Finite Group Databasebeta

← #10826872

Subgroups — #10826872

Order 768.

1950 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2#87239C211yesmaximal, characteristic, complemented
3#84081C212yesmaximal, complemented
4C2×#5077C211yesmaximal, characteristic, complemented
5C2×#5061C211yesmaximal, characteristic, complemented
6#72129C211yesmaximal, characteristic, complemented
7#71769C212yesmaximal, complemented
8#71763C211yesmaximal, characteristic, complemented
9#71731C212yesmaximal, complemented
10#71723C211yesmaximal, characteristic, complemented
11#71591C212yesmaximal, complemented
12#23912—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic, complemented
14C23×C3×D4C2211yescharacteristic
15C2×#1004C2211yescharacteristic
16C3×#535—24no—
17C3×#535—24no—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic, complemented
20C2×C3×#124C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic, complemented
22C2×C3×#121—22no—
23C2×C3×#120—22no—
24C2×C3×#117C2211yescharacteristic, complemented
25C2×#953—22no—
26C2×#942C2212yescomplemented
27#5806—24no—
28#5805C2212yes—
29#5804—22no—
30D4×#67—24no—
31#5798C2211yescharacteristic
32C2×#932C2212yescomplemented
33#5791C2212yes—
34#5239—24no—
35#5238—24no—
36#5225C2212yescomplemented
37C2×#833C2211yescharacteristic
38C2×#833C2211yescharacteristic
39C2×#833C2212yescomplemented
40#5215C2212yescomplemented
41#5207C2212yescomplemented
42#5077—24no—
43#5061—24no—
44C2×#805C2212yescomplemented
45C2×#805C2212yescomplemented
46#4523C2212yescomplemented
47#4523C2212yescomplemented
48#3484—31no—
49C2×#535D311yescharacteristic, complemented
50C2×#455—31no—
51C2×#453—31no—
52#3005—32no—
53#2004—32no—
54#2002—32no—
55#1897—32no—
56#1887—31no—
57#1885—31no—
58#1783—31no—
59C25×C3D412yes—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—24no—
63C2×C3×#41—42no—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—24no—
71C22×C3×D4—42no—
72C22×C3×D4D412yes—
73C22×C3×D4C2311yescharacteristic
74C22×C3×D4C2311yescharacteristic
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—22no—
79C22×#67—24no—
80#1004—22no—
81#1004—22no—
82#1004—42no—
83C2×#237—22no—
84C2×D3×D4—22no—
85C2×#235C2312yes—
86C3×#127—22no—
87C3×#127—22no—
88C3×#127—22no—
89C3×#127D414yescomplemented
90C3×#124—22no—
91C3×#124—22no—
92C3×#124—22no—
93C3×#124D414yescomplemented
94C3×#121—216no—
95C3×#120—216no—
96C3×#117—22no—
97C3×#117—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104#953—28no—
105#953—22no—
106#953—24no—
107C2×#217—24no—
108C2×#217—22no—
109C2×#216—22no—
110#942—24no—
111#942—42no—
112C2×#215—22no—
113#940—22no—
114#939—28no—
115#938—24no—
116#937—24no—
117#936—22no—
118#935—24no—
119D4×Dic3—24no—
120C2×#214—22no—
121#932—24no—
122#932—42no—
123C2×#213—22no—
124#930D414yescomplemented
125#929—22no—
126C4×#67—24no—
127#926C2312yes—
128#924D414yescomplemented
129C2×#208C2312yes—
130C2×#208C2312yes—
131#835—24no—
132#833—22no—
133#833—22no—
134#833—42no—
135#832—24no—
136#830—24no—
137#805—24no—
138#805—24no—
139C2×H(5,𝔽2)—31no—
140C22×#41C2×D311yescharacteristic, complemented
141C23×D4C2×D311yescharacteristic
142C2×#137—32no—
143C2×AGL(1,ℤ/8)—32no—
144#546—64no—
145D42—64no—
146#535—24no—
147#535—24no—
148#535—32no—
