Finite Group Databasebeta

← #10848734

Subgroups — #10848734

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×#5716C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3#83008C211yesmaximal, characteristic, complemented
4C2×#5055C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#72047C212yesmaximal, complemented
7#72043C212yesmaximal, complemented
8#72037C212yesmaximal, complemented
9#71765C211yesmaximal, characteristic, complemented
10#69828C211yesmaximal, characteristic, complemented
11#69807C211yesmaximal, characteristic, complemented
12#25602—31nomaximal, Sylow
13C22×#237—21no—
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#938—21no—
20C2×#938C2212yes—
21C2×#938C2211yescharacteristic
22C2×#936—21no—
23C2×D4×Dic3—21no—
24C2×D4×Dic3C2212yes—
25#5716—24no—
26#5716—24no—
27C2×C42×D3C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C3×#496—24no—
30C2×C3×#97C2211yescharacteristic
31#5229C2211yescharacteristic
32C2×#835C2212yes—
33#5221C2211yescharacteristic
34C2×#832C2212yes—
35C2×#832C2212yes—
36C2×#832C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38#5055—24no—
39#4956—21no—
40#4956—21no—
41#4944—21no—
42#4944—21no—
43#4936C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4551C2212yes—
46#4551C2212yes—
47#4550C2212yes—
48C2×#532—31no—
49C2×#496D311yescharacteristic, complemented
50C2×#489—31no—
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×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2312yes—
64C22×#67—22no—
65C22×#67—21no—
66C23×Dic3—22no—
67C2×#237—28no—
68C2×#237—44no—
69C2×#237—44no—
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6—21no—
74C2×C3×SD16—24no—
75C2×C3×SD16—24no—
76C2×C3×SD16—42no—
77C2×C3×D8—24no—
78C2×C3×D8—24no—
79C2×C3×D8—42no—
80C22×C3×C8C2312yes—
81C3×#127—22no—
82C3×#127D414yescomplemented
83C3×#124—22no—
84C3×#124—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—22no—
89C2×#217—22no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#946—42no—
94#938—28no—
95#938—24no—
96#938—24no—
97#938—42no—
98#936—28no—
99D4×Dic3—28no—
100D4×Dic3—24no—
101D4×Dic3—24no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2312yes—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112C42×D3—22no—
113C42×D3—22no—
114#869—22no—
115#869D414yescomplemented
116C3×#97—22no—
117#835—24no—
118#832—24no—
119#832—42no—
120#832—42no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125(C2×D3)⋊C8—24no—
126#803—22no—
127C22×#41—61no—
128C22×SD16C2×D312yes—
129C22×D8C2×D312yes—
130#532—64no—
131#532—64no—
132C2×C4⋊Q8—31no—
133C2×#127C2×D311yescharacteristic
134C2×#124—61no—
135C2×#124C2×D311yescharacteristic
136C2×C22⋊Q8—31no—
137C2×#121—32no—
138C2×#121—31no—
139C2×#121—61no—
140C2×C4×D4—32no—
141C2×C4×D4—61no—
142C22×C42—31no—
143#496—24no—
144#489—64no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102—32no—
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—22no—
163C24×C3—21no—
164C23×D3C2×D411yescharacteristic
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4C2×D414yes—
185C22×C3×C4—21no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C2×#67—28no—
190C2×#67—44no—
191C2×#67—44no—
192C2×#67—42no—
193C2×#67—42no—
194C2×#67—42no—
195C22×Dic3—28no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—44no—
200C22×Dic3—44no—
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204#237—432no—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—41no—
214C2×C4×D3—41no—
215C2×Dic6—24no—
216C2×Dic6—42no—
217C2×Dic6—42no—
218C3×SD16—48no—
219C3×D8—48no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8#4114yescomplemented
223C3×C22⋊C4—42no—
224C3×C22⋊C4—42no—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227#217—28no—
228#217—28no—
229#217—42no—
230#217—42no—
231#217—42no—
232(C2×D3)⋊C4—42no—
233C4⋊Dic3—24no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C4⋊Dic3C2×D414yes—
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—22no—
246C2×#41—68no—
247C2×#41—124no—
248C2×#41—124no—
249C22×Q8—61no—
250C22×Q8C22×D311yescharacteristic
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—62no—
254C22×D4—61no—
255C22×D4C22×D311yescharacteristic
256C22×D4C22×D312yes—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—62no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—42no—
267C22×C8—32no—
268C22×C8C22×D312yes—
269C4⋊Q8—34no—
270C4⋊Q8—62no—
271#127—22no—
272#127#6714yescomplemented
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—32no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
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—64no—
304#102—64no—
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—24no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3C22×D411yescharacteristic
