Finite Group Databasebeta

← #10822339

Subgroups — #10822339

Order 768.

2706 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic, complemented
3C22×#940C211yesmaximal, characteristic, complemented
4#84086C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6C2×#5131C211yesmaximal, characteristic, complemented
7#72056C214yesmaximal, complemented
8#72044C214yesmaximal, complemented
9#71979C211yesmaximal, characteristic, complemented
10#23430—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C23×#67—22no—
14C23×D12C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)—24no—
16C2×C3×AGL(1,ℤ/8)—24no—
17C2×C3×AGL(1,ℤ/8)—24no—
18C22×C3×SD16C2212yescomplemented
19C22×C3×D8C2212yescomplemented
20C22×C3×M16C2211yescharacteristic
21C2×#949—24no—
22C2×#949—24no—
23C2×#949—24no—
24C22×#215C2212yescomplemented
25C2×#940—24no—
26C2×#940—24no—
27C2×#940—22no—
28C2×#940C2214yes—
29C22×#213C2212yescomplemented
30C2×#926C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5229C2212yescomplemented
34#5221C2212yescomplemented
35#5220—22no—
36#5215—24no—
37C2×#832C2214yescomplemented
38C4×#217—22no—
39#5131—24no—
40C2×#821C2214yescomplemented
41#4550C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C22×AGL(1,ℤ/8)D311yescharacteristic, complemented
44C22×#127—31no—
45C2×#418—31no—
46C2×#418—31no—
47#2297—31no—
48#2014—34no—
49#2011—34no—
50#1728—31no—
51C25×C3—21no—
52C24×D3—21no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41—22no—
58C2×C3×#41D414yescomplemented
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4D414yes—
68C22×C3×D4C2312yes—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C22×#67—216no—
73C22×#67—28no—
74C22×#67—416no—
75C23×Dic3D412yes—
76C22×D12—22no—
77C22×D12—22no—
78C22×D12—22no—
79C22×D12—22no—
80C22×D12—21no—
81C22×D12—22no—
82C22×D12D414yes—
83C22×D12C2312yes—
84C3×AGL(1,ℤ/8)—232no—
85C2×C3×SD16—24no—
86C2×C3×SD16—24no—
87C2×C3×SD16—24no—
88C2×C3×D8—24no—
89C2×C3×D8—24no—
90C2×C3×D8—24no—
91C2×C3×M16—22no—
92C2×C3×M16—22no—
93C2×C3×M16D414yescomplemented
94C22×C3×C8C2312yes—
95C2×#217—21no—
96C2×#217—24no—
97#949—232no—
98C2×#215—24no—
99C2×#215—24no—
100C2×#215—24no—
101#940—216no—
102#940—28no—
103#940—28no—
104C2×#213—24no—
105C2×#213—24no—
106C2×#213—24no—
107#926—22no—
108#926—22no—
109C2×C4⋊Dic3C2312yes—
110C2×C4×Dic3—22no—
111C2×C4×Dic3—22no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D412yes—
114C2×C4×Dic3C2312yes—
115C2×#208—22no—
116C2×#208—22no—
117C2×#208D414yescomplemented
118C22×C3⋊C8C2312yes—
119#832—28no—
120C3⋊#95—24no—
121#821—28no—
122C22×#41C2×D311yescharacteristic
123C23×D4—31no—
124C23×D4—62no—
125C23×D4C2×D311yescharacteristic
126C2×AGL(1,ℤ/8)—24no—
127C2×AGL(1,ℤ/8)—24no—
128C2×AGL(1,ℤ/8)—24no—
129C2×AGL(1,ℤ/8)—64no—
130C2×AGL(1,ℤ/8)—64no—
131C2×AGL(1,ℤ/8)—64no—
132C22×SD16—32no—
133C22×SD16C2×D312yescomplemented
134C22×D8—32no—
135C22×D8C2×D312yescomplemented
136C22×M16—31no—
137C22×M16C2×D311yescharacteristic
138C2×#127—34no—
139C2×#127—62no—
140C2×#127—64no—
141C2×#127—64no—
142C2×#117—31no—
143C22×C42—31no—