149#534—62no—
150C2×#127C2×D311yescharacteristic
151C2×#127C2×D311yescharacteristic, complemented
152C2×#124C2×D311yescharacteristic
153C2×#124C2×D311yescharacteristic, complemented
154#525—32no—
155C2×#121—22no—
156C2×#120—22no—
157C2×#120—62no—
158#518—31no—
159C2×#117C2×D311yescharacteristic, complemented
160#455—64no—
161#453—64no—
162#452—64no—
163#449—64no—
164C2×#104—32no—
165C2×#104—32no—
166#419—32no—
167#418—32no—
168C2×#100—31no—
169C2×#100—31no—
170C2×#100—32no—
171#407—32no—
172#339—32no—
173#339—32no—
174C24×C3—21no—
175C24×C3—24no—
176C24×C3—24no—
177C24×C3—42no—
178C24×C3—84no—
179C24×C3C2×D412yes—
180C24×C3C2×D411yescharacteristic
181C24×C3C2×D412yes—
182C23×D3—22no—
183C3×#41—48no—
184C3×#41—48no—
185C2×C3×Q8—21no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8C2×D412yescomplemented
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—28no—
192C2×C3×D4—28no—
193C2×C3×D4—24no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—22no—
203C2×C3×D4—48no—
204C2×C3×D4—48no—
205C2×C3×D4—42no—
206C2×C3×D4—42no—
207C2×C3×D4—42no—
208C2×C3×D4—84no—
209C2×C3×D4—84no—
210C2×C3×D4C2×D412yes—
211C2×C3×D4C2×D412yes—
212C2×C3×D4C2×D412yescomplemented
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—22no—
216C22×C3×C4—24no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—42no—
221C22×C3×C4—42no—
222C22×C3×C4C2×D414yes—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C2×#67—24no—
226C2×#67—24no—
227C2×#67—22no—
228C2×#67—28no—
229C2×#67—44no—
230C2×#67—44no—
231C2×#67—88no—
232C22×Dic3—22no—
233C22×Dic3—24no—
234#237—44no—
235#237—44no—
236D3×D4—44no—
237D3×D4—44no—
238#235—22no—
239#235—22no—
240#235—22no—
241#235—22no—
242C2×D12C2×D412yes—
243C2×C4×D3—22no—
244C2×Dic6C2×D412yes—
245C3×C4⋊C4—24no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—28no—
248C3×C22⋊C4—28no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42C2×D412yescomplemented
252C3×C42C2×D412yescomplemented
253#217—24no—
254#217—28no—
255#217—24no—
256#217—22no—
257#216—44no—
258#215—44no—
259#214—44no—
260#213—44no—
261(C2×D3)⋊C4—24no—
262C4⋊Dic3C2×D414yes—
263Dic3⋊C4—24no—
264C4×Dic3—22no—
265#208—22no—
266#208—22no—
267#208—22no—
268#208—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C25#6712yes—
272H(5,𝔽2)—62no—
273H(5,𝔽2)—62no—
274H(5,𝔽2)—122no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—24no—
278C2×#41—32no—
279C2×#41—42no—
280C2×#41—62no—
281C22×Q8C22×D311yescharacteristic
282C22×D4—21no—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4—22no—
286C22×D4—22no—
287C22×D4—24no—
288C22×D4—42no—
289C22×D4—62no—
290C22×D4—62no—
291C22×D4—64no—
292C22×D4#6712yes—
293C22×D4C22×D311yescharacteristic
294C22×D4C22×D311yescharacteristic
295C22×D4C22×D311yescharacteristic
296C23×C4—21no—
297C23×C4C22×D311yescharacteristic
298#137—64no—
299#137—122no—
300AGL(1,ℤ/8)—64no—
301AGL(1,ℤ/8)—122no—
302C2×Q16—62no—
303C2×SD16—62no—
304C2×SD16—62no—
305C2×D8—62no—
306C2×M16—32no—
307C2×M16—32no—
308#127—22no—
309#127—22no—
310#127—22no—
311#127—62no—
312#127#6714yescomplemented
313#124—22no—
314#124—22no—
315#124—22no—
316#124—62no—
317#124#6714yescomplemented
318#123—64no—
319#123—62no—
320C22⋊Q8—34no—
321#121—216no—
322#121—34no—
323#121—64no—
324#121—68no—
