Finite Group Databasebeta

← #11659375

Subgroups — #11659375

Order 768.

2882 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87260C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic
3C2×#5521C211yesmaximal, characteristic, complemented
4C3×#2896C211yesmaximal, characteristic, complemented
5C2×#5131C211yesmaximal, characteristic, complemented
6#83747C211yesmaximal, characteristic, complemented
7#82207C211yesmaximal, characteristic, complemented
8#78947C214yesmaximal, complemented
9#78915C214yesmaximal, complemented
10#78866C218yesmaximal, complemented
11#78820C218yesmaximal, complemented
12#60168—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17#5827—22no—
18#5827C2214yescomplemented
19Dic3×#41—22no—
20Dic3×#41C2214yescomplemented
21C2×#949C4116yescomplemented
22C2×#949C2214yes—
23C2×#949C2214yes—
24#5812C2212yescomplemented
25C22×#215C2212yes—
26#5798—22no—
27#5798C2212yescomplemented
28C22×#213C2212yes—
29C2×#926—22no—
30C2×#926C2211yescharacteristic
31C2×#926C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5521—22no—
34#5521—22no—
35#5521C2214yes—
36#5521C2214yescomplemented
37C2×#893C2214yescomplemented
38C2×D3×C22⋊C4—22no—
39C2×C4×D12—22no—
40C3×#419C2212yescomplemented
41C3×#418C2212yescomplemented
42C3×#417C2214yescomplemented
43C3×#416C2214yescomplemented
44#5131C2214yes—
45#5131C2214yescomplemented
46#5122C2214yescomplemented
47Dic3×M16C2214yescomplemented
48C2×#821C2214yescomplemented
49#5101C2214yescomplemented
50#4998C2214yescomplemented
51#4995C2214yescomplemented
52#4964C2214yescomplemented
53#4817C22116yescomplemented
54#4788C22116yescomplemented
55#4785C22116yescomplemented
56#4756C22116yescomplemented
57C22×AGL(1,ℤ/8)—31no—
58#3428—31no—
59C2×#518—31no—
60#2949—34no—
61#2948—34no—
62#2946—38no—
63C4×AGL(1,ℤ/8)—38no—
64#2911—31no—
65#2896D311yescharacteristic, complemented
66C2×#418—31no—
67#2873—31no—
68C24×D3—21no—
69C2×C3×#41—21no—
70C2×C3×#41—21no—
71C2×C3×#41—24no—
72C2×C3×#41C2×C418yescomplemented
73C2×C3×#41C2312yes—
74C2×C3×#41C2312yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C22×D12—21no—
81C22×D12—21no—
82C22×D12—21no—
83C22×D12—24no—
84C22×D12C2×C418yescomplemented
85C22×D12C2312yes—
86C22×D12C2312yes—
87C22×D12C2312yes—
88C22×C4×D3—24no—
89C2×C3×M16—24no—
90C2×C3×M16C2312yes—
91C2×C3×M16C2312yes—
92C22×C3×C8C2312yes—
93C3×#117—21no—
94C3×#117—21no—
95C3×#117C2312yes—
96C3×#117C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42—21no—
100C2×#217—21no—
101C2×#217—24no—
102C2×#217D412yes—
103#949C2×C4132yescomplemented
104#949C2×C4132yescomplemented
105Q8×Dic3—24no—
106Q8×Dic3C2318yescomplemented
107C2×#215C2×C4116yescomplemented
108C2×#215C2318yes—
109D4×Dic3—28no—
110D4×Dic3—28no—
111D4×Dic3—24no—
112D4×Dic3C2318yescomplemented
113C2×#213C2×C4116yescomplemented
114C2×#213C2318yes—
115C2×(C2×D3)⋊C4—24no—
116#926—21no—
