Finite Group Databasebeta

← #13161811

Subgroups — #13161811

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4#11722076C212yesmaximal, complemented
5#11722075C212yesmaximal, complemented
6#10591106—31nomaximal
7#10591088—31nomaximal
8#10591087—31nomaximal
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638—21no—
11C22×C3×C3⋊Dic3—21no—
12C3×#3828—22no—
13C3×#3828C414yescomplemented
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×#634C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)—21no—
18C3×C4⋊(C3⋊Dic3)C2212yes—
19C3×#3748—22no—
20C2×C3×C3⋊(C3⋊C8)C2212yes—
21C2×#3828D311yescharacteristic
22C2×C3×#235—31no—
23C2×C3×#235—31no—
24C2×C3×#235—61no—
25C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
26C2×#3748D311yescharacteristic, complemented
27C2×C3×C4⋊Dic3—31no—
28C2×C3×C4⋊Dic3—31no—
29C2×C3×C4⋊Dic3—61no—
30C2×C3×#208—31no—
31C2×C3×#208—31no—
32C2×C3×#208—61no—
33#63567—32no—
34#63566—32no—
35#63565—32no—
36#63564—32no—
37#63563—32no—
38#63562—32no—
39C23×C33D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C2×C33×C4C2×C412yes—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4D412yes—
44C2×C33×C4C2311yescharacteristic
45C3×#638—24no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3—42no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)C2×C412yescomplemented
51C3×C4×C3⋊D3—24no—
52C3×#634—21no—
53C3×#634C2×C412yescomplemented
54C3×C3⋊(C3⋊C8)—22no—
55#5232—91no—
56#5133—91no—
57#4966—91no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—41no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C2×#638—21no—
66C22×C3⋊Dic3—21no—
67#3828—22no—
68#3828Dic314yescomplemented
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3—21no—
71C2×#634C2×D311yescharacteristic
72C2×C3×#67—62no—
73C2×C3×#67—61no—
74C2×C3×#67—61no—
75C22×C3×Dic3—62no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3—61no—
78C3×#235—34no—
79C3×#235—34no—
80C3×#235—64no—
81C3×#235—64no—
82C3×#235—62no—
83C3×#235—62no—
84C2×C3×D12—31no—
85C2×C3×D12—31no—
86C2×C3×D12—61no—
87C2×C3×C4×D3—62no—
88C2×C3×C4×D3—61no—
89C2×C3×C4×D3—61no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—61no—
93C4⋊(C3⋊Dic3)—21no—
94C4⋊(C3⋊Dic3)C2×D312yes—
95#3748—22no—
96C2×C3⋊(C3⋊C8)C2×D312yes—
97C3×C4⋊Dic3—32no—
98C3×C4⋊Dic3—32no—
99C3×C4⋊Dic3—62no—
100C3×C4⋊Dic3—62no—
101C3×C4⋊Dic3—61no—
102C3×C4⋊Dic3—61no—
103C3×#208—64no—
104C3×#208—62no—
105C3×#208—62no—
106C2×C3×C3⋊C8—32no—
107C2×C3×C3⋊C8—32no—
108C2×C3×C3⋊C8—62no—
109C22×C33—21no—
110C22×C33—21no—
111C22×C33C22⋊C412yes—
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C33×C4C22⋊C412yes—
117C33×C4C22×C411yescharacteristic
118C33×C4C2×D411yescharacteristic
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—42no—
121C2×C3×#41—91no—
122C2×#235—31no—
123C2×#235—31no—
124C2×#235—61no—
125C2×C3×M16—91no—
126C2×C3×C4⋊C4—91no—
127C2×C4⋊Dic3—31no—
128C2×C4⋊Dic3—31no—
129C2×C4⋊Dic3—61no—
130C2×#208—31no—
131C2×#208—31no—
132C2×#208—61no—
133#835—92no—
134#832—92no—
135#821—92no—
136#816—92no—
137#810—92no—
