Finite Group Databasebeta

← C3×#10590483 #13157601

Subgroups — C3×#10590483 #13157601

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64102C211yesmaximal, characteristic
2C3×#63843C211yesmaximal, characteristic, complemented
3C2×C3×#3717C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×#63553C211yesmaximal, characteristic, complemented
8C3×#4852—31nomaximal
9C3×#4788—31nomaximal
10#10590483C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#3717C414yescomplemented
23#64102C2×C311yescharacteristic
24C3×Q8×Dic3—31no—
25C2×C3×#215—31no—
26C2×C3×#204—31no—
27C3×#893—31no—
28C3×#893—31no—
29C3×#893—61no—
30C3×#887—31no—
31#63843C2×C311yescharacteristic, complemented
32C2×#3717C2×C311yescharacteristic
33C32×#106—31no—
34C32×#103—31no—
35C3×#821—31no—
36C3×C8×Dic3—31no—
37C3×C8×Dic3—31no—
38C3×#810—21no—
39C3×#810D311yescharacteristic, complemented
40C3×#810C2×C311yescharacteristic, complemented
41C3×#809—31no—
42C3×#808—21no—
43C3×#808D311yescharacteristic, complemented
44C3×#808C2×C311yescharacteristic, complemented
45#63564C2×C311yescharacteristic
46#63553C2×C311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yes—
54C3×C4×C3⋊D3—42no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C32×Dic6—21no—
58C32×Dic6C2×C412yescomplemented
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×C4×SD16—91no—
62#4852—31no—
63#4788—31no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C4×C3⋊D3—21no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×C4×D3—61no—
71C2×C3×C4×D3—61no—
72C2×C3×C4×D3—62no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C22×C311yescharacteristic
76C2×C32×C8—31no—
77C2×C32×C8—31no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—31no—
82C32×C4⋊C4—41no—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C22×C311yescharacteristic
86#3752—21no—
87C4×C3⋊Dic3C22×C311yescharacteristic
88C3×#215—34no—
89C3×(C2×D3)⋊C4—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×(C2×D3)⋊C4—62no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C3×#204—34no—
109Dic3⋊Dic3—21no—
110Dic32—21no—
111#3717C3×C414yescomplemented
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C33×C4—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—21no—
120C3×C3⋊Dic3SD1612yescomplemented
121C32×Dic3—22no—
122C32×Dic3—41no—
123C2×C3×SD16—91no—
124C3×C4×Q8—91no—
125C3×C4×D4—91no—
126Q8×Dic3—31no—
127C2×#215—31no—
128C2×#204—31no—
129#893—31no—
130#893—31no—
131#893—61no—
132#887—31no—
133C3×#106—31no—
134C3×#106—31no—
135C3×#106—61no—
136C3×#103—31no—
137C3×#103—31no—
138C3×#103—61no—
139C3×#102—91no—
140C3×C4×C8—91no—
141#821—31no—
142C8×Dic3—31no—
143C8×Dic3—31no—
144#810C3×D311yescharacteristic, complemented
145#809—31no—
146#808C3×D311yescharacteristic, complemented
147C22×C3⋊D3—21no—
148C22×C3×D3—62no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C32×Q8—32no—
152C32×Q8—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×C4—61no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C23×C311yescharacteristic
169C2×C3⋊Dic3C3×D411yescharacteristic
170C2×C3⋊Dic3C3×D411yescharacteristic
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)C2×C3×C412yes—
173C4×C3⋊D3—42no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C3×D12—32no—
187C3×D12—32no—
188C3×D12—62no—
189C3×D12—62no—
190C3×D12—61no—
191C3×D12—61no—
192C3×C4×D3—122no—
193C3×C4×D3—122no—
194C3×C4×D3—124no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—22no—
198C3×Dic6—41no—
199C3×Dic6C4×D312yescomplemented
