Finite Group Databasebeta

← C3×#10591556 #13158636

Subgroups — C3×#10591556 #13158636

Order 1728.

918 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic, complemented
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8C3×#5233—31nomaximal
9C3×#4804—31nomaximal
10#10591556C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C32×#239—22no—
13C32×#239C414yescomplemented
14C2×C32×D12—21no—
15C2×C32×D12C2211yescharacteristic
16C2×C32×C4×D3—21no—
17C2×C32×C4×D3C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C3×Dic32—21no—
24C3×D3×C3⋊C8—22no—
25C2×C32×#41—31no—
26C2×C3×#239—21no—
27C2×C3×#239D311yescharacteristic
28C2×C3×#239C2×C311yescharacteristic
29C3×#926—31no—
30C22×C3×C3⋊C8—31no—
31C2×C3×D3×C8—31no—
32C3×#892—31no—
33#63841C2×C311yescharacteristic, complemented
34C2×D3×C3⋊C8C2×C311yescharacteristic, complemented
35C32×#103—31no—
36#63662C2×C311yescharacteristic, complemented
37C3×#835—21no—
38C3×#835—31no—
39C3×#835—31no—
40C3×#835—61no—
41C3×#835D311yescharacteristic, complemented
42C3×#835C2×C311yescharacteristic, complemented
43C3×#832—31no—
44C3×#832—31no—
45C3×#821—31no—
46C3×#809—31no—
47#63563C2×C311yescharacteristic, complemented
48#63562C2×C311yescharacteristic, complemented
49C22×C32×D3—21no—
50C22×C32×D3D411yescharacteristic
51C33×Q8—21no—
52C33×Q8C2×C412yes—
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×D12—24no—
59C32×D12—21no—
60C32×D12C2×C412yes—
61C32×C4×D3—24no—
62C32×C4×D3C2×C412yes—
63C32×C4×D3D412yes—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66C3×#416—91no—
67#5233—31no—
68#4804—31no—
69C32×#41—34no—
70C32×#41—62no—
71C2×C32×Q8—21no—
72C2×C32×Q8—21no—
73C2×C32×Q8—21no—
74C2×C32×Q8—41no—
75C2×C32×Q8C2×D311yescharacteristic
76C2×C32×Q8C2×D311yescharacteristic
77C2×C32×Q8C22×C311yescharacteristic
78C2×C32×D4—31no—
79C2×C32×D4—61no—
80C22×C32×C4—31no—
81C22×C32×C4—61no—
82C3×#239—24no—
83C3×#239—24no—
84C3×#239—22no—
85C3×#239—22no—
86C3×#239Dic314yescomplemented
87C3×#239C3×C414yescomplemented
88C2×C3×D12—22no—
89C2×C3×D12—21no—
90C2×C3×D12—21no—
91C2×C3×D12—21no—
92C2×C3×D12C2×D311yescharacteristic
93C2×C3×D12C22×C311yescharacteristic
94C2×C3×C4×D3—22no—
95C2×C3×C4×D3—21no—
96C2×C3×C4×D3—21no—
97C2×C3×C4×D3—21no—
98C2×C3×C4×D3C2×D311yescharacteristic
99C2×C3×C4×D3C22×C311yescharacteristic
100C2×C32×C8—31no—
101C32×C4⋊C4—31no—
102C4⋊(C3⋊Dic3)C22×C311yescharacteristic
103C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
104C3×#217—61no—
105C3×(C2×D3)⋊C4—61no—
106C3×C4⋊Dic3—21no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3C2×D311yescharacteristic
111C3×C4⋊Dic3C22×C311yescharacteristic
112C3×C4×Dic3—61no—
113C3×C4×Dic3—61no—
114C2×C3×C3⋊C8—21no—
115C2×C3×C3⋊C8—31no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—62no—
