×D411yescharacteristic
171#634—21no—
172#634C2×C3×C412yescomplemented
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—122no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—22no—
189C3×D12—41no—
190C3×D12C4×D312yes—
191C3×D12C2×C3×C412yes—
192C3×C4×D3—122no—
193C3×Dic6—32no—
194C3×Dic6—32no—
195C3×Dic6—62no—
196C3×Dic6—62no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199D3×Dic3—42no—
200C32×C8—32no—
201C32×C8—61no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—22no—
205C3×C3⋊C8—41no—
206C3×C3⋊C8C4×D312yescomplemented
207C3×C3⋊C8C2×C3×C412yescomplemented
208C4×SD16—91noSylow
209C2×C33C4×D411yescharacteristic
210C2×C33C2×SD1611yescharacteristic
211C2×C33#13511yescharacteristic
212C32×D3—42no—
213C2×C3×Q8—91no—
214C2×C3×D4—31no—
215C2×C3×D4—31no—
216C2×C3×D4—61no—
217C22×C3×C4—181no—
218C22×Dic3—61no—
219C2×D12C2×C3×D311yescharacteristic
220C2×C4×D3—61no—
221C2×Dic6—31no—
222C2×Dic6—31no—
223C2×Dic6—61no—
224C3×SD16—94no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—61no—
230C2×C3×C8—61no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—181no—
239C3×C4⋊C4D3211yescharacteristic
240C3×C4⋊C4C2×C3×D311yescharacteristic
241C3×C4⋊C4C2×C3×D311yescharacteristic
242C3×C22⋊C4—181no—
243C3×C42—91no—
244C3×C42—181no—
245#217—61no—
246#214—34no—
247(C2×D3)⋊C4—61no—
248C4⋊Dic3C2×C3×D311yescharacteristic
249Dic3⋊C4—61no—
250Dic3⋊C4—61no—
251Dic3⋊C4—62no—
252C4×Dic3—31no—
253C4×Dic3—31no—
254C4×Dic3—61no—
255C4×Dic3—61no—
256C4×Dic3—61no—
257C4×Dic3—62no—
258C2×C3⋊C8C2×C3×D311yescharacteristic
259C2×C3⋊C8C2×C3×D311yescharacteristic
260#204—34no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—62no—
266C22×C32—121no—
267C22×C32—121no—
268C22×C32D3×D411yescharacteristic
269C22×C32D3×D411yescharacteristic
270C22×C32C2×C3×D411yescharacteristic
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—42no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—81no—
279C2×C3×D3—81no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—42no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4—81no—
293C32×C4—81no—
294C32×C4—82no—
295C32×C4—121no—
296C32×C4C2×C4×D311yescharacteristic
297C32×C4C2×C4×D311yescharacteristic
298C32×C4#23711yescharacteristic
299C32×C4#23911yescharacteristic
300C32×C4C22×C3×C411yescharacteristic
301C32×C4C3×#4111yescharacteristic
302C3⋊Dic3—22no—
303C3⋊Dic3—21no—
304C3⋊Dic3C3×SD1612yescomplemented
305C3×Dic3—32no—
306C3×Dic3—32no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—62no—
310C3×Dic3—64no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C3×Dic3—61no—
314C3×Dic3—61no—
315C3×Dic3—81no—
316C3×Dic3—121no—
317C2×SD16—91no—
318C4×Q8—91no—
319C4×D4—91no—
320#106—31no—
321#103—91no—
322#102—31no—
323C4×C8—91no—
324C33C4×SD1611yescharacteristic, complemented, Sylow
325C23×C3—61no—
326C23×C3—61no—
327C23×C3—121no—
328C22×D3—21no—
329C3×Q8—92no—
330C3×Q8—181no—
331C3×D4—32no—
332C3×D4—32no—
333C3×D4—62no—
334C3×D4—61no—
335C3×D4—61no—
336C3×D4—121no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—121no—
353C2×C3×C4—181no—
354C2×C3×C4—362no—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C22×C3×D311yescharacteristic
357C2×C3×C4C22×C3×D311yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—31no—
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—62no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—122no—
369D12—21no—
370D12C3×C4×D312yes—
371C4×D3—122no—
372Dic6—32no—
373Dic6—32no—
374Dic6—62no—
375Dic6—62no—
376Dic6—61no—
377Dic6—61no—
378C3×C8—32no—
379C3×C8—32no—
380C3×C8—62no—
381C3×C8—61no—
382C3×C8—61no—
383C3×C8—121no—
384C3⋊C8—21no—
385C3⋊C8C3×C4×D312yescomplemented
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32—41no—
398C2×C32—122no—
399C2×C32#67⋊C411yescharacteristic
