Finite Group Databasebeta

← #10943999

Subgroups — #10943999

Order 768.

1608 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C211yesmaximal, characteristic, complemented
2C2×#5238C212yesmaximal, complemented
3C2×#5027C211yesmaximal, characteristic, complemented
4C2×#5015C211yesmaximal, characteristic, complemented
5#82081C211yesmaximal, characteristic, complemented
6#71633C212yesmaximal, complemented
7#71626C211yesmaximal, characteristic, complemented
8#71613C212yesmaximal, complemented
9#71604C211yesmaximal, characteristic, complemented
10#71586C212yesmaximal, complemented
11#71530C211yesmaximal, characteristic, complemented
12#32904—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C3×C4×#41—24no—
15C3×C4×#41—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×C42—21no—
25C22×C3×C42C2211yescharacteristic
26C22×#215C2212yes—
27C22×#213C2212yes—
28C2×#930—21no—
29C2×#930—21no—
30C2×#930C2212yes—
31C2×#874C2211yescharacteristic
32#5238—28no—
33C2×#835C2212yes—
34C2×#832C2212yes—
35C2×#830C2212yes—
36#5027—24no—
37#5015—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40#4939—21no—
41#4939—21no—
42#4939C2212yes—
43#4935C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4529C2212yes—
47#4529C2212yes—
48#4524C2212yes—
49C2×C4×#41D311yescharacteristic, complemented
50C2×#460—31no—
51C2×#459—31no—
52C2×#449—32no—
53#2856—31no—
54#1880—32no—
55#1879—32no—
56#1875—32no—
57#1803—31no—
58#1802—31no—
59#1797—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×C4D411yescharacteristic
73C23×C3×C4D411yescharacteristic
74C23×C3×C4C2311yescharacteristic
75C22×#67—22no—
76C22×D12—21no—
77C22×D12—21no—
78C22×D12C2312yes—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8D414yescomplemented
83C3×C4×D4—28no—
84C3×C4×D4—28no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D414yescomplemented
89C3×#117—28no—
90C3×#117—22no—
91C3×#117—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42—28no—
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42—22no—
104C2×C3×C42—22no—
105C2×C3×C42D411yescharacteristic
106C2×C3×C42D411yescharacteristic
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—
117#930—44no—
118#930—44no—
119C2×(C2×D3)⋊C4—22no—
120C2×(C2×D3)⋊C4—22no—
121C2×(C2×D3)⋊C4—22no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3—21no—
124C2×C4⋊Dic3C2312yes—
125C22×C3⋊C8C2312yes—
126C22×C3⋊C8C2312yes—
127#874—22no—
128#874—22no—
129#835—24no—
130#832—24no—
131#830—24no—
132#809—24no—
133#809—24no—
134C4⋊(C3⋊C8)—22no—
135C4⋊(C3⋊C8)—22no—
136C22×#41C2×D311yescharacteristic
137C22×SD16—32no—
138C22×D8—32no—
139C2×#127—31no—
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
143C4×#41—24no—
144C4×#41—24no—
145C2×C4×Q8C2×D311yescharacteristic
146C2×C4×Q8C2×D311yescharacteristic
147C2×C4×D4—21no—
148C2×C4×D4—21no—
149C2×C4×D4C2×D311yescharacteristic
150C2×C4×D4C2×D311yescharacteristic
151C2×#117—21no—
152C2×#117C2×D311yescharacteristic
153C22×C42—21no—
154C22×C42C2×D311yescharacteristic
155#460—64no—
156#459—64no—
157#449—68no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102—32no—
164C2×C22⋊C8—32no—
165#390—32no—
166#390—61no—
167#390—61no—
168#384—31no—
169#340—32no—
170#340—32no—
171#336—32no—
172C24×C3—21no—
173C24×C3C2×D411yescharacteristic
174C23×D3—22no—
175C3×#41—28no—
176C3×#41—28no—
177C3×#41—28no—
178C3×#41—28no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8C2×D412yes—
183C2×C3×Q8C2×D412yes—
184C2×C3×Q8#4112yes—
185C2×C3×D4—24no—
186C2×C3×D4—24no—
187C2×C3×D4—24no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4C2×D412yes—
192C2×C3×D4C2×D412yes—
193C2×C3×D4#4112yes—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—24no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—24no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—24no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—41no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4112yes—
223C22×C3×C4C2411yescharacteristic
224C2×#67—44no—
225C2×#67—44no—
226C2×#67—44no—
227C22×Dic3—22no—
228C2×D12—22no—
229C2×D12—22no—
230C2×D12—22no—
231C2×D12—22no—
232C2×D12—22no—
