Finite Group Databasebeta

← #10822266

Subgroups — #10822266

Order 768.

2290 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D8C211yesmaximal, characteristic, complemented
2C2×#5805C211yesmaximal, characteristic, complemented
3C22×#932C211yesmaximal, characteristic, complemented
4#84044C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6C2×#5113C211yesmaximal, characteristic, complemented
7#72045C214yesmaximal, complemented
8#72044C214yesmaximal, complemented
9#71965C211yesmaximal, characteristic, complemented
10#23422—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C22×C3×D8—24no—
15C22×C3×D8—24no—
16C22×C3×D8—24no—
17C22×C3×D8C2214yes—
18C23×C3×C8C2211yescharacteristic
19C2×#953—22no—
20#5805—24no—
21#5805—24no—
22C2×#940C2212yescomplemented
23C2×#938—22no—
24C2×#936C2212yescomplemented
25C22×#214C2212yescomplemented
26C2×#932—24no—
27C2×#932—24no—
28C2×#932—24no—
29C22×#213C2212yescomplemented
30C2×#926C2211yescharacteristic
31C2×#926C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5247—22no—
34#5221—22no—
35#5221C2214yescomplemented
36#5215—24no—
37C2×#832C2214yescomplemented
38#5113—24no—
39C2×#821C2212yescomplemented
40C2×#816C2212yescomplemented
41#4550C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C23×D8D311yescharacteristic, complemented
44C2×#535—31no—
45C2×#418—31no—
46C2×#416—31no—
47#2309—31no—
48#2012—34no—
49#2011—34no—
50#1729—31no—
51C25×C3—21no—
52C25×C3—21no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4D414yes—
66C22×C3×D4D414yescomplemented
67C22×C3×D4C2312yes—
68C22×C3×D4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C22×#67—24no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235—22no—
77C2×#235D414yes—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C2×C3×D8—232no—
82C2×C3×D8—28no—
83C2×C3×D8—28no—
84C2×C3×D8—28no—
85C22×C3×C8—22no—
86C22×C3×C8—22no—
87C22×C3×C8D414yescomplemented
88C22×C3×C8C2312yes—
89#953—216no—
90C2×#217—24no—
91C2×#217—21no—
92C2×#217—28no—
93C2×#217D412yes—
94#940—24no—
95#940—24no—
96#938—216no—
97#936—24no—
98#936—24no—
99C2×#214—24no—
100C2×#214—24no—
101C2×#214—24no—
102#932—232no—
103C2×#213—24no—
104C2×#213—24no—
105C2×#213—24no—
106#926—22no—
107#926—22no—
108#926—22no—
109#926—22no—
110C2×C4⋊Dic3D412yes—
111C2×C4⋊Dic3C2312yes—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2312yes—
114C2×#208—22no—
115C2×#208—22no—
116C2×#208D414yescomplemented
117C22×C3⋊C8C2312yes—
118#832—28no—
119#821—24no—
120#816—24no—
121C22×#41—31no—
122C23×D4C2×D311yescharacteristic
123C23×D4C2×D311yescharacteristic
124C2×AGL(1,ℤ/8)—64no—
125C2×AGL(1,ℤ/8)—64no—
126C2×AGL(1,ℤ/8)—64no—
127C22×SD16—32no—
128C22×D8—24no—
129C22×D8—24no—
130C22×D8—24no—
131C22×D8—32no—
132C22×D8C2×D314yes—
133C22×M16—31no—
134C23×C8C2×D311yescharacteristic
135#535—64no—
136#535—64no—
137C2×#127—32no—
138C2×#124—32no—
139C2×#121—62no—
140C2×#120—62no—
141C2×#117—31no—
142C2×#117—31no—
143#418—64no—
144#416—64no—
145C2×#103—32no—
146C2×#102—34no—
147C2×#102—32no—
148#390—34no—
149#390—62no—
150#379—62no—
151#336—34no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—42no—
157C24×C3—42no—
158C24×C3—42no—
159C24×C3—42no—
160C24×C3—84no—
161C24×C3—84no—
162C24×C3C2×D411yescharacteristic
163C23×D3—21no—
164C2×C3×D4—28no—
165C2×C3×D4—28no—
166C2×C3×D4—28no—
167C2×C3×D4—24no—
168C2×C3×D4—28no—
169C2×C3×D4—24no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
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—44no—
181C2×C3×D4—42no—
182C2×C3×D4—44no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4C2×D414yes—
189C2×C3×D4C2×D414yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—216no—