118C2×C3×C3⋊C8—61no—
119C2×C3×C3⋊C8C2×D311yescharacteristic
120C2×C3×C3⋊C8C22×C311yescharacteristic
121C3×D3×C8—62no—
122(C2×D3)⋊Dic3—21no—
123Dic32—21no—
124D3×C3⋊C8—22no—
125C22×C33C2×D411yescharacteristic
126C2×C32×D3—21no—
127C2×C32×D3—21no—
128C2×C32×D3—22no—
129C2×C32×D3—41no—
130C2×C32×D3—41no—
131C2×C32×D3C22⋊C412yes—
132C33×C4—22no—
133C33×C4C22×C411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C3×C3⋊Dic3—41no—
136C32×Dic3—41no—
137C32×Dic3C22⋊C412yes—
138C2×C3×#41—31no—
139C2×C3×#41—31no—
140C2×C3×#41—61no—
141C2×#239C3×D311yescharacteristic
142C22×C3×C8—91no—
143C3×#117—91no—
144#926—31no—
145C22×C3⋊C8—31no—
146C2×D3×C8—31no—
147#892—31no—
148C3×#103—31no—
149C3×#103—31no—
150C3×#103—61no—
151C3×#103—91no—
152C3×#102—91no—
153C3×#102—91no—
154#835—31no—
155#835—31no—
156#835—61no—
157#835C3×D311yescharacteristic, complemented
158#832—31no—
159#832—31no—
160#821—31no—
161#809—31no—
162C23×C32—31no—
163C23×C32—61no—
164C22×C3×D3—22no—
165C22×C3×D3—21no—
166C22×C3×D3—21no—
167C22×C3×D3—21no—
168C22×C3×D3#6711yescharacteristic
169C22×C3×D3C3×D411yescharacteristic
170C32×Q8—21no—
171C32×Q8—22no—
172C32×Q8—21no—
173C32×Q8—21no—
174C32×Q8—22no—
175C32×Q8—22no—
176C32×Q8—22no—
177C32×Q8—42no—
178C32×Q8—41no—
179C32×Q8—41no—
180C32×Q8—42no—
181C32×Q8C4×D312yes—
182C32×Q8C2×Dic312yes—
183C32×Q8C2×C3×C412yes—
184C32×D4—32no—
185C32×D4—64no—
186C32×D4—61no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—31no—
195C2×C32×C4—32no—
196C2×C32×C4—32no—
197C2×C32×C4—41no—
198C2×C32×C4—41no—
199C2×C32×C4—41no—
200C2×C32×C4—42no—
201C2×C32×C4—64no—
202C2×C32×C4—61no—
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205C2×C32×C4C23×C311yescharacteristic
206C2×C3⋊Dic3—21no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—21no—
210C2×C3×Dic3—41no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—121no—
214C2×C3×Dic3#6711yescharacteristic
215C2×C3×Dic3C3×D411yescharacteristic
216C3×D12—28no—
217C3×D12—22no—
218C3×D12—24no—
219C3×D12—24no—
220C3×D12—21no—
221C3×D12—21no—
222C3×D12—22no—
223C3×D12C2×Dic312yes—
224C3×D12C2×C3×C412yes—
225C3×C4×D3—28no—
226C3×C4×D3—24no—
227C3×C4×D3—24no—
228C3×C4×D3—22no—
229C3×C4×D3—22no—
230C3×C4×D3C2×Dic312yes—
231C3×C4×D3#6712yes—
232C3×C4×D3C2×C3×C412yes—
233C3×C4×D3C3×D412yes—
234C32×C8—61no—
235C3⋊(C3⋊C8)—21no—
236C3×C3⋊C8—21no—
237C3×C3⋊C8—21no—
238C3×C3⋊C8—41no—
239C3×C3⋊C8—61no—
240C3×C3⋊C8—61no—
241C3×C3⋊C8—121no—
242#416—91noSylow
243C2×C33C2×C22⋊C411yescharacteristic
244C2×C33#13511yescharacteristic
245C2×C33#13511yescharacteristic
246C32×D3—22no—
247C32×D3—42no—
248C3×#41—34no—
249C3×#41—34no—
250C3×#41—64no—
251C3×#41—62no—
252C3×#41—62no—
253C3×#41—64no—
254C2×C3×Q8—21no—
255C2×C3×Q8—21no—
