Finite Group Databasebeta

← #10776607

Subgroups — #10776607

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3558C211yesmaximal, characteristic, complemented
2#83287C211yesmaximal, characteristic, complemented
3#83146C211yesmaximal, characteristic, complemented
4C2×#5058C211yesmaximal, characteristic, complemented
5#71952C214yesmaximal, complemented
6#71948C214yesmaximal, complemented
7#71798C211yesmaximal, characteristic, complemented
8#71746C211yesmaximal, characteristic, complemented
9#71718C211yesmaximal, characteristic, complemented
10#19014—31nomaximal, Sylow
11C3×#562C2212yescomplemented
12C3×H(3,𝔽4)C2212yescomplemented
13C3×#559—21no—
14C3×#548—21no—
15C3×#546—22no—
16C3×#546—21no—
17C3×D42—21no—
18C3×#542—21no—
19C3×#538—22no—
20C3×#538—21no—
21C3×#535—21no—
22C3×#534—21no—
23C3×#532C2211yescharacteristic
24C3×#529C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C2×#949—22no—
27C22×#215—21no—
28C22×#213—21no—
29C2×#874C2211yescharacteristic, complemented
30#5085—21no—
31#5073—21no—
32#5067C2212yescomplemented
33#5058—22no—
34#5058C2212yescomplemented
35#5058C2214yescomplemented
36#5058C2214yescomplemented
37#5046—21no—
38#5036—21no—
39#5027—21no—
40#5015—21no—
41#4926C2211yescharacteristic, complemented
42C4×#208C2211yescharacteristic, complemented
43C2×#803C2211yescharacteristic, complemented
44#4543C2212yescomplemented
45#4540C2212yescomplemented
46#4540C2214yescomplemented
47#4540C2214yescomplemented
48#3558D311yescharacteristic, complemented
49#3217—31no—
50#3151—31no—
51C2×#496—31no—
52#1656—34no—
53#1655—34no—
54#1530—31no—
55#1527—31no—
56#1524—31no—
57C2×C3×#41—21no—
58C2×C3×#41D412yes—
59C22×C3×Q8D411yescharacteristic
60C22×C3×D4—22no—
61C22×C3×D4—21no—
62C22×C3×D4D411yescharacteristic
63C22×D12—21no—
64C22×D12—21no—
65C22×D12—21no—
66C3×C4⋊Q8D411yescharacteristic
67C3×#127—21no—
68C3×#127D411yescharacteristic
69C3×#126—22no—
70C3×#126—21no—
71C3×#125C2311yescharacteristic
72C3×#124—21no—
73C3×#124—22no—
74C3×#124—22no—
75C3×#124—22no—
76C3×#124—21no—
77C3×#124—21no—
78C3×#124D412yes—
79C3×#124C2311yescharacteristic
80C3×#124C2312yes—
81C3×#124C2312yes—
82C3×#123—22no—
83C3×#123—22no—
84C3×#123—21no—
85C3×#123—22no—
86C3×#123—22no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8D411yescharacteristic
91C3×#121—22no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—21no—
95C3×#121—22no—
96C3×#121—22no—
97C3×#121—21no—
98C3×#121—21no—
99C3×#121D411yescharacteristic
100C3×#120—22no—
101C3×#120—22no—
102C3×#120—22no—
103C3×#120—22no—
104C3×#120—22no—
105C3×C4×Q8—21no—
106C3×C4×Q8D411yescharacteristic
107C3×C4×D4—22no—
108C3×C4×D4—21no—
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4D411yescharacteristic
114C3×#117—21no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C42C2311yescharacteristic
119#949—48no—
120C2×#215—24no—
121C2×#215—24no—
122C2×#215—22no—
123C2×#215—44no—
124C2×#213—24no—
125C2×#213—24no—
126C2×#213—22no—
127C2×#213—44no—
128C2×#208—21no—
129C22×C3⋊C8—21no—
130#874—21no—
131#874—21no—
132#874—21no—
133#874C2312yes—
134#830—21no—
135#809—22no—
136#809—22no—
137C4⋊(C3⋊C8)—21no—
138#803—21no—
139#803C2312yes—
140#803C2312yes—
141#803C2312yes—
142C4×C3⋊C8—21no—
143C2×AGL(1,ℤ/8)—62no—
144C22×SD16—61no—
145C22×D8—61no—
146#562C2×D312yescomplemented
147H(3,𝔽4)C2×D312yescomplemented
148#559—21no—
149#548—21no—
150#546—22no—
151#546—21no—
152D42—21no—
153#542—21no—
154#538—22no—
155#538—21no—
156#535—21no—
157#534—21no—
158#532C2×D311yescharacteristic
159C2×#127—31no—
160#529C2×D311yescharacteristic
161C2×#124C2×D311yescharacteristic
162#496—34no—
163#496—34no—
164#496—32no—
165#496—62no—
166#493—61no—
167#492—61no—
168#490—32no—
169#460—61no—
170#459—61no—
171#450—61no—
172#447—61no—
173#432—31no—