200C2×#67—42no—
201C2×#67—42no—
202C2×#67—88no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3C2×D414yes—
206#235—28no—
207#235—28no—
208#235—44no—
209#235—44no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12C2×D412yes—
216C2×C4×D3—24no—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6C2×D412yes—
225C3×D8—232no—
226C2×C3×C8—28no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8C2×D414yes—
230#217—216no—
231#217—24no—
232#217—42no—
233#217—88no—
234#214—216no—
235#213—216no—
236C4⋊Dic3—24no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241C4×Dic3—42no—
242#208—28no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8C2×D414yes—
246C25—21no—
247C25—21no—
248C2×#41—34no—
249C2×#41—62no—
250C2×#41—62no—
251C2×#41—62no—
252C2×#41—62no—
253C2×#41—62no—
254C22×Q8—31no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—31no—
268C22×D4—64no—
269C22×D4—61no—
270C22×D4#6714yes—
271C22×D4#6714yescomplemented
272C22×D4C22×D312yes—
273C22×D4C22×D312yes—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276AGL(1,ℤ/8)—632no—
277C2×SD16—64no—
278C2×SD16—64no—
279C2×SD16—64no—
280C2×D8—232no—
281C2×D8—28no—
282C2×D8—28no—
283C2×D8—28no—
284C2×D8—64no—
285C2×D8—64no—
286C2×D8—64no—
287C2×M16—34no—
288C2×M16—62no—
289C2×M16—62no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—32no—
293C22×C8#6714yescomplemented
294C22×C8C22×D312yes—
295#127—64no—
296#127—64no—
297#124—64no—
298#124—64no—
299#121—616no—
300#120—616no—
301#117—62no—
302#117—62no—
303#117—62no—
304#117—62no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C22⋊C4—32no—
308C2×C22⋊C4—64no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—68no—
311C2×C42—32no—
312C2×C42—61no—
313#103—64no—
314#102—68no—
315#102—64no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—48no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—48no—
327C23×C3—88no—
328C23×C3—88no—
329C23×C3—88no—
330C23×C3—88no—
331C23×C3—88no—
332C23×C3—88no—
333C23×C3#12012yes—
334C23×C3#12112yes—
335C23×C3#12712yes—
336C23×C3C22×D411yescharacteristic
337C22×D3—22no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—44no—
341C3×D4—24no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—48no—
350C3×D4—48no—
351C2×C3×C4#12014yes—
352C2×C3×C4#12012yes—
353C2×C3×C4#12014yes—
354C2×C3×C4#12114yes—
355C2×C3×C4#12114yes—
356C2×C3×C4#12112yes—
357C2×C3×C4#12712yes—
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364#67—48no—
365#67—816no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—28no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—44no—
373D12—22no—
374D12—22no—
375D12—41no—
376D12—41no—
377C4×D3—48no—
378Dic6—22no—
379Dic6—22no—
380Dic6—41no—
381Dic6—41no—
382C3×C8—24no—
383C3⋊C8—24no—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—24no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—61no—
393C24—84no—
394C24—84no—
395C24D3×D411yescharacteristic
396#41—68no—
397#41—68no—
398#41—124no—
399#41—124no—
400C2×Q8—32no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—32no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—64no—
432C2×D4—616no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—122no—
439C2×D4—122no—
440C2×D4—248no—
441C2×D4C2×#6714yes—
442C2×D4C2×#6714yes—
443C22×C4—34no—
444C22×C4—61no—
445C22×C4—64no—
446C22×C4—62no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4D3×D412yes—
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D412yes—
452C22×C4#23712yes—
453C22×C4C2×#6712yes—
454C22×C4C2×#6712yes—
455C22×C4C2×#6712yes—
456C22×C4C23×D311yescharacteristic
457SD16—616no—
458D8—232no—