256C2×C3×Q8—21no—
257C2×C3×Q8—41no—
258C2×C3×Q8D3211yescharacteristic
259C2×C3×Q8C2×C3×D311yescharacteristic
260C2×C3×Q8C2×C3×D311yescharacteristic
261C2×C3×D4—31no—
262C2×C3×D4—31no—
263C2×C3×D4—61no—
264C2×C3×D4—61no—
265C2×C3×D4—61no—
266C2×C3×D4—62no—
267C22×C3×C4—31no—
268C22×C3×C4—31no—
269C22×C3×C4—61no—
270C22×C3×C4—61no—
271C22×C3×C4—61no—
272C22×C3×C4—62no—
273#239—22no—
274#239C3×Dic314yescomplemented
275C2×D12—21no—
276C2×D12C2×C3×D311yescharacteristic
277C2×C4×D3—21no—
278C2×C4×D3C2×C3×D311yescharacteristic
279C2×C3×C8—31no—
280C2×C3×C8—31no—
281C2×C3×C8—61no—
282C2×C3×C8—91no—
283C2×C3×C8—182no—
284C3×C4⋊C4—31no—
285C3×C4⋊C4—31no—
286C3×C4⋊C4—61no—
287C3×C4⋊C4—91no—
288C3×C22⋊C4—181no—
289C3×C42—181no—
290#217—61no—
291(C2×D3)⋊C4—61no—
292C4⋊Dic3—31no—
293C4⋊Dic3—31no—
294C4⋊Dic3—61no—
295C4⋊Dic3C2×C3×D311yescharacteristic
296C4×Dic3—61no—
297C4×Dic3—61no—
298C2×C3⋊C8—31no—
299C2×C3⋊C8—31no—
300C2×C3⋊C8—61no—
301C2×C3⋊C8—62no—
302C2×C3⋊C8C2×C3×D311yescharacteristic
303D3×C8—62no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—32no—
308C22×C32—41no—
309C22×C32—62no—
310C22×C32—61no—
311C22×C32—61no—
312C22×C32—121no—
313C22×C32—121no—
314C22×C32D3×D411yescharacteristic
315C22×C32C2×#6711yescharacteristic
316C22×C32C2×C3×D411yescharacteristic
317C2×C3×D3—21no—
318C2×C3×D3—21no—
319C2×C3×D3—21no—
320C2×C3×D3—21no—
321C2×C3×D3—24no—
322C2×C3×D3—22no—
323C2×C3×D3—22no—
324C2×C3×D3—22no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—42no—
328C2×C3×D3—42no—
329C2×C3×D3—41no—
330C2×C3×D3—41no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3#21712yes—
334C2×C3×D3C3×C22⋊C412yes—
335C32×C4—22no—
336C32×C4—24no—
337C32×C4—22no—
338C32×C4—22no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—32no—
346C32×C4—41no—
347C32×C4—41no—
348C32×C4—42no—
349C32×C4—42no—
350C32×C4—44no—
351C32×C4—121no—
352C32×C4C2×C4×D311yescharacteristic
353C32×C4D3×D411yescharacteristic
354C32×C4C22×Dic311yescharacteristic
355C32×C4C2×#6711yescharacteristic
356C32×C4C22×C3×C411yescharacteristic
357C32×C4C2×C3×D411yescharacteristic
358C3⋊Dic3—41no—
359C3×Dic3—22no—
360C3×Dic3—41no—
361C3×Dic3—41no—
362C3×Dic3—81no—
363C3×Dic3—121no—
364C3×Dic3—121no—
365C3×Dic3—241no—
366C3×Dic3#21712yes—
367C3×Dic3C3×C22⋊C412yes—
368C2×#41—31no—
369C22×C8—91no—
370#117—91no—
371#103—31no—
372#103—91no—
373#102—91no—
374#102—91no—
375C33#41611yescharacteristic, complemented, Sylow
376C23×C3—31no—
377C23×C3—31no—
378C23×C3—61no—
379C23×C3—62no—
380C23×C3—61no—
381C23×C3—61no—
382C22×D3—21no—
383C22×D3C3×#6711yescharacteristic
384C3×Q8—22no—
385C3×Q8—21no—
386C3×Q8—21no—
387C3×Q8—21no—
388C3×Q8—22no—
389C3×Q8—22no—
390C3×Q8—22no—
391C3×Q8—41no—
392C3×Q8—41no—
393C3×Q8—42no—
394C3×Q8—42no—
