Finite Group Databasebeta

← C3×#10591570 #13158648

Subgroups — C3×#10591570 #13158648

Order 1728.

720 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3C3×#63898C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#5165—31nomaximal
9C3×#4819—31nomaximal
10#10591570C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C3×#3752—21no—
16C3×#3752—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#215—22no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3714—22no—
24C2×C32×SD16—31no—
25#64156C2×C311yescharacteristic, complemented
26C3×#947—31no—
27C3×#947—31no—
28C3×#947—61no—
29C2×C3×#215—21no—
30C2×C3×#215D311yescharacteristic, complemented
31C2×C3×#215C2×C311yescharacteristic, complemented
32C3×D4×Dic3—31no—
33C2×C3×#213—31no—
34C2×C3×D24—31no—
35C3×#893—31no—
36#63898C2×C311yescharacteristic, complemented
37C2×#3714C2×C311yescharacteristic
38C32×#103—31no—
39C3×#835—21no—
40C3×#835D311yescharacteristic, complemented
41C3×#835C2×C311yescharacteristic, complemented
42C3×#821—31no—
43C3×Dic3⋊C8—31no—
44C3×Dic3⋊C8—31no—
45C3×#809—31no—
46#63572C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C2×C32×Dic3—21no—
59C32×D12—21no—
60C32×D12D412yes—
61C3×D3×Dic3—42no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#5165—31no—
66#4819—31no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C2×C32×D4—31no—
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C22×C3×Dic3—61no—
77C2×C3×D12—21no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C3×C4×D3—61no—
84C2×D3×Dic3—21no—
85C32×SD16—62no—
86C2×C32×C8—31no—
87C2×C32×C8—31no—
88C32×C4⋊C4—31no—
89#3752—21no—
90#3752—21no—
91C4×C3⋊Dic3C22×C311yescharacteristic
92C3×#217—61no—
93C3×#215—24no—
94C3×#215—22no—
95C3×#215—22no—
96C3×#213—62no—
97C3×(C2×D3)⋊C4—61no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—61no—
100C3×(C2×D3)⋊C4—61no—
101C3×(C2×D3)⋊C4—61no—
102C3×(C2×D3)⋊C4—62no—
103C3×(C2×D3)⋊C4—62no—
104C3×C4⋊Dic3—21no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106C3×C4⋊Dic3C22×C311yescharacteristic
107C3×C4×Dic3—31no—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—61no—
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8C2×D311yescharacteristic
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×D24—62no—
117(C2×D3)⋊Dic3—21no—
118#3714—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C32×D3—22no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C33×C4—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3—21no—
131C32×Dic3—41no—
132C2×C3×SD16—31no—
133C2×C3×SD16—31no—
134C2×C3×SD16—61no—
135C2×C3×D8—91no—
136C3×#124—91no—
137C3×C4×D4—91no—
138#947—31no—
139#947—31no—
140#947—61no—
141C2×#215C3×D311yescharacteristic, complemented
142D4×Dic3—31no—
143C2×#213—31no—
144C2×D24—31no—
145#893—31no—
146C3×C4⋊C8—91no—
147C3×#103—31no—
148C3×#103—31no—
149C3×#103—61no—
150C3×#102—91no—
151#835C3×D311yescharacteristic, complemented
152#821—31no—
153Dic3⋊C8—31no—
154Dic3⋊C8—31no—
155#809—31no—
156C23×C32—61no—
157C22×C3⋊D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C22×C3×D3—62no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—22no—
167C32×Q8—22no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×Q8—21no—
171C32×Q8—21no—
172C32×Q8—41no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×Q8—41no—
176C32×Q8—42no—
177C32×Q8—42no—
178C32×D4—32no—
179C32×D4—61no—
180C2×C32×C4—21no—
181C2×C32×C4—22no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—42no—
191C2×C32×C4—61no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C23×C311yescharacteristic
195C2×C3⋊Dic3C3×D411yescharacteristic
196C2×C3⋊Dic3C3×D411yescharacteristic
197C4⋊(C3⋊D3)—21no—
198C4⋊(C3⋊D3)—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—41no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—122no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—22no—
212C3×D12—41no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×D12—121no—
