Finite Group Databasebeta

← C3×#10590830 #13157796

Subgroups — C3×#10590830 #13157796

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3744C211yesmaximal, characteristic, complemented
2C2×C32×D3×C8C211yesmaximal, characteristic, complemented
3C3×#63909C211yesmaximal, characteristic
4C3×#63857C211yesmaximal, characteristic, complemented
5C32×#817C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#5114—31nomaximal
9C3×#4864—31nomaximal
10#10590830C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C33×C8C2211yescharacteristic
16C3×#3744—22no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C32×D3×C8—22no—
21C3×Dic3⋊Dic3—21no—
22C3×#3722—21no—
23C3×(C2×D3)⋊Dic3—21no—
24C22×C32×C8—31no—
25C2×#3744C2×C311yescharacteristic, complemented
26C3×#930—31no—
27C3×#924—31no—
28C2×C3×#204—31no—
29C2×C3×#204—31no—
30C2×C3×#204—61no—
31C2×C3×D3×C8—21no—
32C2×C3×D3×C8D311yescharacteristic, complemented
33C2×C3×D3×C8C2×C311yescharacteristic, complemented
34C3×#897—31no—
35C3×#895—31no—
36#63909C2×C311yescharacteristic
37#63857C2×C311yescharacteristic, complemented
38C32×#106—31no—
39C3×#821—31no—
40C3×#817—21no—
41C3×#817D311yescharacteristic, complemented
42C3×#817C2×C311yescharacteristic, complemented
43C3×#816—31no—
44C3×#810—31no—
45C3×#809—31no—
46#63567C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3D411yescharacteristic
59C32×C4×D3—21no—
60C32×C4×D3—21no—
61C3×#626—42no—
62C33×C8D412yescomplemented
63C32×C3⋊C8—21no—
64C3×#465—91no—
65#5114—31no—
66#4864—31no—
67C22×C32×C4—31no—
68C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
69C2×#634C22×C311yescharacteristic
70C2×C3×#67—61no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×C3×C4×D3C22×C311yescharacteristic
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×#626—21no—
82C2×C32×C8—21no—
83C2×C32×C8—21no—
84C2×C32×C8—21no—
85C2×C32×C8—31no—
86C2×C32×C8—41no—
87C2×C32×C8—62no—
88C2×C32×C8C2×D311yescharacteristic
89C2×C32×C8C2×D311yescharacteristic
90C2×C32×C8C22×C311yescharacteristic
91C32×C4⋊C4—31no—
92C32×C4⋊C4—31no—
93#3744—22no—
94C3×#217—61no—
95C3×(C2×D3)⋊C4—61no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—21no—
100C3×C4⋊Dic3—61no—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4⋊Dic3C22×C311yescharacteristic
104C3×C4⋊Dic3C22×C311yescharacteristic
105C3×Dic3⋊C4—61no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C3×#204—64no—
110C3×#204—62no—
111C3×#204—62no—
112C3×D3×C8—24no—
113C3×D3×C8—22no—
114C3×D3×C8—22no—
115Dic3⋊Dic3—21no—
116#3722—21no—
117(C2×D3)⋊Dic3—21no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C32×D3—21no—
123C2×C32×D3—21no—
124C2×C32×D3SD1612yescomplemented
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—41no—
128C32×Dic3—21no—
129C32×Dic3—41no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C22×C3×C8—31no—
133C22×C3×C8—31no—
134C22×C3×C8—61no—
135C3×C22⋊Q8—91no—
136C3×#121—91no—
137#930—31no—
138#924—31no—
139C2×#204—31no—
140C2×#204—31no—
141C2×#204—61no—
142C2×D3×C8C3×D311yescharacteristic, complemented
143#897—31no—
144#895—31no—
145C3×#106—31no—
146C3×#106—31no—
147C3×#106—61no—
148C3×#103—91no—
149C3×#102—91no—
150#821—31no—
151#817C3×D311yescharacteristic, complemented
152#816—31no—
153#810—31no—
154#809—31no—
155C23×C32—31no—
156C22×C3⋊D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C22×C3×D3D1211yescharacteristic
162C22×C3×D3C3×D411yescharacteristic
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—31no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C2×C3⋊Dic3—21no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178#634—21no—
179#634—21no—
180C3×#67—122no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—41no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3D1211yescharacteristic
192C2×C3×Dic3C3×D411yescharacteristic
193C3×D12—62no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—121no—
199C3×D12—122no—
200C3×C4×D3—22no—
201C3×C4×D3—22no—
202C3×C4×D3—21no—
203C3×C4×D3—21no—
204C3×C4×D3—21no—
205C3×C4×D3—21no—
206C3×Dic6—62no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—121no—
