Finite Group Databasebeta

← #11675344

Subgroups — #11675344

Order 768.

1852 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85848C211yesmaximal, characteristic, complemented
2#85829C211yesmaximal, characteristic, complemented
3C3×#2943C211yesmaximal, characteristic, complemented
4#83901C212yesmaximal, complemented
5#82526C211yesmaximal, characteristic, complemented
6#82509C211yesmaximal, characteristic, complemented
7#79851C214yesmaximal, complemented
8#79816C214yesmaximal, complemented
9#79753C214yesmaximal, complemented
10#79752C214yesmaximal, complemented
11#79197C212yesmaximal, complemented
12#78827C212yesmaximal, complemented
13#78820C212yesmaximal, complemented
14C2×#4713C211yesmaximal, characteristic, complemented
15#78480C211yesmaximal, characteristic, complemented
16#61337—31nomaximal, Sylow
17C2×C3×#135C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C3×C4×#41C2211yescharacteristic
20#5830C2212yes—
21#5827C2212yes—
22C2×#949C2212yes—
23#5585C2214yes—
24#5569C2214yescomplemented
25#5553C2214yes—
26#5550—24no—
27#5539—24no—
28#5539C2214yescomplemented
29#5535—21no—
30#5535—21no—
31#5521C2212yes—
32#5519—22no—
33#5519C2212yes—
34#5477—22no—
35C2×#874C2211yescharacteristic
36#5470—21no—
37#5470C2211yescharacteristic
38C2×#870—21no—
39C2×#870C2211yescharacteristic
40C3×#442C2212yescomplemented
41C3×#441C2212yescomplemented
42C3×#440C2212yescomplemented
43C3×#440C2212yescomplemented
44C3×#427C2211yescharacteristic
45C3×#416C2211yescharacteristic
46C3×#416C2211yescharacteristic
47C2×C3×#97C2211yescharacteristic
48#5232C2212yes—
49#5185C2214yescomplemented
50#5165C2214yescomplemented
51#5163C2214yescomplemented
52#5150C2214yescomplemented
53C2×#821C2212yes—
54C2×#821C2212yes—
55#5102C2212yes—
56#5101C2212yes—
57#5027C2214yes—
58#5022C2214yescomplemented
59#5015C2214yes—
60#5009C2214yescomplemented
61#5001C2212yes—
62#4995C2212yes—
63#4924C2211yescharacteristic
64#4891C2214yescomplemented
65#4890C2214yescomplemented
66#4871C2214yescomplemented
67#4870C2214yescomplemented
68#4817C2214yescomplemented
69#4800C2214yescomplemented
70#4794C2214yescomplemented
71#4788C2214yescomplemented
72#4785C2214yescomplemented
73#4764C2214yescomplemented
74#4760C2214yescomplemented
75#4756C2214yescomplemented
76#4713C2214yes—
77#4713C2214yes—
78#4701C2214yescomplemented
79#4701C2214yescomplemented
80#3508—31no—
81#3504—31no—
82#3371—34no—
83#3370—34no—
84#3368—34no—
85#3366—34no—
86#3136—32no—
87C2×#490—31no—
88#3081—32no—
89#3080—31no—
90#3045—32no—
91#3038—31no—
92#2961—31no—
93#2946—32no—
94#2943D311yescharacteristic, complemented
95C2×C3×#41C2311yescharacteristic
96C2×C3×#41C2311yescharacteristic
97C2×#235—24no—
98C22×D12—22no—
99C22×D12C2312yes—
100C3×#135—24no—
101C2×C3×Q16C2311yescharacteristic
102C2×C3×SD16C2311yescharacteristic
103C2×C3×SD16C2311yescharacteristic
104C2×C3×D8C2311yescharacteristic
105C22×C3×C8C2311yescharacteristic
106C22×C3×C8C2311yescharacteristic
107C3×C4×Q8C2312yes—
108C3×C4×Q8C2312yes—
109C3×C4×D4—22no—
110C3×C4×D4—22no—
111C3×C4×D4C2312yes—
112C3×C4×D4C2312yes—
113C3×#117—21no—
114C3×#117—21no—
115C3×#117C2311yescharacteristic
116C3×#117C2311yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120#949—28no—
121#947C2314yes—
122Q8×Dic3C2314yes—
123C2×#215C2314yes—
124#940C2314yes—
125#937—24no—
126D4×Dic3—24no—
127D4×Dic3C2314yes—
128C2×#213C2314yes—
129#930—22no—