400C2×C32#89311yescharacteristic
401C2×C32#91211yescharacteristic
402C2×C32D3×SD1611yescharacteristic
403C2×C32D3×SD1611yescharacteristic
404C2×C32#91611yescharacteristic
405C2×C32C3×C4×D411yescharacteristic
406C2×C32C2×C3×SD1611yescharacteristic
407C2×C32C3×#13511yescharacteristic
408C3×D3—42no—
409C3×D3—42no—
410C3×D3—82no—
411C2×Q8—91no—
412C2×D4—31no—
413C22×C4—181no—
414SD16—94no—
415C2×C8—31no—
416C2×C8—31no—
417C4⋊C4—31no—
418C4⋊C4—181no—
419C4⋊C4C3×D3211yescharacteristic
420C22⋊C4—181no—
421C42—91no—
422C42—181no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—122no—
430C22×C3—121no—
431C22×C3—121no—
432C22×C3—121no—
433C22×C3—121no—
434C22×C3—241no—
435C22×C3—241no—
436C22×C3#381111yescharacteristic
437C22×C3C3×D3×D411yescharacteristic
438C22×C3C3×D3×D411yescharacteristic
439C2×D3—22no—
440C2×D3—41no—
441C2×D3—41no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—42no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4—82no—
457C3×C4—92no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—182no—
461C3×C4—181no—
462C3×C4—241no—
463C3×C4#379611yescharacteristic
464C3×C4C4×D3211yescharacteristic
465C3×C4C2×C3×C4×D311yescharacteristic
466C3×C4C2×C3×C4×D311yescharacteristic
467C3×C4C3×#23711yescharacteristic
468C3×C4C3×#23911yescharacteristic
469Dic3—32no—
470Dic3—32no—
471Dic3—41no—
472Dic3—62no—
473Dic3—64no—
474Dic3—62no—
475Dic3—62no—
476Dic3—61no—
477Dic3—61no—
478Dic3—121no—
479C32—21no—
480C32—21no—
481C32—21no—
482C32—41no—
483C32#475811yescharacteristic, complemented
484C32#485211yescharacteristic, complemented
485C32C3×C4×SD1611yescharacteristic, complemented
486C23—61no—
487Q8—92no—
488Q8—181no—
489D4—32no—
490D4—61no—
491C2×C4—21no—
492C2×C4—61no—
493C2×C4—91no—
494C2×C4—91no—
495C2×C4—181no—
496C2×C4—362no—
497C2×C4C2×C3×D3211yescharacteristic
498C8—32no—
499C8—61no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—122no—
513C2×C3—122no—
514C2×C3—242no—
515C2×C3#6388111yescharacteristic
516C2×C3#6393611yescharacteristic
517C2×C3#6394011yescharacteristic
518C2×C3C3×#67⋊C411yescharacteristic
519C2×C3C3×#89311yescharacteristic
520C2×C3C3×#91211yescharacteristic
521C2×C3C3×D3×SD1611yescharacteristic
522C2×C3C3×D3×SD1611yescharacteristic
523C2×C3C3×#91611yescharacteristic
524D3—42no—
525C22—62no—
526C22—121no—
527C22—121no—
528C22C3×#381111yescharacteristic
529C4—41no—
530C4—92no—
531C4—121no—
532C4—182no—
533C4—181no—
534C4C3×#379611yescharacteristic
535C4C3×C4×D3211yescharacteristic
536C3—21no—
537C3—21no—
538C3—21no—
539C3—41no—
540C3#1059046011yescharacteristic, direct product factor
541C3C3×#475811yescharacteristic, complemented
542C3C3×#485211yescharacteristic, complemented
543C2—122no—
544C2C3×#6388111yescharacteristic
545C2C3×#6393611yescharacteristic
546C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×C4×SD16
LCSG ▷ 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
5. C32×#808C2semidirect
6. C3×#63567C2semidirect
7. C3×#63553C2semidirect
10. #10590460C3direct (N × H)
22. C3×#3716C4semidirect
23. #64096C2×C3semidirect
43. C3×#808D3semidirect
44. C3×#808C2×C3semidirect
45. #63567C2×C3semidirect
46. #63553C2×C3semidirect
54. C3×#634C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
111. #3716C3×C4semidirect
121. C3×C3⋊Dic3SD16semidirect
146. #808C3×D3semidirect
172. #634C2×C3×C4semidirect
206. C3×C3⋊C8C4×D3semidirect
207. C3×C3⋊C8C2×C3×C4semidirect
304. C3⋊Dic3C3×SD16semidirect
324. C33C4×SD16semidirect
385. C3⋊C8C3×C4×D3semidirect
483. C32#4758semidirect
484. C32#4852semidirect
485. C32C3×C4×SD16semidirect
540. C3#10590460direct (N × H)
541. C3C3×#4758semidirect
542. C3C3×#4852semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#638982
C2×C3×#37162
C32×#8092
C32×#8082
C3×#635672
C3×#635532
C3×#48523
C3×#47583
#105904603