200C3×Dic6C2×C3×C412yescomplemented
201C32×C8—32no—
202C32×C8—61no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—22no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8C4×D312yescomplemented
208C3×C3⋊C8C2×C3×C412yescomplemented
209C4×SD16—91noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33C2×SD1611yescharacteristic
212C2×C33#13511yescharacteristic
213C3×C3⋊D3—42no—
214C2×C3×Q8—31no—
215C2×C3×Q8—31no—
216C2×C3×Q8—61no—
217C2×C3×D4—91no—
218C22×C3×C4—181no—
219C2×D12—31no—
220C2×D12—31no—
221C2×D12—61no—
222C2×C4×D3—61no—
223C2×C4×D3—61no—
224C2×C4×D3—62no—
225C2×Dic6C2×C3×D311yescharacteristic
226C3×SD16—94no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—61no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4—181no—
241C3×C4⋊C4D3211yescharacteristic
242C3×C4⋊C4C2×C3×D311yescharacteristic
243C3×C4⋊C4C2×C3×D311yescharacteristic
244C3×C22⋊C4—181no—
245C3×C42—91no—
246C3×C42—181no—
247#215—34no—
248(C2×D3)⋊C4—61no—
249(C2×D3)⋊C4—61no—
250(C2×D3)⋊C4—62no—
251C4⋊Dic3—61no—
252C4⋊Dic3C2×C3×D311yescharacteristic
253Dic3⋊C4—61no—
254C4×Dic3—31no—
255C4×Dic3—31no—
256C4×Dic3—61no—
257C4×Dic3—61no—
258C4×Dic3—61no—
259C2×C3⋊C8C2×C3×D311yescharacteristic
260C2×C3⋊C8C2×C3×D311yescharacteristic
261#204—34no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32C2×C3×D411yescharacteristic
269C2×C3⋊D3—22no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3×D3—64no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—122no—
276C2×C3×D3—122no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C2×C3×D3—121no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—42no—
291C32×C4—41no—
292C32×C4—41no—
293C32×C4—62no—
294C32×C4—81no—
295C32×C4—81no—
296C32×C4—82no—
297C32×C4—121no—
298C32×C4C2×C4×D311yescharacteristic
299C32×C4C2×C4×D311yescharacteristic
300C32×C4#23711yescharacteristic
301C32×C4#23911yescharacteristic
302C32×C4C22×C3×C411yescharacteristic
303C32×C4C3×#4111yescharacteristic
304C3⋊Dic3—21no—
305C3⋊Dic3C3×SD1612yescomplemented
306C3×Dic3—22no—
307C3×Dic3—22no—
308C3×Dic3—32no—
309C3×Dic3—32no—
310C3×Dic3—42no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—62no—
314C3×Dic3—61no—
315C3×Dic3—61no—
316C3×Dic3—81no—
317C3×Dic3—121no—
318C2×SD16—91no—
319C4×Q8—91no—
320C4×D4—91no—
321#106—31no—
322#103—31no—
323#102—91no—
324C4×C8—91no—
325C33C4×SD1611yescharacteristic, complemented, Sylow
326C23×C3—181no—
327C22×D3—62no—
328C22×D3—61no—
329C22×D3—61no—
330C3×Q8—32no—
331C3×Q8—32no—
332C3×Q8—62no—
333C3×Q8—61no—
334C3×Q8—61no—
335C3×Q8—121no—
336C3×D4—92no—
337C3×D4—181no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4—121no—
356C2×C3×C4—121no—
357C2×C3×C4—362no—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D311yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369D12—32no—
370D12—32no—
371D12—62no—
372D12—62no—
373D12—61no—
374D12—61no—
375C4×D3—122no—
376C4×D3—122no—
377C4×D3—124no—
378Dic6—21no—
379Dic6C3×C4×D312yescomplemented
380C3×C8—32no—
381C3×C8—32no—
382C3×C8—62no—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—121no—
386C3⋊C8—21no—
387C3⋊C8C3×C4×D312yescomplemented
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32#67⋊C411yescharacteristic
401C2×C32#89311yescharacteristic
402C2×C32D3×SD1611yescharacteristic
403C2×C32D3×SD1611yescharacteristic
404C2×C32#91611yescharacteristic
405C2×C32#91911yescharacteristic
406C2×C32C3×C4×D411yescharacteristic
407C2×C32C2×C3×SD1611yescharacteristic
408C2×C32C3×#13511yescharacteristic
409C3⋊D3—42no—
410C3×D3—124no—
411C3×D3—122no—
