Finite Group Databasebeta

← #11675346

Subgroups — #11675346

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×#3504C211yesmaximal, characteristic, complemented
2#86261C211yesmaximal, characteristic, complemented
3#83691C214yesmaximal, complemented
4#83690C214yesmaximal, complemented
5#83639C214yesmaximal, complemented
6#83638C214yesmaximal, complemented
7#83474C212yesmaximal, complemented
8#83470C212yesmaximal, complemented
9#83329C211yesmaximal, characteristic, complemented
10#83017C211yesmaximal, characteristic, complemented
11C2×#5067C211yesmaximal, characteristic, complemented
12#82565C212yesmaximal, complemented
13#82564C212yesmaximal, complemented
14#82526C211yesmaximal, characteristic, complemented
15#82506C211yesmaximal, characteristic, complemented
16#61337—31nomaximal, Sylow
17C3×#557C2214yescomplemented
18C3×#554—24no—
19C3×#554C2214yescomplemented
20C3×#544—22no—
21C3×#544C2212yes—
22C3×#533—21no—
23C3×#533C2211yescharacteristic
24C2×C3×#125—21no—
25C2×C3×#125C2211yescharacteristic
26C3×#520C2212yes—
27C3×C4×#41C2211yescharacteristic
28C2×#950C2211yescharacteristic
29C2×#949C2212yes—
30#5725—22no—
31#5697—21no—
32#5697—21no—
33#5630—24no—
34#5585C2214yes—
35#5577C2214yes—
36#5525C2212yes—
37#5521C2212yes—
38C2×#874C2211yescharacteristic
39C4×#235C2211yescharacteristic
40#5233C2211yescharacteristic
41#5089C2214yescomplemented
42#5088C2214yescomplemented
43#5087C2214yescomplemented
44#5084C2214yescomplemented
45#5068C2214yescomplemented
46#5067C2214yes—
47#5067C2214yes—
48#5067C2214yescomplemented
49#5066C2214yescomplemented
50#5066C2214yescomplemented
51#5066C2214yescomplemented
52#5063C2214yescomplemented
53#5030C2212yescomplemented
54#5028C2214yescomplemented
55#5027C2214yes—
56#5027C2214yescomplemented
57#5026C2212yescomplemented
58#5026C2214yescomplemented
59C4×#215C2214yescomplemented
60#5024C2214yescomplemented
61#5024C2214yescomplemented
62#5018C2212yescomplemented
63#5016C2214yescomplemented
64#5015C2214yes—
65#5015C2214yescomplemented
66#5014C2212yescomplemented
67#5014C2214yescomplemented
68C4×#213C2214yescomplemented
69#5002C2211yescharacteristic
70#5001C2212yes—
71#4999C2212yes—
72#4998C2211yescharacteristic
73#4997C2212yes—
74#4966C2212yes—
75C2×#809C2212yes—
76C2×#809C2212yes—
77#4964C2212yes—
78#4924C2211yescharacteristic
79C2×#803C2211yescharacteristic
80#3508—31no—
81#3504D311yescharacteristic, complemented
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×#41C2311yescharacteristic
96C2×#239—24no—
97C2×#235C2311yescharacteristic
98C22×D12—22no—
99C22×D12C2312yes—
100C3×C4⋊Q8—24no—
101C3×C4⋊Q8—22no—
102C3×C4⋊Q8D412yes—
103C3×C4⋊Q8C2314yes—
104C3×#125—22no—
105C3×#125—24no—
106C3×#125—24no—
107C3×#125—21no—
108C3×#125—21no—
109C3×#125D411yescharacteristic
110C3×#125D411yescharacteristic
111C3×#125C2314yes—
112C3×#125C2314yes—
113C3×C22⋊Q8—24no—
114C3×C22⋊Q8—24no—
115C3×C22⋊Q8—24no—
116C3×C22⋊Q8—24no—
117C3×C22⋊Q8D414yes—
118C3×C4×Q8—24no—
119C3×C4×Q8C2314yes—
120C3×C4×Q8C2312yes—
