Finite Group Databasebeta

← C3×#10590600 #13157617

Subgroups — C3×#10590600 #13157617

Order 1728.

786 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C211yesmaximal, characteristic
2C32×#892C211yesmaximal, characteristic
3C2×C3×#3711C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#5004—31nomaximal
9C3×#4802—31nomaximal
10#10590600C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#627—21no—
15C2×C3×D3×Dic3—21no—
16C3×D3×Dic6—22no—
17C3×D3×Dic6C414yescomplemented
18C33×C4⋊C4C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×(C2×D3)⋊C4—21no—
21C32×C4⋊Dic3C2211yescharacteristic
22C32×C4×Dic3—21no—
23C2×C32×C3⋊C8C2211yescharacteristic
24C3×#3711—22no—
25C2×C3×D3×Q8—31no—
26C22×C3×Dic6—31no—
27C2×D3×Dic6C2×C311yescharacteristic
28C32×#117—31no—
29C2×C3×#208—31no—
30C2×C3×ΓL(1,𝔽25)—31no—
31C3×#892—21no—
32C3×#892D311yescharacteristic
33C3×#892C2×C311yescharacteristic
34C2×#3711C2×C311yescharacteristic
35C32×#103—31no—
36#63637C2×C311yescharacteristic, complemented
37C3×#835—31no—
38C3×#816—31no—
39C3×#810—21no—
40C3×#810—31no—
41C3×#810—31no—
42C3×#810—31no—
43C3×#810—31no—
44C3×#810—61no—
45C3×#810D311yescharacteristic, complemented
46C3×#810C2×C311yescharacteristic, complemented
47#63567C2×C311yescharacteristic, complemented
48#63565C2×C311yescharacteristic, complemented
49C22×C32×D3D411yescharacteristic
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634C2×C412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×C4×D3C2×C412yes—
59C32×C4×D3D412yes—
60C32×Dic6—21no—
61C32×Dic6C2×C412yes—
62C3×#627—24no—
63C3×D3×Dic3—24no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66C3×#419—91no—
67#5004—31no—
68#4802—31no—
69C2×C32×Q8—31no—
70C22×C32×C4—31no—
71C2×#634C22×C311yescharacteristic
72C22×C3×Dic3—61no—
73C3×D3×Q8—34no—
74C3×D3×Q8—62no—
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3C2×D311yescharacteristic
78C2×C3×C4×D3C22×C311yescharacteristic
79C2×C3×Dic6—21no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—34no—
83C2×C3×Dic6—62no—
84C2×C3×Dic6—61no—
85C2×C3×Dic6—61no—
86C2×C3×Dic6—61no—
87C2×C3×Dic6C2×D311yescharacteristic
88C2×C3×Dic6C22×C311yescharacteristic
89C2×#627—21no—
90C2×D3×Dic3—21no—
91D3×Dic6—22no—
92D3×Dic6C3×C414yescomplemented
93C2×C32×C8—31no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4—21no—
96C32×C4⋊C4—21no—
97C32×C4⋊C4—31no—
98C32×C4⋊C4—41no—
99C32×C4⋊C4C2×D311yescharacteristic
100C32×C4⋊C4C2×D311yescharacteristic
101C32×C4⋊C4C22×C311yescharacteristic
102C32×C22⋊C4—61no—
103C32×C42—61no—
104C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
105C3×(C2×D3)⋊C4—22no—
106C3×(C2×D3)⋊C4—21no—
107C3×(C2×D3)⋊C4—21no—
108C3×C4⋊Dic3—21no—
109C3×C4⋊Dic3C2×D311yescharacteristic
110C3×C4⋊Dic3C22×C311yescharacteristic
111C3×C4×Dic3—22no—
112C3×C4×Dic3—21no—
113C3×C4×Dic3—21no—
114C3×#208—62no—
115C2×C3×C3⋊C8—21no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—61no—
119C2×C3×C3⋊C8C2×D311yescharacteristic
120C2×C3×C3⋊C8C22×C311yescharacteristic
121C3×ΓL(1,𝔽25)—62no—
122#3711—22no—
123C22×C33C2×D411yescharacteristic