130#930—22no—
131#930—24no—
132#926—22no—
133#926C2312yes—
134C2×C4⋊Dic3—22no—
135C2×C4⋊Dic3C2312yes—
136#924—24no—
137#924D414yes—
138C2×Dic3⋊C4—22no—
139C2×Dic3⋊C4—24no—
140C2×#208C2312yes—
141#894—24no—
142#894—24no—
143#893C2314yes—
144#893C2314yes—
145#890—24no—
146#890C2314yes—
147#888—24no—
148#888C2314yes—
149#884—24no—
150#884—24no—
151D3×C22⋊C4—24no—
152#878—24no—
153#878—24no—
154#878—24no—
155#875—24no—
156#874—21no—
157#874—21no—
158#874C2314yes—
159C4×D12—22no—
160C4×D12—22no—
161C4×D12—24no—
162#870—21no—
163#870—21no—
164#870—22no—
165#870—24no—
166#870D411yescharacteristic
167#870D411yescharacteristic
168#870C2314yes—
169#869—22no—
170#869D412yes—
171C4×Dic6—24no—
172C4×Dic6C2314yes—
173C3×#107C2312yes—
174C3×#106C2312yes—
175C3×#103C2312yes—
176C3×#103C2312yes—
177C3×#102C2312yes—
178C3×#102C2312yes—
179C3×#97C2312yes—
180C3×#97C2312yes—
181#835C2314yes—
182#832C2314yes—
183#821C2314yes—
184#821C2314yes—
185#821C2314yes—
186#821C2314yes—
187#815C2314yes—
188Dic3⋊C8C2314yes—
189Dic3⋊C8C2314yes—
190C8×Dic3C2314yes—
191#809C2314yes—
192#809C2314yes—
193#808C2314yes—
194#807C2314yes—
195C4⋊(C3⋊C8)C2314yes—
196C2×AGL(1,ℤ/8)—32no—
197C2×#135C2×D311yescharacteristic
198#559—34no—
199#557—34no—
200#554—34no—
201#554—64no—
202#550—34no—
203#546—64no—
204#544—32no—
205#544—62no—
206#540—61no—
207#540—61no—
208#535—32no—
209#533—31no—
210#533—61no—
211#532—62no—
212C2×#127—31no—
213C2×#125—31no—
214C2×#125—61no—
215#520—32no—
216#518—32no—
217C4×#41C2×D311yescharacteristic
218C4×#41C2×D311yescharacteristic
219#490—34no—
220#490—34no—
221#490—34no—
222#489—34no—
223#488—34no—
224#486—34no—
225#479—34no—
226#478—34no—
227#478—34no—
228#478—34no—
229#476—34no—
230#474—34no—
231#464—34no—
232#463—34no—
233#460—34no—
234#460—34no—
235#459—34no—
236#459—34no—
237#446—34no—
238#446—34no—
239#442—34no—
240#442C2×D312yescomplemented
241#441C2×D312yescomplemented
242#440—34no—
243#440C2×D312yescomplemented
244#440C2×D312yescomplemented
245C4×SD16—34no—
246C4×D8—34no—
247#428—32no—
248#427C2×D311yescharacteristic
249#424—32no—
250#423—31no—
251#418—32no—
252#417—32no—
253#416C2×D311yescharacteristic
254#416C2×D311yescharacteristic
255C2×#102—32no—
256C2×#102—32no—
257#405—32no—
258C2×#97C2×D311yescharacteristic
259C23×D3—22no—
260C3×#41—22no—
261C3×#41—22no—
262C3×#41—22no—
263C3×#41#4114yescomplemented
264C2×C3×Q8C2411yescharacteristic
265C2×C3×Q8C2411yescharacteristic
266C2×C3×D4—21no—
267C2×C3×D4—21no—
268C2×C3×D4C2411yescharacteristic
269C2×C3×D4C2411yescharacteristic
270C22×C3×C4—21no—
271C22×C3×C4—21no—
272C22×C3×C4—22no—
273C22×C3×C4—22no—
274C22×C3×C4C2×D412yes—
275C22×C3×C4C2411yescharacteristic
276C2×#67—24no—
277C2×#67—24no—
278C22×Dic3—22no—
279C22×Dic3—24no—
280#235—88no—
281C2×D12—24no—
282C2×D12—24no—
283C2×D12—22no—
284C2×D12—24no—
285C2×D12—24no—
286C2×D12—84no—
287C2×D12C2414yes—
288C2×C4×D3—24no—
289C2×C4×D3—24no—
290C2×Dic6—24no—
291C2×Dic6C2×D414yes—
292C3×Q16—22no—
293C3×SD16—22no—
294C3×SD16—22no—
295C3×D8—22no—
296C2×C3×C8—22no—
297C2×C3×C8—22no—
298C2×C3×C8C2412yes—
299C2×C3×C8C2411yescharacteristic
300C2×C3×C8C2411yescharacteristic
301C3×C4⋊C4—22no—
302C3×C4⋊C4—22no—
303C3×C4⋊C4C2412yes—
304C3×C4⋊C4C2412yes—
305C3×C22⋊C4—21no—
306C3×C22⋊C4—21no—
307C3×C22⋊C4—22no—
308C3×C22⋊C4—22no—
309C3×C42—22no—
310C3×C42—21no—
311C3×C42—21no—
312C3×C42—22no—