219C3×D12#6712yes—
220C3×D12C3×D412yes—
221C3×C4×D3—122no—
222D3×Dic3—42no—
223C32×C8—61no—
224C32×C8—61no—
225C3×C3⋊C8—22no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—21no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—41no—
231#463—91noSylow
232C2×C33#12111yescharacteristic
233C2×C33#13511yescharacteristic
234C2×C33AGL(1,ℤ/8)11yescharacteristic
235C3×C3⋊D3—81no—
236C32×D3—42no—
237C2×C3×Q8—21no—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—41no—
241C2×C3×Q8D3211yescharacteristic
242C2×C3×Q8C2×C3×D311yescharacteristic
243C2×C3×Q8C2×C3×D311yescharacteristic
244C2×C3×D4—31no—
245C2×C3×D4—31no—
246C2×C3×D4—61no—
247C2×C3×D4—91no—
248C22×C3×C4—181no—
249C22×Dic3—61no—
250C2×D12—31no—
251C2×D12—31no—
252C2×D12—61no—
253C2×D12C2×C3×D311yescharacteristic
254C2×C4×D3—61no—
255C3×SD16—62no—
256C3×SD16—62no—
257C3×SD16—64no—
258C3×D8—182no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—61no—
264C2×C3×C8—61no—
265C3×C4⋊C4—31no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—61no—
268C3×C22⋊C4—181no—
269C3×C22⋊C4—181no—
270C3×C22⋊C4—181no—
271C3×C42—91no—
272#217—61no—
273#215—22no—
274#213—62no—
275(C2×D3)⋊C4—61no—
276(C2×D3)⋊C4—61no—
277(C2×D3)⋊C4—61no—
278(C2×D3)⋊C4—61no—
279(C2×D3)⋊C4—61no—
280(C2×D3)⋊C4—62no—
281(C2×D3)⋊C4—62no—
282C4⋊Dic3C2×C3×D311yescharacteristic
283C4×Dic3—31no—
284C4×Dic3—31no—
285C4×Dic3—61no—
286C2×C3⋊C8C2×C3×D311yescharacteristic
287C2×C3⋊C8C2×C3×D311yescharacteristic
288D24—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—62no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32D3×D411yescharacteristic
297C22×C32D3×D411yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3⋊D3—41no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3×D3—22no—
303C2×C3×D3—22no—
304C2×C3×D3—42no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—81no—
310C2×C3×D3—81no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—41no—
327C32×C4—41no—
328C32×C4—42no—
329C32×C4—42no—
330C32×C4—42no—
331C32×C4—44no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4—121no—
336C32×C4#23511yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4#23711yescharacteristic
339C32×C4C2×#6711yescharacteristic
340C32×C4C2×C3×D411yescharacteristic
341C32×C4C3×#4111yescharacteristic
342C3⋊Dic3—21no—
343C3⋊Dic3—21no—
344C3×Dic3—41no—
345C3×Dic3—41no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—61no—
349C3×Dic3—61no—
350C3×Dic3—81no—
351C3×Dic3—121no—
352C3×Dic3—121no—
353C2×SD16—31no—
354C2×D8—91no—
355#124—91no—
356C4×D4—91no—
357C4⋊C8—91no—
358#103—31no—
359#102—91no—
360C33#46311yescharacteristic, complemented, Sylow
361C23×C3—61no—
362C23×C3—61no—
363C23×C3—121no—
364C23×C3—181no—
365C22×D3—21no—
366C22×D3—62no—
367C22×D3—61no—
368C22×D3—61no—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—21no—
372C3×Q8—21no—
373C3×Q8—21no—
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—21no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×Q8—41no—
381C3×Q8—42no—
382C3×Q8—42no—
383C3×D4—32no—
384C3×D4—32no—
385C3×D4—62no—
386C3×D4—61no—
387C3×D4—61no—
388C3×D4—121no—
389C3×D4—181no—
390C3×D4—181no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—91no—
405C2×C3×C4—91no—
406C2×C3×C4—121no—
407C2×C3×C4—362no—
408C2×C3×C4C2×D3211yescharacteristic
409C2×C3×C4C22×C3×D311yescharacteristic
410C2×C3×C4C22×C3×D311yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—31no—
415C2×Dic3—31no—
416C2×Dic3—61no—
417C2×Dic3—61no—
418C2×Dic3—122no—
419D12—21no—
420D12—61no—
421D12—61no—
422D12—61no—
423D12—61no—
424D12—121no—
425D12—121no—
426D12C3×#6712yes—
427C4×D3—122no—
428C3×C8—62no—
429C3×C8—61no—
430C3×C8—61no—
431C3×C8—61no—
432C3×C8—61no—
433C3×C8—121no—
434C3⋊C8—21no—
435C3⋊C8—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—122no—
449C2×C32#88411yescharacteristic
450C2×C32#91411yescharacteristic
451C2×C32#91411yescharacteristic