395C3×Q8D3×Dic312yes—
396C3×Q8C3×C4×D312yes—
397C3×Q8C2×C3×Dic312yes—
398C3×D4—32no—
399C3×D4—32no—
400C3×D4—62no—
401C3×D4—64no—
402C3×D4—64no—
403C3×D4—68no—
404C3×D4—62no—
405C3×D4—61no—
406C3×D4—61no—
407C2×C3×C4—22no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—31no—
415C2×C3×C4—31no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—42no—
423C2×C3×C4—41no—
424C2×C3×C4—62no—
425C2×C3×C4—62no—
426C2×C3×C4—61no—
427C2×C3×C4—64no—
428C2×C3×C4—64no—
429C2×C3×C4—68no—
430C2×C3×C4—61no—
431C2×C3×C4—61no—
432C2×C3×C4—121no—
433C2×C3×C4—181no—
434C2×C3×C4C2×D3211yescharacteristic
435C2×C3×C4C22×C3×D311yescharacteristic
436C2×C3×C4C22×C3×D311yescharacteristic
437C2×Dic3—21no—
438C2×Dic3—61no—
439C2×Dic3—61no—
440C2×Dic3—121no—
441C2×Dic3C3×#6711yescharacteristic
442D12—24no—
443D12—21no—
444D12C2×C3×Dic312yes—
445C4×D3—24no—
446C4×D3C2×C3×Dic312yes—
447C4×D3C3×#6712yes—
448C3×C8—61no—
449C3×C8—61no—
450C3×C8—121no—
451C3×C8—181no—
452C3⋊C8—21no—
453C3⋊C8—61no—
454C3⋊C8—61no—
455C3⋊C8—121no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—41no—
466C2×C32—41no—
467C2×C32—41no—
468C2×C32—62no—
469C2×C32—122no—
470C2×C32D3×C22⋊C411yescharacteristic
471C2×C32#91611yescharacteristic
472C2×C32#91911yescharacteristic
473C2×C32#95011yescharacteristic
474C2×C32#95011yescharacteristic
475C2×C32C2×#21711yescharacteristic
476C2×C32C2×C3×C22⋊C411yescharacteristic
477C2×C32C3×#13511yescharacteristic
478C2×C32C3×#13511yescharacteristic
479C3×D3—22no—
480C3×D3—22no—
481C3×D3—42no—
482C3×D3—44no—
483C3×D3—42no—
484C3×D3—42no—
485#41—34no—
486#41—62no—
487C2×Q8C3×D3211yescharacteristic
488C2×D4—31no—
489C2×D4—61no—
490C22×C4—31no—
491C22×C4—61no—
492C2×C8—31no—
493C2×C8—91no—
494C2×C8—182no—
495C4⋊C4—31no—
496C4⋊C4—91no—
497C22⋊C4—181no—
498C42—181no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—32no—
503C22×C3—32no—
504C22×C3—41no—
505C22×C3—62no—
506C22×C3—64no—
507C22×C3—61no—
508C22×C3—61no—
509C22×C3—61no—
510C22×C3—61no—
511C22×C3—62no—
512C22×C3—62no—
513C22×C3—121no—
514C22×C3—121no—
515C22×C3—122no—
516C22×C3—122no—
517C22×C3—121no—
518C22×C3—121no—
519C22×C3—121no—
520C22×C3—121no—
521C22×C3D3×#6711yescharacteristic
522C22×C3C3×D3×D411yescharacteristic
523C22×C3C2×C3×#6711yescharacteristic
524C2×D3—21no—
525C2×D3—21no—
526C2×D3—22no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3C3×#21712yes—
530C3×C4—24no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—32no—
541C3×C4—32no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—44no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4—62no—
548C3×C4—121no—
549C3×C4—121no—
550C3×C4—241no—
551C3×C4—361no—
552C3×C4C2×D3×Dic311yescharacteristic
553C3×C4D3×#6711yescharacteristic
554C3×C4C2×C3×C4×D311yescharacteristic