313C3×C42C2×D411yescharacteristic
314C3×C42C2×D411yescharacteristic
315C3×C42C2412yes—
316#217—24no—
317#217C2×D414yes—
318#215—28no—
319#213—28no—
320(C2×D3)⋊C4—24no—
321(C2×D3)⋊C4—24no—
322(C2×D3)⋊C4—24no—
323(C2×D3)⋊C4—24no—
324C4⋊Dic3—22no—
325C4⋊Dic3—24no—
326C4⋊Dic3—24no—
327C4⋊Dic3C2×D414yes—
328C4⋊Dic3C2×D412yes—
329C4⋊Dic3C2×D412yes—
330C4⋊Dic3C2414yes—
331Dic3⋊C4—24no—
332Dic3⋊C4—24no—
333Dic3⋊C4—24no—
334Dic3⋊C4—24no—
335Dic3⋊C4C2×D414yes—
336Dic3⋊C4C2×D414yes—
337C4×Dic3—24no—
338C4×Dic3C2414yes—
339#208—24no—
340C2×C3⋊C8C2414yes—
341C2×#41—64no—
342C2×#41C22×D311yescharacteristic
343C2×#41C22×D311yescharacteristic
344C22×D4—32no—
345C22×D4—62no—
346AGL(1,ℤ/8)—68no—
347#135—24no—
348C2×Q16C22×D311yescharacteristic
349C2×SD16—34no—
350C2×SD16C22×D311yescharacteristic
351C2×SD16C22×D311yescharacteristic
352C2×D8—34no—
353C2×D8C22×D311yescharacteristic
354C2×M16—32no—
355C22×C8C22×D311yescharacteristic
356C22×C8C22×D311yescharacteristic
357C4⋊Q8—34no—
358C4⋊Q8—32no—
359C4⋊Q8—62no—
360C4⋊Q8—64no—
361#127—34no—
362#127—34no—
363#127—61no—
364#127—61no—
365#125—34no—
366#125—31no—
367#125—31no—
368#125—34no—
369#125—62no—
370#125—64no—
371#125—64no—
372#125—61no—
373#125—61no—
374#124—34no—
375#124—64no—
376#123—64no—
377#123—64no—
378C22⋊Q8—34no—
379C22⋊Q8—64no—
380C22⋊Q8—64no—
381C22⋊Q8—64no—
382C22⋊Q8—64no—
383#121—64no—
384#121—64no—
385#121—62no—
386#121—62no—
387#121—64no—
388#120—64no—
389C4×Q8—34no—
390C4×Q8—34no—
391C4×Q8—64no—
392C4×Q8C22×D312yes—
393C4×Q8C22×D312yes—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—34no—
397C4×D4—34no—
398C4×D4—34no—
399C4×D4—62no—
400C4×D4—62no—
401C4×D4—64no—
402C4×D4—64no—
403C4×D4C22×D312yes—
404C4×D4C22×D312yes—
405#117—21no—
406#117—21no—
407#117—32no—
408#117—62no—
409#117C22×D311yescharacteristic
410#117C22×D311yescharacteristic
411C2×C4⋊C4—32no—
412C2×C4⋊C4—62no—
413C2×C4⋊C4—64no—
414C2×C4⋊C4—62no—
415C2×C22⋊C4—64no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42C22×D311yescharacteristic
419#107—34no—
420#107C22×D312yes—
421#106—34no—
422#106C22×D312yes—
423C4⋊C8—34no—
424C4⋊C8—34no—
425C4⋊C8—34no—
426#103—34no—
427#103C22×D312yes—
428#103C22×D312yes—
429#102—34no—
430#102—34no—
431#102—34no—
432#102—34no—
433#102—34no—
434#102—34no—
435#102—34no—
436#102C22×D312yes—
437#102C22×D312yes—
438#97—34no—
439#97C22×D312yes—
440#97C22×D312yes—
441C4×C8—34no—
442C23×C3—21no—
443C23×C3—21no—
444C23×C3C22×D411yescharacteristic
445C22×D3—24no—
446C22×D3—44no—
447C22×D3—82no—
448C22×D3—82no—
449C3×Q8—21no—
450C3×Q8—21no—
451C3×Q8—21no—
452C3×Q8C2×#4112yes—
453C3×D4—24no—
454C3×D4—21no—
455C3×D4—21no—
456C3×D4—21no—
457C3×D4—42no—
458C3×D4C2×#4112yes—
459C2×C3×C4—21no—
460C2×C3×C4—24no—
461C2×C3×C4—22no—
462C2×C3×C4—21no—
463C2×C3×C4—22no—
464C2×C3×C4—42no—
465C2×C3×C4—44no—
466C2×C3×C4—42no—
467C2×C3×C4—82no—
468C2×C3×C4C22×D412yes—
469C2×C3×C4C22×D412yes—
470C2×C3×C4C22×D411yescharacteristic
471C2×C3×C4C2×#4112yes—
472C2×C3×C42−1+412yes—
473C2×C3×C4C2511yescharacteristic
474#67—84no—
475#67—84no—
476C2×Dic3—24no—
477C2×Dic3—84no—
478C2×Dic3—82no—
479C2×Dic3C22×D414yes—
480C2×Dic3C22×D414yes—
481D12—24no—
482D12—24no—
483D12—48no—
484D12—44no—
485D12—84no—
486C4×D3—84no—
487C4×D3—84no—
488Dic6—44no—
489Dic6—84no—
490C3×C8—21no—
491C3×C8—21no—
492C3×C8—22no—
493C3⋊C8—24no—
494C24—62no—
495#41—22no—
496#41—22no—
497#41—22no—
498#41—248no—