117#926—21no—
118#926—24no—
119#926—24no—
120#926—24no—
121#926D412yes—
122#926D412yes—
123#926C2312yes—
124#926C2312yes—
125#926C2312yes—
126#926C2312yes—
127C2×C4⋊Dic3—24no—
128C2×C4⋊Dic3D412yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4×Dic3—21no—
131C2×C4×Dic3—24no—
132C2×C4×Dic3C2312yes—
133C2×#208C2×C418yescomplemented
134C2×#208C2312yes—
135C2×#208C2312yes—
136C22×C3⋊C8C2312yes—
137#893—28no—
138#893C23116yes—
139D3×C22⋊C4—28no—
140D3×C22⋊C4—28no—
141C4×D12—28no—
142C4×D12—28no—
143C3×#103C2318yescomplemented
144C3×#102C2318yescomplemented
145#821C23116yes—
146#815C2318yescomplemented
147C8×Dic3C2318yescomplemented
148#808C2318yescomplemented
149#807C2318yescomplemented
150C22×#41C2×D311yescharacteristic
151C23×D4—31no—
152C2×AGL(1,ℤ/8)—316no—
153C2×AGL(1,ℤ/8)—34no—
154C2×AGL(1,ℤ/8)—34no—
155C22×SD16—32no—
156C22×D8—32no—
157C22×M16—31no—
158C22×M16C2×D311yescharacteristic
159#520—34no—
160#520—62no—
161#519—32no—
162#518—32no—
163#518—34no—
164#518—34no—
165#518—62no—
166#518—62no—
167#518—62no—
168C4×#41—34no—
169C4×#41—62no—
170C2×C4×D4—34no—
171C2×C4×D4—62no—
172C2×#117—31no—
173C2×#117—31no—
174C2×#117—62no—
175C2×#117C2×D311yescharacteristic
176C22×C22⋊C4—62no—
177#442—316no—
178#440—316no—
179C4×SD16—316no—
180C4×D8—316no—
181#428—34no—
182#428—34no—
183#427—34no—
184#419C2×D312yescomplemented
185#418—34no—
186#418—34no—
187#418C2×D312yescomplemented
188#417C2×D314yescomplemented
189#416C2×D314yescomplemented
190C2×#102—34no—
191#405—34no—
192#405—34no—
193C4×M16—34no—
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C23×D3—22no—
197C23×D3—28no—
198C23×D3—22no—
199C23×D3—22no—
200C23×D3—84no—
201C3×#41—28no—
202C3×#41—28no—
203C3×#41—24no—
204C3×#41—24no—
205C3×#41C22×C418yes—
206C3×#41C22×C418yescomplemented
207C2×C3×Q8—21no—
208C2×C3×Q8—22no—
209C2×C3×Q8C22×C414yescomplemented
210C2×C3×Q8C2412yes—
211C2×C3×D4—28no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4C22×C414yescomplemented
217C2×C3×D4C2412yes—
218C22×C3×C4—21no—
219C22×C3×C4—28no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4C22×C414yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4#4114yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—22no—
232C22×Dic3—28no—
233C22×Dic3C2×D414yes—
234C2×D12—28no—
235C2×D12—28no—
236C2×D12—28no—
237C2×D12—22no—
238C2×D12—22no—
239C2×D12—22no—
240C2×D12—24no—
241C2×D12—24no—
242C2×D12—24no—
243C2×D12C22×C418yes—
244C2×D12C22×C418yes—
245C2×D12C22×C418yes—
246C2×D12C2414yes—
247C2×C4×D3—216no—
248C2×C4×D3—88no—
249C3×M16—24no—
250C3×M16—24no—
251C2×C3×C8—24no—
252C2×C3×C8C2414yes—
253C3×C4⋊C4—22no—
254C3×C4⋊C4C2414yes—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C42—22no—
258C3×C42—22no—
259#217—22no—
260#217—22no—
261#217—28no—
262#217C2×D414yes—
263#217C2×D414yes—
264#215C22×C4132yescomplemented
265#213C22×C4132yescomplemented