412C3×D3—122no—
413C2×Q8—31no—
414C2×D4—91no—
415C22×C4—181no—
416SD16—94no—
417C2×C8—31no—
418C2×C8—31no—
419C4⋊C4—31no—
420C4⋊C4—181no—
421C4⋊C4C3×D3211yescharacteristic
422C22⋊C4—181no—
423C42—91no—
424C42—181no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—41no—
429C22×C3—182no—
430C22×C3—361no—
431C22×C3—361no—
432C22×C3#381111yescharacteristic
433C22×C3C3×D3×D411yescharacteristic
434C22×C3C3×D3×D411yescharacteristic
435C2×D3—64no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—121no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4—82no—
461C3×C4—92no—
462C3×C4—122no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4—181no—
466C3×C4—241no—
467C3×C4#379611yescharacteristic
468C3×C4C4×D3211yescharacteristic
469C3×C4C2×C3×C4×D311yescharacteristic
470C3×C4C2×C3×C4×D311yescharacteristic
471C3×C4C3×#23711yescharacteristic
472C3×C4C3×#23911yescharacteristic
473Dic3—22no—
474Dic3—32no—
475Dic3—32no—
476Dic3—41no—
477Dic3—62no—
478Dic3—61no—
479Dic3—61no—
480Dic3—121no—
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32#478811yescharacteristic, complemented
486C32#485211yescharacteristic, complemented
487C32C3×C4×SD1611yescharacteristic, complemented
488C23—181no—
489Q8—32no—
490Q8—61no—
491D4—92no—
492D4—181no—
493C2×C4—21no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—91no—
497C2×C4—91no—
498C2×C4—362no—
499C2×C4C2×C3×D3211yescharacteristic
500C8—32no—
501C8—61no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—362no—
515C2×C3#6388111yescharacteristic
516C2×C3#6393011yescharacteristic
517C2×C3#6395011yescharacteristic
518C2×C3C3×#67⋊C411yescharacteristic
519C2×C3C3×#89311yescharacteristic
520C2×C3C3×D3×SD1611yescharacteristic
521C2×C3C3×D3×SD1611yescharacteristic
522C2×C3C3×#91611yescharacteristic
523C2×C3C3×#91911yescharacteristic
524D3—124no—
525D3—122no—
526D3—122no—
527C22—182no—
528C22—361no—
529C22—361no—
530C22C3×#381111yescharacteristic
531C4—41no—
532C4—62no—
533C4—92no—
534C4—121no—
535C4—181no—
536C4C3×#379611yescharacteristic
537C4C3×C4×D3211yescharacteristic
538C3—21no—
539C3—21no—
540C3—21no—
541C3—41no—
542C3#1059048311yescharacteristic, direct product factor
543C3C3×#478811yescharacteristic, complemented
544C3C3×#485211yescharacteristic, complemented
545C2—362no—
546C2C3×#6388111yescharacteristic
547C2C3×#6393011yescharacteristic
548C2C3×#6395011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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
2. C3×#63843C2semidirect
4. C32×#810C2semidirect
5. C32×#808C2semidirect
7. C3×#63553C2semidirect
10. #10590483C3direct (N × H)
22. C3×#3717C4semidirect
31. #63843C2×C3semidirect
39. C3×#810D3semidirect
40. C3×#810C2×C3semidirect
43. C3×#808D3semidirect
44. C3×#808C2×C3semidirect
46. #63553C2×C3semidirect
58. C32×Dic6C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
111. #3717C3×C4semidirect
120. C3×C3⋊Dic3SD16semidirect
144. #810C3×D3semidirect
146. #808C3×D3semidirect
199. C3×Dic6C4×D3semidirect
200. C3×Dic6C2×C3×C4semidirect
207. C3×C3⋊C8C4×D3semidirect
208. C3×C3⋊C8C2×C3×C4semidirect
305. C3⋊Dic3C3×SD16semidirect
325. C33C4×SD16semidirect
379. Dic6C3×C4×D3semidirect
387. C3⋊C8C3×C4×D3semidirect
485. C32#4788semidirect
486. C32#4852semidirect
487. C32C3×C4×SD16semidirect
542. C3#10590483direct (N × H)
543. C3C3×#4788semidirect
544. C3C3×#4852semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
C3×#638432
C2×C3×#37172
C32×#8102
C32×#8082
C3×#635642
C3×#635532
C3×#48523
C3×#47883
#105904833