459D8—616no—
460M16—68no—
461C2×C8—28no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—34no—
465C2×C8—62no—
466C2×C8—62no—
467C2×C8C2×#6714yes—
468C4⋊C4—64no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C22⋊C4—616no—
472C22⋊C4—64no—
473C22⋊C4—122no—
474C22⋊C4—248no—
475C42—62no—
476C42—62no—
477C42—122no—
478C22×C3—24no—
479C22×C3—24no—
480C22×C3—24no—
481C22×C3—44no—
482C22×C3—44no—
483C22×C3—48no—
484C22×C3—48no—
485C22×C3—816no—
486C22×C3—816no—
487C22×C3—84no—
488C22×C3—84no—
489C22×C3—84no—
490C22×C3—84no—
491C22×C3#39214yes—
492C22×C3C2×#12011yescharacteristic
493C22×C3C2×#12111yescharacteristic
494C22×C3C2×#12711yescharacteristic
495C22×C3#57614yes—
496C2×D3—42no—
497C2×D3—44no—
498C2×D3—82no—
499C2×D3—82no—
500C3×C4#39214yes—
501C3×C4C2×#12011yescharacteristic
502C3×C4C2×#12011yescharacteristic
503C3×C4C2×#12111yescharacteristic
504C3×C4C2×#12111yescharacteristic
505Dic3—42no—
506Dic3—44no—
507Dic3—82no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—62no—
520C23—88no—
521C23—88no—
522C23—88no—
523C23—88no—
524C23—88no—
525C23—88no—
526C23—124no—
527C23—121no—
528C23—121no—
529C23#93712yes—
530C23#93912yes—
531C23#94012yes—
532C23C2×D3×D411yescharacteristic
533Q8—62no—
534Q8—62no—
535Q8—121no—
536Q8—121no—
537D4—24no—
538D4—24no—
539D4—24no—
540D4—24no—
541D4—48no—
542D4—48no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—62no—
548D4—62no—
549D4—128no—
550D4—121no—
551D4—121no—
552D4—2416no—
553C2×C4—68no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—128no—
559C2×C4—124no—
560C2×C4—122no—
561C2×C4#93712yes—
562C2×C4#93714yes—
563C2×C4#93814yes—
564C2×C4#93912yes—
565C2×C4#93914yes—
566C2×C4#94012yes—
567C2×C4#95314yes—
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4C22×#6711yescharacteristic
572C2×C4C22×#6711yescharacteristic
573C2×C4C22×#6711yescharacteristic
574C8—24no—
575C8—64no—
576C2×C3—24no—
577C2×C3—84no—
578C2×C3—84no—
579C2×C3C2×#39211yescharacteristic
580C2×C3#300812yes—
581C2×C3#304112yes—
582C2×C3#315512yes—
583D3—82no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—48no—
591C22—816no—
592C22—816no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—122no—
598C22—124no—
599C22—242no—
600C22—242no—
601C22#522314yes—
602C22#575714yes—
603C22C2×#93711yescharacteristic
604C22C2×#93911yescharacteristic
605C22C2×#94011yescharacteristic
606C4—122no—
607C4—124no—
608C4—242no—
609C4#522314yes—
610C4C2×#93711yescharacteristic
611C4C2×#93811yescharacteristic
612C4C2×#93911yescharacteristic
613C4C2×#95311yescharacteristic
614C3#2342211yescharacteristic, complemented, Sylow
615C2—24no—
616C2—84no—
617C2—84no—
618C2—242no—
619C2#8382612yes—
620C2#8383012yes—
621C2#8383312yes—
622C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×D8C2
socleC23×C3C22×D4
hypercenterC23×D8D3
hypocenterC3#23422
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C23×C3×D8C2semidirect
2. C2×#5805C2semidirect
3. C22×#932C2semidirect
4. #84044C2semidirect
5. C2×#5215C2semidirect
6. C2×#5113C2semidirect
7. #72045C2semidirect
8. #72044C2semidirect
9. #71965C2semidirect
22. C2×#940C22semidirect
24. C2×#936C22semidirect
25. C22×#214C22semidirect
29. C22×#213C22semidirect
35. #5221C22semidirect
37. C2×#832C22semidirect
39. C2×#821C22semidirect
40. C2×#816C22semidirect
41. #4550C22semidirect
43. C23×D8D3semidirect
66. C22×C3×D4D4semidirect
87. C22×C3×C8D4semidirect
116. C2×#208D4semidirect
271. C22×D4#67semidirect
293. C22×C8#67semidirect
614. C3#23422semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D82
C2×#58052
C22×#9322
#840442
C2×#52152
C2×#51132
#720452
#720442
#719652
#234223