Finite Group Databasebeta

← C3×#10590561 #13157548

Subgroups — C3×#10590561 #13157548

Order 1728.

612 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#887C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic
4C2×C3×#3718C211yesmaximal, characteristic
5C3×#63637C211yesmaximal, characteristic
6C32×#810C211yesmaximal, characteristic
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
8C3×#5031—31nomaximal
9C3×#4795—31nomaximal
10#10590561C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×C4×Dic3—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#3719—22no—
23C3×#3718—22no—
24C32×C4×Q8—31no—
25C3×#944—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C2×C3×Dic12—31no—
30C3×#887—21no—
31C3×#887D311yescharacteristic
32C3×#887C2×C311yescharacteristic
33C3×#869—31no—
34#63866C2×C311yescharacteristic
35C2×#3719C2×C311yescharacteristic
36C2×#3718C2×C311yescharacteristic
37C32×#103—31no—
38#63637C2×C311yescharacteristic
39C3×Dic3⋊C8—31no—
40C3×#810—21no—
41C3×#810—31no—
42C3×#810—31no—
43C3×#810—61no—
44C3×#810D311yescharacteristic
45C3×#810C2×C311yescharacteristic
46C3×C4⋊(C3⋊C8)—31no—
47Dic3⋊(C3⋊C8)C2×C311yescharacteristic
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63C3×#462—91no—
64#5031—31no—
65#4795—31no—
66C2×C32×Q8—31no—
67C2×C32×Q8—31no—
68C2×#634C22×C311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C2×D311yescharacteristic
76C2×C3×Dic6C22×C311yescharacteristic
77C2×C3×Dic6C22×C311yescharacteristic
78C2×C32×C8—31no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—41no—
83C32×C4⋊C4—61no—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C22×C311yescharacteristic
87C32×C42—31no—
88C32×C42—61no—
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C3×#216—62no—
91C3×#216—62no—
92C3×#216—62no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×Dic3⋊C4—22no—
96C3×Dic3⋊C4—21no—
97C3×Dic3⋊C4—21no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3—21no—
104C3×C4×Dic3C2×D311yescharacteristic
105C3×C4×Dic3C22×C311yescharacteristic
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112C3×Dic12—62no—
113Dic3⋊Dic3—21no—
114Dic3⋊Dic3—21no—
115#3719—22no—
116#3718—22no—
117C22×C33C2×D411yescharacteristic
118C33×C4—41no—
119C33×C4C2×D411yescharacteristic
120C33×C4#4111yescharacteristic
121C3×C3⋊Dic3—41no—
122C32×Dic3—22no—
123C32×Dic3—21no—
124C32×Dic3—41no—
125C32×Dic3Q1612yescomplemented
126C2×C3×Q16—91no—
127C2×C3×Q16—91no—
128C3×C4⋊Q8—91no—
129C3×C4×Q8—31no—
130C3×C4×Q8—31no—
131C3×C4×Q8—61no—
132#944—31no—
133C2×#216—31no—
134C2×#216—31no—
135C2×#216—31no—
136C2×Dic12—31no—
137#887C3×D311yescharacteristic
138#869—31no—
139C3×C4⋊C8—91no—
140C3×#103—31no—
141C3×#103—31no—
142C3×#103—61no—
143C3×#103—91no—
144Dic3⋊C8—31no—
145#810—31no—
146#810—31no—
147#810—61no—
148#810C3×D311yescharacteristic
149C4⋊(C3⋊C8)—31no—
150C32×Q8—32no—
151C32×Q8—61no—
152C32×Q8—61no—
153C32×Q8—61no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4—41no—
166C2×C32×C4—42no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C23×C311yescharacteristic
172C2×C3⋊Dic3—21no—
173#634—21no—
174#634—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—121no—
186C2×C3×Dic3#6711yescharacteristic
187C2×C3×Dic3#6711yescharacteristic
188C2×C3×Dic3C3×D411yescharacteristic
189C2×C3×Dic3C3×D411yescharacteristic
