Finite Group Databasebeta

← C3×#10591346 #13158285

Subgroups — C3×#10591346 #13158285

Order 1728.

1106 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
1C22×C32×Dic6C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic
3C2×C3×Dic3⋊Dic3C212yesmaximal
4C32×C3⋊#95C211yesmaximal, characteristic
5C3×#63579C212yesmaximal
6C3×#5230—31nomaximal
7C3×#4656—31nomaximal
8#10591346C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C2×C32×Dic6—22no—
15C2×C32×Dic6—24no—
16C3×#3750—22no—
17C3×Dic3⋊Dic3—24no—
18C22×C32×Q8—31no—
19C22×C3×Dic6—21no—
20C22×C3×Dic6D311yescharacteristic
21C22×C3×Dic6C2×C311yescharacteristic
22C2×#3750C2×C311yescharacteristic
23C2×C3×C4⋊Dic3—32no—
24C2×C3×Dic3⋊C4—31no—
25C2×C3×Dic3⋊C4—31no—
26C2×C3×Dic3⋊C4—32no—
27C2×C3×Dic3⋊C4—61no—
28C2×Dic3⋊Dic3C2×C312yes—
29C32×#95—31no—
30C3×C3⋊#95—21no—
31C3×C3⋊#95—32no—
32C3×C3⋊#95—32no—
33C3×C3⋊#95D311yescharacteristic
34C3×C3⋊#95C2×C311yescharacteristic
35#63579C2×C312yes—
36C23×C33C2311yescharacteristic
37C2×C33×C4—22no—
38C2×C33×C4D412yes—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3Q812yescomplemented
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3D414yes—
49C2×C32×Dic3Q814yescomplemented
50C32×Dic6—28no—
51C3×#395—91no—
52#5230—31no—
53#4656—31no—
54C2×C32×Q8—64no—
55C2×C32×Q8—62no—
56C22×C32×C4—21no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—32no—
60C22×C32×C4—32no—
61C22×C32×C4—41no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C22×C3⋊Dic3C22×C311yescharacteristic
66C22×C3⋊Dic3C22×C311yescharacteristic
67C22×C3×Dic3—22no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—61no—
74C22×C3×Dic3—61no—
75C22×C3×Dic3C2×D312yes—
76C22×C3×Dic3C2×D312yes—
77C22×C3×Dic3C22×C312yes—
78C22×C3×Dic3C22×C312yes—
79C2×C3×Dic6—24no—
80C2×C3×Dic6—28no—
81C2×C3×Dic6—22no—
82C2×C3×Dic6—22no—
83C2×C3×Dic6—24no—
84C2×C3×Dic6—24no—
85#3750—22no—
86C3×C4⋊Dic3—64no—
87C3×Dic3⋊C4—64no—
88C3×Dic3⋊C4—64no—
89C3×Dic3⋊C4—62no—
90C3×Dic3⋊C4—62no—
91Dic3⋊Dic3—24no—
92C22×C33C2×D411yescharacteristic
93C22×C33C2×D412yes—
94C22×C33C2×Q812yes—
95C22×C33C2×Q811yescharacteristic
96C22×C33#4111yescharacteristic
97C33×C4—22no—
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3—41no—
100C32×Dic3—24no—
101C32×Dic3—24no—
102C22×C3×Q8—31no—
103C22×C3×Q8—31no—
104C22×C3×Q8—61no—
105C22×Dic6C3×D311yescharacteristic
106C2×C3×C4⋊C4—92no—
107C2×C3×C4⋊C4—91no—
108C2×C4⋊Dic3—32no—
109C2×Dic3⋊C4—31no—
110C2×Dic3⋊C4—31no—
111C2×Dic3⋊C4—32no—
112C2×Dic3⋊C4—61no—
113C3×#95—31no—
114C3×#95—31no—
115C3×#95—61no—
116C3×#95—92no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C3⋊#95C3×D311yescharacteristic
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C23×C32C22×D311yescharacteristic
125C23×C32C22×D311yescharacteristic
126C23×C32C23×C311yescharacteristic
127C32×Q8—68no—
128C2×C32×C4—22no—
129C2×C32×C4—24no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—34no—
136C2×C32×C4—34no—
137C2×C32×C4—42no—
138C2×C32×C4—42no—
139C2×C32×C4—42no—
140C2×C32×C4—44no—
141C2×C32×C4—62no—
142C2×C32×C4—62no—
143C2×C32×C4—62no—
144C2×C32×C4—62no—
145C2×C32×C4D1212yes—
146C2×C32×C4#6712yes—
147C2×C32×C4C3×D412yes—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3C3×Q812yescomplemented
153C2×C3×Dic3—24no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—42no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—122no—
182C2×C3×Dic3Dic614yescomplemented
183C2×C3×Dic3#6714yes—
184C2×C3×Dic3C3×D414yes—
185C2×C3×Dic3C3×Q814yescomplemented
186C3×Dic6—216no—