124C2×C32×D3—21no—
125C2×C32×D3—21no—
126C2×C32×D3C22⋊C412yes—
127C33×C4—41no—
128C33×C4C22×C411yescharacteristic
129C33×C4C2×D411yescharacteristic
130C3×C3⋊Dic3—22no—
131C32×Dic3—22no—
132C32×Dic3—41no—
133C32×Dic3C22⋊C412yes—
134C22×C3×Q8—91no—
135C2×D3×Q8—31no—
136C22×Dic6—31no—
137C2×C3×M16—91no—
138C3×#117—31no—
139C3×#117—31no—
140C3×#117—61no—
141C2×#208—31no—
142C2×ΓL(1,𝔽25)—31no—
143#892C3×D311yescharacteristic
144C3×#103—31no—
145C3×#103—31no—
146C3×#103—61no—
147C3×#103—91no—
148C3×#103—91no—
149C3×#103—91no—
150#835—31no—
151#816—31no—
152#810—31no—
153#810—31no—
154#810—31no—
155#810—31no—
156#810—61no—
157#810C3×D311yescharacteristic, complemented
158C23×C32—31no—
159C22×C3×D3—21no—
160C22×C3×D3#6711yescharacteristic
161C22×C3×D3C3×D411yescharacteristic
162C32×Q8—32no—
163C32×Q8—61no—
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—31no—
172C2×C32×C4—32no—
173C2×C32×C4—32no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—42no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C311yescharacteristic
183C2×C3⋊Dic3—21no—
184#634—21no—
185#634C2×C3×C412yes—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—121no—
197C2×C3×Dic3#6711yescharacteristic
198C2×C3×Dic3C3×D411yescharacteristic
199C3×C4×D3—22no—
200C3×C4×D3—22no—
201C3×C4×D3—64no—
202C3×C4×D3C4×D312yes—
203C3×C4×D3D1212yes—
204C3×C4×D3C2×C3×C412yes—
205C3×C4×D3C3×D412yes—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—22no—
209C3×Dic6—32no—
210C3×Dic6—32no—
211C3×Dic6—41no—
212C3×Dic6—62no—
213C3×Dic6—64no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—64no—
217C3×Dic6—121no—
218C3×Dic6C4×D312yes—
219C3×Dic6C2×C3×C412yes—
220#627—24no—
221D3×Dic3—24no—
222C32×C8—61no—
223C3⋊(C3⋊C8)—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—41no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—121no—
230#419—91noSylow
231C2×C33C2×C22⋊C411yescharacteristic
232C2×C33#13711yescharacteristic
233C2×C33#13711yescharacteristic
234C32×D3—22no—
235C2×C3×Q8—31no—
236C2×C3×Q8—31no—
237C2×C3×Q8—61no—
238C2×C3×Q8—94no—
239C2×C3×Q8—91no—
240C2×C3×Q8—181no—
241C2×C3×Q8—182no—
242C22×C3×C4—31no—
243C22×C3×C4—31no—
244C22×C3×C4—61no—
245C22×C3×C4—181no—
246C22×Dic3—61no—
247D3×Q8—34no—
248D3×Q8—62no—
249C2×C4×D3—61no—
250C2×C4×D3C2×C3×D311yescharacteristic
251C2×Dic6—31no—
252C2×Dic6—31no—
253C2×Dic6—34no—
254C2×Dic6—61no—
255C2×Dic6—62no—
256C2×Dic6—61no—
257C2×Dic6—61no—
258C2×Dic6C2×C3×D311yescharacteristic
259C3×M16—182no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—61no—
263C2×C3×C8—91no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—41no—
270C3×C4⋊C4—61no—
271C3×C4⋊C4D3211yescharacteristic
272C3×C4⋊C4C2×C3×D311yescharacteristic
273C3×C4⋊C4C2×C3×D311yescharacteristic
274C3×C22⋊C4—61no—
275C3×C22⋊C4—61no—
276C3×C22⋊C4—62no—
277C3×C42—61no—
278C3×C42—61no—
279C3×C42—62no—
280(C2×D3)⋊C4—21no—
281C4⋊Dic3C2×C3×D311yescharacteristic
282C4×Dic3—21no—
283#208—62no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8C2×C3×D311yescharacteristic
288ΓL(1,𝔽25)—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—32no—