190C3×Dic6—22no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—22no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—22no—
199C3×Dic6—41no—
200C3×Dic6—62no—
201C3×Dic6—61no—
202C3×Dic6—61no—
203C3×Dic6—61no—
204C3×Dic6—61no—
205C3×Dic6—121no—
206C3×Dic6D1212yes—
207C3×Dic6C3×D412yes—
208C32×C8—61no—
209C3⋊(C3⋊C8)—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—41no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—61no—
215C3×C3⋊C8—121no—
216#462—91noSylow
217C2×C33#12111yescharacteristic
218C2×C33C2×Q1611yescharacteristic
219C2×C33#13711yescharacteristic
220C2×C3×Q8—31no—
221C2×C3×Q8—31no—
222C2×C3×Q8—31no—
223C2×C3×Q8—31no—
224C2×C3×Q8—61no—
225C2×C3×Q8—61no—
226C2×C3×Q8—91no—
227C2×Dic6—31no—
228C2×Dic6—31no—
229C2×Dic6—61no—
230C2×Dic6C2×C3×D311yescharacteristic
231C2×Dic6C2×C3×D311yescharacteristic
232C3×Q16—182no—
233C3×Q16—182no—
234C2×C3×C8—31no—
235C2×C3×C8—31no—
236C2×C3×C8—61no—
237C2×C3×C8—91no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4—181no—
246C3×C4⋊C4—181no—
247C3×C4⋊C4D3211yescharacteristic
248C3×C4⋊C4C2×C3×D311yescharacteristic
249C3×C4⋊C4C2×C3×D311yescharacteristic
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253C3×C42—61no—
254C3×C42—61no—
255C3×C42—62no—
256#216—62no—
257#216—62no—
258#216—62no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261Dic3⋊C4—21no—
262Dic3⋊C4—61no—
263Dic3⋊C4—61no—
264C4×Dic3—21no—
265C4×Dic3C2×C3×D311yescharacteristic
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—61no—
269C2×C3⋊C8C2×C3×D311yescharacteristic
270Dic12—62no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—41no—
275C22×C32D3×D411yescharacteristic
276C22×C32C2×#6711yescharacteristic
277C22×C32C2×C3×D411yescharacteristic
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—32no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—42no—
289C32×C4—41no—
290C32×C4—41no—
291C32×C4—62no—
292C32×C4—61no—
293C32×C4—81no—
294C32×C4—81no—
295C32×C4—82no—
296C32×C4—121no—
297C32×C4C2×D1211yescharacteristic
298C32×C4#23511yescharacteristic
299C32×C4#23511yescharacteristic
300C32×C4D3×D411yescharacteristic
301C32×C4C2×C3×D411yescharacteristic
302C32×C4C3×#4111yescharacteristic
303C3⋊Dic3—41no—
304C3×Dic3—24no—
305C3×Dic3—22no—
306C3×Dic3—21no—
307C3×Dic3—22no—
308C3×Dic3—21no—
309C3×Dic3—22no—
310C3×Dic3—41no—
311C3×Dic3—42no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—122no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3#21612yescomplemented
318C3×Dic3C3×Q1612yescomplemented
319C2×Q16—91no—
320C2×Q16—91no—
321C4⋊Q8—91no—
322C4×Q8—31no—
323C4⋊C8—91no—
324#103—31no—
325#103—91no—
326C33#46211yescharacteristic, complemented, Sylow
327C3×Q8—32no—
328C3×Q8—32no—
329C3×Q8—62no—
330C3×Q8—62no—
331C3×Q8—62no—
332C3×Q8—61no—
333C3×Q8—61no—
334C3×Q8—61no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—121no—
339C3×Q8—181no—
340C3×Q8—181no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—121no—
364C2×C3×C4—181no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C22×C3×D311yescharacteristic
367C2×C3×C4C22×C3×D311yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—121no—
373C2×Dic3C3×#6711yescharacteristic
374C2×Dic3C3×#6711yescharacteristic
375Dic6—21no—
376Dic6—21no—
377Dic6—21no—
378Dic6—62no—
379Dic6—61no—
380Dic6—61no—
381Dic6—61no—
382Dic6—61no—
383Dic6—121no—
384Dic6C3×D1212yes—
385C3×C8—61no—
386C3×C8—61no—