233C2×D12—22no—
234C2×D12—42no—
235C2×D12—42no—
236C2×D12—42no—
237C2×D12—42no—
238C2×D12—42no—
239C2×D12—42no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2×D412yes—
245C3×C4⋊C4C2×D412yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C42—24no—
250C3×C42—24no—
251C3×C42—22no—
252C3×C42—24no—
253C3×C42—22no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42C2×D412yes—
257C3×C42C2×D412yes—
258#215—48no—
259#213—48no—
260(C2×D3)⋊C4—44no—
261(C2×D3)⋊C4—44no—
262(C2×D3)⋊C4—44no—
263C4⋊Dic3—22no—
264C4⋊Dic3—22no—
265C4⋊Dic3—42no—
266C4⋊Dic3—42no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×#41—22no—
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—22no—
278C2×#41D1214yescomplemented
279C22×Q8C22×D311yescharacteristic
280C22×D4—21no—
281C22×D4—32no—
282C22×D4—62no—
283C22×D4—61no—
284C22×D4—61no—
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—21no—
289C23×C4#6711yescharacteristic
290C23×C4#6711yescharacteristic
291C23×C4C22×D311yescharacteristic
292C2×SD16—64no—
293C2×SD16—64no—
294C2×SD16—64no—
295C2×D8—64no—
296C2×D8—64no—
297C2×D8—64no—
298C22×C8—32no—
299C22×C8—32no—
300#127—62no—
301#127—62no—
302#121—64no—
303#121—64no—
304#121—124no—
305#121—124no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8#6714yescomplemented
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4#6714yescomplemented
316#117—28no—
317#117—22no—
318#117—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—62no—
330C2×C22⋊C4—62no—
331C2×C22⋊C4—62no—
332C2×C42—28no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42#6711yescharacteristic
339C2×C42#6711yescharacteristic
340C2×C42C22×D311yescharacteristic
341C2×C42C22×D311yescharacteristic
342C4⋊C8—62no—
343C4⋊C8—62no—
344#103—64no—
345#102—64no—
346#102—64no—
347#102—64no—
348C22⋊C8—64no—
349C23×C3—21no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—21no—
353C23×C3—22no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3#12112yes—
363C23×C3C22×D411yescharacteristic
364C22×D3—42no—
365C22×D3—42no—
366C22×D3—42no—
367C22×D3—42no—
368C22×D3—42no—
369C22×D3—42no—
370C22×D3—42no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8#12114yescomplemented
375C3×D4—216no—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4#12114yescomplemented
380C2×C3×C4—216no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—44no—
400C2×C3×C4—48no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—44no—
406C2×C3×C4—48no—
407C2×C3×C4#12012yes—
408C2×C3×C4#12012yes—
409C2×C3×C4#12112yes—
410C2×C3×C4#13512yes—
411C2×C3×C4AGL(1,ℤ/8)12yes—
412C2×C3×C4AGL(1,ℤ/8)12yes—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419#67—88no—
420C2×Dic3—42no—
421C2×Dic3—42no—
422C2×Dic3—42no—
423D12—42no—
424D12—42no—
425D12—42no—
426D12—42no—
427D12—84no—
428D12—84no—
429C3⋊C8—42no—
430C3⋊C8—42no—
431C24—21no—
432C24—62no—
433C24C2×#6711yescharacteristic
434#41—28no—
435#41—28no—
436#41—28no—
437#41—28no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×D1212yes—
442C2×Q8#23512yes—
443C2×Q8C2×#6712yes—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—62no—
453C2×D4—62no—
454C2×D4—62no—
455C2×D4—62no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—124no—
459C2×D4—124no—
460C2×D4—124no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—122no—
464C2×D4—122no—
465C2×D4C2×D1212yes—
466C2×D4#23512yes—
467C2×D4C2×#6712yes—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—41no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—62no—
491C22×C4C2×D1212yes—
492C22×C4#23512yes—
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6711yescharacteristic
496C22×C4C2×#6711yescharacteristic
497C22×C4C2×#6711yescharacteristic
498C22×C4C23×D311yescharacteristic
499SD16—128no—
500D8—128no—
501C2×C8—62no—
502C2×C8—62no—
503C2×C8—62no—
504C2×C8—62no—
505C2×C8—62no—
506C2×C8—62no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—62no—
512C4⋊C4—62no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4C2×#6712yes—
516C4⋊C4C2×#6712yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—124no—
521C22⋊C4—124no—