293C22×C32—41no—
294C22×C32—61no—
295C22×C32—61no—
296C22×C32D3×D411yescharacteristic
297C22×C32C2×#6711yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
302C2×C3×D3—21no—
303C2×C3×D3—22no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3(C2×D3)⋊C412yes—
307C2×C3×D3C3×C22⋊C412yes—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—32no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4—42no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—62no—
322C32×C4—81no—
323C32×C4—81no—
324C32×C4—82no—
325C32×C4—121no—
326C32×C4C2×C4×D311yescharacteristic
327C32×C4C2×C4×D311yescharacteristic
328C32×C4C2×D1211yescharacteristic
329C32×C4D3×D411yescharacteristic
330C32×C4C22×C3×C411yescharacteristic
331C32×C4C2×C3×D411yescharacteristic
332C3⋊Dic3—22no—
333C3×Dic3—22no—
334C3×Dic3—22no—
335C3×Dic3—22no—
336C3×Dic3—42no—
337C3×Dic3—42no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C3×Dic3—122no—
343C3×Dic3(C2×D3)⋊C412yes—
344C3×Dic3C3×C22⋊C412yes—
345C22×Q8—91no—
346C2×M16—91no—
347#117—31no—
348#103—31no—
349#103—91no—
350#103—91no—
351#103—91no—
352C33#41911yescharacteristic, complemented, Sylow
353C23×C3—31no—
354C23×C3—31no—
355C23×C3—61no—
356C22×D3C3×#6711yescharacteristic
357C3×Q8—32no—
358C3×Q8—32no—
359C3×Q8—62no—
360C3×Q8—61no—
361C3×Q8—61no—
362C3×Q8—92no—
363C3×Q8—121no—
364C3×Q8—184no—
365C3×Q8—181no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—32no—
376C2×C3×C4—32no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—62no—
384C2×C3×C4—62no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—62no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—121no—
392C2×C3×C4—184no—
393C2×C3×C4—181no—
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C22×C3×D311yescharacteristic
396C2×C3×C4C22×C3×D311yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—64no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402C2×Dic3—121no—
403C2×Dic3C3×#6711yescharacteristic
404C4×D3—64no—
405C4×D3C3×C4×D312yes—
406C4×D3C3×D1212yes—
407Dic6—21no—
408Dic6—32no—
409Dic6—32no—
410Dic6—62no—
411Dic6—64no—
412Dic6—61no—
413Dic6—61no—
414Dic6—64no—
415Dic6—121no—
416Dic6C3×C4×D312yes—
417C3×C8—61no—
418C3×C8—61no—
419C3×C8—121no—
420C3×C8—181no—
421C3⋊C8—21no—
422C3⋊C8—61no—
423C3⋊C8—61no—
424C3⋊C8—121no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—62no—
438C2×C32D3×C22⋊C411yescharacteristic
439C2×C32#91511yescharacteristic
440C2×C32#91811yescharacteristic
441C2×C32C2×(C2×D3)⋊C411yescharacteristic
442C2×C32#95111yescharacteristic
443C2×C32#95111yescharacteristic
444C2×C32C2×C3×C22⋊C411yescharacteristic
445C2×C32C3×#13711yescharacteristic
446C2×C32C3×#13711yescharacteristic
447C3×D3—22no—
448C3×D3—22no—
449C3×D3—42no—
450C2×Q8—31no—
451C2×Q8—94no—
452C2×Q8—91no—
453C2×Q8—181no—
454C2×Q8—182no—
455C22×C4—31no—
456C22×C4—181no—
457M16—182no—
458C2×C8—31no—
459C2×C8—91no—
460C4⋊C4—31no—
461C4⋊C4C3×D3211yescharacteristic
462C22⋊C4—61no—
463C42—61no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—32no—
468C22×C3—32no—
469C22×C3—41no—