387C3×C8—121no—
388C3×C8—181no—
389C3⋊C8—21no—
390C3⋊C8—61no—
391C3⋊C8—61no—
392C3⋊C8—121no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—41no—
405C2×C32#88411yescharacteristic
406C2×C32#91511yescharacteristic
407C2×C32D3×Q1611yescharacteristic
408C2×C32#93011yescharacteristic
409C2×C32C2×#21611yescharacteristic
410C2×C32#95111yescharacteristic
411C2×C32C3×#12111yescharacteristic
412C2×C32C2×C3×Q1611yescharacteristic
413C2×C32C3×#13711yescharacteristic
414C2×Q8—31no—
415C2×Q8—31no—
416C2×Q8—91no—
417Q16—182no—
418Q16—182no—
419C2×C8—31no—
420C2×C8—91no—
421C4⋊C4—61no—
422C4⋊C4—181no—
423C4⋊C4—181no—
424C4⋊C4C3×D3211yescharacteristic
425C42—31no—
426C42—61no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3D3×#6711yescharacteristic
432C22×C3C3×D3×D411yescharacteristic
433C22×C3C2×C3×#6711yescharacteristic
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—42no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—64no—
450C3×C4—62no—
451C3×C4—61no—
452C3×C4—62no—
453C3×C4—61no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4—82no—
457C3×C4—121no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—361no—
462C3×C4#379811yescharacteristic
463C3×C4D3×D1211yescharacteristic
464C3×C4C2×C3×D1211yescharacteristic
465C3×C4C3×#23511yescharacteristic
466C3×C4C3×#23511yescharacteristic
467C3×C4C3×D3×D411yescharacteristic
468Dic3—22no—
469Dic3—21no—
470Dic3—41no—
471Dic3—122no—
472Dic3—121no—
473Dic3—121no—
474Dic3C3×#21612yescomplemented
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#479511yescharacteristic, complemented
480C32#503111yescharacteristic, complemented
481C32C3×#46211yescharacteristic, complemented
482Q8—32no—
483Q8—61no—
484Q8—61no—
485Q8—61no—
486Q8—181no—
487Q8—181no—
488C2×C4—21no—
489C2×C4—31no—
490C2×C4—31no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—181no—
494C2×C4C2×C3×D3211yescharacteristic
495C8—61no—
496C8—181no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3#6390611yescharacteristic
510C2×C3#6392911yescharacteristic
511C2×C3D3×#21611yescharacteristic
512C2×C3C3×#88411yescharacteristic
513C2×C3C3×#91511yescharacteristic
514C2×C3C3×D3×Q1611yescharacteristic
515C2×C3C3×#93011yescharacteristic
516C2×C3C2×C3×#21611yescharacteristic
517C2×C3C3×#95111yescharacteristic
518C22C3×D3×#6711yescharacteristic
519C4—32no—
520C4—41no—
521C4—62no—
522C4—61no—
523C4—121no—
524C4—361no—
525C4C3×#379811yescharacteristic
526C4C3×D3×D1211yescharacteristic
527C3—21no—
528C3—21no—
529C3—21no—
530C3—41no—
531C3#1059056111yescharacteristic, direct product factor
532C3C3×#479511yescharacteristic, complemented
533C3C3×#503111yescharacteristic, complemented
534C2C3×#6390611yescharacteristic
535C2C3×#6392911yescharacteristic
536C2C3×D3×#21611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#462
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
10. #10590561C3direct (N × H)
125. C32×Dic3Q16semidirect
317. C3×Dic3#216semidirect
318. C3×Dic3C3×Q16semidirect
326. C33#462semidirect
474. Dic3C3×#216semidirect
479. C32#4795semidirect
480. C32#5031semidirect
481. C32C3×#462semidirect
531. C3#10590561direct (N × H)
532. C3C3×#4795semidirect
533. C3C3×#5031semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
C3×#638662
C2×C3×#37192
C2×C3×#37182
C3×#636372
C32×#8102
C3×Dic3⋊(C3⋊C8)2
C3×#50313
C3×#47953
#105905613