Finite Group Databasebeta

← #11675341

Subgroups — #11675341

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
1C3×#3508C211yesmaximal, characteristic, complemented
2#86282C211yesmaximal, characteristic, complemented
3#83718C214yesmaximal, complemented
4#83714C214yesmaximal, complemented
5#83587C214yesmaximal, complemented
6#83581C214yesmaximal, complemented
7#83385C212yesmaximal, complemented
8#83380C212yesmaximal, complemented
9#83329C211yesmaximal, characteristic, complemented
10#83165C211yesmaximal, characteristic, complemented
11C2×#5071C211yesmaximal, characteristic, complemented
12#82539C212yesmaximal, complemented
13#82537C212yesmaximal, complemented
14#82509C211yesmaximal, characteristic, complemented
15#82506C211yesmaximal, characteristic, complemented
16#61337—31nomaximal, Sylow
17C3×#559C2214yes—
18C3×#550C2214yes—
19C3×#546—24no—
20C3×#540—21no—
21C3×#540—21no—
22C3×#535C2212yes—
23C3×#533C2211yescharacteristic
24C3×#532—22no—
25C2×C3×#127C2211yescharacteristic
26C3×#518C2212yes—
27C3×C4×#41C2211yescharacteristic
28C2×#950C2211yescharacteristic
29C2×#932C2212yes—
30#5729—22no—
31#5688—21no—
32D3×#125—21no—
33#5636—24no—
34#5579C2214yescomplemented
35#5567C2214yescomplemented
36#5523C2212yes—
37#5519C2212yes—
38C4×#235C2211yescharacteristic
39C2×#870C2211yescharacteristic
40#5233C2211yescharacteristic
41#5232C2212yes—
42#5215C2212yes—
43C2×#832C2212yes—
44C2×#832C2212yes—
45#5075C2214yes—
46#5075C2214yescomplemented
47#5072C2214yes—
48#5072C2214yescomplemented
49#5071C2214yes—
50#5071C2214yes—
51#5071C2214yescomplemented
52#5070C2214yescomplemented
53#5057C2214yescomplemented
54#5056C2214yescomplemented
55#5055C2214yescomplemented
56#5054C2214yescomplemented
57#5030C2212yescomplemented
58#5026C2212yescomplemented
59#5024C2214yescomplemented
60#5024C2214yescomplemented
61#5022C2214yescomplemented
62#5018C2214yescomplemented
63#5018C2212yescomplemented
64C4×#214C2214yescomplemented
65#5014C2214yescomplemented
66#5014C2212yescomplemented
67C4×#213C2214yescomplemented
68#5009C2214yescomplemented
69#5009C2214yescomplemented
70#5009C2214yescomplemented
71#5008C2214yescomplemented
72#5007C2214yescomplemented
73#5002C2211yescharacteristic
74#4999C2212yes—
75#4998C2211yescharacteristic
76#4997C2212yes—
77#4995C2212yes—
78#4924C2211yescharacteristic
79C2×#803C2211yescharacteristic
80#3508D311yescharacteristic, complemented
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#2943—31no—
95C2×C3×#41—24no—
96C2×C3×#41C2311yescharacteristic
97C22×C3×D4—22no—
98C22×C3×D4C2312yes—
99C2×#235C2311yescharacteristic
100C3×C4⋊Q8D412yes—
101C3×#127—21no—
102C3×#127—21no—
103C3×#127C2314yes—
104C3×#127C2314yes—
105C3×#125D411yescharacteristic
106C3×#125D411yescharacteristic
107C3×#124—24no—
108C3×#124C2314yes—
109C3×#123—24no—
110C3×#123—24no—
111C3×C22⋊Q8D414yes—