121C3×C4×Q8C2314yes—
122C3×C4×D4—22no—
123C3×C4×D4—24no—
124C3×C4×D4C2314yes—
125C3×C4×D4C2312yes—
126C3×#117—22no—
127C3×#117—21no—
128C3×#117C2311yescharacteristic
129C3×#117C2312yes—
130C2×C3×C4⋊C4—22no—
131C2×C3×C4⋊C4—22no—
132C2×C3×C4⋊C4—24no—
133C2×C3×C4⋊C4C2312yes—
134C2×C3×C42—21no—
135C2×C3×C42C2311yescharacteristic
136#950—24no—
137#949—28no—
138#947—24no—
139C2×#216C2311yescharacteristic
140C2×#215C2314yes—
141C2×#215C2311yescharacteristic
142C2×#214C2311yescharacteristic
143C2×#213C2314yes—
144C2×#213C2311yescharacteristic
145C4×#67—22no—
146#926C2311yescharacteristic
147C2×#208C2312yes—
148C22×C3⋊C8C2311yescharacteristic
149C22×C3⋊C8C2311yescharacteristic
150#895—22no—
151#895—22no—
152#895—24no—
153#895—24no—
154#895—24no—
155#894—24no—
156#894—24no—
157#893—22no—
158#893—22no—
159#893—24no—
160#882—24no—
161D3×C22⋊C4—24no—
162#875C2314yes—
163#874—21no—
164#874—21no—
165#874C2314yes—
166#874C2314yes—
167C4×D12C2312yes—
168C4×D12C2314yes—
169C4×D12C2314yes—
170#872—21no—
171C42×D3—21no—
172C4×Dic6C2312yes—
173#835C2312yes—
174#832C2312yes—
175#810C2314yes—
176#810C2312yes—
177#809C2314yes—
178#809C2314yes—
179#809C2314yes—
180#809C2314yes—
181#809C2314yes—
182#809C2314yes—
183#809C2314yes—
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#559—34no—
199#557C2×D314yescomplemented
200#554—24no—
201#554C2×D314yescomplemented
202#550—34no—
203#546—64no—
204#544—22no—
205#544C2×D312yes—
206#540—61no—
207#540—61no—
208#535—32no—
209#533—21no—
210#533C2×D311yescharacteristic
211#532—62no—
212C2×#127—31no—
213C2×#125—21no—
214C2×#125C2×D311yescharacteristic
215#520C2×D312yes—
216#518—32no—
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—34no—
244#440—32no—
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—
259C23×D3—22no—
260C3×#41—22no—
261C3×#41#4114yescomplemented
262C2×C3×Q8—24no—
263C2×C3×Q8C2×D414yes—
264C2×C3×Q8C2411yescharacteristic
265C2×C3×D4—21no—
266C2×C3×D4C2411yescharacteristic
267C22×C3×C4—21no—
268C22×C3×C4—22no—
269C22×C3×C4—22no—
270C22×C3×C4—24no—
271C22×C3×C4C2×D412yes—
272C22×C3×C4C2411yescharacteristic
273C2×#67—21no—
274#239—88no—
275#235—22no—
276#235—22no—
277C2×D12—24no—
278C2×D12—24no—
279C2×D12—22no—
280C2×D12—24no—
281C2×D12—24no—
282C2×D12—24no—
283C2×D12—24no—
284C2×D12—84no—
285C2×D12C2411yescharacteristic
286C2×D12C2414yes—
287C2×C4×D3—21no—
288C2×C4×D3—22no—
289C2×C4×D3—24no—
290C2×C4×D3—24no—
291C2×Dic6C2411yescharacteristic
292C3×C4⋊C4—24no—
293C3×C4⋊C4—22no—
294C3×C4⋊C4—24no—
295C3×C4⋊C4—24no—
296C3×C4⋊C4—24no—
297C3×C4⋊C4—24no—
298C3×C4⋊C4—22no—
299C3×C4⋊C4—24no—
300C3×C4⋊C4C2×D412yes—
301C3×C4⋊C4C2×D412yes—
302C3×C4⋊C4C2×D414yes—
303C3×C4⋊C4C2×D414yes—
304C3×C4⋊C4C2×D414yes—
305C3×C4⋊C4C2414yes—
306C3×C4⋊C4C2412yes—
307C3×C22⋊C4—21no—
308C3×C22⋊C4—22no—
309C3×C22⋊C4—24no—
310C3×C22⋊C4C2×D414yes—
311C3×C42—21no—
312C3×C42—22no—
313C3×C42—24no—
314C3×C42C2×D411yescharacteristic
315C3×C42C2×D411yescharacteristic
316C3×C42C2414yes—
317C3×C42C2412yes—