499#41#23514yescomplemented
500C2×Q8—34no—
501C2×Q8—64no—
502C2×Q8C23×D311yescharacteristic
503C2×Q8C23×D311yescharacteristic
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—34no—
507C2×D4—64no—
508C2×D4—64no—
509C2×D4—64no—
510C2×D4—62no—
511C2×D4—64no—
512C2×D4—64no—
513C2×D4—64no—
514C2×D4—244no—
515C2×D4C23×D311yescharacteristic
516C2×D4C23×D311yescharacteristic
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—64no—
522C22×C4—64no—
523C22×C4—64no—
524C22×C4—62no—
525C22×C4D3×D412yes—
526C22×C4C23×D311yescharacteristic
527Q16—22no—
528SD16—22no—
529SD16—22no—
530SD16—68no—
531D8—22no—
532D8—68no—
533M16—64no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—34no—
537C2×C8C23×D312yes—
538C2×C8C23×D311yescharacteristic
539C2×C8C23×D311yescharacteristic
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—32no—
543C4⋊C4—34no—
544C4⋊C4—32no—
545C4⋊C4—34no—
546C4⋊C4—34no—
547C4⋊C4—34no—
548C4⋊C4—64no—
549C4⋊C4—64no—
550C4⋊C4—64no—
551C4⋊C4—64no—
552C4⋊C4—62no—
553C4⋊C4—64no—
554C4⋊C4—64no—
555C4⋊C4C23×D312yes—
556C4⋊C4C23×D312yes—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—34no—
562C22⋊C4—64no—
563C22⋊C4—64no—
564C22⋊C4—64no—
565C22⋊C4—64no—
566C22⋊C4—64no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—22no—
571C42—34no—
572C42—64no—
573C42D3×D411yescharacteristic
574C42D3×D411yescharacteristic
575C42C23×D312yes—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—42no—
579C22×C3—42no—
580C22×C3—41no—
581C22×C3#54812yes—
582C22×C3C23×D411yescharacteristic
583C2×D3—44no—
584C2×D3—44no—
585C2×D3—44no—
586C2×D3—164no—
587C3×C4—22no—
588C3×C4—22no—
589C3×C4—22no—
590C3×C4—42no—
591C3×C4—42no—
592C3×C4—41no—
593C3×C4#54812yes—
594C3×C4C22×#4111yescharacteristic
595C3×C4C2×2−1+411yescharacteristic
596Dic3—24no—
597Dic3—24no—
598Dic3—44no—
599C23—21no—
600C23—21no—
601C23—64no—
602C23—124no—
603C23—242no—
604C23—242no—
605C23C2×D3×D411yescharacteristic
606Q8—21no—
607Q8—21no—
608Q8—21no—
609Q8—124no—
610Q8—244no—
611Q8C2×#23512yes—
612D4—24no—
613D4—21no—
614D4—21no—
615D4—21no—
616D4—42no—
617D4—64no—
618D4—64no—
619D4—128no—
620D4—124no—
621D4—244no—
622D4—244no—
623D4—244no—
624D4C2×#23512yes—
625C2×C4—21no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—34no—
631C2×C4—34no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—64no—
636C2×C4—82no—
637C2×C4—244no—
638C2×C4—244no—
639C2×C4—244no—
640C2×C4—242no—
641C2×C4C2×#23512yes—
642C2×C4C2×D3×D412yes—
643C2×C4C2×D3×D412yes—
644C2×C4C2×D3×D411yescharacteristic
645C2×C4#101012yes—
646C2×C4C24×D311yescharacteristic
647C8—21no—
648C8—21no—
649C8—22no—
650C8—64no—
651C2×C3—22no—
652C2×C3—42no—
653C2×C3—81no—
654C2×C3C2×#54811yescharacteristic
655C2×C3#358512yes—
656D3—84no—
657C22—22no—
658C22—22no—
659C22—42no—
660C22—42no—
661C22—41no—
662C22—124no—
663C22—124no—
664C22—124no—
665C22—484no—
666C22#555212yes—
667C22C22×D3×D411yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—64no—
675C4—64no—
676C4—124no—
677C4#555212yes—
678C4C22×#23511yescharacteristic
679C4C2×#101011yescharacteristic
680C3#6133711yescharacteristic, complemented, Sylow
681C2—22no—
682C2—42no—
683C2—81no—
684C2—244no—
685C2C2×#555211yescharacteristic
686C2#8720012yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2943C2
socleC22×C3C23×D4
hypercenter#2943D3
hypocenterC3#61337
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2943