212#626—42no—
213C32×C8—22no—
214C32×C8—22no—
215C32×C8—22no—
216C32×C8—42no—
217C32×C8—61no—
218C32×C8D1212yescomplemented
219C32×C8#6712yescomplemented
220C32×C8C3×D412yescomplemented
221C3×C3⋊C8—22no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224#465—91noSylow
225C2×C33#12111yescharacteristic
226C2×C33C2×SD1611yescharacteristic
227C2×C33#13511yescharacteristic
228C3×C3⋊D3—81no—
229C32×D3—22no—
230C2×C3×Q8—91no—
231C2×C3×D4—91no—
232C2×C3×D4—181no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—61no—
236C2×#67—61no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—61no—
240C2×D12—61no—
241C2×C4×D3C2×C3×D311yescharacteristic
242C2×Dic6—31no—
243C2×Dic6—31no—
244C2×Dic6—61no—
245C3×SD16—182no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—41no—
252C2×C3×C8—61no—
253C2×C3×C8—64no—
254C2×C3×C8—62no—
255C2×C3×C8—62no—
256C2×C3×C8D3211yescharacteristic
257C2×C3×C8C2×C3×D311yescharacteristic
258C2×C3×C8C2×C3×D311yescharacteristic
259C3×C4⋊C4—31no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—181no—
266C3×C22⋊C4—181no—
267C3×C22⋊C4—181no—
268#217—61no—
269(C2×D3)⋊C4—61no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—61no—
272C4⋊Dic3—61no—
273C4⋊Dic3C2×C3×D311yescharacteristic
274C4⋊Dic3C2×C3×D311yescharacteristic
275Dic3⋊C4—61no—
276C2×C3⋊C8C2×C3×D311yescharacteristic
277#204—64no—
278#204—62no—
279#204—62no—
280D3×C8—22no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—32no—
285C22×C32—41no—
286C22×C32—61no—
287C22×C32—61no—
288C22×C32C2×D1211yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32C2×C3×D411yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3×D3—22no—
295C2×C3×D3—22no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—21no—
300C2×C3×D3—22no—
301C2×C3×D3—122no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—241no—
309C2×C3×D3—241no—
310C2×C3×D3#20412yescomplemented
311C2×C3×D3C3×SD1612yescomplemented
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—61no—
321C32×C4—121no—
322C32×C4—121no—
323C32×C4C2×D1211yescharacteristic
324C32×C4#23511yescharacteristic
325C32×C4#23911yescharacteristic
326C32×C4C2×#6711yescharacteristic
327C32×C4C2×C3×D411yescharacteristic
328C32×C4C3×#4111yescharacteristic
329C3⋊Dic3—41no—
330C3×Dic3—22no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—41no—
336C3×Dic3—41no—
337C3×Dic3—81no—
338C3×Dic3—81no—
339C3×Dic3—122no—
340C3×Dic3—121no—
341C3×Dic3—121no—
342C2×SD16—91no—
343C22×C8—31no—
344C22⋊Q8—91no—
345#121—91no—
346#106—31no—
347#103—91no—
348#102—91no—
349C33#46511yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—61no—
353C23×C3—181no—
354C22×D3—61no—
355C22×D3—61no—
356C22×D3—121no—
357C22×D3C3×D1211yescharacteristic
358C3×Q8—181no—
359C3×Q8—181no—
360C3×D4—181no—
361C3×D4—181no—
362C3×D4—362no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—41no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—121no—
381C2×C3×C4—121no—
382C2×C3×C4—181no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C22×C3×D311yescharacteristic
385C2×C3×C4C22×C3×D311yescharacteristic
386#67—122no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—121no—
392C2×Dic3C3×D1211yescharacteristic
393D12—62no—
394D12—61no—
395D12—61no—
396D12—61no—
397D12—61no—
398D12—121no—
399D12—122no—
400C4×D3—21no—
401C4×D3—21no—
402Dic6—62no—
403Dic6—61no—
404Dic6—61no—
405Dic6—61no—
406Dic6—61no—
407Dic6—121no—
408C3×C8—22no—
409C3×C8—22no—
410C3×C8—22no—
411C3×C8—42no—
412C3×C8—62no—
413C3×C8—61no—
414C3×C8—61no—
415C3×C8#62612yescomplemented
416C3×C8C3×D1212yescomplemented
417C3×C8C3×#6712yescomplemented
418C3⋊C8—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—62no—
432C2×C32#89511yescharacteristic
433C2×C32C2×#20411yescharacteristic
434C2×C32#90411yescharacteristic
435C2×C32D3×SD1611yescharacteristic
436C2×C32#91611yescharacteristic
437C2×C32#93011yescharacteristic
438C2×C32C3×#12111yescharacteristic
439C2×C32C2×C3×SD1611yescharacteristic
440C2×C32C3×#13511yescharacteristic
441C3⋊D3—81no—
442C3×D3—24no—