452C2×C32#91611yescharacteristic
453C2×C32#91911yescharacteristic
454C2×C32#93911yescharacteristic
455C2×C32C3×#12111yescharacteristic
456C2×C32C3×#13511yescharacteristic
457C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
458C3⋊D3—81no—
459C3×D3—42no—
460C3×D3—42no—
461C3×D3—82no—
462C3×D3—242no—
463C3×D3—241no—
464C3×D3—241no—
465C2×Q8C3×D3211yescharacteristic
466C2×D4—31no—
467C2×D4—91no—
468C22×C4—181no—
469SD16—62no—
470D8—182no—
471C2×C8—31no—
472C2×C8—31no—
473C4⋊C4—31no—
474C22⋊C4—181no—
475C22⋊C4—181no—
476C22⋊C4—181no—
477C42—91no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—41no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—122no—
485C22×C3—121no—
486C22×C3—121no—
487C22×C3—121no—
488C22×C3—121no—
489C22×C3—241no—
490C22×C3—241no—
491C22×C3—361no—
492C22×C3—361no—
493C22×C3—361no—
494C22×C3#381111yescharacteristic
495C22×C3C3×D3×D411yescharacteristic
496C22×C3C3×D3×D411yescharacteristic
497C2×D3—22no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—122no—
501C2×D3—122no—
502C2×D3—122no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—121no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—121no—
525C3×C4—121no—
526C3×C4—181no—
527C3×C4—181no—
528C3×C4—241no—
529C3×C4#380411yescharacteristic
530C3×C4D3×#6711yescharacteristic
531C3×C4C3×#23511yescharacteristic
532C3×C4C3×D3×D411yescharacteristic
533C3×C4C3×#23711yescharacteristic
534C3×C4C2×C3×#6711yescharacteristic
535Dic3—41no—
536Dic3—61no—
537Dic3—61no—
538Dic3—61no—
539Dic3—61no—
540Dic3—121no—
541Dic3—121no—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#481911yescharacteristic, complemented
547C32#516511yescharacteristic, complemented
548C32C3×#46311yescharacteristic, complemented
549C23—61no—
550C23—181no—
551Q8—21no—
552Q8—21no—
553D4—32no—
554D4—61no—
555D4—181no—
556D4—181no—
557C2×C4—21no—
558C2×C4—61no—
559C2×C4—91no—
560C2×C4—91no—
561C2×C4—362no—
562C2×C4C2×C3×D3211yescharacteristic
563C8—61no—
564C8—61no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—122no—
578C2×C3—122no—
579C2×C3—242no—
580C2×C3—721no—
581C2×C3#6394111yescharacteristic
582C2×C3#6395011yescharacteristic
583C2×C3#6397111yescharacteristic
584C2×C3C3×#88411yescharacteristic
585C2×C3C3×#91411yescharacteristic
586C2×C3C3×#91411yescharacteristic
587C2×C3C3×#91611yescharacteristic
588C2×C3C3×#91911yescharacteristic
589C2×C3C3×#93911yescharacteristic
590D3—42no—
591D3—242no—
592D3—241no—
593D3—241no—
594C22—62no—
595C22—121no—
596C22—121no—
597C22—361no—
598C22—361no—
599C22—361no—
600C22C3×#381111yescharacteristic
601C4—41no—
602C4—121no—
603C4—181no—
604C4—181no—
605C4C3×#380411yescharacteristic
606C4C3×D3×#6711yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059157011yescharacteristic, direct product factor
612C3C3×#481911yescharacteristic, complemented
613C3C3×#516511yescharacteristic, complemented
614C2—122no—
615C2—721no—
616C2C3×#6394111yescharacteristic
617C2C3×#6395011yescharacteristic
618C2C3×#6397111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypercenterC2×C3×Q8D32
hypocenterC32C3×#463
LCSG ▷ C2×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
1. C3×#64156C2semidirect
2. C2×C32×#215C2semidirect
3. C3×#63898C2semidirect
5. C32×#835C2semidirect
6. C3×#63572C2semidirect
7. C3×#63564C2semidirect
10. #10591570C3direct (N × H)
25. #64156C2×C3semidirect
30. C2×C3×#215D3semidirect
31. C2×C3×#215C2×C3semidirect
36. #63898C2×C3semidirect
40. C3×#835D3semidirect
41. C3×#835C2×C3semidirect
46. #63572C2×C3semidirect
47. #63564C2×C3semidirect
141. C2×#215C3×D3semidirect
151. #835C3×D3semidirect
360. C33#463semidirect
546. C32#4819semidirect
547. C32#5165semidirect
548. C32C3×#463semidirect
611. C3#10591570direct (N × H)
612. C3C3×#4819semidirect
613. C3C3×#5165semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C32×#2152
C3×#638982
C2×C3×#37142
C32×#8352
C3×#635722
C3×#635642
C3×#51653
C3×#48193
#105915703