266(C2×D3)⋊C4—216no—
267C4⋊Dic3—22no—
268C4⋊Dic3—28no—
269C4⋊Dic3C2×D414yes—
270C4⋊Dic3C2×D414yes—
271C4⋊Dic3C2414yes—
272C4×Dic3—22no—
273C4×Dic3—22no—
274C4×Dic3—28no—
275C4×Dic3C2×D414yes—
276C4×Dic3C2414yes—
277#208C22×C418yes—
278#208C22×C418yescomplemented
279C2×C3⋊C8C22×C418yescomplemented
280C2×C3⋊C8C2414yes—
281C25—61no—
282C2×#41—21no—
283C2×#41—21no—
284C2×#41—24no—
285C2×#41C4×D318yescomplemented
286C2×#41C22×D312yes—
287C2×#41C22×D312yes—
288C22×Q8C22×D311yescharacteristic
289C22×D4—21no—
290C22×D4—38no—
291C22×D4—32no—
292C22×D4—32no—
293C22×D4—32no—
294C22×D4—61no—
295C22×D4—61no—
296C22×D4—61no—
297C22×D4—64no—
298C22×D4C22×D311yescharacteristic
299C23×C4—21no—
300C23×C4—64no—
301C23×C4C22×D311yescharacteristic
302AGL(1,ℤ/8)—332no—
303AGL(1,ℤ/8)—332no—
304C2×SD16—316no—
305C2×SD16—38no—
306C2×D8—316no—
307C2×D8—38no—
308C2×M16—24no—
309C2×M16—38no—
310C2×M16—32no—
311C2×M16—32no—
312C2×M16C22×D312yes—
313C2×M16C22×D312yes—
314C22×C8—32no—
315C22×C8C22×D312yes—
316C4×Q8—38no—
317C4×Q8—64no—
318C4×D4—38no—
319C4×D4—316no—
320C4×D4—68no—
321C4×D4—68no—
322C4×D4—68no—
323C4×D4—68no—
324C4×D4—64no—
325C4×D4—68no—
326#117—21no—
327#117—21no—
328#117—32no—
329#117—32no—
330#117—32no—
331#117—32no—
332#117—32no—
333#117—32no—
334#117—64no—
335#117—64no—
336#117—64no—
337#117—61no—
338#117—61no—
339#117C22×D312yes—
340#117C22×D312yes—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—64no—
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—32no—
347C2×C22⋊C4—68no—
348C2×C22⋊C4—68no—
349C2×C22⋊C4—64no—
350C2×C22⋊C4—64no—
351C2×C22⋊C4—61no—
352C2×C42—21no—
353C2×C42—32no—
354C2×C42—64no—
355C2×C42—61no—
356#107—38no—
357#106—38no—
358#103C22×D318yescomplemented
359#102—316no—
360#102C22×D318yescomplemented
361#97—38no—
362C4×C8—38no—
363C23×C3—22no—
364C23×C3—24no—
365C23×C3—42no—
366C23×C3—82no—
367C23×C3C4×D414yes—
368C23×C3C22×D411yescharacteristic
369C23×C3C22×D411yescharacteristic
370C23×C3C22×D411yescharacteristic
371C22×D3—28no—
372C22×D3—24no—
373C22×D3—28no—
374C22×D3—28no—
375C22×D3—48no—
376C22×D3—84no—
377C22×D3—84no—
378C22×D3—816no—
379C3×Q8—24no—
380C3×Q8—22no—
381C3×Q8C23×C414yes—
382C3×D4—28no—
383C3×D4—28no—
384C3×D4—24no—
385C3×D4—22no—
386C3×D4C23×C414yes—
387C2×C3×C4—24no—
388C2×C3×C4—22no—
389C2×C3×C4—28no—
390C2×C3×C4—28no—
391C2×C3×C4—24no—
392C2×C3×C4—44no—
393C2×C3×C4C4×D414yes—
394C2×C3×C4C4×D414yes—
395C2×C3×C4C4×D414yes—
396C2×C3×C4C23×C412yes—
397C2×C3×C4C23×C412yes—
398C2×C3×C4C23×C412yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4112yes—
403C2×C3×C4C2×#4112yes—
404C2×C3×C4C2×#4112yes—
405C2×C3×C4C2511yescharacteristic
406C2×Dic3—24no—
407C2×Dic3—44no—
408C2×Dic3—88no—
409C2×Dic3—82no—
410C2×Dic3C22×D418yes—
411D12—28no—
412D12—28no—
413D12—28no—
414D12—24no—
415D12C23×C418yes—
416C4×D3—816no—
417C3×C8—24no—
418C3⋊C8C23×C418yes—
419C24—21no—
