Finite Group Databasebeta

← #10944092

Subgroups — #10944092

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×#5523C211yesmaximal, characteristic, complemented
2#82286C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×#4843C211yesmaximal, characteristic, complemented
5C2×#4734C212yesmaximal, complemented
6C3×#1798C211yesmaximal, characteristic, complemented
7#70486C211yesmaximal, characteristic, complemented
8#70480C211yesmaximal, characteristic, complemented
9#69932C212yesmaximal, complemented
10#68686C212yesmaximal, complemented
11#68673C212yesmaximal, complemented
12#32913—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#930C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×D24C2212yes—
18C22×#204C2212yes—
19#5523—24no—
20#5523—24no—
21C2×#892—21no—
22C2×D3×C4⋊C4—21no—
23C2×#67⋊C4—21no—
24C2×#67⋊C4—21no—
25C2×#874C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#821C2212yes—
36C2×#816C2212yes—
37#4991—21no—
38#4991—21no—
39#4960—21no—
40#4960—21no—
41#4939C2212yes—
42#4936C2211yescharacteristic
43#4848—24no—
44#4843—24no—
45#4734—28no—
46C3×#336C2212yes—
47#4551C2212yes—
48#4551C2212yes—
49C2×#520—31no—
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#1798D311yescharacteristic, complemented
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×#67—22no—
63C23×Dic3—21no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235D414yescomplemented
70C22×D12—21no—
71C22×D12—21no—
72C22×D12C2312yes—
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×C4×D3—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2312yes—
79C22×C3×C8C2312yes—
80C3×#117—22no—
81C3×#117—22no—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88#930—24no—
89#930—24no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×D24—24no—
106C2×D24—24no—
107C2×D24—24no—
108C2×#204—24no—
109C2×#204—24no—
110C2×#204—24no—
111#892—28no—
112D3×C4⋊C4—28no—
113#67⋊C4—28no—
114#67⋊C4—28no—
115#874—22no—
116#874—22no—
117C4×D12—22no—
118C4×D12—22no—
119C4×D12—22no—
120C4×D12D414yescomplemented
121C4×Dic6—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6D414yescomplemented
125C3×C4⋊C8—22no—
126C3×C4⋊C8—22no—
127C3×C22⋊C8—24no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#821—24no—
131#821—24no—
132#821—24no—
133#816—24no—
134C22×#41—31no—
135C22×SD16—32no—
136C22×D8—32no—
137C2×#127—31no—
138C2×#121—32no—
139#520—64no—
140#520—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×C4×D4—61no—
147C2×#117—61no—
148C2×#117C2×D311yescharacteristic
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#460—64no—
152#459—64no—
153#449—68no—
154C2×C4⋊C8C2×D311yescharacteristic
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#102—32no—
158C2×#102—32no—
159C2×#102—32no—
160C2×C22⋊C8C2×D312yes—
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#336C2×D312yes—
170C24×C3C2×D411yescharacteristic
171C23×D3—21no—
172C23×D3—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4#4112yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—24no—
187C2×#67—24no—
188C2×#67—44no—
189C2×#67—82no—
190C2×#67—82no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—24no—
195C22×Dic3—22no—
196C22×Dic3—82no—
197#235—28no—
198#235—28no—
199#235—28no—
200#235—28no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—42no—
211C2×D12—42no—
212C2×D12—82no—
213C2×D12—82no—
214C2×D12C2×D412yes—
215C2×D12C2×D412yes—
216C2×D12#4112yes—
217C2×C4×D3—24no—
218C2×C4×D3—24no—
219C2×C4×D3—24no—
220C2×C4×D3—24no—
221C2×C4×D3—24no—
222C2×C4×D3—82no—
223C2×C4×D3—82no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C2×D412yes—
228C2×Dic6C2×D412yes—
229C2×Dic6#4112yes—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42C2×D412yes—
243C3×C42C2×D412yes—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—44no—
247(C2×D3)⋊C4—82no—
248(C2×D3)⋊C4—82no—
249C4⋊Dic3—24no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—22no—
253C4⋊Dic3—22no—
254C4⋊Dic3C2×D412yes—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—24no—
257Dic3⋊C4—24no—
258C4×Dic3—24no—
259C4×Dic3—24no—
260D24—48no—
261#204—48no—
262C2×#41—34no—
263C2×#41—62no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C22×Q8—31no—
269C22×D4—31no—
270C22×D4—32no—
271C22×D4—61no—
272C22×D4—62no—
273C22×D4—61no—
274C22×D4—61no—
275C23×C4—61no—
276C23×C4—61no—
277C23×C4—61no—
278C23×C4—61no—
279C23×C4C22×D311yescharacteristic
280C2×SD16—64no—
281C2×SD16—64no—
282C2×SD16—64no—
283C2×D8—64no—
284C2×D8—64no—
285C2×D8—64no—
286C22×C8C22×D312yes—
287C22×C8C22×D312yes—
288#127—62no—
289#127—62no—
290#121—64no—
291#121—64no—
292C4×Q8—34no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×D4—34no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—68no—
300C4×D4—68no—
301C4×D4—62no—
302#117—22no—
303#117—22no—
304#117—68no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—68no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4D1211yescharacteristic
315C2×C4⋊C4D1211yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—62no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4D1211yescharacteristic
322C2×C22⋊C4D1211yescharacteristic
323C2×C42—61no—
324C2×C42—61no—
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327C4⋊C8—22no—
328C4⋊C8—22no—
329#103—64no—
330#102—64no—
331#102—64no—
332#102—64no—