187C3×Dic6—28no—
188C3×Dic6—28no—
189#395—91noSylow
190C2×C33#12011yescharacteristic
191C2×C33C22⋊Q812yes—
192C2×C33C22⋊Q811yescharacteristic
193C2×C33C4⋊Q811yescharacteristic
194C2×C33C4⋊Q812yes—
195C2×C3×Q8—64no—
196C2×C3×Q8—64no—
197C2×C3×Q8—68no—
198C2×C3×Q8—64no—
199C2×C3×Q8—62no—
200C2×C3×Q8—62no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—32no—
205C22×C3×C4—32no—
206C22×C3×C4—32no—
207C22×C3×C4—32no—
208C22×C3×C4—41no—
209C22×C3×C4—62no—
210C22×C3×C4—62no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4D3211yescharacteristic
214C22×C3×C4C2×C3×D311yescharacteristic
215C22×C3×C4C2×C3×D311yescharacteristic
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—61no—
221C22×Dic3—61no—
222C22×Dic3C2×C3×D312yes—
223C22×Dic3C2×C3×D312yes—
224C2×Dic6—22no—
225C2×Dic6—24no—
226C3×C4⋊C4—184no—
227C3×C4⋊C4—182no—
228C4⋊Dic3—64no—
229Dic3⋊C4—64no—
230Dic3⋊C4—64no—
231Dic3⋊C4—62no—
232Dic3⋊C4—62no—
233C22×C32—22no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—21no—
240C22×C32—22no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—41no—
249C22×C32—42no—
250C22×C32—42no—
251C22×C32—41no—
252C22×C32—41no—
253C22×C32C2×Dic612yes—
254C22×C32C2×D1211yescharacteristic
255C22×C32D3×D412yes—
256C22×C32#23711yescharacteristic
257C22×C32D3×Q812yes—
258C22×C32D3×Q811yescharacteristic
259C22×C32D3×Q811yescharacteristic
260C22×C32#23911yescharacteristic
261C22×C32C2×#6711yescharacteristic
262C22×C32C2×#6712yes—
263C22×C32C2×C3×D411yescharacteristic
264C22×C32C2×C3×D412yes—
265C22×C32C2×C3×Q812yes—
266C22×C32C2×C3×Q811yescharacteristic
267C22×C32C3×#4111yescharacteristic
268C32×C4—22no—
269C32×C4—24no—
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—42no—
273C32×C4—42no—
274C32×C4—44no—
275C32×C4—64no—
276C32×C4—64no—
277C3⋊Dic3—22no—
278C3⋊Dic3—41no—
279C3×Dic3—28no—
280C3×Dic3—24no—
281C3×Dic3—24no—
282C3×Dic3—24no—
283C3×Dic3—24no—
284C3×Dic3—44no—
285C3×Dic3—64no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C3×Dic3—122no—
289C3×Dic3—121no—
290C3×Dic3—121no—
291C22×Q8—31no—
292C2×C4⋊C4—92no—
293C2×C4⋊C4—91no—
294#95—31no—
295#95—92no—
296C33#39511yescharacteristic, complemented, Sylow
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—41no—
301C23×C3C2×D3211yescharacteristic
302C23×C3C22×C3×D311yescharacteristic
303C23×C3C22×C3×D311yescharacteristic
304C3×Q8—616no—
305C3×Q8—68no—
306C3×Q8—68no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—34no—
315C2×C3×C4—34no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—44no—
321C2×C3×C4—42no—
322C2×C3×C4—64no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—92no—
335C2×C3×C4—122no—
336C2×C3×C4—122no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4—182no—
341C2×C3×C4#62612yes—
342C2×C3×C4C3×D1212yes—
343C2×C3×C4C3×#6712yes—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—32no—
349C2×Dic3—32no—
350C2×Dic3—62no—
351C2×Dic3—62no—
352C2×Dic3—62no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—62no—
358C2×Dic3—61no—
359C2×Dic3—62no—
360C2×Dic3—61no—
361C2×Dic3—62no—
362C2×Dic3—122no—
363C2×Dic3C3×Dic614yescomplemented
364C2×Dic3C3×#6714yes—
365Dic6—28no—
366C2×C32—22no—
367C2×C32—21no—
368C2×C32—22no—
369C2×C32—21no—
370C2×C32—22no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—22no—
374C2×C32—21no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—42no—
384C2×C32—42no—
385C2×C32—41no—
386C2×C32#86911yescharacteristic
387C2×C32#87812yes—
388C2×C32#88211yescharacteristic
389C2×C32#88812yes—
390C2×C32#89711yescharacteristic