470C22×C3—62no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3D3×#6711yescharacteristic
478C22×C3C3×D3×D411yescharacteristic
479C22×C3C2×C3×#6711yescharacteristic
480C2×D3—21no—
481C2×D3—21no—
482C2×D3C3×(C2×D3)⋊C412yes—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—32no—
490C3×C4—32no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—62no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—82no—
503C3×C4—122no—
504C3×C4—122no—
505C3×C4—121no—
506C3×C4—121no—
507C3×C4—182no—
508C3×C4C4×D3211yescharacteristic
509C3×C4D3×D1211yescharacteristic
510C3×C4C2×C3×C4×D311yescharacteristic
511C3×C4C2×C3×C4×D311yescharacteristic
512C3×C4C2×C3×D1211yescharacteristic
513C3×C4C3×D3×D411yescharacteristic
514Dic3—22no—
515Dic3—41no—
516Dic3—62no—
517Dic3—62no—
518Dic3—122no—
519Dic3C3×(C2×D3)⋊C412yes—
520C32—21no—
521C32—21no—
522C32—21no—
523C32—41no—
524C32#480211yescharacteristic, complemented
525C32#500411yescharacteristic, complemented
526C32C3×#41911yescharacteristic, complemented
527C23—31no—
528Q8—32no—
529Q8—61no—
530Q8—92no—
531Q8—184no—
532Q8—181no—
533C2×C4—21no—
534C2×C4—31no—
535C2×C4—32no—
536C2×C4—32no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—184no—
540C2×C4—181no—
541C2×C4C2×C3×D3211yescharacteristic
542C8—61no—
543C8—181no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—41no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—62no—
557C2×C3—62no—
558C2×C3—122no—
559C2×C3D3×(C2×D3)⋊C411yescharacteristic
560C2×C3#6392911yescharacteristic
561C2×C3#6394311yescharacteristic
562C2×C3C3×D3×C22⋊C411yescharacteristic
563C2×C3C3×#91511yescharacteristic
564C2×C3C3×#91811yescharacteristic
565C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
566C2×C3C3×#95111yescharacteristic
567C2×C3C3×#95111yescharacteristic
568D3—22no—
569C22—32no—
570C22—61no—
571C22—61no—
572C22C3×D3×#6711yescharacteristic
573C4—32no—
574C4—41no—
575C4—62no—
576C4—121no—
577C4—182no—
578C4C3×C4×D3211yescharacteristic
579C4C3×D3×D1211yescharacteristic
580C3—21no—
581C3—21no—
582C3—21no—
583C3—41no—
584C3#1059060011yescharacteristic, direct product factor
585C3C3×#480211yescharacteristic, complemented
586C3C3×#500411yescharacteristic, complemented
587C2—62no—
588C2C3×D3×(C2×D3)⋊C411yescharacteristic
589C2C3×#6392911yescharacteristic
590C2C3×#6394311yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#419
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
4. C3×#63637C2semidirect
5. C32×#810C2semidirect
6. C3×#63567C2semidirect
7. C3×#63565C2semidirect
10. #10590600C3direct (N × H)
17. C3×D3×Dic6C4semidirect
36. #63637C2×C3semidirect
45. C3×#810D3semidirect
46. C3×#810C2×C3semidirect
47. #63567C2×C3semidirect
48. #63565C2×C3semidirect
92. D3×Dic6C3×C4semidirect
157. #810C3×D3semidirect
352. C33#419semidirect
524. C32#4802semidirect
525. C32#5004semidirect
526. C32C3×#419semidirect
584. C3#10590600direct (N × H)
585. C3C3×#4802semidirect
586. C3C3×#5004semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C32×#8922
C2×C3×#37112
C3×#636372
C32×#8102
C3×#635672
C3×#635652
C3×#50043
C3×#48023
#105906003