Finite Group Databasebeta

← #10944090

Subgroups — #10944090

Order 768.

1506 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic, complemented
2C2×#5238C212yesmaximal, complemented
3C2×#5027C211yesmaximal, characteristic, complemented
4C2×#5015C211yesmaximal, characteristic, complemented
5#82286C211yesmaximal, characteristic, complemented
6#71633C212yesmaximal, complemented
7#71627C211yesmaximal, characteristic, complemented
8#71613C212yesmaximal, complemented
9#71605C211yesmaximal, characteristic, complemented
10#71583C211yesmaximal, characteristic, complemented
11#71560C212yesmaximal, complemented
12#32913—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C3×#520—24no—
15C3×#520—24no—
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×#215C2212yes—
27C22×#213C2212yes—
28C2×#930C2212yes—
29C2×#874C2211yescharacteristic
30#5238—28no—
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×#830C2212yes—
34#5027—24no—
35#5015—24no—
36#4991—21no—
37#4991—21no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40#4960—21no—
41#4960—21no—
42#4939C2212yes—
43#4936C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4529C2212yes—
47#4529C2212yes—
48#4524C2212yes—
49C2×#520D311yescharacteristic, complemented
50C2×#460—31no—
51C2×#459—31no—
52C2×#449—32no—
53#2619—31no—
54#1880—32no—
55#1879—32no—
56#1874—32no—
57#1805—31no—
58#1804—31no—
59#1798—31no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41D414yescomplemented
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C22×#67—22no—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2312yes—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8D414yescomplemented
82C3×C4×D4—28no—
83C3×C4×D4—28no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4D414yescomplemented
88C3×#117—28no—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—28no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×#215—24no—
110C2×#215—24no—
111C2×#215—24no—
112C2×#213—24no—
113C2×#213—24no—
114C2×#213—24no—
115#930—24no—
116#930—24no—
117C2×(C2×D3)⋊C4—22no—
118C2×(C2×D3)⋊C4—22no—
119C2×(C2×D3)⋊C4—22no—
120C2×C4⋊Dic3C2312yes—
121C22×C3⋊C8C2312yes—
122C22×C3⋊C8C2312yes—
123#874—22no—
124#874—22no—
125#835—24no—
126#832—24no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#830—24no—
130#809—24no—
131#809—24no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—22no—
134C22×#41C2×D311yescharacteristic
135C22×SD16—32no—
136C22×D8—32no—
137C2×#127—31no—
138C2×#121—32no—
139#520—24no—
140#520—24no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4C2×D311yescharacteristic
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—21no—
148C2×#117C2×D311yescharacteristic
149C22×C4⋊C4—21no—
150C22×C4⋊C4C2×D311yescharacteristic
151#460—64no—
152#459—64no—
153#449—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×#103—32no—
157C2×#102—32no—
158C2×#102—32no—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#394—61no—
162#394—61no—
163#392—61no—
164#392—61no—
165#390—32no—
166#382—31no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—21no—
171C24×C3C2×D411yescharacteristic
172C23×D3—22no—
173C3×#41—28no—
174C3×#41—28no—
175C3×#41—28no—
176C3×#41—28no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×Q8#4112yes—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4C2×D412yes—
190C2×C3×D4C2×D412yes—
191C2×C3×D4#4112yes—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—24no—
203C22×C3×C4—24no—
204C22×C3×C4—24no—
205C22×C3×C4—82no—
206C22×C3×C4—82no—
207C22×C3×C4—82no—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4#4112yes—
215C22×C3×C4C2411yescharacteristic
216C2×#67—44no—
217C2×#67—82no—
218C2×#67—82no—
219C22×Dic3—22no—
220C2×D12—22no—
221C2×D12—22no—
222C2×D12—22no—
223C2×D12—22no—
224C2×D12—22no—
225C2×D12—22no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12—82no—
229C2×D12—82no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4C2×D412yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C42—24no—
244C3×C42—24no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42C2×D412yes—
248C3×C42C2×D412yes—
249#215—48no—
250#213—48no—
251(C2×D3)⋊C4—44no—
252(C2×D3)⋊C4—82no—
253(C2×D3)⋊C4—82no—
254C4⋊Dic3—22no—
255C4⋊Dic3—22no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41D1214yescomplemented
268C22×Q8C22×D311yescharacteristic
269C22×D4—21no—
270C22×D4—32no—
271C22×D4—62no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4C22×D311yescharacteristic
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4C22×D311yescharacteristic
280C2×SD16—64no—
281C2×SD16—64no—
282C2×SD16—64no—
283C2×D8—64no—
284C2×D8—64no—
285C2×D8—64no—
286C22×C8—32no—
287C22×C8—32no—
288#127—62no—
289#127—62no—
290#121—64no—
291#121—64no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8#6714yescomplemented
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4#6714yescomplemented
302#117—28no—
303#117—22no—
304#117—22no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4#6711yescharacteristic
313C2×C4⋊C4#6711yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—62no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4#6711yescharacteristic
322C2×C22⋊C4#6711yescharacteristic
323C2×C42—21no—