138#809—92no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—21no—
142C23×C32—41no—
143C23×C32D1211yescharacteristic
144C23×C32#6711yescharacteristic
145C23×C32#6711yescharacteristic
146C22×C3⋊D3—21no—
147C22×C3×D3—62no—
148C22×C3×D3—61no—
149C22×C3×D3—61no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—22no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—42no—
163C2×C32×C4—42no—
164C2×C32×C4—41no—
165C2×C32×C4—41no—
166C2×C32×C4C4×D312yes—
167C2×C32×C4C4×D312yes—
168C2×C32×C4D1211yescharacteristic
169C2×C32×C4D1212yes—
170C2×C32×C4C2×Dic312yes—
171C2×C32×C4#6711yescharacteristic
172C2×C32×C4#6712yes—
173C2×C32×C4#6711yescharacteristic
174C2×C32×C4#6712yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178#638—24no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—22no—
181C2×C3⋊Dic3—42no—
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)C2×Dic312yescomplemented
184C4×C3⋊D3—24no—
185#634—21no—
186#634C2×Dic312yescomplemented
187C3×#67—68no—
188C3×#67—64no—
189C3×#67—64no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—124no—
197C2×C3×Dic3—122no—
198C2×C3×Dic3—122no—
199C3×D12—32no—
200C3×D12—32no—
201C3×D12—62no—
202C3×D12—62no—
203C3×D12—61no—
204C3×D12—61no—
205C3×C4×D3—68no—
206C3×C4×D3—64no—
207C3×C4×D3—64no—
208C3×Dic6—32no—
209C3×Dic6—32no—
210C3×Dic6—62no—
211C3×Dic6—62no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3⋊(C3⋊C8)—22no—
215C3×C3⋊C8—64no—
216C3×C3⋊C8—62no—
217C3×C3⋊C8—62no—
218#417—271noSylow
219C2×C33—22no—
220C2×C33C2×C22⋊C411yescharacteristic
221C2×C33AGL(1,ℤ/8)11yescharacteristic
222C2×C33#13711yescharacteristic
223C3×C3⋊D3—42no—
224C3×#41—94no—
225C3×#41—182no—
226C2×C3×Q8—91no—
227C2×C3×D4—91no—
228C2×C3×D4—181no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—41no—
233C22×C3×C4—181no—
234C22×C3×C4—181no—
235C22×C3×C4D3211yescharacteristic
236C22×C3×C4D3211yescharacteristic
237C22×C3×C4D3211yescharacteristic
238C2×#67—62no—
239C2×#67—61no—
240C2×#67—61no—
241C22×Dic3—62no—
242C22×Dic3—61no—
243C22×Dic3—61no—
244#235—34no—
245#235—34no—
246#235—64no—
247#235—64no—
248#235—62no—
249#235—62no—
250C2×D12—31no—
251C2×D12—31no—
252C2×D12—61no—
253C2×C4×D3—62no—
254C2×C4×D3—61no—
255C2×C4×D3—61no—
256C2×Dic6—31no—
257C2×Dic6—31no—
258C2×Dic6—61no—
259C3×M16—182no—
260C2×C3×C8—92no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—181no—
263C4⋊Dic3—32no—
264C4⋊Dic3—32no—
265C4⋊Dic3—62no—
266C4⋊Dic3—62no—
267C4⋊Dic3—61no—
268C4⋊Dic3—61no—
269#208—64no—
270#208—62no—
271#208—62no—
272C2×C3⋊C8—32no—
273C2×C3⋊C8—32no—
274C2×C3⋊C8—62no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—22no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—42no—
290C22×C32—41no—
291C22×C32—41no—
292C22×C32—41no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32(C2×D3)⋊C412yes—
298C22×C32(C2×D3)⋊C412yes—
299C22×C32#21712yes—
300C22×C32C2×D1211yescharacteristic
301C22×C32C2×#6711yescharacteristic
302C22×C32C2×#6711yescharacteristic
303C2×C3⋊D3—22no—
304C2×C3⋊D3—41no—
305C2×C3⋊D3—41no—
306C2×C3×D3—64no—