325#120—216no—
326#120—64no—
327#120—68no—
328#120—62no—
329#120—64no—
330C4×D4—64no—
331C4×D4—64no—
332#117—22no—
333#117—22no—
334#117—32no—
335C2×C4⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—64no—
340C2×C22⋊C4—62no—
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343#104—64no—
344#104—64no—
345#103—64no—
346#102—64no—
347#100—62no—
348#100—62no—
349#100—122no—
350C22⋊C8—64no—
351C23×C3—22no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—42no—
360C23×C3—48no—
361C23×C3—44no—
362C23×C3—44no—
363C23×C3—48no—
364C23×C3—44no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—84no—
368C23×C3—88no—
369C23×C3—88no—
370C23×C3#12012yes—
371C23×C3#12014yes—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—44no—
375C22×D3—84no—
376C3×Q8—24no—
377C3×Q8—42no—
378C3×D4—24no—
379C3×D4—24no—
380C3×D4—24no—
381C3×D4—48no—
382C3×D4—48no—
383C3×D4—42no—
384C3×D4—42no—
385C3×D4—42no—
386C3×D4—88no—
387C3×D4—88no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—24no—
391C2×C3×C4—24no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—48no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4#12012yes—
401C2×C3×C4C22×D412yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4H(5,𝔽2)12yes—
405#67—48no—
406#67—48no—
407#67—42no—
408#67—42no—
409#67—88no—
410C2×Dic3—24no—
411C2×Dic3—22no—
412C2×Dic3—44no—
413C2×Dic3—84no—
414D12—22no—
415D12—22no—
416C4×D3—42no—
417C4×D3—42no—
418Dic6—22no—
419Dic6—22no—
420C3⋊C8—42no—
421C3⋊C8—42no—
422C24—21no—
423C24—24no—
424C24—24no—
425C24—42no—
426C24—62no—
427C24—84no—
428C24C2×#6712yes—
429C24C2×#6711yescharacteristic
430C24C2×#6712yes—
431#41—48no—
432#41—48no—
433#41—62no—
434#41—62no—
435#41—62no—
436#41—62no—
437#41—124no—
438#41—124no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—32no—
443C2×Q8C2×#6712yescomplemented
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—28no—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—32no—
459C2×D4—48no—
460C2×D4—48no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—64no—
465C2×D4—64no—
466C2×D4—62no—
467C2×D4—68no—
468C2×D4—84no—
469C2×D4—84no—
470C2×D4—124no—
471C2×D4—124no—
472C2×D4—124no—
473C2×D4—124no—
474C2×D4—248no—
475C2×D4C2×#6712yes—
476C2×D4C2×#6712yes—
477C2×D4C2×#6712yescomplemented
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—62no—
488C22×C4—64no—
489C22×C4—62no—
490C22×C4D3×D414yes—
491C22×C4C2×#6711yescharacteristic
492C22×C4C23×D311yescharacteristic
493Q16—124no—
494SD16—124no—
495SD16—124no—
496D8—124no—
497M16—62no—
498M16—62no—
499M16—62no—
500M16—62no—
501C2×C8—62no—
502C2×C8—62no—
503C4⋊C4—24no—
504C4⋊C4—34no—
505C4⋊C4—64no—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—64no—
510C22⋊C4—68no—
511C22⋊C4—64no—
512C22⋊C4—62no—
513C22⋊C4—64no—
514C42—21no—
515C42—21no—
516C42—62no—
517C42C2×#6712yescomplemented
518C42C2×#6712yescomplemented
519C22×C3—22no—
520C22×C3—24no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—44no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—44no—
528C22×C3—44no—
529C22×C3—44no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—44no—
533C22×C3—82no—
534C22×C3—82no—
535C22×C3—82no—
536C22×C3—816no—
537C22×C3—82no—
538C22×C3C2×#12011yescharacteristic