144#418—64no—
145#418—64no—
146C2×#102—34no—
147C2×#102—34no—
148#391—32no—
149#390—32no—
150#386—62no—
151C4×C22⋊C4—62no—
152#336—34no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—21no—
156C24×C3—48no—
157C24×C3—42no—
158C24×C3—42no—
159C24×C3C2×D411yescharacteristic
160C23×D3—24no—
161C23×D3—22no—
162C23×D3—48no—
163C23×D3—42no—
164C23×D3—42no—
165C3×#41—28no—
166C3×#41—28no—
167C3×#41—44no—
168C3×#41—44no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C2×D412yescomplemented
175C2×C3×D4—24no—
176C2×C3×D4—28no—
177C2×C3×D4—24no—
178C2×C3×D4—28no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—44no—
192C2×C3×D4—44no—
193C2×C3×D4—42no—
194C2×C3×D4—42no—
195C2×C3×D4C2×D414yes—
196C2×C3×D4C2×D412yescomplemented
197C22×C3×C4—21no—
198C22×C3×C4—24no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4#4112yes—
208C22×C3×C4C2411yescharacteristic
209C2×#67—232no—
210C2×#67—464no—
211C2×#67—416no—
212C2×#67—416no—
213C22×Dic3—24no—
214C22×Dic3—28no—
215C22×Dic3—22no—
216C22×Dic3C2×D414yes—
217C2×D12—28no—
218C2×D12—28no—
219C2×D12—22no—
220C2×D12—22no—
221C2×D12—22no—
222C2×D12—22no—
223C2×D12—24no—
224C2×D12—22no—
225C2×D12—44no—
226C2×D12—44no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12C2×D414yes—
230C3×SD16—216no—
231C3×D8—216no—
232C3×M16—28no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8C2×D414yescomplemented
236#217—42no—
237#217—48no—
238#215—216no—
239#213—216no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4×Dic3—24no—
243C4×Dic3—22no—
244C4×Dic3—22no—
245C4×Dic3—42no—
246#208—28no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8C2×D414yescomplemented
250C25—21no—
251C25—61no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41#6714yescomplemented
258C22×Q8C22×D311yescharacteristic
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—34no—
267C22×D4—32no—
268C22×D4—68no—
269C22×D4—616no—
270C22×D4—62no—
271C22×D4—61no—
272C22×D4—62no—
273C22×D4—62no—
274C22×D4—62no—
275C22×D4—62no—
276C22×D4—1216no—
277C22×D4#6714yes—
278C22×D4C22×D312yes—
279C22×D4C22×D311yescharacteristic
280C23×C4—21no—
281C23×C4—32no—
282C23×C4C22×D311yescharacteristic
283AGL(1,ℤ/8)—232no—
284AGL(1,ℤ/8)—632no—
285C2×SD16—24no—
286C2×SD16—24no—
287C2×SD16—24no—
288C2×SD16—64no—
289C2×SD16—64no—
290C2×SD16—64no—
291C2×D8—24no—
292C2×D8—24no—
293C2×D8—24no—
294C2×D8—64no—
295C2×D8—64no—
296C2×D8—64no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—34no—
300C2×M16—62no—
301C2×M16—62no—
302C2×M16#6714yescomplemented
303C22×C8—32no—
304C22×C8C22×D312yes—
305#127—68no—
306#127—616no—
307#127—68no—
308#117—62no—
309#117—62no—
310C2×C4⋊C4—32no—
311C2×C22⋊C4—64no—
312C2×C22⋊C4—61no—
313C2×C42—32no—
314C2×C42—32no—
315C2×C42—62no—
316C2×C42—62no—
317C2×C42—61no—
318#102—68no—
319#102—68no—
320#95—64no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—22no—
324C23×C3—416no—
325C23×C3—41no—
326C23×C3—42no—
327C23×C3—41no—
328C23×C3—42no—
329C23×C3—44no—
330C23×C3—48no—
331C23×C3—48no—
332C23×C3—88no—
333C23×C3—88no—
334C23×C3#12012yes—
335C23×C3#12112yes—