522C22⋊C4—124no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42C2×#6712yes—
531C42C2×#6712yes—
532C22×C3—22no—
533C22×C3—24no—
534C22×C3—22no—
535C22×C3—22no—
536C22×C3—41no—
537C22×C3—42no—
538C22×C3—41no—
539C22×C3—42no—
540C22×C3—41no—
541C22×C3—42no—
542C22×C3—41no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3C2×#12011yescharacteristic
547C22×C3C2×#12111yescharacteristic
548C22×C3#52511yescharacteristic
549C22×C3C2×#13511yescharacteristic
550C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
551C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
552C22×C3#57511yescharacteristic
553C2×D3—82no—
554C2×D3—82no—
555C2×D3—82no—
556C2×D3—82no—
557C2×D3—82no—
558C2×D3—82no—
559C2×D3—82no—
560C3×C4—22no—
561C3×C4—24no—
562C3×C4—22no—
563C3×C4—22no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4#44914yescomplemented
568C3×C4C2×#12111yescharacteristic
569C3×C4C2×#12111yescharacteristic
570C3×C4#52511yescharacteristic
571C3×C4#52511yescharacteristic
572Dic3—82no—
573C23—22no—
574C23—21no—
575C23—22no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—122no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23#93012yes—
594C23C22×#6711yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—22no—
598Q8#93014yescomplemented
599D4—216no—
600D4—22no—
601D4—22no—
602D4—22no—
603D4—122no—
604D4—122no—
605D4—122no—
606D4—122no—
607D4—244no—
608D4—248no—
609D4—244no—
610D4#93014yescomplemented
611C2×C4—216no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4#93012yes—
642C2×C4#94912yes—
643C2×C4#94912yes—
644C2×C4#95012yes—
645C2×C4#95312yes—
646C2×C4#95312yes—
647C2×C4C22×D1211yescharacteristic
648C2×C4C2×#23511yescharacteristic
649C2×C4C2×#23511yescharacteristic
650C2×C4C2×#23511yescharacteristic
651C2×C4C22×#6711yescharacteristic
652C2×C4C22×#6711yescharacteristic
653C8—122no—
654C8—122no—
655C2×C3—42no—
656C2×C3—44no—
657C2×C3#284711yescharacteristic
658C2×C3C2×#44911yescharacteristic
659C2×C3#300611yescharacteristic
660C2×C3#303511yescharacteristic
661C2×C3#303811yescharacteristic
662C2×C3#310611yescharacteristic
663C2×C3#311511yescharacteristic
664D3—162no—
665C22—22no—
666C22—24no—
667C22—22no—
668C22—22no—
669C22—42no—
670C22—42no—
671C22—41no—
672C22—42no—
673C22—41no—
674C22—42no—
675C22—41no—
676C22—42no—
677C22—41no—
678C22—42no—
679C22—242no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22—242no—
684C22—242no—
685C22—242no—
686C22C2×#93011yescharacteristic
687C22#579111yescharacteristic
688C22C2×#94911yescharacteristic
689C22C2×#94911yescharacteristic
690C22C2×#95011yescharacteristic
691C22#582211yescharacteristic
692C22C2×#95311yescharacteristic
693C4—22no—
694C4—24no—
695C4—22no—
696C4—22no—
697C4—41no—
698C4—42no—
699C4—42no—
700C4—242no—
701C4#523814yescomplemented
702C4C2×#93011yescharacteristic
703C4C2×#93011yescharacteristic
704C4#579111yescharacteristic
705C4#579111yescharacteristic
706C3#3290411yescharacteristic, complemented, Sylow
707C2—42no—
708C2—44no—
709C2—482no—
710C2#8252511yescharacteristic
711C2#8252611yescharacteristic
712C2#8252911yescharacteristic
713C2#8253111yescharacteristic
714C2#8403211yescharacteristic
715C2C2×#523811yescharacteristic
716C2#8408211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×#41C2
socleC23×C3C22×D4
hypercenterC2×C4×#41D3
hypocenterC3#32904
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×#41C2semidirect
2. C2×#5238C2semidirect
3. C2×#5027C2semidirect
4. C2×#5015C2semidirect
5. #82081C2semidirect
6. #71633C2semidirect
7. #71626C2semidirect
8. #71613C2semidirect
9. #71604C2semidirect
10. #71586C2semidirect
11. #71530C2semidirect
49. C2×C4×#41D3semidirect
65. C2×C3×#41D4semidirect
82. C3×C4×Q8D4semidirect
88. C3×C4×D4D4semidirect
278. C2×#41D12semidirect
309. C4×Q8#67semidirect
315. C4×D4#67semidirect
374. C3×Q8#121semidirect
379. C3×D4#121semidirect
567. C3×C4#449semidirect
598. Q8#930semidirect
610. D4#930semidirect
701. C4#5238semidirect
706. C3#32904semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C2×#52382
C2×#50272
C2×#50152
#820812
#716332
#716262
#716132
#716042
#715862
#715302
#329043