555C3×C4C3×D3×D411yescharacteristic
556C3×C4C22×C3×Dic311yescharacteristic
557C3×C4C2×C3×#6711yescharacteristic
558Dic3—41no—
559Dic3—121no—
560Dic3—121no—
561Dic3—241no—
562Dic3C3×#21712yes—
563C32—21no—
564C32—21no—
565C32—21no—
566C32—41no—
567C32#480411yescharacteristic, complemented
568C32#523311yescharacteristic, complemented
569C32C3×#41611yescharacteristic, complemented
570C23—31no—
571C23—61no—
572Q8—21no—
573Q8C3×D3×Dic312yes—
574D4—32no—
575D4—64no—
576D4—61no—
577C2×C4—21no—
578C2×C4—31no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—64no—
582C2×C4—61no—
583C2×C4—181no—
584C2×C4C2×C3×D3211yescharacteristic
585C8—61no—
586C8—181no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—62no—
600C2×C3—62no—
601C2×C3—122no—
602C2×C3—124no—
603C2×C3—122no—
604C2×C3—122no—
605C2×C3#6394711yescharacteristic
606C2×C3#6394911yescharacteristic
607C2×C3D3×#21711yescharacteristic
608C2×C3C3×D3×C22⋊C411yescharacteristic
609C2×C3C3×#91611yescharacteristic
610C2×C3C3×#91911yescharacteristic
611C2×C3C3×#95011yescharacteristic
612C2×C3C3×#95011yescharacteristic
613C2×C3C2×C3×#21711yescharacteristic
614D3—22no—
615D3—42no—
616C22—32no—
617C22—61no—
618C22—61no—
619C22—62no—
620C22—121no—
621C22—121no—
622C22C3×D3×#6711yescharacteristic
623C4—22no—
624C4—32no—
625C4—121no—
626C4—361no—
627C4C2×C3×D3×Dic311yescharacteristic
628C4C3×D3×#6711yescharacteristic
629C3—21no—
630C3—21no—
631C3—21no—
632C3—41no—
633C3#1059155611yescharacteristic, direct product factor
634C3C3×#480411yescharacteristic, complemented
635C3C3×#523311yescharacteristic, complemented
636C2—62no—
637C2—122no—
638C2C3×#6394711yescharacteristic
639C2C3×#6394911yescharacteristic
640C2C3×D3×#21711yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×#239C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×Q8D32
hypocenterC32C3×#416
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#63841C2semidirect
3. C2×C3×D3×C3⋊C8C2semidirect
4. C3×#63662C2semidirect
5. C32×#835C2semidirect
6. C3×#63563C2semidirect
7. C3×#63562C2semidirect
10. #10591556C3direct (N × H)
13. C32×#239C4semidirect
33. #63841C2×C3semidirect
34. C2×D3×C3⋊C8C2×C3semidirect
36. #63662C2×C3semidirect
41. C3×#835D3semidirect
42. C3×#835C2×C3semidirect
47. #63563C2×C3semidirect
48. #63562C2×C3semidirect
86. C3×#239Dic3semidirect
87. C3×#239C3×C4semidirect
157. #835C3×D3semidirect
274. #239C3×Dic3semidirect
375. C33#416semidirect
567. C32#4804semidirect
568. C32#5233semidirect
569. C32C3×#416semidirect
633. C3#10591556direct (N × H)
634. C3C3×#4804semidirect
635. C3C3×#5233semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C3×#638412
C2×C3×D3×C3⋊C82
C3×#636622
C32×#8352
C3×#635632
C3×#635622
C3×#52333
C3×#48043
#105915563