336C23×C3#12712yes—
337C23×C3C22×D411yescharacteristic
338C22×D3—24no—
339C22×D3—416no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—88no—
345C22×D3—88no—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—24no—
353C3×D4—24no—
354C3×D4—48no—
355C3×D4—42no—
356C3×D4—42no—
357C3×D4—41no—
358C3×D4—41no—
359C3×D4—48no—
360C2×C3×C4—22no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—48no—
364C2×C3×C4#12014yes—
365C2×C3×C4#12012yes—
366C2×C3×C4#12014yes—
367C2×C3×C4#12114yes—
368C2×C3×C4#12114yes—
369C2×C3×C4#12112yes—
370C2×C3×C4#12712yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377#67—464no—
378C2×Dic3—216no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—42no—
382C2×Dic3—44no—
383C2×Dic3AGL(1,ℤ/8)18yescomplemented
384D12—24no—
385D12—24no—
386D12—42no—
387D12—42no—
388D12—48no—
389C3×C8—24no—
390C3⋊C8—24no—
391C24—24no—
392C24—21no—
393C24—22no—
394C24—48no—
395C24—42no—
396C24—42no—
397C24—64no—
398C24—62no—
399C24—128no—
400C24—122no—
401C24—122no—
402C24D3×D411yescharacteristic
403#41—28no—
404#41—28no—
405#41—44no—
406#41—44no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×#6712yescomplemented
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—34no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—632no—
435C2×D4—68no—
436C2×D4—64no—
437C2×D4—68no—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—62no—
443C2×D4—1216no—
444C2×D4—1264no—
445C2×D4—1216no—
446C2×D4—122no—
447C2×D4—124no—
448C2×D4—122no—
449C2×D4—124no—
450C2×D4C2×#6714yes—
451C2×D4C2×#6712yescomplemented
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—34no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—68no—
458C22×C4—64no—
459C22×C4—62no—
460C22×C4D3×D412yes—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D412yes—
463C22×C4#23712yes—
464C22×C4C2×#6712yes—
465C22×C4C2×#6712yes—
466C22×C4C2×#6712yes—
467C22×C4C23×D311yescharacteristic
468SD16—216no—
469SD16—616no—
470D8—216no—
471D8—616no—
472M16—28no—
473M16—68no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—34no—
477C2×C8—62no—
478C2×C8—62no—
479C2×C8C2×#6714yescomplemented
480C4⋊C4—62no—
481C4⋊C4—62no—
482C22⋊C4—128no—
483C22⋊C4—122no—
484C42—64no—
485C42—62no—
486C42—62no—
487C42—122no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—42no—
492C22×C3—44no—
493C22×C3—48no—
494C22×C3—44no—
495C22×C3—816no—
496C22×C3—82no—
497C22×C3—84no—
498C22×C3—82no—
499C22×C3—84no—
500C22×C3#39214yes—
501C22×C3C2×#12011yescharacteristic
502C22×C3C2×#12111yescharacteristic
503C22×C3C2×#12711yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)14yes—
505C2×D3—44no—
506C2×D3—48no—
507C2×D3—816no—
508C2×D3—84no—
509C2×D3—84no—
510C3×C4—42no—
511C3×C4#39214yes—
512C3×C4C2×#12011yescharacteristic
513C3×C4C2×#12011yescharacteristic
514C3×C4C2×#12111yescharacteristic
515C3×C4C2×#12111yescharacteristic
516Dic3—28no—
517Dic3—82no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—416no—
522C23—48no—
523C23—44no—
524C23—48no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—42no—
529C23—64no—
530C23—88no—
