Finite Group Databasebeta

← #10944085

Subgroups — #10944085

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×#5574C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×#5172C212yesmaximal, complemented
4#83189C211yesmaximal, characteristic, complemented
5C2×#5072C211yesmaximal, characteristic, complemented
6#72048C212yesmaximal, complemented
7#71627C211yesmaximal, characteristic, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70476C211yesmaximal, characteristic, complemented
10#70208C212yesmaximal, complemented
11#70200C212yesmaximal, complemented
12#32913—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×Q8×Dic3C2211yescharacteristic
18C2×#938C2212yes—
19C22×#213C2212yes—
20#5574—24no—
21#5574—24no—
22C2×#893—21no—
23C2×#893C2211yescharacteristic
24C2×D3×C4⋊C4—21no—
25C2×D3×C4⋊C4C2211yescharacteristic
26C2×C4×D12—21no—
27C2×C4×D12C2211yescharacteristic
28C2×#872—21no—
29C2×#872C2211yescharacteristic
30C3×#460—24no—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#102C2212yes—
33C2×#835C2212yes—
34#5223—21no—
35#5223—21no—
36#5222—21no—
37#5222—21no—
38#5221C2212yes—
39C2×#832C2212yes—
40#5172—28no—
41C2×#821C2212yes—
42C2×(C2×D3)⋊C8C2212yes—
43#5072—24no—
44#4936C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×#520—31no—
50C2×#460D311yescharacteristic, complemented
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—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2312yes—
64C22×#67—22no—
65C2×#239—22no—
66C2×#239—22no—
67C2×#239—22no—
68C2×#239—22no—
69C2×#239—22no—
70C2×#239D414yescomplemented
71C22×D12—21no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C22×C3×C8C2312yes—
82C3×#127—22no—
83C3×#127—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8D414yescomplemented
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—22no—
91C2×#217—22no—
92C2×#217—22no—
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95#938—24no—
96#938—24no—
97C2×#213—24no—
98C2×#213—24no—
99C2×#213—24no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×(C2×D3)⋊C4D411yescharacteristic
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113#893—28no—
114#893—22no—
115#893—22no—
116D3×C4⋊C4—28no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—22no—
119C4×D12—28no—
120C4×D12—22no—
121C4×D12D414yescomplemented
122#872—28no—
123#872—22no—
124#872—22no—
125C3×C4⋊C8—22no—
126C3×#102—24no—
127#835—24no—
128#832—24no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#821—24no—
132(C2×D3)⋊C8—24no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×SD16C2×D312yes—
136C22×D8—32no—
137C2×#127C2×D311yescharacteristic
138C2×#121—32no—
139#520—64no—
140#520—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—31no—
148C2×#117—61no—
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#460—24no—
152#459—64no—
153#449—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#102—32no—
158C2×#102—32no—
159C2×#102C2×D312yes—
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—22no—
171C23×D3—21no—
172C23×D3C2×D411yescharacteristic
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8C2×D412yes—
178C2×C3×Q8#4112yes—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—82no—
188C2×C3×D4—82no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×#67—44no—
195C2×#67—82no—
196C2×#67—82no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202#239—28no—
203#239—28no—
204#239—28no—
205#239—28no—
206C2×D12—24no—
207C2×D12—24no—
208C2×D12—24no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C2×D412yes—
213C2×D12C2×D412yes—
214C2×D12#4112yes—
215C2×C4×D3—24no—
216C2×C4×D3—24no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—24no—
222C2×C4×D3—24no—
223C2×C4×D3—24no—
224C2×C4×D3—24no—
225C2×C4×D3—82no—
226C2×C4×D3—82no—
227C2×C4×D3—82no—
228C2×C4×D3C2×D412yes—
229C2×C4×D3#4112yes—
230C3×SD16—48no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C2×D412yes—
237C3×C42—24no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240#217—44no—
241#217—82no—
242#217—82no—
243#213—48no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—22no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—22no—
251C4⋊Dic3—22no—
252C4⋊Dic3C2×D412yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256C4×Dic3—24no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×#41—34no—
263C2×#41—62no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C22×Q8C22×D311yescharacteristic
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—31no—
272C22×D4—61no—
273C22×D4—62no—
274C22×D4C22×D312yes—
275C23×C4—31no—
276C23×C4—61no—
277C23×C4—61no—
278C23×C4—61no—
279C23×C4—61no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×D8—64no—
284C2×D8—64no—
285C2×D8—64no—
286C22×C8—32no—
287C22×C8C22×D312yes—
288#127—22no—
289#127—22no—
290#121—64no—
291#121—64no—
292C4×Q8—22no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8#6714yescomplemented
296C4×D4—34no—
297C4×D4—68no—
298C4×D4—62no—
299C4×D4—68no—
300C4×D4—62no—
301C4×D4—62no—
302#117—68no—
303#117—62no—
304#117—62no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—68no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4—62no—
322C2×C22⋊C4—62no—
323C2×C42—21no—
324C2×C42—31no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C4⋊C8—22no—
328C4⋊C8—62no—
329#103—64no—
330#102—24no—
331#102—64no—
332#102—64no—
333C22⋊C8—64no—
334#95—61no—