333C22⋊C8—24no—
334#95—61no—
335#95—61no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3#12112yes—
341C23×C3C22×D411yescharacteristic
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4#12012yes—
367C2×C3×C4#12012yes—
368C2×C3×C4#12112yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375#67—216no—
376#67—88no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—84no—
388D12—22no—
389D12—22no—
390D12—22no—
391D12—42no—
392D12—42no—
393D12—42no—
394D12—42no—
395D12—84no—
396D12—84no—
397D12#12114yes—
398C4×D3—216no—
399C4×D3—84no—
400C4×D3—84no—
401Dic6—22no—
402Dic6—22no—
403Dic6—22no—
404Dic6#12114yes—
405C3×C8—42no—
406C3×C8—42no—
407C24—61no—
408C24—62no—
409C24C2×D1211yescharacteristic
410#41—68no—
411#41—68no—
412#41—68no—
413#41—68no—
414C2×Q8—32no—
415C2×Q8—32no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×D4—32no—
421C2×D4—32no—
422C2×D4—32no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—64no—
426C2×D4—61no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—61no—
431C2×D4—62no—
432C2×D4—61no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—124no—
436C2×D4—122no—
437C2×D4—122no—
438C2×D4—242no—
439C2×D4—242no—
440C2×D4—242no—
441C2×D4—242no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—64no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—61no—
450C22×C4—64no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—64no—
454C22×C4—64no—
455C22×C4—62no—
456C22×C4—242no—
457C22×C4—242no—
458C22×C4—242no—
459C22×C4C2×D1211yescharacteristic
460C22×C4C2×D1211yescharacteristic
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4C2×D1211yescharacteristic
464C22×C4D3×D412yes—
465C22×C4#23912yes—
466C22×C4C23×D311yescharacteristic
467SD16—128no—
468D8—128no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—32no—
478C4⋊C4—32no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4—64no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—64no—
489C22⋊C4—64no—
490C22⋊C4—124no—
491C22⋊C4—242no—
492C22⋊C4—242no—
493C42—21no—
494C42—21no—
495C42—64no—
496C42—64no—
497C42D3×D412yes—
498C42D3×D412yes—
499C22×C3—22no—
500C22×C3—24no—
501C22×C3—42no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3C2×#12011yescharacteristic
505C22×C3C2×#12111yescharacteristic
506C22×C3#52511yescharacteristic
507C22×C3#57612yes—
508C22×C3#57712yes—
509C2×D3—24no—
510C2×D3—44no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—44no—
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—22no—
523C3×C4—22no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4C2×#12111yescharacteristic
527C3×C4C2×#12111yescharacteristic
528C3×C4#52511yescharacteristic
529C3×C4#52511yescharacteristic
530Dic3—24no—
531Dic3—42no—
532Dic3—42no—
533Dic3—82no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—21no—
538C23—62no—
539C23—62no—
540C23—62no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23#89512yes—
551C23C22×D1211yescharacteristic
552Q8—34no—
553Q8—62no—
554Q8—62no—
555Q8—62no—
556D4—34no—
557D4—616no—
558D4—62no—
559D4—62no—
560D4—62no—
561D4—122no—
562D4—122no—
563D4—122no—
564D4—122no—
565D4—244no—
566D4—244no—
567D4—248no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—616no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—244no—
592C2×C4—244no—
593C2×C4—244no—
594C2×C4#88212yes—
595C2×C4#88212yes—
596C2×C4#89512yes—
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×D3×D411yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×#23911yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C8—42no—
604C8—42no—
605C2×C3—42no—
606C2×C3#284711yescharacteristic
607C2×C3#300811yescharacteristic
608C2×C3#301011yescharacteristic
609C2×C3#304512yes—
610C2×C3#311912yes—
611D3—44no—
612D3—162no—
613C22—22no—
614C22—24no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—64no—
619C22—124no—
620C22—122no—
621C22—122no—
622C22—124no—
623C22—242no—
624C22—242no—
625C22—242no—
626C22—242no—
627C22—242no—
628C22—242no—
629C22—242no—
630C22C2×#88211yescharacteristic
631C22C2×#89511yescharacteristic
632C22#552411yescharacteristic
633C22#575112yes—
634C22#575212yes—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—64no—
641C4—122no—
642C4—122no—
643C4—242no—
644C4D3×#12111yescharacteristic
645C4D3×#12111yescharacteristic
646C4#562711yescharacteristic
647C4#562711yescharacteristic
648C3#3291311yescharacteristic, complemented, Sylow
649C2—42no—
650C2—124no—
651C2—482no—
652C2#7404811yescharacteristic
653C2#7867011yescharacteristic
654C2#7867811yescharacteristic
655C2#7908812yes—
656C2#7938412yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1798C2
socleC23×C3C22×D4
hypercenter#1798D3
hypocenterC3#32913
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
1. C2×#5523C2semidirect
2. #82286C2semidirect
3. C2×#4848C2semidirect
4. C2×#4843C2semidirect
5. C2×#4734C2semidirect
6. C3×#1798C2semidirect
7. #70486C2semidirect
8. #70480C2semidirect
9. #69932C2semidirect
10. #68686C2semidirect
11. #68673C2semidirect
59. #1798D3semidirect
69. C2×#235D4semidirect
120. C4×D12D4semidirect
124. C4×Dic6D4semidirect
648. C3#32913semidirect

Maximal subgroups

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

subgroupindex
C2×#55232
#822862
C2×#48482
C2×#48432
C2×#47342
C3×#17982
#704862
#704802
#699322
#686862
#686732
#329133