112C3×#121—22no—
113C3×#121—22no—
114C3×#121—24no—
115C3×#121—24no—
116C3×#121—24no—
117C3×#120—24no—
118C3×C4×Q8C2312yes—
119C3×C4×Q8C2314yes—
120C3×C4×D4—22no—
121C3×C4×D4—24no—
122C3×C4×D4—22no—
123C3×C4×D4—22no—
124C3×C4×D4C2314yes—
125C3×C4×D4C2314yes—
126C3×C4×D4C2312yes—
127C3×#117—21no—
128C3×#117C2311yescharacteristic
129C3×#117C2312yes—
130C2×C3×C22⋊C4—24no—
131C2×C3×C42—21no—
132C2×C3×C42C2311yescharacteristic
133#950—24no—
134#946—24no—
135#944—24no—
136C2×#216C2311yescharacteristic
137C2×#215C2311yescharacteristic
138C2×#214C2314yes—
139C2×#214C2311yescharacteristic
140#932—28no—
141C2×#213C2314yes—
142C2×#213C2311yescharacteristic
143C4×#67—22no—
144#926C2311yescharacteristic
145C2×C4⋊Dic3C2312yes—
146C2×Dic3⋊C4—22no—
147C2×#208C2312yes—
148C22×C3⋊C8C2311yescharacteristic
149C22×C3⋊C8C2311yescharacteristic
150#896—24no—
151#896—24no—
152#892—22no—
153D3×C4⋊C4—24no—
154D3×C4⋊C4—22no—
155#890—22no—
156#889—24no—
157#889—24no—
158#888—22no—
159#887—24no—
160#67⋊C4—24no—
161#878—24no—
162C4×D12C2312yes—
163C4×D12C2314yes—
164#872—21no—
165C42×D3—21no—
166#870—21no—
167#870—21no—
168#870C2314yes—
169#870C2314yes—
170#869C2314yes—
171C4×Dic6C2312yes—
172C4×Dic6C2314yes—
173#835C2314yes—
174#835C2312yes—
175#832C2314yes—
176#832C2314yes—
177#832C2314yes—
178#832C2314yes—
179#832C2314yes—
180#832C2314yes—
181#832C2314yes—
182#832C2312yes—
183#810C2312yes—
184#809C2312yes—
185#808C2314yes—
186#808C2312yes—
187#807C2314yes—
188#807C2312yes—
189C4⋊(C3⋊C8)C2314yes—
190C4⋊(C3⋊C8)C2314yes—
191C4⋊(C3⋊C8)C2314yes—
192#803C2312yes—
193#803C2312yes—
194#803C2314yes—
195C4×C3⋊C8C2314yes—
196C2×AGL(1,ℤ/8)—32no—
197C2×#135—31no—
198#559C2×D314yes—
199#557—34no—
200#554—34no—
201#554—64no—
202#550C2×D314yes—
203#546—24no—
204#544—32no—
205#544—62no—
206#540—21no—
207#540—21no—
208#535C2×D312yes—
209#533—61no—
210#533C2×D311yescharacteristic
211#532—22no—
212C2×#127C2×D311yescharacteristic
213C2×#125—31no—
214C2×#125—61no—
215#520—32no—
216#518C2×D312yes—
217C4×#41—31no—
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—32no—
240#442—34no—
241#441—32no—
242#440—32no—
243#440—32no—
244#440—34no—
245C4×SD16—34no—
246C4×D8—34no—
247#428—32no—
248#427—31no—
249#424—32no—
250#423—31no—
251#418—32no—
252#417—32no—
253#416—31no—
254#416—31no—
255C2×#102—32no—
256C2×#102—32no—
257#405—32no—
258C2×#97—31no—
259C24×C3—22no—
260C3×#41—22no—
261C3×#41—22no—
262C3×#41—88no—
263C2×C3×Q8C2×D414yes—
264C2×C3×Q8C2411yescharacteristic
265C2×C3×D4—21no—
266C2×C3×D4—22no—
267C2×C3×D4—24no—
268C2×C3×D4—24no—
269C2×C3×D4—24no—
270C2×C3×D4—24no—
271C2×C3×D4—24no—
272C2×C3×D4—24no—
273C2×C3×D4—84no—
274C2×C3×D4C2414yes—
275C2×C3×D4C2411yescharacteristic
276C22×C3×C4—21no—
277C22×C3×C4—22no—
278C22×C3×C4—24no—
279C22×C3×C4—24no—
280C22×C3×C4C2×D412yes—
281C22×C3×C4C2411yescharacteristic