335#95—61no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3C22×D411yescharacteristic
344C22×D3—21no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—21no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—22no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3#12112yes—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8#12114yes—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4—42no—
361C3×D4—42no—
362C3×D4—84no—
363C3×D4—84no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—42no—
371C2×C3×C4#12012yes—
372C2×C3×C4#12012yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379#67—88no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3#12112yes—
392D12—216no—
393D12—22no—
394D12—22no—
395D12—22no—
396D12#12114yes—
397C4×D3—216no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—84no—
403C4×D3—84no—
404C4×D3—84no—
405C3×C8—42no—
406C3⋊C8—42no—
407C24—22no—
408C24—31no—
409C24—61no—
410#41—68no—
411#41—68no—
412#41—68no—
413#41—68no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8D3×D412yes—
418C2×Q8#23712yes—
419C2×Q8C2×#6712yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—32no—
427C2×D4—32no—
428C2×D4—32no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—64no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—124no—
440C2×D4—242no—
441C2×D4—242no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—31no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—64no—
460C22×C4—62no—
461C22×C4—242no—
462C22×C4—242no—
463C22×C4—242no—
464C22×C4D3×D411yescharacteristic
465C22×C4D3×D411yescharacteristic
466C22×C4C23×D311yescharacteristic
467SD16—48no—
468D8—128no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—62no—
473C2×C8—62no—
474C2×C8—62no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—32no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—62no—
481C4⋊C4—62no—
482C4⋊C4—61no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C4⋊C4C2×#6712yes—
486C22⋊C4—64no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—64no—
490C22⋊C4—124no—
491C22⋊C4—242no—
492C22⋊C4—242no—
493C42—24no—
494C42—64no—
495C42—61no—
496C42—61no—
497C42D3×D412yes—
498C42C2×#6712yes—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3—82no—
502C22×C3—82no—
503C22×C3—82no—
504C22×C3—82no—
505C22×C3—82no—
506C22×C3C2×#12011yescharacteristic
507C22×C3C2×#12111yescharacteristic
508C22×C3#52511yescharacteristic
509C22×C3#57612yes—
510C22×C3#57712yes—
511C2×D3—22no—
512C2×D3—24no—
513C2×D3—24no—
514C2×D3—42no—
515C2×D3—44no—
516C2×D3—41no—
517C2×D3—42no—
518C2×D3—41no—
519C2×D3—42no—
520C2×D3—44no—
521C3×C4—24no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—42no—
525C3×C4C2×#12111yescharacteristic
526C3×C4C2×#12111yescharacteristic
527C3×C4#52511yescharacteristic
528C3×C4#52511yescharacteristic
529Dic3—22no—
530Dic3—41no—
531Dic3—42no—
532Dic3—41no—
533Dic3—82no—
534C23—32no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—62no—
543C23—62no—
544C23—62no—
545C23—61no—
546C23—62no—
547C23—62no—
548C23—61no—
549C23—122no—
550C23—122no—
551C23C2×D3×D411yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8#93814yes—
556D4—34no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—616no—
562D4—62no—
563D4—62no—
564D4—62no—
565D4—84no—
566D4—84no—
567D4—248no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—32no—
575C2×C4—42no—
576C2×C4—616no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—61no—
586C2×C4—62no—
587C2×C4—61no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—244no—
593C2×C4—244no—
594C2×C4—244no—
595C2×C4#93712yes—
596C2×C4#93712yes—
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×D3×D411yescharacteristic
599C2×C4C2×#23711yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4C22×#6711yescharacteristic
603C8—42no—
604C8—122no—
605C2×C3—162no—
606C2×C3#284711yescharacteristic
607C2×C3#300811yescharacteristic
608C2×C3#301011yescharacteristic
609C2×C3#304512yes—
610C2×C3#311912yes—
611D3—42no—
612D3—44no—
613C22—64no—
614C22—64no—
615C22—62no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—124no—
624C22—121no—
625C22—122no—
626C22—121no—
627C22—122no—
628C22—122no—
629C22—124no—
630C22D3×#12111yescharacteristic
631C22#560611yescharacteristic
632C22#576212yes—
633C22#577412yes—
634C22C2×#93711yescharacteristic
635C4—24no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—62no—
640C4—121no—
641C4—121no—
642C4—122no—
643C4—242no—
644C4D3×#12111yescharacteristic
645C4#560611yescharacteristic
646C4C2×#93811yescharacteristic
647C4#582811yescharacteristic
648C3#3291311yescharacteristic, complemented, Sylow
649C2—122no—
650C2—124no—
651C2—162no—
652C2#8036412yes—
653C2#8038612yes—
654C2#8272311yescharacteristic
655C2#8386011yescharacteristic
656C2#8394711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
hypercenterC2×#460D3
hypocenterC3#32913
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×#5574C2semidirect
2. C2×C3×#460C2semidirect
3. C2×#5172C2semidirect
4. #83189C2semidirect
5. C2×#5072C2semidirect
6. #72048C2semidirect
7. #71627C2semidirect
8. #70480C2semidirect
9. #70476C2semidirect
10. #70208C2semidirect
11. #70200C2semidirect
50. C2×#460D3semidirect
70. C2×#239D4semidirect
85. C3×C4×Q8D4semidirect
121. C4×D12D4semidirect
295. C4×Q8#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×#55742
C2×C3×#4602
C2×#51722
#831892
C2×#50722
#720482
#716272
#704802
#704762
#702082
#702002
#329133