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3AGL(1,ℤ/8)12yes—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×D4—24no—
339C3×D4—24no—
340C3×D4—48no—
341C3×D4—48no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—42no—
345C3×D4—42no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4#12012yes—
354C2×C3×C4#12012yes—
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4112yes—
358C2×C3×C4H(5,𝔽2)12yes—
359#67—416no—
360#67—84no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—24no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—416no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3AGL(1,ℤ/8)12yes—
380C4×D3—28no—
381C4×D3—44no—
382C4×D3—42no—
383C4×D3—42no—
384Dic6—48no—
385C3×C8—24no—
386C3⋊C8—42no—
387C24—22no—
388C24—21no—
389C24—31no—
390#41—1232no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—64no—
396C2×Q8—122no—
397C2×Q8—122no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—68no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—122no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4C2×#6714yes—
420C22×C4—21no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—62no—
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—48no—
444D8—48no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8#23714yescomplemented
451C4⋊C4—34no—
452C4⋊C4—64no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—122no—
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—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3C2×#12011yescharacteristic
486C22×C3#53211yescharacteristic
487C22×C3#53511yescharacteristic
488C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
490C22×C3#57511yescharacteristic
491C22×C3#57511yescharacteristic
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—41no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—81no—
502C3×C4#54612yes—
503C3×C4#55012yes—
504Dic3—22no—
505Dic3—41no—
506Dic3—44no—
507Dic3—41no—
508Dic3—81no—
509Dic3—81no—
510C23—24no—
511C23—32no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23C2×D3×D411yescharacteristic
532Q8—42no—
533Q8—42no—
534Q8—128no—
535D4—24no—
536D4—24no—
537D4—48no—
538D4—48no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—1216no—
544D4—244no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—32no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—68no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—64no—
563C2×C4—124no—
564C2×C4—1216no—
565C2×C4—122no—
566C2×C4—122no—
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×C4C2×#23712yes—
579C2×C4#100412yes—
580C2×C4C22×#6711yescharacteristic
581C8—24no—
582C8—122no—
583C2×C3—84no—
584C2×C3—161no—
585C2×C3#243111yescharacteristic
586C2×C3#300311yescharacteristic
587C2×C3#300511yescharacteristic
588C2×C3#317111yescharacteristic
589C2×C3#318011yescharacteristic
590C2×C3#321311yescharacteristic
591C2×C3#321411yescharacteristic
592D3—42no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—62no—
597C22—62no—
598C22—62no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—121no—
611C22—121no—
612C22—121no—
613C22—121no—
614C22#571511yescharacteristic
615C22#571611yescharacteristic
616C22D3×AGL(1,ℤ/8)11yescharacteristic
617C22D3×AGL(1,ℤ/8)11yescharacteristic
618C22#577311yescharacteristic
619C22#577311yescharacteristic
620C22C2×#93711yescharacteristic
621C4—22no—
622C4—22no—
623C4—62no—
624C4—81no—
625C4—124no—
626C4—121no—
627C4—121no—
628C4—241no—
629C4—241no—
630C4#571812yes—
631C4#580612yes—
632C3#2560211yescharacteristic, complemented, Sylow
633C2—84no—
634C2—122no—
635C2—161no—
636C2#8183611yescharacteristic
637C2#8183811yescharacteristic
638C2#8184911yescharacteristic
639C2#8185011yescharacteristic
640C2#8318711yescharacteristic
641C2#8394411yescharacteristic
642C2#8394511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#496C2
socleC23×C3C22×D4
hypercenterC2×#496D3
hypocenterC3#25602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#496

Complements

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

NH ≅ G/Ntype
1. C2×#5716C2semidirect
2. C2×C3×#496C2semidirect
3. #83008C2semidirect
4. C2×#5055C2semidirect
5. #72054C2semidirect
6. #72047C2semidirect
7. #72043C2semidirect
8. #72037C2semidirect
9. #71765C2semidirect
10. #69828C2semidirect
11. #69807C2semidirect
49. C2×#496D3semidirect
82. C3×#127D4semidirect
115. #869D4semidirect
222. C2×C3×C8#41semidirect
272. #127#67semidirect
450. C2×C8#237semidirect
632. C3#25602semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×C3×#4962
#830082
C2×#50552
#720542
#720472
#720432
#720372
#717652
#698282
#698072
#256023