174C4×M16—31no—
175C2×#97—31no—
176#330—32no—
177#329—32no—
178#329—34no—
179#329—34no—
180C24×C3—22no—
181C24×C3—21no—
182C23×D3—41no—
183C3×#41—44no—
184C3×#41—82no—
185C2×C3×Q8—21no—
186C2×C3×Q8—42no—
187C2×C3×Q8C2×D411yescharacteristic
188C2×C3×Q8C2×D412yes—
189C2×C3×Q8C2×D412yes—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
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—42no—
201C2×C3×D4—82no—
202C2×C3×D4—84no—
203C2×C3×D4C2×D411yescharacteristic
204C2×C3×D4C2×D412yes—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—42no—
218C2×D12—42no—
219C2×D12—44no—
220C2×D12—41no—
221C2×D12—41no—
222C2×D12—41no—
223C2×D12—82no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4C2×D411yescharacteristic
233C3×C4⋊C4C2×D411yescharacteristic
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C42—22no—
251C3×C42—21no—
252C3×C42C2×D411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42C2411yescharacteristic
256#215—48no—
257#215—48no—
258#215—82no—
259#213—48no—
260#213—48no—
261#213—82no—
262#208—24no—
263C2×C3⋊C8—24no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8—22no—
266C2×C3⋊C8—22no—
267C2×#41—21no—
268C2×#41#6712yes—
269C22×Q8#6711yescharacteristic
270C22×D4—22no—
271C22×D4—21no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—61no—
275C22×D4#6711yescharacteristic
276AGL(1,ℤ/8)—128no—
277C2×SD16—64no—
278C2×SD16—64no—
279C2×SD16—62no—
280C2×SD16—124no—
281C2×D8—64no—
282C2×D8—64no—
283C2×D8—62no—
284C2×D8—124no—
285C2×M16—61no—
286C22×C8—61no—
287C4⋊Q8#6711yescharacteristic
288#127—21no—
289#127—32no—
290#127—61no—
291#127—61no—
292#127—61no—
293#127#6711yescharacteristic
294#126—22no—
295#126—21no—
296#125C22×D311yescharacteristic
297#124—21no—
298#124—22no—
299#124—22no—
300#124—22no—
301#124—21no—
302#124—21no—
303#124#6712yes—
304#124C22×D312yes—
305#124C22×D311yescharacteristic
306#124C22×D312yes—
307#123—22no—
308#123—22no—
309#123—21no—
310#123—22no—
311#123—22no—
312C22⋊Q8—21no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8#6711yescharacteristic
316#121—22no—
317#121—22no—
318#121—22no—
319#121—21no—
320#121—22no—
321#121—22no—
322#121—21no—
323#121—21no—
324#121#6711yescharacteristic
325#120—22no—
326#120—22no—
327#120—22no—
328#120—22no—
329#120—22no—
330C4×Q8—21no—
331C4×Q8#6711yescharacteristic
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4#6711yescharacteristic
339#117—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42C22×D311yescharacteristic
344C4⋊C8—61no—
345#102—62no—
346#102—62no—
347C22⋊C8—61no—
348#97—32no—
349#97—32no—
350#97—32no—
351#97—61no—
352C4×C8—61no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—42no—
357C23×C3—81no—
358C23×C3—81no—
359C23×C3—82no—
360C23×C3—82no—
361C23×C3—82no—
362C23×C3#12011yescharacteristic
363C22×D3—42no—
364C22×D3—82no—
365C22×D3—82no—
366C22×D3—82no—
367C3×Q8—42no—
368C3×Q8—42no—
369C3×Q8—41no—
370C3×Q8—42no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—41no—
374C3×D4—42no—
375C3×D4—44no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—81no—
379C3×D4—82no—
380C3×D4—82no—
381C3×D4—82no—
382C3×D4—82no—
383C3×D4—82no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—42no—
393C2×C3×C4—82no—
394C2×C3×C4—81no—
395C2×C3×C4—82no—
396C2×C3×C4—82no—
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4#12011yescharacteristic
399C2×C3×C4#12011yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403D12—42no—
404D12—44no—
405D12—42no—
406D12—42no—
407D12—42no—
408D12—82no—
409D12—82no—
410D12—82no—
411C3⋊C8—24no—
412C3⋊C8—24no—
413C3⋊C8—42no—
414C24—22no—
415C24—21no—
416C24—121no—
417#41—44no—
418#41—82no—
419C2×Q8—21no—
420C2×Q8—42no—
421C2×Q8C2×#6711yescharacteristic
422C2×Q8C2×#6712yes—