282C2×#67—21no—
283C22×Dic3—22no—
284#235—22no—
285#235#4114yescomplemented
286C2×D12C2411yescharacteristic
287C2×C4×D3—21no—
288C2×C4×D3—22no—
289C2×C4×D3—24no—
290C2×Dic6—24no—
291C2×Dic6C2411yescharacteristic
292C3×C4⋊C4—22no—
293C3×C4⋊C4C2×D414yes—
294C3×C4⋊C4C2×D412yes—
295C3×C4⋊C4C2×D414yes—
296C3×C4⋊C4C2×D414yes—
297C3×C4⋊C4C2×D412yes—
298C3×C4⋊C4C2412yes—
299C3×C22⋊C4—21no—
300C3×C22⋊C4—24no—
301C3×C22⋊C4—24no—
302C3×C22⋊C4—22no—
303C3×C22⋊C4—24no—
304C3×C22⋊C4—24no—
305C3×C22⋊C4C2×D414yes—
306C3×C42—21no—
307C3×C42—22no—
308C3×C42C2×D411yescharacteristic
309C3×C42C2×D411yescharacteristic
310C3×C42C2414yes—
311C3×C42C2412yes—
312#217—21no—
313#216—22no—
314#215—22no—
315#214—28no—
316#214—22no—
317#213—28no—
318#213—22no—
319(C2×D3)⋊C4—22no—
320(C2×D3)⋊C4—24no—
321C4⋊Dic3—24no—
322C4⋊Dic3—22no—
323C4⋊Dic3C2412yes—
324C4⋊Dic3C2414yes—
325Dic3⋊C4—22no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—24no—
330Dic3⋊C4—24no—
331C4×Dic3—22no—
332C4×Dic3—24no—
333C4×Dic3—21no—
334#208—24no—
335C2×C3⋊C8—22no—
336C2×C3⋊C8—22no—
337C2×C3⋊C8C2412yes—
338C2×C3⋊C8C2414yes—
339C2×C3⋊C8C2411yescharacteristic
340C2×C3⋊C8C2411yescharacteristic
341C2×#41—24no—
342C2×#41—31no—
343C2×#41C22×D311yescharacteristic
344C22×D4—22no—
345C22×D4C22×D312yes—
346AGL(1,ℤ/8)—68no—
347#135—64no—
348C2×Q16—31no—
349C2×SD16—31no—
350C2×SD16—31no—
351C2×SD16—34no—
352C2×D8—31no—
353C2×D8—34no—
354C2×M16—32no—
355C22×C8—31no—
356C22×C8—31no—
357C4⋊Q8—34no—
358C4⋊Q8—64no—
359C4⋊Q8—62no—
360C4⋊Q8#6712yes—
361#127—21no—
362#127—21no—
363#127C22×D314yes—
364#127C22×D314yes—
365#125—34no—
366#125—34no—
367#125—62no—
368#125—64no—
369#125—64no—
370#125—61no—
371#125—61no—
372#125#6711yescharacteristic
373#125#6711yescharacteristic
374#124—24no—
375#124C22×D314yes—
376#123—24no—
377#123—24no—
378C22⋊Q8—64no—
379C22⋊Q8—64no—
380C22⋊Q8—64no—
381C22⋊Q8—64no—
382C22⋊Q8#6714yes—
383#121—22no—
384#121—22no—
385#121—24no—
386#121—24no—
387#121—24no—
388#120—24no—
389C4×Q8—32no—
390C4×Q8—34no—
391C4×Q8—64no—
392C4×Q8C22×D312yes—
393C4×Q8C22×D314yes—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—24no—
398C4×D4—32no—
399C4×D4—34no—
400C4×D4—62no—
401C4×D4—64no—
402C4×D4C22×D312yes—
403C4×D4C22×D314yes—
404C4×D4C22×D314yes—
405#117—21no—
406#117—31no—
407#117—62no—
408#117—61no—
409#117C22×D311yescharacteristic
410#117C22×D312yes—
411C2×C4⋊C4—32no—
412C2×C4⋊C4—62no—
413C2×C4⋊C4—62no—
414C2×C4⋊C4—64no—
415C2×C22⋊C4—24no—
416C2×C42—21no—
417C2×C42—61no—
418C2×C42C22×D311yescharacteristic
419#107—32no—
420#107—34no—
421#106—32no—
422#106—34no—
423C4⋊C8—34no—
424C4⋊C8—34no—
425C4⋊C8—34no—
426#103—32no—
427#103—34no—
428#103—32no—
429#102—34no—
430#102—34no—
431#102—34no—
432#102—34no—
433#102—32no—
434#102—34no—
435#102—34no—
436#102—34no—
437#102—32no—
438#97—32no—
439#97—32no—