531C23—88no—
532C23—1216no—
533C23—128no—
534C23—128no—
535C23—122no—
536C23—122no—
537C23—248no—
538C23—248no—
539C23#93712yes—
540C23#93912yes—
541C23#94012yes—
542C23C2×D3×D411yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547D4—22no—
548D4—24no—
549D4—22no—
550D4—24no—
551D4—48no—
552D4—48no—
553D4—42no—
554D4—42no—
555D4—41no—
556D4—41no—
557D4—64no—
558D4—64no—
559D4—1264no—
560D4—128no—
561D4—122no—
562D4—122no—
563C2×C4—22no—
564C2×C4—38no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—48no—
568C2×C4—616no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—124no—
572C2×C4—122no—
573C2×C4#93712yes—
574C2×C4#93714yes—
575C2×C4#93814yes—
576C2×C4#93912yes—
577C2×C4#93914yes—
578C2×C4#94012yes—
579C2×C4#95314yes—
580C2×C4C2×D3×D411yescharacteristic
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×#23711yescharacteristic
583C2×C4C22×#6711yescharacteristic
584C2×C4C22×#6711yescharacteristic
585C2×C4C22×#6711yescharacteristic
586C8—24no—
587C8—64no—
588C2×C3—24no—
589C2×C3—82no—
590C2×C3—84no—
591C2×C3C2×#39211yescharacteristic
592C2×C3#300312yes—
593C2×C3#303812yes—
594C2×C3#315112yes—
595D3—84no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—48no—
603C22—816no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—124no—
609C22—128no—
610C22—2416no—
611C22—244no—
612C22—244no—
613C22#522314yes—
614C22D3×AGL(1,ℤ/8)14yes—
615C22C2×#93711yescharacteristic
616C22C2×#93911yescharacteristic
617C22C2×#94011yescharacteristic
618C4—42no—
619C4—68no—
620C4—242no—
621C4#522314yes—
622C4C2×#93711yescharacteristic
623C4C2×#93811yescharacteristic
624C4C2×#93911yescharacteristic
625C4C2×#95311yescharacteristic
626C3#2343011yescharacteristic, complemented, Sylow
627C2—24no—
628C2—84no—
629C2—82no—
630C2—244no—
631C2#8393212yes—
632C2#8393712yes—
633C2#8394412yes—
634C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×AGL(1,ℤ/8)C2
socleC23×C3C22×D4
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23430
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×C3×AGL(1,ℤ/8)C2semidirect
2. C22×#949C2semidirect
3. C22×#940C2semidirect
4. #84086C2semidirect
5. C2×#5215C2semidirect
6. C2×#5131C2semidirect
7. #72056C2semidirect
8. #72044C2semidirect
9. #71979C2semidirect
18. C22×C3×SD16C22semidirect
19. C22×C3×D8C22semidirect
24. C22×#215C22semidirect
29. C22×#213C22semidirect
33. #5229C22semidirect
34. #5221C22semidirect
37. C2×#832C22semidirect
40. C2×#821C22semidirect
41. #4550C22semidirect
43. C22×AGL(1,ℤ/8)D3semidirect
58. C2×C3×#41D4semidirect
93. C2×C3×M16D4semidirect
117. C2×#208D4semidirect
133. C22×SD16C2×D3semidirect
135. C22×D8C2×D3semidirect
174. C2×C3×Q8C2×D4semidirect
196. C2×C3×D4C2×D4semidirect
235. C2×C3×C8C2×D4semidirect
249. C2×C3⋊C8C2×D4semidirect
257. C2×#41#67semidirect
302. C2×M16#67semidirect
383. C2×Dic3AGL(1,ℤ/8)semidirect
412. C2×Q8C2×#67semidirect
451. C2×D4C2×#67semidirect
479. C2×C8C2×#67semidirect
626. C3#23430semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
C22×#9492
C22×#9402
#840862
C2×#52152
C2×#51312
#720562
#720442
#719792
#234303