318#217—21no—
319#216—22no—
320#215—28no—
321#215—22no—
322#214—22no—
323#213—28no—
324#213—22no—
325(C2×D3)⋊C4—22no—
326(C2×D3)⋊C4—24no—
327(C2×D3)⋊C4—24no—
328(C2×D3)⋊C4—24no—
329(C2×D3)⋊C4—24no—
330C4⋊Dic3C2412yes—
331Dic3⋊C4—22no—
332C4×Dic3—22no—
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—31no—
342C2×#41—64no—
343C2×#41C22×D311yescharacteristic
344C22×D4—32no—
345C22×D4—62no—
346AGL(1,ℤ/8)—68no—
347#135—64no—
348C2×Q16—31no—
349C2×SD16—31no—
350C2×SD16—34no—
351C2×SD16—31no—
352C2×D8—31no—
353C2×D8—34no—
354C2×M16—32no—
355C22×C8—31no—
356C22×C8—31no—
357C4⋊Q8—24no—
358C4⋊Q8—22no—
359C4⋊Q8#6712yes—
360C4⋊Q8C22×D314yes—
361#127—34no—
362#127—34no—
363#127—61no—
364#127—61no—
365#125—22no—
366#125—24no—
367#125—24no—
368#125—21no—
369#125—21no—
370#125#6711yescharacteristic
371#125#6711yescharacteristic
372#125C22×D314yes—
373#125C22×D314yes—
374#124—34no—
375#124—64no—
376#123—64no—
377#123—64no—
378C22⋊Q8—24no—
379C22⋊Q8—24no—
380C22⋊Q8—24no—
381C22⋊Q8—24no—
382C22⋊Q8#6714yes—
383#121—62no—
384#121—62no—
385#121—64no—
386#121—64no—
387#121—64no—
388#120—64no—
389C4×Q8—24no—
390C4×Q8—32no—
391C4×Q8C22×D312yes—
392C4×Q8C22×D314yes—
393C4×Q8C22×D314yes—
394C4×D4—22no—
395C4×D4—24no—
396C4×D4—32no—
397C4×D4—34no—
398C4×D4—34no—
399C4×D4—62no—
400C4×D4—62no—
401C4×D4—62no—
402C4×D4—64no—
403C4×D4C22×D312yes—
404C4×D4C22×D314yes—
405#117—22no—
406#117—21no—
407#117—31no—
408#117—61no—
409#117C22×D311yescharacteristic
410#117C22×D312yes—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—24no—
414C2×C4⋊C4C22×D312yes—
415C2×C22⋊C4—64no—
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—32no—
428#103—34no—
429#102—32no—
430#102—34no—
431#102—34no—
432#102—34no—
433#102—34no—
434#102—32no—
435#102—34no—
436#102—34no—
437#102—34no—
438#97—32no—
439#97—32no—
440#97—34no—
441C4×C8—34no—
442C23×C3—21no—
443C23×C3C22×D411yescharacteristic
444C22×D3—24no—
445C22×D3—21no—
446C22×D3—44no—
447C22×D3—82no—
448C22×D3—82no—
449C3×Q8—21no—
450C3×Q8—44no—
451C3×Q8—84no—
452C3×Q8C2×#4112yes—
453C3×D4—24no—
454C3×D4—21no—
455C3×D4C2×#4112yes—
456C2×C3×C4—21no—
457C2×C3×C4—24no—
458C2×C3×C4—22no—
459C2×C3×C4—24no—
460C2×C3×C4—42no—
461C2×C3×C4—84no—
462C2×C3×C4—82no—
463C2×C3×C4—82no—
464C2×C3×C4C22×D412yes—
465C2×C3×C4C22×D412yes—
466C2×C3×C4C22×D414yes—
467C2×C3×C4C22×D414yes—
468C2×C3×C4C22×D411yescharacteristic
469C2×C3×C4C2×#4112yes—
470C2×C3×C42−1+412yes—
471C2×C3×C4C2511yescharacteristic
472#67—42no—
473C2×Dic3—21no—
474C2×Dic3—22no—
475D12—24no—
476D12—21no—
477D12—24no—
478D12—21no—
479D12—48no—
480D12—44no—
481D12—84no—
482D12—84no—
483D12—84no—
484C4×D3—44no—
485C4×D3—42no—
486C4×D3—84no—
487C4×D3—84no—
488Dic6—21no—
489Dic6—21no—
490C3⋊C8—21no—
491C3⋊C8—21no—
492C3⋊C8—24no—
493C3⋊C8—22no—
494C24—62no—
495#41—22no—
496#41—62no—
497#41—62no—
498#41—248no—
499#41#23914yescomplemented
500C2×Q8—24no—
501C2×Q8—31no—
502C2×Q8C2×#6714yes—
503C2×Q8C23×D311yescharacteristic
504C2×D4—21no—
505C2×D4—34no—
506C2×D4—31no—