443C3×D3—22no—
444C3×D3—22no—
445C3×D3—242no—
446C3×D3—241no—
447C3×D3—241no—
448C2×Q8—91no—
449C2×D4—91no—
450C2×D4—181no—
451C22×C4—31no—
452SD16—182no—
453C2×C8—31no—
454C2×C8—62no—
455C2×C8C3×D3211yescharacteristic
456C4⋊C4—31no—
457C4⋊C4—31no—
458C4⋊C4—181no—
459C22⋊C4—181no—
460C22⋊C4—181no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—32no—
465C22×C3—32no—
466C22×C3—41no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—361no—
475C22×C3—361no—
476C22×C3—361no—
477C22×C3D3×D1211yescharacteristic
478C22×C3C2×C3×D1211yescharacteristic
479C22×C3C3×D3×D411yescharacteristic
480C2×D3—21no—
481C2×D3—21no—
482C2×D3—122no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—121no—
489C2×D3—241no—
490C2×D3—241no—
491C2×D3C3×#20412yescomplemented
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—62no—
501C3×C4—61no—
502C3×C4—61no—
503C3×C4—121no—
504C3×C4—121no—
505C3×C4—121no—
506C3×C4—121no—
507C3×C4—241no—
508C3×C4—241no—
509C3×C4—361no—
510C3×C4#379911yescharacteristic
511C3×C4C2×#62611yescharacteristic
512C3×C4C2×C3×D1211yescharacteristic
513C3×C4C3×#23511yescharacteristic
514C3×C4C3×#23911yescharacteristic
515C3×C4C2×C3×#6711yescharacteristic
516Dic3—21no—
517Dic3—41no—
518Dic3—41no—
519Dic3—122no—
520Dic3—121no—
521Dic3—121no—
522C32—21no—
523C32—21no—
524C32—21no—
525C32—41no—
526C32#486411yescharacteristic, complemented
527C32#511411yescharacteristic, complemented
528C32C3×#46511yescharacteristic, complemented
529C23—31no—
530C23—181no—
531Q8—181no—
532Q8—181no—
533D4—181no—
534D4—181no—
535D4—362no—
536C2×C4—31no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—181no—
542C2×C4C2×C3×D3211yescharacteristic
543C8—61no—
544C8C3×#62612yescomplemented
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—21no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—41no—
557C2×C3—64no—
558C2×C3—62no—
559C2×C3—62no—
560C2×C3—721no—
561C2×C3D3×#20411yescharacteristic
562C2×C3#6380611yescharacteristic
563C2×C3#6390911yescharacteristic
564C2×C3C3×#89511yescharacteristic
565C2×C3C2×C3×#20411yescharacteristic
566C2×C3C3×#90411yescharacteristic
567C2×C3C3×D3×SD1611yescharacteristic
568C2×C3C3×#91611yescharacteristic
569C2×C3C3×#93011yescharacteristic
570D3—22no—
571D3—242no—
572D3—241no—
573D3—241no—
574C22—32no—
575C22—61no—
576C22—61no—
577C22—361no—
578C22—361no—
579C22—361no—
580C22C3×D3×D1211yescharacteristic
581C4—61no—
582C4—121no—
583C4—121no—
584C4—361no—
585C4C3×#379911yescharacteristic
586C4C2×C3×#62611yescharacteristic
587C3—21no—
588C3—21no—
589C3—21no—
590C3—41no—
591C3#1059083011yescharacteristic, direct product factor
592C3C3×#486411yescharacteristic, complemented
593C3C3×#511411yescharacteristic, complemented
594C2—62no—
595C2—721no—
596C2C3×D3×#20411yescharacteristic
597C2C3×#6380611yescharacteristic
598C2C3×#6390911yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC3×#63909C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3744C2semidirect
2. C2×C32×D3×C8C2semidirect
4. C3×#63857C2semidirect
5. C32×#817C2semidirect
6. C3×#63567C2semidirect
7. C3×#63564C2semidirect
10. #10590830C3direct (N × H)
25. C2×#3744C2×C3semidirect
32. C2×C3×D3×C8D3semidirect
33. C2×C3×D3×C8C2×C3semidirect
37. #63857C2×C3semidirect
41. C3×#817D3semidirect
42. C3×#817C2×C3semidirect
46. #63567C2×C3semidirect
47. #63564C2×C3semidirect
62. C33×C8D4semidirect
124. C2×C32×D3SD16semidirect
142. C2×D3×C8C3×D3semidirect
151. #817C3×D3semidirect
218. C32×C8D12semidirect
219. C32×C8#67semidirect
220. C32×C8C3×D4semidirect
310. C2×C3×D3#204semidirect
311. C2×C3×D3C3×SD16semidirect
349. C33#465semidirect
415. C3×C8#626semidirect
416. C3×C8C3×D12semidirect
417. C3×C8C3×#67semidirect
491. C2×D3C3×#204semidirect
526. C32#4864semidirect
527. C32#5114semidirect
528. C32C3×#465semidirect
544. C8C3×#626semidirect
591. C3#10590830direct (N × H)
592. C3C3×#4864semidirect
593. C3C3×#5114semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
C2×C32×D3×C82
C3×#639092
C3×#638572
C32×#8172
C3×#635672
C3×#635642
C3×#51143
C3×#48643
#105908303