440#97—34no—
441C4×C8—34no—
442C23×C3—24no—
443C23×C3—21no—
444C23×C3—44no—
445C23×C3—82no—
446C23×C3—82no—
447C23×C3C22×D411yescharacteristic
448C22×D3—21no—
449C3×Q8—21no—
450C3×Q8—21no—
451C3×Q8—44no—
452C3×D4—24no—
453C3×D4—21no—
454C3×D4—24no—
455C3×D4—21no—
456C3×D4—48no—
457C3×D4—42no—
458C3×D4—44no—
459C3×D4—84no—
460C3×D4—84no—
461C3×D4—84no—
462C2×C3×C4—21no—
463C2×C3×C4—22no—
464C2×C3×C4—42no—
465C2×C3×C4—44no—
466C2×C3×C4—42no—
467C2×C3×C4—84no—
468C2×C3×C4—84no—
469C2×C3×C4C22×D412yes—
470C2×C3×C4C22×D414yes—
471C2×C3×C4C22×D412yes—
472C2×C3×C4C22×D414yes—
473C2×C3×C4C22×D411yescharacteristic
474C2×C3×C4C2×#4112yes—
475C2×C3×C42−1+412yes—
476C2×C3×C4C2511yescharacteristic
477#67—24no—
478C2×Dic3—21no—
479C2×Dic3—22no—
480C2×Dic3—24no—
481C2×Dic3—82no—
482D12—21no—
483D12C2×#4112yes—
484C4×D3—24no—
485C4×D3—82no—
486C4×D3—84no—
487Dic6—21no—
488Dic6—84no—
489Dic6C2×#4112yes—
490C3⋊C8—21no—
491C3⋊C8—21no—
492C3⋊C8—24no—
493C3⋊C8—22no—
494C24—22no—
495#41—22no—
496#41—22no—
497#41—34no—
498#41—62no—
499#41—88no—
500C2×Q8—31no—
501C2×Q8—64no—
502C2×Q8C2×#6714yes—
503C2×Q8C23×D311yescharacteristic
504C2×D4—21no—
505C2×D4—24no—
506C2×D4—24no—
507C2×D4—22no—
508C2×D4—24no—
509C2×D4—24no—
510C2×D4—24no—
511C2×D4—24no—
512C2×D4—31no—
513C2×D4—61no—
514C2×D4—84no—
515C2×D4C23×D314yes—
516C2×D4C23×D311yescharacteristic
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—24no—
520C22×C4—24no—
521C22×C4—61no—
522C22×C4—62no—
523C22×C4—62no—
524C22×C4—64no—
525C22×C4C2×#6712yes—
526C22×C4C23×D311yescharacteristic
527Q16—62no—
528SD16—62no—
529SD16—68no—
530SD16—62no—
531D8—62no—
532D8—68no—
533M16—64no—
534C2×C8—32no—
535C2×C8—34no—
536C2×C8—31no—
537C2×C8—31no—
538C2×C8—62no—
539C2×C8—62no—
540C4⋊C4—22no—
541C4⋊C4—34no—
542C4⋊C4—32no—
543C4⋊C4—62no—
544C4⋊C4—64no—
545C4⋊C4—64no—
546C4⋊C4—64no—
547C4⋊C4—64no—
548C4⋊C4—64no—
549C4⋊C4—62no—
550C4⋊C4—64no—
551C4⋊C4C2×#6712yes—
552C4⋊C4C2×#6714yes—
553C4⋊C4C2×#6712yes—
554C4⋊C4C2×#6714yes—
555C4⋊C4C2×#6714yes—
556C4⋊C4C23×D312yes—
557C22⋊C4—21no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—22no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—61no—
564C22⋊C4—62no—
565C22⋊C4—64no—
566C22⋊C4C2×#6714yes—
567C42—21no—
568C42—22no—
569C42—61no—
570C42—62no—
571C42—64no—
572C42C2×#6711yescharacteristic
573C42C2×#6711yescharacteristic
574C42C23×D314yes—
575C42C23×D312yes—
576C22×C3—22no—
577C22×C3—42no—
578C22×C3—44no—
579C22×C3—44no—
580C22×C3—41no—
581C22×C3—44no—
582C22×C3—164no—
583C22×C3#54812yes—
584C22×C3C23×D411yescharacteristic
585C2×D3—22no—
586C2×D3—42no—
587C3×C4—24no—
588C3×C4—22no—
589C3×C4—24no—
590C3×C4—22no—
591C3×C4—42no—
592C3×C4—41no—
593C3×C4#54812yes—
594C3×C4C22×#4111yescharacteristic
595C3×C4C2×2−1+411yescharacteristic
596Dic3—22no—
597Dic3—42no—
598Dic3—44no—
599C23—24no—