507C2×D4—61no—
508C2×D4—64no—
509C2×D4—64no—
510C2×D4—62no—
511C2×D4—64no—
512C2×D4—64no—
513C2×D4—64no—
514C2×D4—64no—
515C2×D4—244no—
516C2×D4C23×D311yescharacteristic
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—24no—
521C22×C4—61no—
522C22×C4—62no—
523C22×C4—64no—
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—24no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—24no—
548C4⋊C4—32no—
549C4⋊C4—62no—
550C4⋊C4C2×#6712yes—
551C4⋊C4C2×#6714yes—
552C4⋊C4C2×#6712yes—
553C4⋊C4C2×#6714yes—
554C4⋊C4C2×#6714yes—
555C4⋊C4C23×D314yes—
556C4⋊C4C23×D312yes—
557C22⋊C4—21no—
558C22⋊C4—22no—
559C22⋊C4—24no—
560C22⋊C4—61no—
561C22⋊C4—64no—
562C22⋊C4—64no—
563C22⋊C4—62no—
564C22⋊C4—64no—
565C22⋊C4—64no—
566C22⋊C4C2×#6714yes—
567C42—21no—
568C42—22no—
569C42—24no—
570C42—61no—
571C42—62no—
572C42C2×#6711yescharacteristic
573C42C2×#6711yescharacteristic
574C42C23×D314yes—
575C42C23×D312yes—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—42no—
579C22×C3#54812yes—
580C22×C3C23×D411yescharacteristic
581C2×D3—42no—
582C2×D3—44no—
583C2×D3—44no—
584C2×D3—41no—
585C2×D3—44no—
586C2×D3—164no—
587C3×C4—22no—
588C3×C4—24no—
589C3×C4—24no—
590C3×C4—22no—
591C3×C4—22no—
592C3×C4—44no—
593C3×C4—42no—
594C3×C4#54812yes—
595C3×C4C22×#4111yescharacteristic
596C3×C4C2×2−1+411yescharacteristic
597Dic3—42no—
598Dic3—41no—
599C23—21no—
600C23—64no—
601C23—61no—
602C23—124no—
603C23—242no—
604C23—242no—
605C23C22×#6711yescharacteristic
606Q8—21no—
607Q8—44no—
608Q8—61no—
609Q8—61no—
610Q8—84no—
611Q8C2×#23912yes—
612D4—24no—
613D4—21no—
614D4—64no—
615D4—61no—
616D4—64no—
617D4—61no—
618D4—128no—
619D4—122no—
620D4—124no—
621D4—244no—
622D4—244no—
623D4—244no—
624D4C2×#23912yes—
625C2×C4—21no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—42no—
630C2×C4—61no—
631C2×C4—62no—
632C2×C4—82no—
633C2×C4—84no—
634C2×C4—82no—
635C2×C4—124no—
636C2×C4—122no—
637C2×C4—244no—
638C2×C4—244no—
639C2×C4C2×#23912yes—
640C2×C4#100712yes—
641C2×C4C22×#6712yes—
642C2×C4C22×#6714yes—
643C2×C4C22×#6712yes—
644C2×C4C22×#6714yes—
645C2×C4C22×#6711yescharacteristic
646C2×C4C24×D311yescharacteristic
647C8—61no—
648C8—61no—
649C8—64no—
650C8—62no—
651C2×C3—22no—
652C2×C3—42no—
653C2×C3C2×#54811yescharacteristic
654C2×C3#358512yes—
655D3—81no—
656D3—84no—
657C22—22no—
658C22—22no—
659C22—42no—
660C22—122no—
661C22—124no—
662C22—124no—
663C22—121no—
664C22—124no—
665C22—484no—
666C22#581712yes—
667C22C23×#6711yescharacteristic
668C4—22no—
669C4—22no—
670C4—24no—
671C4—24no—
672C4—22no—
673C4—44no—
674C4—42no—
675C4—122no—
676C4—121no—
677C4#581712yes—
678C4C22×#23911yescharacteristic
679C4C2×#100711yescharacteristic
680C3#6133711yescharacteristic, complemented, Sylow
681C2—22no—
682C2—42no—
683C2—244no—
684C2—241no—
685C2C2×#581711yescharacteristic
686C2#8729812yes—
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3504C2
socleC22×C3C23×D4
hypercenter#3504D3
hypocenterC3#61337
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533 ◁ #3504