539C22×C3C2×#12011yescharacteristic
540C22×C3#53411yescharacteristic
541C2×D3—42no—
542C2×D3—44no—
543C2×D3—164no—
544C3×C4—22no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4D4212yes—
549C3×C4D4212yes—
550Dic3—42no—
551Dic3—44no—
552C23—22no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—48no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—48no—
566C23—41no—
567C23—41no—
568C23—62no—
569C23—84no—
570C23—88no—
571C23—88no—
572C23—124no—
573C23—244no—
574C23#93712yes—
575C23#95314yes—
576C23C22×#6711yescharacteristic
577Q8—24no—
578Q8—42no—
579Q8—62no—
580Q8—62no—
581D4—24no—
582D4—24no—
583D4—24no—
584D4—48no—
585D4—48no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—62no—
590D4—62no—
591D4—88no—
592D4—88no—
593D4—128no—
594D4—128no—
595D4—122no—
596D4—122no—
597D4—248no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—124no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—244no—
616C2×C4#93712yes—
617C2×C4C2×D3×D412yes—
618C2×C4#100412yes—
619C2×C4C22×#6711yescharacteristic
620C2×C4C22×#6711yescharacteristic
621C8—122no—
622C8—122no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—44no—
626C2×C3—42no—
627C2×C3—82no—
628C2×C3—82no—
629C2×C3#240311yescharacteristic
630C2×C3#301912yes—
631D3—82no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—22no—
636C22—22no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—816no—
650C22—82no—
651C22—122no—
652C22—124no—
653C22—484no—
654C22C2×#93711yescharacteristic
655C22#580411yescharacteristic
656C22C2×#95311yescharacteristic
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—122no—
662C4—124no—
663C4D4×#6712yes—
664C4D4×#6712yes—
665C3#2391211yescharacteristic, complemented, Sylow
666C2—21no—
667C2—21no—
668C2—42no—
669C2—44no—
670C2—82no—
671C2—82no—
672C2—242no—
673C2#8339412yes—
674C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#535C2
socleC22×C3C2×#120
hypercenterC2×#535D3
hypocenterC3#23912
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. #87239C2semidirect
3. #84081C2semidirect
4. C2×#5077C2semidirect
5. C2×#5061C2semidirect
6. #72129C2semidirect
7. #71769C2semidirect
8. #71763C2semidirect
9. #71731C2semidirect
10. #71723C2semidirect
11. #71591C2semidirect
13. C22×C3×#41C22semidirect
19. C2×C3×#127C22semidirect
21. C2×C3×#124C22semidirect
24. C2×C3×#117C22semidirect
26. C2×#942C22semidirect
32. C2×#932C22semidirect
36. #5225C22semidirect
39. C2×#833C22semidirect
40. #5215C22semidirect
41. #5207C22semidirect
44. C2×#805C22semidirect
45. C2×#805C22semidirect
46. #4523C22semidirect
47. #4523C22semidirect
49. C2×#535D3semidirect
89. C3×#127D4semidirect
93. C3×#124D4semidirect
124. #930D4semidirect
128. #924D4semidirect
140. C22×#41C2×D3semidirect
151. C2×#127C2×D3semidirect
153. C2×#124C2×D3semidirect
159. C2×#117C2×D3semidirect
188. C2×C3×Q8C2×D4semidirect
212. C2×C3×D4C2×D4semidirect
251. C3×C42C2×D4semidirect
252. C3×C42C2×D4semidirect
312. #127#67semidirect
317. #124#67semidirect
443. C2×Q8C2×#67semidirect
477. C2×D4C2×#67semidirect
517. C42C2×#67semidirect
518. C42C2×#67semidirect
665. C3#23912semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
#872392
#840812
C2×#50772
C2×#50612
#721292
#717692
#717632
#717312
#717232
#715912
#239123