420C24—62no—
421C24—68no—
422C24—62no—
423C24—62no—
424C24—244no—
425C24D3×D411yescharacteristic
426#41—28no—
427#41—28no—
428#41—24no—
429#41—24no—
430#41C2×C4×D318yes—
431#41C2×C4×D318yescomplemented
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8C2×C4×D314yescomplemented
435C2×Q8C23×D312yes—
436C2×D4—28no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—38no—
442C2×D4—38no—
443C2×D4—38no—
444C2×D4—34no—
445C2×D4—68no—
446C2×D4—68no—
447C2×D4—68no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4—64no—
452C2×D4—64no—
453C2×D4—64no—
454C2×D4C2×C4×D314yescomplemented
455C2×D4C23×D312yes—
456C22×C4—21no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—34no—
462C22×C4—616no—
463C22×C4—68no—
464C22×C4—62no—
465C22×C4—248no—
466C22×C4C2×C4×D314yes—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4#23714yes—
471C22×C4C23×D311yescharacteristic
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D311yescharacteristic
474SD16—332no—
475D8—332no—
476M16—24no—
477M16—24no—
478M16—38no—
479M16—38no—
480C2×C8—24no—
481C2×C8—38no—
482C2×C8—34no—
483C2×C8C23×D314yes—
484C4⋊C4—22no—
485C4⋊C4—34no—
486C4⋊C4—34no—
487C4⋊C4—34no—
488C4⋊C4—68no—
489C4⋊C4—62no—
490C4⋊C4C23×D314yes—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—34no—
494C22⋊C4—34no—
495C22⋊C4—616no—
496C22⋊C4—68no—
497C22⋊C4—62no—
498C22⋊C4—62no—
499C42—22no—
500C42—22no—
501C42—34no—
502C42—34no—
503C42—68no—
504C42—62no—
505C42—62no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—24no—
510C22×C3—44no—
511C22×C3—44no—
512C22×C3—84no—
513C22×C3C2×C4×D412yes—
514C22×C3C2×C4×D412yes—
515C22×C3C2×C4×D412yes—
516C22×C3C23×D411yescharacteristic
517C2×D3—28no—
518C2×D3—416no—
519C2×D3—48no—
520C2×D3—88no—
521C2×D3—88no—
522C3×C4—24no—
523C3×C4—44no—
524C3×C4C2×C4×D412yes—
525C3×C4C2×C4×D412yes—
526C3×C4C2×C4×D412yes—
527C3×C4C24×C411yescharacteristic
528C3×C4C22×#4111yescharacteristic
529Dic3—28no—
530Dic3—44no—
531C23—22no—
532C23—24no—
533C23—42no—
534C23—64no—
535C23—68no—
536C23—68no—
537C23—68no—
538C23—82no—
539C23—128no—
540C23—244no—
541C23—244no—
542C23—2416no—
543C23#67⋊C414yes—
544C23C2×D3×D411yescharacteristic
545C23C2×D3×D411yescharacteristic
546C23C2×D3×D411yescharacteristic
547Q8—22no—
548Q8—24no—
549Q8C22×C4×D314yes—
550D4—28no—
551D4—28no—
552D4—22no—
553D4—24no—
554D4—38no—
555D4—64no—
556D4—68no—
557D4—68no—
558D4—68no—
559D4C22×C4×D314yes—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—38no—
566C2×C4—44no—
567C2×C4—64no—
568C2×C4—124no—
569C2×C4—2416no—
570C2×C4—248no—
571C2×C4—242no—
572C2×C4#67⋊C414yes—
573C2×C4#67⋊C414yes—
574C2×C4#67⋊C414yes—
575C2×C4C22×C4×D312yes—
576C2×C4C22×C4×D312yes—
577C2×C4C22×C4×D312yes—
578C2×C4C2×D3×D411yescharacteristic
579C2×C4C2×D3×D411yescharacteristic
580C2×C4C2×D3×D411yescharacteristic
581C2×C4C2×#23712yes—
582C2×C4C2×#23712yes—
583C2×C4C2×#23712yes—
