Finite Group Databasebeta

← #10943992

Subgroups — #10943992

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×C4×#237C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic, complemented
3C2×#5158C212yesmaximal, complemented
4#83192C211yesmaximal, characteristic, complemented
5C2×#5075C211yesmaximal, characteristic, complemented
6#72037C212yesmaximal, complemented
7#71604C211yesmaximal, characteristic, complemented
8#70485C211yesmaximal, characteristic, complemented
9#70475C211yesmaximal, characteristic, complemented
10#70213C212yesmaximal, complemented
11#70201C212yesmaximal, complemented
12#32904—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#938—21no—
18C2×#938—21no—
19C2×#938C2212yes—
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2212yes—
22C2×C4×#67—21no—
23C22×C4×Dic3—21no—
24C4×#237—24no—
25C4×#237—24no—
26C2×#892C2211yescharacteristic
27C2×#887C2211yescharacteristic
28C2×#67⋊C4—21no—
29C2×#877—21no—
30C2×C42×D3C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C3×#459—24no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#102C2212yes—
35#5221—21no—
36#5221—21no—
37#5221C2212yes—
38C2×#832C2212yes—
39C2×#832C2212yes—
40#5158—28no—
41C2×(C2×D3)⋊C8C2212yes—
42C2×#816C2212yes—
43#5075—24no—
44#4935C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×C4×#41—31no—
50C2×#460—31no—
51C2×#459D311yescharacteristic, complemented
52C2×#449—32no—
53#2856—31no—
54#1880—32no—
55#1879—32no—
56#1875—32no—
57#1803—31no—
58#1802—31no—
59#1797—31no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2312yes—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C22×#67—21no—
66C22×#67—22no—
67C23×Dic3—21no—
68C23×Dic3—21no—
69C2×#237—22no—
70C2×#237—22no—
71C2×#237—22no—
72C2×#237—22no—
73C2×#237—22no—
74C2×#237D414yescomplemented
75C22×C4×D3D411yescharacteristic
76C22×C4×D3D411yescharacteristic
77C22×C4×D3C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C2×C3×D8—24no—
80C2×C3×D8—24no—
81C2×C3×D8—24no—
82C22×C3×C8C2312yes—
83C3×#127—22no—
84C3×#127—22no—
85C3×C4×D4—22no—
86C3×C4×D4D414yescomplemented
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—22no—
92C2×#217—22no—
93C2×#217—22no—
94#938—24no—
95#938—24no—
96#938—44no—
97#938—44no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100C2×#214—24no—
101C2×#214—24no—
102C2×#214—24no—
103C4×#67—28no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—28no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2312yes—
119#892—22no—
120#892—22no—
121#887—22no—
122#887—22no—
123#67⋊C4—28no—
124#877—28no—
125C42×D3—22no—
126C42×D3—22no—
127C4×Dic6—22no—
128C4×Dic6D414yescomplemented
129C3×C4⋊C8—22no—
130C3×#102—24no—
131#832—24no—
132#832—24no—
133(C2×D3)⋊C8—24no—
134#816—24no—
135C4⋊(C3⋊C8)—22no—
136C22×#41—31no—
137C22×SD16—32no—
138C22×D8C2×D312yes—
139C2×#127C2×D311yescharacteristic
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
143C4×#41—64no—
144C4×#41—64no—
145C2×C4×Q8—31no—
146C2×C4×Q8—31no—
147C2×C4×D4—31no—
148C2×C4×D4—61no—
149C2×C4×D4—61no—
150C2×C4×D4C2×D311yescharacteristic
151C2×#117—31no—
152C2×#117—61no—
153C22×C42—31no—
154C22×C42—61no—
155#460—64no—
156#459—24no—
157#449—68no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102C2×D312yes—
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×C3—22no—
174C23×D3C2×D411yescharacteristic
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4C2×D412yes—
191C2×C3×D4C2×D412yes—
192C2×C3×D4#4112yes—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—24no—
202C2×#67—24no—
203C2×#67—44no—
204C2×#67—44no—
205C2×#67—44no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—21no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3—42no—
214C22×Dic3—42no—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218#237—28no—
219#237—28no—
220#237—28no—