324C2×C42—21no—
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327C4⋊C8—62no—
328C4⋊C8—62no—
329#103—64no—
330#102—64no—
331#102—64no—
332#102—64no—
333C22⋊C8—64no—
334#95—61no—
335#95—61no—
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—22no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3#12112yes—
346C23×C3C22×D411yescharacteristic
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8#12114yes—
358C3×D4—216no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4#12114yes—
363C2×C3×C4—216no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—84no—
384C2×C3×C4—84no—
385C2×C3×C4—84no—
386C2×C3×C4#12012yes—
387C2×C3×C4#12012yes—
388C2×C3×C4#12112yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395#67—88no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399D12—42no—
400D12—42no—
401D12—42no—
402D12—42no—
403D12—84no—
404D12—84no—
405C3⋊C8—42no—
406C3⋊C8—42no—
407C24—21no—
408C24—62no—
409C24C2×#6711yescharacteristic
410#41—28no—
411#41—28no—
412#41—28no—
413#41—28no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8C2×D1212yes—
418C2×Q8#23512yes—
419C2×Q8C2×#6712yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—62no—
431C2×D4—62no—
432C2×D4—124no—
433C2×D4—122no—
434C2×D4—122no—
435C2×D4—242no—
436C2×D4—242no—
437C2×D4—242no—
438C2×D4—242no—
439C2×D4C2×D1212yes—
440C2×D4#23512yes—
441C2×D4C2×#6712yes—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—62no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4C2×D1212yes—
460C22×C4#23512yes—
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6711yescharacteristic
464C22×C4C2×#6711yescharacteristic
465C22×C4C2×#6711yescharacteristic
466C22×C4C23×D311yescharacteristic
467SD16—128no—
468D8—128no—
469C2×C8—62no—
470C2×C8—62no—
471C2×C8—62no—
472C2×C8—62no—
473C2×C8—62no—
474C2×C8—62no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4C2×#6712yes—
485C4⋊C4C2×#6712yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—124no—
491C22⋊C4—242no—
492C22⋊C4—242no—
493C42—24no—
494C42—24no—
495C42—21no—
496C42—21no—
497C42C2×#6712yes—
498C42C2×#6712yes—
499C22×C3—22no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—42no—
503C22×C3—44no—
504C22×C3—41no—
505C22×C3—42no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—44no—
509C22×C3C2×#12011yescharacteristic
510C22×C3C2×#12111yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3#57612yes—
513C22×C3#57712yes—
514C2×D3—82no—
515C2×D3—82no—
516C2×D3—82no—
517C2×D3—82no—
518C2×D3—82no—
519C2×D3—82no—
520C2×D3—82no—
521C3×C4—22no—
522C3×C4—24no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—41no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—41no—
529C3×C4C2×#12111yescharacteristic
530C3×C4C2×#12111yescharacteristic
531C3×C4#52511yescharacteristic
532C3×C4#52511yescharacteristic
533Dic3—82no—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23#93012yes—
551C23C22×#6711yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8#93014yes—
556D4—216no—
557D4—22no—
558D4—22no—
559D4—22no—
560D4—122no—
561D4—122no—
562D4—122no—
563D4—122no—
564D4—244no—
565D4—244no—
566D4—248no—
567D4#93014yes—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4#93012yes—
595C2×C4#95312yes—
596C2×C4#95312yes—
597C2×C4C22×D1211yescharacteristic
598C2×C4C2×#23511yescharacteristic
599C2×C4C2×#23511yescharacteristic
600C2×C4C2×#23511yescharacteristic
601C2×C4C22×#6711yescharacteristic
602C2×C4C22×#6711yescharacteristic
603C8—122no—
604C8—122no—
605C2×C3—42no—
606C2×C3—44no—
607C2×C3#284711yescharacteristic
608C2×C3#300811yescharacteristic
609C2×C3#301011yescharacteristic
610C2×C3#304512yes—
611C2×C3#311912yes—
612D3—162no—
613C22—22no—
614C22—24no—
615C22—24no—
616C22—44no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—41no—
622C22—42no—
623C22—242no—
624C22—242no—
625C22—242no—
626C22—242no—
627C22—242no—
628C22—242no—
629C22—242no—
630C22C2×#93011yescharacteristic
631C22#579111yescharacteristic
632C22#583512yes—
633C22#583712yes—
634C22C2×#95311yescharacteristic
635C4—22no—
636C4—24no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—242no—
644C4C2×#93011yescharacteristic
645C4C2×#93011yescharacteristic
646C4#579111yescharacteristic
647C4#579111yescharacteristic
648C3#3291311yescharacteristic, complemented, Sylow
649C2—42no—
650C2—44no—
651C2—482no—
652C2#8256512yes—
653C2#8256812yes—
654C2#8403211yescharacteristic
655C2#8410111yescharacteristic
656C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#520C2
socleC23×C3C22×D4
hypercenterC2×#520D3
hypocenterC3#32913
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×C3×#520C2semidirect
2. C2×#5238C2semidirect
3. C2×#5027C2semidirect
4. C2×#5015C2semidirect
5. #82286C2semidirect
6. #71633C2semidirect
7. #71627C2semidirect
8. #71613C2semidirect
9. #71605C2semidirect
10. #71583C2semidirect
11. #71560C2semidirect
49. C2×#520D3semidirect
65. C2×C3×#41D4semidirect
81. C3×C4×Q8D4semidirect
87. C3×C4×D4D4semidirect
267. C2×#41D12semidirect
295. C4×Q8#67semidirect
301. C4×D4#67semidirect
648. C3#32913semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×#52382
C2×#50272
C2×#50152
#822862
#716332
#716272
#716132
#716052
#715832
#715602
#329133