600C23—21no—
601C23—44no—
602C23—61no—
603C23—82no—
604C23—82no—
605C23C22×#6711yescharacteristic
606Q8—21no—
607Q8—21no—
608Q8—32no—
609Q8—44no—
610Q8—61no—
611Q8—244no—
612D4—24no—
613D4—21no—
614D4—24no—
615D4—21no—
616D4—32no—
617D4—48no—
618D4—42no—
619D4—44no—
620D4—64no—
621D4—61no—
622D4—84no—
623D4—84no—
624D4—84no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—61no—
631C2×C4—64no—
632C2×C4—62no—
633C2×C4—64no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—242no—
637C2×C4—244no—
638C2×C4—242no—
639C2×C4D3×#4112yes—
640C2×C4#101012yes—
641C2×C4C22×#6712yes—
642C2×C4C22×#6712yes—
643C2×C4C22×#6714yes—
644C2×C4C22×#6714yes—
645C2×C4C22×#6711yescharacteristic
646C2×C4C24×D311yescharacteristic
647C8—61no—
648C8—61no—
649C8—64no—
650C8—62no—
651C2×C3—22no—
652C2×C3—81no—
653C2×C3—84no—
654C2×C3C2×#54811yescharacteristic
655C2×C3#358512yes—
656D3—42no—
657C22—22no—
658C22—41no—
659C22—44no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—62no—
664C22—122no—
665C22—164no—
666C22#583112yes—
667C22C23×#6711yescharacteristic
668C4—22no—
669C4—22no—
670C4—24no—
671C4—24no—
672C4—41no—
673C4—42no—
674C4—62no—
675C4—124no—
676C4—122no—
677C4#583112yes—
678C4C2×D3×#4111yescharacteristic
679C4C2×#101011yescharacteristic
680C3#6133711yescharacteristic, complemented, Sylow
681C2—22no—
682C2—84no—
683C2—81no—
684C2—122no—
685C2C2×#583111yescharacteristic
686C2#8727712yes—
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3508C2
socleC22×C3C23×D4
hypercenter#3508D3
hypocenterC3#61337
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. C3×#3508C2semidirect
2. #86282C2semidirect
3. #83718C2semidirect
4. #83714C2semidirect
5. #83587C2semidirect
6. #83581C2semidirect
7. #83385C2semidirect
8. #83380C2semidirect
9. #83329C2semidirect
10. #83165C2semidirect
11. C2×#5071C2semidirect
12. #82539C2semidirect
13. #82537C2semidirect
14. #82509C2semidirect
15. #82506C2semidirect
34. #5579C22semidirect
35. #5567C22semidirect
46. #5075C22semidirect
48. #5072C22semidirect
51. #5071C22semidirect
52. #5070C22semidirect
53. #5057C22semidirect
54. #5056C22semidirect
55. #5055C22semidirect
56. #5054C22semidirect
57. #5030C22semidirect
58. #5026C22semidirect
59. #5024C22semidirect
60. #5024C22semidirect
61. #5022C22semidirect
62. #5018C22semidirect
63. #5018C22semidirect
64. C4×#214C22semidirect
65. #5014C22semidirect
66. #5014C22semidirect
67. C4×#213C22semidirect
68. #5009C22semidirect
69. #5009C22semidirect
70. #5009C22semidirect
71. #5008C22semidirect
72. #5007C22semidirect
80. #3508D3semidirect
285. #235#41semidirect
680. C3#61337semidirect

Maximal subgroups

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

subgroupindex
C3×#35082
#862822
#837182
#837142
#835872
#835812
#833852
#833802
#833292
#831652
C2×#50712
#825392
#825372
#825092
#825062
#613373