221#237—28no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—22no—
231C2×C4×D3—41no—
232C2×C4×D3C2×D412yes—
233C2×C4×D3#4112yes—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×D412yes—
238C2×Dic6C2×D412yes—
239C2×Dic6#4112yes—
240C3×D8—48no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C2×C3×C8—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D412yes—
246C3×C22⋊C4—24no—
247C3×C42C2×D412yes—
248C3×C42C2×D412yes—
249#217—24no—
250#217—44no—
251#217—44no—
252#217—44no—
253#214—48no—
254(C2×D3)⋊C4—24no—
255C4⋊Dic3—21no—
256C4⋊Dic3—22no—
257C4⋊Dic3—22no—
258C4⋊Dic3—42no—
259C4⋊Dic3—42no—
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263C4×Dic3—24no—
264C4×Dic3—24no—
265C4×Dic3—24no—
266C4×Dic3—22no—
267C4×Dic3—22no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×#41—34no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—62no—
279C22×Q8—31no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—61no—
283C22×D4—62no—
284C22×D4C22×D312yes—
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—61no—
291C23×C4—61no—
292C2×SD16—64no—
293C2×SD16—64no—
294C2×SD16—64no—
295C2×D8—24no—
296C2×D8—24no—
297C2×D8—24no—
298C22×C8—32no—
299C22×C8C22×D312yes—
300#127—22no—
301#127—22no—
302#121—64no—
303#121—64no—
304#121—124no—
305#121—124no—
306C4×Q8—34no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×D4—22no—
311C4×D4—68no—
312C4×D4—62no—
313C4×D4—68no—
314C4×D4—62no—
315C4×D4#6714yescomplemented
316#117—68no—
317#117—62no—
318#117—62no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—62no—
330C2×C22⋊C4—62no—
331C2×C22⋊C4—62no—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—68no—
338C2×C42—61no—
339C2×C42—62no—
340C2×C42—62no—
341C2×C42C22×D311yescharacteristic
342C4⋊C8—22no—
343C4⋊C8—62no—
344#103—64no—
345#102—24no—
346#102—64no—
347#102—64no—
348C22⋊C8—64no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—42no—
357C23×C3—42no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3—42no—
363C23×C3C22×D411yescharacteristic
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3#12112yes—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—84no—
379C3×D4—84no—
380C3×D4#12114yescomplemented
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—48no—
387C2×C3×C4#12012yes—
388C2×C3×C4#12012yes—
389C2×C3×C4#13512yes—
390C2×C3×C4AGL(1,ℤ/8)12yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397#67—216no—
398#67—88no—
399C2×Dic3—216no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—48no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—42no—
415C2×Dic3—42no—
416C2×Dic3—42no—
417C2×Dic3#12112yes—
418C2×Dic3AGL(1,ℤ/8)12yes—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—44no—
424C4×D3—44no—
425Dic6—22no—
426Dic6—22no—
427Dic6—22no—
428Dic6#12114yescomplemented
429C3×C8—42no—
430C3⋊C8—42no—
431C24—21no—
432C24—22no—
433C24—31no—
434#41—68no—
435#41—68no—
436#41—68no—
437#41—68no—
438C2×Q8—32no—
439C2×Q8—32no—
440C2×Q8—32no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—64no—
460C2×D4—64no—
461C2×D4—64no—
462C2×D4—124no—
463C2×D4—124no—
464C2×D4—124no—
465C2×D4D3×D412yes—
466C2×D4#23712yes—
467C2×D4C2×#6712yes—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—31no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—62no—
478C22×C4—64no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—64no—
487C22×C4—61no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—62no—
493C22×C4—121no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4D3×D411yescharacteristic
497C22×C4D3×D411yescharacteristic
498C22×C4C23×D311yescharacteristic
499SD16—128no—
500D8—48no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—62no—
505C2×C8—62no—
506C2×C8—62no—
507C4⋊C4—21no—
508C4⋊C4—32no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—61no—
512C4⋊C4—62no—
513C4⋊C4—62no—
514C4⋊C4—122no—
515C4⋊C4—122no—
516C4⋊C4C2×#6712yes—