Complements

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

NH ≅ G/Ntype
1. #85848C2semidirect
2. #85829C2semidirect
3. C3×#2943C2semidirect
4. #83901C2semidirect
5. #82526C2semidirect
6. #82509C2semidirect
7. #79851C2semidirect
8. #79816C2semidirect
9. #79753C2semidirect
10. #79752C2semidirect
11. #79197C2semidirect
12. #78827C2semidirect
13. #78820C2semidirect
14. C2×#4713C2semidirect
15. #78480C2semidirect
24. #5569C22semidirect
28. #5539C22semidirect
40. C3×#442C22semidirect
41. C3×#441C22semidirect
42. C3×#440C22semidirect
43. C3×#440C22semidirect
49. #5185C22semidirect
50. #5165C22semidirect
51. #5163C22semidirect
52. #5150C22semidirect
58. #5022C22semidirect
60. #5009C22semidirect
64. #4891C22semidirect
65. #4890C22semidirect
66. #4871C22semidirect
67. #4870C22semidirect
68. #4817C22semidirect
69. #4800C22semidirect
70. #4794C22semidirect
71. #4788C22semidirect
72. #4785C22semidirect
73. #4764C22semidirect
74. #4760C22semidirect
75. #4756C22semidirect
78. #4701C22semidirect
79. #4701C22semidirect
94. #2943D3semidirect
240. #442C2×D3semidirect
241. #441C2×D3semidirect
243. #440C2×D3semidirect
244. #440C2×D3semidirect
263. C3×#41#41semidirect
499. #41#235semidirect
680. C3#61337semidirect

Maximal subgroups

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

subgroupindex
#858482
#858292
C3×#29432
#839012
#825262
#825092
#798512
#798162
#797532
#797522
#791972
#788272
#788202
C2×#47132
#784802
#613373