307C2×C3×D3—62no—
308C2×C3×D3—62no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—121no—
315C32×C4—22no—
316C32×C4—22no—
317C32×C4—22no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—42no—
325C32×C4—41no—
326C32×C4—41no—
327C32×C4(C2×D3)⋊C412yes—
328C32×C4(C2×D3)⋊C412yes—
329C32×C4#21712yes—
330C32×C4C2×C4×D311yescharacteristic
331C32×C4C2×C4×D311yescharacteristic
332C32×C4C2×D1211yescharacteristic
333C32×C4C22×Dic311yescharacteristic
334C32×C4C2×#6711yescharacteristic
335C32×C4C2×#6711yescharacteristic
336C3⋊Dic3—22no—
337C3⋊Dic3—42no—
338C3×Dic3—64no—
339C3×Dic3—62no—
340C3×Dic3—62no—
341C3×Dic3—124no—
342C3×Dic3—122no—
343C3×Dic3—122no—
344C2×#41—91no—
345C2×M16—91no—
346C2×C4⋊C4—91no—
347#103—272no—
348#102—272no—
349C33#41711yescharacteristic, complemented, Sylow
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3—181no—
355C23×C3#62511yescharacteristic
356C23×C3#62611yescharacteristic
357C23×C3#62611yescharacteristic
358C22×D3—62no—
359C22×D3—61no—
360C22×D3—61no—
361C3×Q8—92no—
362C3×Q8—181no—
363C3×D4—92no—
364C3×D4—184no—
365C3×D4—181no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—181no—
383C2×C3×C4—184no—
384C2×C3×C4—182no—
385C2×C3×C4—362no—
386C2×C3×C4D3×Dic312yes—
387C2×C3×C4D3×Dic312yes—
388C2×C3×C4#62412yes—
389C2×C3×C4#62511yescharacteristic
390C2×C3×C4#62512yes—
391C2×C3×C4#62611yescharacteristic
392C2×C3×C4#62612yes—
393C2×C3×C4#62611yescharacteristic
394C2×C3×C4#62612yes—
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398#67—68no—
399#67—64no—
400#67—64no—
401C2×Dic3—62no—
402C2×Dic3—61no—
403C2×Dic3—61no—
404C2×Dic3—64no—
405C2×Dic3—62no—
406C2×Dic3—62no—
407C2×Dic3—124no—
408C2×Dic3—122no—
409C2×Dic3—122no—
410D12—32no—
411D12—32no—
412D12—62no—
413D12—62no—
414D12—61no—
415D12—61no—
416C4×D3—68no—
417C4×D3—64no—
418C4×D3—64no—
419Dic6—32no—
420Dic6—32no—
421Dic6—62no—
422Dic6—62no—
423Dic6—61no—
424Dic6—61no—
425C3×C8—182no—
426C3⋊C8—64no—
427C3⋊C8—62no—
428C3⋊C8—62no—
429C2×C32—22no—
430C2×C32—24no—
431C2×C32—22no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—41no—
443C2×C32—41no—
444C2×C32—41no—
445C2×C32—42no—
446C2×C32—42no—
447C2×C32—44no—
448C2×C32#90811yescharacteristic
449C2×C32#90911yescharacteristic
450C2×C32C2×(C2×D3)⋊C411yescharacteristic
451C2×C32C2×(C2×D3)⋊C411yescharacteristic
452C2×C32#93211yescharacteristic
453C2×C32#94211yescharacteristic
454C2×C32#94911yescharacteristic
455C2×C32#95111yescharacteristic
456C2×C32C2×#21711yescharacteristic
457C3⋊D3—42no—
458C3×D3—124no—
459C3×D3—122no—
460C3×D3—122no—
461#41—94no—
462#41—182no—
463C2×Q8—91no—
464C2×D4—91no—
465C2×D4—181no—
466C22×C4—181no—
467C22×C4—181no—
468C22×C4#111311yescharacteristic
469M16—182no—
470C2×C8—92no—
471C4⋊C4—92no—
472C4⋊C4—181no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—182no—
496C22×C3—361no—
497C22×C3—361no—
498C22×C3(C2×D3)⋊Dic312yes—
499C22×C3(C2×D3)⋊Dic312yes—
500C22×C3#372212yes—
501C22×C3C2×#62511yescharacteristic
502C22×C3C2×#62611yescharacteristic
503C22×C3C2×#62611yescharacteristic
