Finite Group Databasebeta

← #10848741

Subgroups — #10848741

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic, complemented
2C2×C3×#489C211yesmaximal, characteristic, complemented
3#83008C211yesmaximal, characteristic, complemented
4C2×#5058C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70276C212yesmaximal, complemented
7#70272C212yesmaximal, complemented
8#70208C212yesmaximal, complemented
9#70204C212yesmaximal, complemented
10#69823C211yesmaximal, characteristic, complemented
11#69807C211yesmaximal, characteristic, complemented
12#25602—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×#947—21no—
17C2×#946C2211yescharacteristic
18C22×#215C2212yes—
19C2×#938C2211yescharacteristic
20C22×#213C2212yes—
21#5725—24no—
22#5725—24no—
23C2×#895—21no—
24C2×#895C2212yes—
25C2×#893—21no—
26C2×#893C2212yes—
27C2×#874C2211yescharacteristic
28C2×C42×D3C2211yescharacteristic
29C3×#489—24no—
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#97C2211yescharacteristic
33#5229C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×#821C2212yes—
36C2×#821C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38#5058—24no—
39#4956—21no—
40#4956—21no—
41#4944—21no—
42#4944—21no—
43#4936C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#532—31no—
49C2×#496—31no—
50C2×#489D311yescharacteristic, complemented
51#2850—31no—
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1961—31no—
56#1879—32no—
57#1877—32no—
58#1749—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C22×#67—21no—
63C2×#239—28no—
64C2×#239—44no—
65C2×#239—44no—
66C22×D12—22no—
67C22×D12—21no—
68C22×D12—21no—
69C22×D12C2312yes—
70C22×C4×D3—22no—
71C22×C4×D3D411yescharacteristic
72C22×C4×D3D411yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C22×C3×C8C2312yes—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8D414yescomplemented
77C3×#124—22no—
78C3×#124—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85C2×#217—21no—
86C2×#217—21no—
87#947—28no—
88#946—42no—
89C2×#215—24no—
90C2×#215—24no—
91C2×#215—42no—
92#938—42no—
93C2×#213—24no—
94C2×#213—24no—
95C2×#213—42no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106#895—28no—
107#895—24no—
108#895—24no—
109#893—28no—
110#893—24no—
111#893—24no—
112#874—22no—
113#874D414yescomplemented
114C42×D3—22no—
115C42×D3—22no—
116C3×#103—24no—
117C3×#102—24no—
118C3×#97—22no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123#821—42no—
124#821—42no—
125(C2×D3)⋊C8—24no—
126#803—22no—
127C22×#41—61no—
128C22×SD16—32no—
129C22×D8—32no—
130#532—64no—
131#532—64no—
132C2×C4⋊Q8C2×D311yescharacteristic
133C2×#127—31no—
134C2×#124—61no—
135C2×#124C2×D311yescharacteristic
136C2×C22⋊Q8—31no—
137C2×#121—31no—
138C2×#121—32no—
139C2×#121—61no—
140C2×C4×D4—32no—
141C2×C4×D4—61no—
142C22×C42—31no—
143#496—64no—
144#489—24no—
145C2×#103C2×D312yes—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102C2×D312yes—
149C2×C22⋊C8—32no—
150C2×#97—31no—
151C2×#97C2×D311yescharacteristic
152#391—31no—
153#390—31no—
154#388—61no—
155#388—61no—
156#386—61no—
157#386—61no—
158#382—31no—
159#340—32no—
160#340—32no—
161#336—32no—
162C24×C3—21no—
163C23×D3—22no—
164C23×D3C2×D411yescharacteristic
165C2×C3×Q8—24no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—42no—
171C2×C3×Q8—42no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×#67—42no—
182C2×#67—42no—
183C2×#67—42no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189#239—432no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—28no—
195C2×D12—24no—
196C2×D12—24no—
197C2×D12—22no—
198C2×D12—42no—
199C2×D12—42no—
200C2×D12—42no—
201C2×D12—42no—
202C2×D12—44no—
203C2×D12—44no—
204C2×D12C2×D414yes—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—28no—
209C2×C4×D3—22no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—41no—
215C2×C4×D3—41no—
216C2×C4×D3—44no—
217C2×C4×D3—44no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4C2×D414yes—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C42C2×D412yes—
227C3×C42C2×D412yes—
228#217—42no—
229#217—42no—
230#217—42no—
231#215—48no—
232#213—48no—
233(C2×D3)⋊C4—28no—
234(C2×D3)⋊C4—28no—
235(C2×D3)⋊C4—42no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238Dic3⋊C4—42no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8#4114yescomplemented
246C2×#41—68no—
247C2×#41—124no—
248C2×#41—124no—
249C22×Q8—21no—
250C22×Q8C22×D311yescharacteristic
251C22×D4—32no—
252C22×D4—62no—
253C22×D4—61no—
254C22×D4—61no—
255C22×D4—61no—
256C22×D4C22×D311yescharacteristic
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—62no—
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C2×D8—64no—
265C2×D8—64no—
266C2×D8—122no—
267C22×C8—32no—
268C22×C8C22×D312yes—
269C4⋊Q8—22no—
270C4⋊Q8#6714yescomplemented
271#127—34no—
272#127—62no—
273#124—22no—
274#124—22no—
275#124—68no—
276C22⋊Q8—122no—
277#121—64no—
278#121—68no—
279#121—64no—
280#121—122no—
281C4×D4—64no—
282C4×D4—68no—
283C4×D4—64no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D312yes—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—62no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—62no—