391C2×C32#94412yes—
392C2×C32#94411yescharacteristic
393C2×C32#94611yescharacteristic
394C2×C32#94612yes—
395C2×C32#95311yescharacteristic
396C2×C32C3×#12011yescharacteristic
397C2×C32C3×C22⋊Q812yes—
398C2×C32C3×C22⋊Q811yescharacteristic
399C2×C32C3×C4⋊Q812yes—
400C2×C32C3×C4⋊Q811yescharacteristic
401C2×Q8—64no—
402C2×Q8—62no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—91no—
406C22×C4—91no—
407C22×C4C3×D3211yescharacteristic
408C4⋊C4—184no—
409C4⋊C4—182no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—42no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—42no—
430C22×C3D3×Dic612yes—
431C22×C3#379611yescharacteristic
432C22×C3#379711yescharacteristic
433C22×C3C2×#62611yescharacteristic
434C22×C3D3×#6712yes—
435C22×C3C2×C3×Dic612yes—
436C22×C3C2×C3×D1211yescharacteristic
437C22×C3C3×D3×D412yes—
438C22×C3C3×#23711yescharacteristic
439C22×C3C3×D3×Q812yes—
440C22×C3C3×D3×Q811yescharacteristic
441C22×C3C3×D3×Q811yescharacteristic
442C22×C3C3×#23911yescharacteristic
443C22×C3C2×C3×#6711yescharacteristic
444C22×C3C2×C3×#6712yes—
445C3×C4—24no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—44no—
452C3×C4—68no—
453C3×C4—64no—
454C3×C4—64no—
455C3×C4—64no—
456C3×C4—64no—
457C3×C4—124no—
458C3×C4—182no—
459C3×C4—361no—
460Dic3—24no—
461Dic3—24no—
462Dic3—64no—
463Dic3—62no—
464Dic3—62no—
465Dic3—122no—
466Dic3—121no—
467Dic3—121no—
468C32—21no—
469C32—21no—
470C32—21no—
471C32—41no—
472C32#465611yescharacteristic, complemented
473C32#523011yescharacteristic, complemented
474C32C3×#39511yescharacteristic, complemented
475C23C2×C3×D3211yescharacteristic
476Q8—68no—
477C2×C4—22no—
478C2×C4—34no—
479C2×C4—34no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—92no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—182no—
489C2×C4C3×#62612yes—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—22no—
505C2×C3—41no—
506C2×C3—42no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—42no—
510C2×C3#6383912yes—
511C2×C3#6386611yescharacteristic
512C2×C3#6387111yescharacteristic
513C2×C3#6388712yes—
514C2×C3#6397711yescharacteristic
515C2×C3C3×#86911yescharacteristic
516C2×C3C3×#87812yes—
517C2×C3C3×#88211yescharacteristic
518C2×C3C3×#88812yes—
519C2×C3C3×#89711yescharacteristic
520C2×C3C3×#94412yes—
521C2×C3C3×#94411yescharacteristic
522C2×C3C3×#94611yescharacteristic
523C2×C3C3×#94612yes—
524C2×C3C3×#95311yescharacteristic
525C22C3×D3×Dic612yes—
526C22C3×#379611yescharacteristic
527C22C3×#379711yescharacteristic
528C22C2×C3×#62611yescharacteristic
529C22C3×D3×#6712yes—
530C4—22no—
531C4—64no—
532C4—64no—
533C4—182no—
534C4—361no—
535C3—21no—
536C3—21no—
537C3—21no—
538C3—41no—
539C3#1059134611yescharacteristic, direct product factor
540C3C3×#465611yescharacteristic, complemented
541C3C3×#523011yescharacteristic, complemented
542C2C3×#6383912yes—
543C2C3×#6386611yescharacteristic
544C2C3×#6387111yescharacteristic
545C2C3×#6388712yes—
546C2C3×#6397711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10591346C3direct (N × H)
43. C2×C3×C3⋊Dic3Q8semidirect
49. C2×C32×Dic3Q8semidirect
152. C2×C3⋊Dic3C3×Q8semidirect
182. C2×C3×Dic3Dic6semidirect
185. C2×C3×Dic3C3×Q8semidirect
296. C33#395semidirect
363. C2×Dic3C3×Dic6semidirect
472. C32#4656semidirect
473. C32#5230semidirect
474. C32C3×#395semidirect
539. C3#10591346direct (N × H)
540. C3C3×#4656semidirect
541. C3C3×#5230semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C2×C3×#37502
C2×C3×Dic3⋊Dic32
C32×C3⋊#952
C3×#635792
C3×#52303
C3×#46563
#105913463