584C2×C4C24×D311yescharacteristic
585C8—24no—
586C8—38no—
587C2×C3—22no—
588C2×C3—22no—
589C2×C3—22no—
590C2×C3—44no—
591C2×C3C22×C4×D411yescharacteristic
592C2×C3#358512yes—
593D3—48no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—68no—
601C22—84no—
602C22—1216no—
603C22—128no—
604C22—248no—
605C22—248no—
606C22C2×#67⋊C412yes—
607C22C2×#67⋊C412yes—
608C22C2×#67⋊C412yes—
609C22C22×D3×D411yescharacteristic
610C4—24no—
611C4—44no—
612C4—68no—
613C4—124no—
614C4C2×#67⋊C412yes—
615C4C2×#67⋊C412yes—
616C4C2×#67⋊C412yes—
617C4C23×C4×D311yescharacteristic
618C4C22×#23711yescharacteristic
619C3#6016811yescharacteristic, complemented, Sylow
620C2—22no—
621C2—22no—
622C2—22no—
623C2—44no—
624C2—128no—
625C2C22×#67⋊C411yescharacteristic
626C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2896C2
socleC22×C3C23×D4
prime radicalC22×#949C2
cosocleC2×C3×C4C25
hypercenter#2896D3
hypocenterC3#60168
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
1. #87260C2semidirect
3. C2×#5521C2semidirect
4. C3×#2896C2semidirect
5. C2×#5131C2semidirect
6. #83747C2semidirect
7. #82207C2semidirect
8. #78947C2semidirect
9. #78915C2semidirect
10. #78866C2semidirect
11. #78820C2semidirect
18. #5827C22semidirect
20. Dic3×#41C22semidirect
21. C2×#949C4semidirect
24. #5812C22semidirect
27. #5798C22semidirect
36. #5521C22semidirect
37. C2×#893C22semidirect
40. C3×#419C22semidirect
41. C3×#418C22semidirect
42. C3×#417C22semidirect
43. C3×#416C22semidirect
45. #5131C22semidirect
46. #5122C22semidirect
47. Dic3×M16C22semidirect
48. C2×#821C22semidirect
49. #5101C22semidirect
50. #4998C22semidirect
51. #4995C22semidirect
52. #4964C22semidirect
53. #4817C22semidirect
54. #4788C22semidirect
55. #4785C22semidirect
56. #4756C22semidirect
65. #2896D3semidirect
72. C2×C3×#41C2×C4semidirect
84. C22×D12C2×C4semidirect
103. #949C2×C4semidirect
104. #949C2×C4semidirect
106. Q8×Dic3C23semidirect
107. C2×#215C2×C4semidirect
112. D4×Dic3C23semidirect
113. C2×#213C2×C4semidirect
133. C2×#208C2×C4semidirect
143. C3×#103C23semidirect
144. C3×#102C23semidirect
146. #815C23semidirect
147. C8×Dic3C23semidirect
148. #808C23semidirect
149. #807C23semidirect
184. #419C2×D3semidirect
187. #418C2×D3semidirect
188. #417C2×D3semidirect
189. #416C2×D3semidirect
206. C3×#41C22×C4semidirect
209. C2×C3×Q8C22×C4semidirect
216. C2×C3×D4C22×C4semidirect
264. #215C22×C4semidirect
265. #213C22×C4semidirect
278. #208C22×C4semidirect
279. C2×C3⋊C8C22×C4semidirect
285. C2×#41C4×D3semidirect
358. #103C22×D3semidirect
360. #102C22×D3semidirect
431. #41C2×C4×D3semidirect
434. C2×Q8C2×C4×D3semidirect
454. C2×D4C2×C4×D3semidirect
619. C3#60168semidirect

Maximal subgroups

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

subgroupindex
#872602
C22×#9492
C2×#55212
C3×#28962
C2×#51312
#837472
#822072
#789472
#789152
#788662
#788202
#601683