504C2×D3—64no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—122no—
508C2×D3—122no—
509C2×D3—121no—
510C2×D3—121no—
511C2×D3—121no—
512C2×D3—121no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—22no—
520C3×C4—22no—
521C3×C4—22no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—42no—
525C3×C4—182no—
526C3×C4—362no—
527C3×C4(C2×D3)⋊Dic312yes—
528C3×C4(C2×D3)⋊Dic312yes—
529C3×C4#372212yes—
530C3×C4C2×D3×Dic311yescharacteristic
531C3×C4C2×D3×Dic311yescharacteristic
532C3×C4C2×#62411yescharacteristic
533C3×C4C2×#62511yescharacteristic
534C3×C4C2×#62611yescharacteristic
535C3×C4C2×#62611yescharacteristic
536Dic3—64no—
537Dic3—62no—
538Dic3—62no—
539Dic3—124no—
540Dic3—122no—
541Dic3—122no—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#496611yescharacteristic, complemented
547C32#513311yescharacteristic, complemented
548C32#523211yescharacteristic, complemented
549C23—181no—
550C23#629511yescharacteristic
551Q8—92no—
552Q8—181no—
553D4—92no—
554D4—184no—
555D4—181no—
556C2×C4—181no—
557C2×C4—184no—
558C2×C4—182no—
559C2×C4—362no—
560C2×C4#629412yes—
561C2×C4#629512yes—
562C2×C4#629511yescharacteristic
563C2×C4C2×#111311yescharacteristic
564C8—182no—
565C2×C3—24no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—22no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—42no—
579C2×C3—42no—
580C2×C3—44no—
581C2×C3—41no—
582C2×C3—41no—
583C2×C3—41no—
584C2×C3—362no—
585C2×C3#6382311yescharacteristic
586C2×C3#6382511yescharacteristic
587C2×C3#6382711yescharacteristic
588C2×C3#6383011yescharacteristic
589C2×C3#6383111yescharacteristic
590C2×C3#6383311yescharacteristic
591C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
592C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
593C2×C3C2×#372211yescharacteristic
594D3—124no—
595D3—122no—
596D3—122no—
597C22—21no—
598C22—21no—
599C22—182no—
600C22—361no—
601C22—361no—
602C22#8881412yes—
603C22C2×#629511yescharacteristic
604C4—182no—
605C4—362no—
606C4#8881412yes—
607C4C2×#629411yescharacteristic
608C4C2×#629511yescharacteristic
609C3—21no—
610C3—21no—
611C3—21no—
612C3—41no—
613C3#1059108711yescharacteristic, complemented
614C3#1059108811yescharacteristic, complemented
615C3#1059110611yescharacteristic, complemented
616C2—22no—
617C2—362no—
618C2#1172366111yescharacteristic
619C2#1172366311yescharacteristic
620C2C2×#8881411yescharacteristic
subgroupG/H
centerC22C2×#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C2×C3×#3748C2semidirect
4. #11722076C2semidirect
5. #11722075C2semidirect
13. C3×#3828C4semidirect
25. C2×C4⋊(C3⋊Dic3)D3semidirect
26. C2×#3748D3semidirect
50. C3×C4⋊(C3⋊D3)C2×C4semidirect
53. C3×#634C2×C4semidirect
68. #3828Dic3semidirect
183. C4⋊(C3⋊D3)C2×Dic3semidirect
186. #634C2×Dic3semidirect
349. C33#417semidirect
546. C32#4966semidirect
547. C32#5133semidirect
548. C32#5232semidirect
613. C3#10591087semidirect
614. C3#10591088semidirect
615. C3#10591106semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37482
#117220762
#117220752
#105911063
#105910883
#105910873