Complements

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

NH ≅ G/Ntype
1. C3×#3504C2semidirect
2. #86261C2semidirect
3. #83691C2semidirect
4. #83690C2semidirect
5. #83639C2semidirect
6. #83638C2semidirect
7. #83474C2semidirect
8. #83470C2semidirect
9. #83329C2semidirect
10. #83017C2semidirect
11. C2×#5067C2semidirect
12. #82565C2semidirect
13. #82564C2semidirect
14. #82526C2semidirect
15. #82506C2semidirect
17. C3×#557C22semidirect
19. C3×#554C22semidirect
41. #5089C22semidirect
42. #5088C22semidirect
43. #5087C22semidirect
44. #5084C22semidirect
45. #5068C22semidirect
48. #5067C22semidirect
49. #5066C22semidirect
50. #5066C22semidirect
51. #5066C22semidirect
52. #5063C22semidirect
53. #5030C22semidirect
54. #5028C22semidirect
56. #5027C22semidirect
57. #5026C22semidirect
58. #5026C22semidirect
59. C4×#215C22semidirect
60. #5024C22semidirect
61. #5024C22semidirect
62. #5018C22semidirect
63. #5016C22semidirect
65. #5015C22semidirect
66. #5014C22semidirect
67. #5014C22semidirect
68. C4×#213C22semidirect
81. #3504D3semidirect
199. #557C2×D3semidirect
201. #554C2×D3semidirect
261. C3×#41#41semidirect
499. #41#239semidirect
680. C3#61337semidirect

Maximal subgroups

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

subgroupindex
C3×#35042
#862612
#836912
#836902
#836392
#836382
#834742
#834702
#833292
#830172
C2×#50672
#825652
#825642
#825262
#825062
#613373