301C2×C42—62no—
302C2×C42C22×D311yescharacteristic
303#103—24no—
304#102—24no—
305#102—122no—
306#102—122no—
307C22⋊C8—64no—
308#97—22no—
309#97—62no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3C22×D411yescharacteristic
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—24no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3AGL(1,ℤ/8)12yes—
336C3×Q8—48no—
337C3×Q8—42no—
338C3×Q8—42no—
339C3×D4—42no—
340C3×D4—42no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—24no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4#12012yes—
352C2×C3×C4#12012yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4112yes—
356C2×C3×C4H(5,𝔽2)12yes—
357#67—84no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3AGL(1,ℤ/8)12yes—
373D12—24no—
374D12—24no—
375D12—48no—
376D12—48no—
377D12—416no—
378D12—42no—
379D12—42no—
380C4×D3—28no—
381C4×D3—416no—
382C4×D3—44no—
383C4×D3—42no—
384C4×D3—42no—
385C3×C8—42no—
386C3⋊C8—24no—
387C24—21no—
388C24—31no—
389C24—62no—
390#41—1232no—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—34no—
403C2×D4—68no—
404C2×D4—64no—
405C2×D4—64no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—122no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4—122no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—122no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—68no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4—124no—
439C22×C4—124no—
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D411yescharacteristic
442C22×C4C23×D311yescharacteristic
443SD16—128no—
444D8—128no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—34no—
449C2×C8—62no—
450C2×C8—62no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—122no—
456C4⋊C4C2×#6714yes—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—68no—
460C22⋊C4—68no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C42—62no—
466C42—62no—
467C42—61no—
468C42—61no—
469C42D3×D412yes—
470C42C2×#6712yes—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3C2×#12011yescharacteristic
479C22×C3#53211yescharacteristic
480C22×C3#53511yescharacteristic
481C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
483C22×C3#57511yescharacteristic
484C22×C3#57511yescharacteristic
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—44no—
495C2×D3—82no—
496C2×D3—82no—
497C2×D3—82no—
498C2×D3—82no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—44no—
502C3×C4—81no—
503C3×C4#54612yes—
504C3×C4#55012yes—
505Dic3—22no—
506Dic3—41no—
507Dic3—41no—
508Dic3—81no—
509Dic3—81no—
510C23—32no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23—61no—
523C23—61no—
524C23—64no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23C2×D3×D411yescharacteristic
532Q8—48no—
533Q8—42no—
534Q8—42no—
535D4—42no—
536D4—42no—
537D4—64no—
538D4—64no—
539D4—128no—
540D4—128no—
541D4—1216no—
542D4—122no—
543D4—122no—
544D4—244no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—32no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—68no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—124no—
566C2×C4—1216no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4#93712yes—
576C2×C4#93712yes—
577C2×C4C2×D3×D411yescharacteristic
578C2×C4D3×#4112yes—
579C2×C4#100912yes—
580C2×C4C22×#6711yescharacteristic
581C8—42no—
582C8—64no—
583C2×C3—161no—
584C2×C3#243111yescharacteristic
585C2×C3#300311yescharacteristic
586C2×C3#300511yescharacteristic
587C2×C3#317111yescharacteristic
588C2×C3#318011yescharacteristic
589C2×C3#321311yescharacteristic
590C2×C3#321411yescharacteristic
591D3—42no—
592D3—84no—
593C22—62no—
594C22—62no—
595C22—62no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—124no—
608C22—124no—
609C22—124no—
610C22—242no—
611C22—242no—
612C22—242no—
613C22—242no—
614C22#566811yescharacteristic
615C22#567411yescharacteristic
616C22D3×AGL(1,ℤ/8)11yescharacteristic
617C22D3×AGL(1,ℤ/8)11yescharacteristic
618C22#577711yescharacteristic
619C22#577711yescharacteristic
620C22C2×#93711yescharacteristic
621C4—22no—
622C4—22no—
623C4—44no—
624C4—62no—
625C4—81no—
626C4—121no—
627C4—121no—
628C4—241no—
629C4—241no—
630C4#567612yes—
631C4#583412yes—
632C3#2560211yescharacteristic, complemented, Sylow
633C2—122no—
634C2—161no—
635C2—244no—
636C2#8166411yescharacteristic
637C2#8166811yescharacteristic
638C2#8168411yescharacteristic
639C2#8168511yescharacteristic
640C2#8295811yescharacteristic
641C2#8394411yescharacteristic
642C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
hypercenterC2×#489D3
hypocenterC3#25602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#5725C2semidirect
2. C2×C3×#489C2semidirect
3. #83008C2semidirect
4. C2×#5058C2semidirect
5. #71788C2semidirect
6. #70276C2semidirect
7. #70272C2semidirect
8. #70208C2semidirect
9. #70204C2semidirect
10. #69823C2semidirect
11. #69807C2semidirect
50. C2×#489D3semidirect
76. C3×C4⋊Q8D4semidirect
113. #874D4semidirect
245. C2×C3⋊C8#41semidirect
270. C4⋊Q8#67semidirect
632. C3#25602semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×C3×#4892
#830082
C2×#50582
#717882
#702762
#702722
#702082
#702042
#698232
#698072
#256023