517C22⋊C4—24no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C22⋊C4—124no—
522C22⋊C4—124no—
523C42—64no—
524C42—64no—
525C42—62no—
526C42—62no—
527C42—64no—
528C42—61no—
529C42—61no—
530C42D3×D412yes—
531C42C2×#6712yes—
532C22×C3—24no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3—82no—
540C22×C3—82no—
541C22×C3—82no—
542C22×C3—82no—
543C22×C3—82no—
544C22×C3—82no—
545C22×C3—82no—
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—22no—
554C2×D3—22no—
555C2×D3—22no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—41no—
560C3×C4—22no—
561C3×C4—42no—
562C3×C4#44914yescomplemented
563C3×C4C2×#12111yescharacteristic
564C3×C4C2×#12111yescharacteristic
565C3×C4#52511yescharacteristic
566C3×C4#52511yescharacteristic
567Dic3—24no—
568Dic3—22no—
569Dic3—22no—
570Dic3—41no—
571Dic3—42no—
572Dic3—82no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—32no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23C2×D3×D411yescharacteristic
595Q8—34no—
596Q8—62no—
597Q8—62no—
598Q8—62no—
599D4—22no—
600D4—22no—
601D4—22no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—616no—
607D4—84no—
608D4—84no—
609D4—248no—
610D4#93814yescomplemented
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—32no—
615C2×C4—32no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—48no—
619C2×C4—616no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—61no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—62no—
634C2×C4—61no—
635C2×C4—124no—
636C2×C4—128no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—124no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4#91112yes—
644C2×C4#91212yes—
645C2×C4#93712yes—
646C2×C4#93712yes—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4C22×#6711yescharacteristic
653C8—42no—
654C8—122no—
655C2×C3—44no—
656C2×C3—162no—
657C2×C3#284711yescharacteristic
658C2×C3C2×#44911yescharacteristic
659C2×C3#300611yescharacteristic
660C2×C3#303511yescharacteristic
661C2×C3#303811yescharacteristic
662C2×C3#310611yescharacteristic
663C2×C3#311511yescharacteristic
664D3—42no—
665C22—24no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—62no—
673C22—62no—
674C22—62no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—121no—
683C22—121no—
684C22—121no—
685C22—121no—
686C22D3×#12111yescharacteristic
687C22#560611yescharacteristic
688C22C2×#91111yescharacteristic
689C22C2×#91211yescharacteristic
690C22D3×AGL(1,ℤ/8)11yescharacteristic
691C22#577311yescharacteristic
692C22C2×#93711yescharacteristic
693C4—22no—
694C4—42no—
695C4—62no—
696C4—64no—
697C4—62no—
698C4—121no—
699C4—122no—
700C4—242no—
701C4#515814yescomplemented
702C4D3×#12111yescharacteristic
703C4#560611yescharacteristic
704C4C2×#93811yescharacteristic
705C4#582811yescharacteristic
706C3#3290411yescharacteristic, complemented, Sylow
707C2—44no—
708C2—122no—
709C2—162no—
710C2#8031311yescharacteristic
711C2#8031511yescharacteristic
712C2#8033111yescharacteristic
713C2#8033311yescharacteristic
714C2#8272311yescharacteristic
715C2C2×#515811yescharacteristic
716C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#32904
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×C4×#237C2semidirect
2. C2×C3×#459C2semidirect
3. C2×#5158C2semidirect
4. #83192C2semidirect
5. C2×#5075C2semidirect
6. #72037C2semidirect
7. #71604C2semidirect
8. #70485C2semidirect
9. #70475C2semidirect
10. #70213C2semidirect
11. #70201C2semidirect
51. C2×#459D3semidirect
74. C2×#237D4semidirect
86. C3×C4×D4D4semidirect
128. C4×Dic6D4semidirect
315. C4×D4#67semidirect
380. C3×D4#121semidirect
428. Dic6#121semidirect
562. C3×C4#449semidirect
610. D4#938semidirect
701. C4#5158semidirect
706. C3#32904semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2372
C2×C3×#4592
C2×#51582
#831922
C2×#50752
#720372
#716042
#704852
#704752
#702132
#702012
#329043