423C2×Q8C2×#6712yes—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—82no—
439C2×D4—84no—
440C2×D4—122no—
441C2×D4—122no—
442C2×D4—124no—
443C2×D4—121no—
444C2×D4—121no—
445C2×D4—121no—
446C2×D4—242no—
447C2×D4C2×#6711yescharacteristic
448C2×D4C2×#6712yes—
449C2×D4C2×#6712yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C2×#6711yescharacteristic
458SD16—128no—
459SD16—128no—
460SD16—242no—
461D8—128no—
462D8—128no—
463D8—242no—
464M16—64no—
465C2×C8—64no—
466C2×C8—62no—
467C2×C8—62no—
468C2×C8—62no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4C2×#6711yescharacteristic
478C4⋊C4C2×#6711yescharacteristic
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—21no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C42—22no—
496C42—21no—
497C42C2×#6711yescharacteristic
498C42C2×#6711yescharacteristic
499C42C2×#6711yescharacteristic
500C42C23×D311yescharacteristic
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—41no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—162no—
513C22×C3—164no—
514C22×C3#45312yes—
515C22×C3C2×#12011yescharacteristic
516C2×D3—82no—
517C2×D3—82no—
518C2×D3—82no—
519C2×D3—164no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—42no—
528C3×C4—42no—
529C3×C4—41no—
530C3×C4#45312yes—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—42no—
535C23—81no—
536C23—81no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—122no—
541C23—242no—
542C23—242no—
543C23—242no—
544C23#95311yescharacteristic
545Q8—42no—
546Q8—42no—
547Q8—41no—
548Q8—42no—
549D4—42no—
550D4—42no—
551D4—41no—
552D4—42no—
553D4—44no—
554D4—42no—
555D4—42no—
556D4—81no—
557D4—82no—
558D4—82no—
559D4—82no—
560D4—82no—
561D4—82no—
562D4—122no—
563D4—124no—
564D4—122no—
565D4—122no—
566D4—122no—
567D4—242no—
568D4—242no—
569D4—242no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4#95311yescharacteristic
584C2×C4#95311yescharacteristic
585C2×C4#95311yescharacteristic
586C2×C4C22×#6711yescharacteristic
587C2×C4C22×#6711yescharacteristic
588C2×C4C22×#6711yescharacteristic
589C8—64no—
590C8—64no—
591C8—122no—
592C2×C3—22no—
593C2×C3—82no—
594C2×C3—81no—
595C2×C3—82no—
596C2×C3C2×#45311yescharacteristic
597C2×C3C2×#45311yescharacteristic
598C2×C3#302111yescharacteristic
599D3—162no—
600C22—21no—
601C22—21no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—162no—
612C22—164no—
613C22—242no—
614C22—242no—
615C22—242no—
616C22—484no—
617C22#524112yes—
618C22C2×#95311yescharacteristic
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4#524112yes—
630C3#1901411yescharacteristic, complemented, Sylow
631C2—22no—
632C2—81no—
633C2—82no—
634C2—82no—
635C2—482no—
636C2C2×#524111yescharacteristic
637C2C2×#524111yescharacteristic
638C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3558C2
socleC22×C3C2×#120
hypercenter#3558D3
hypocenterC3#19014
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3558

Complements

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

NH ≅ G/Ntype
1. C3×#3558C2semidirect
2. #83287C2semidirect
3. #83146C2semidirect
4. C2×#5058C2semidirect
5. #71952C2semidirect
6. #71948C2semidirect
7. #71798C2semidirect
8. #71746C2semidirect
9. #71718C2semidirect
11. C3×#562C22semidirect
12. C3×H(3,𝔽4)C22semidirect
29. C2×#874C22semidirect
32. #5067C22semidirect
34. #5058C22semidirect
35. #5058C22semidirect
36. #5058C22semidirect
41. #4926C22semidirect
42. C4×#208C22semidirect
43. C2×#803C22semidirect
44. #4543C22semidirect
45. #4540C22semidirect
46. #4540C22semidirect
47. #4540C22semidirect
48. #3558D3semidirect
146. #562C2×D3semidirect
147. H(3,𝔽4)C2×D3semidirect
630. C3#19014semidirect

Maximal subgroups

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

subgroupindex
C3×#35582
#832872
#831462
C2×#50582
#719522
#719482
#717982
#717462
#717182
#190143