Finite Group Databasebeta

← #11675292

Subgroups — #11675292

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85803C211yesmaximal, characteristic, complemented
2#85788C211yesmaximal, characteristic
3C2×C3×#440C211yesmaximal, characteristic, complemented
4#83849C212yesmaximal, complemented
5#82441C211yesmaximal, characteristic, complemented
6#82296C211yesmaximal, characteristic, complemented
7#79758C214yesmaximal, complemented
8#79756C214yesmaximal, complemented
9#79754C214yesmaximal, complemented
10#79752C214yesmaximal, complemented
11#79162C212yesmaximal, complemented
12#78774C212yesmaximal, complemented
13#78737C212yesmaximal, complemented
14#78497C211yesmaximal, characteristic, complemented
15#78480C211yesmaximal, characteristic, complemented
16#61334—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20#5812C2212yes—
21C2×#942C2212yes—
22#5805C2212yes—
23#5791—22no—
24#5570—21no—
25#5570C2212yescomplemented
26#5569—22no—
27#5569C2214yescomplemented
28Q8×Dic6—21no—
29Q8×Dic6C2212yescomplemented
30#5563—21no—
31#5559C2214yes—
32#5558—22no—
33#5553C2212yes—
34#5551—21no—
35#5542C2212yes—
36#5504C2212yes—
37#5500—22no—
38#5500C2212yes—
39#5494—22no—
40#5477C2211yescharacteristic
41#5470—22no—
42#5470C2211yescharacteristic
43#5470C2211yescharacteristic
44C3×#440C2214yes—
45C3×#440C2214yes—
46C2×C3×#107C2211yescharacteristic
47C2×C3×#103C2211yescharacteristic
48C2×C3×#102C2211yescharacteristic
49C2×C3×#97C2211yescharacteristic
50#5215C2212yes—
51#5166C2214yes—
52#5166C2214yes—
53#5163C2214yes—
54#5163C2214yes—
55#5113C2212yes—
56#5113C2212yes—
57#5102C2212yes—
58#5101C2212yes—
59#5031C2212yescomplemented
60#5028C2214yescomplemented
61#5027C2212yescomplemented
62#5009C2214yes—
63#5008C2212yes—
64#5007C2212yes—
65#4966C2212yes—
66#4964C2212yes—
67#4924C2211yescharacteristic
68#4891C2214yescomplemented
69#4890C2214yescomplemented
70#4877C2214yescomplemented
71#4875C2214yescomplemented
72#4817C2214yescomplemented
73#4790C2214yescomplemented
74#4789C2214yescomplemented
75#4788C2214yescomplemented
76#4768C2214yes—
77#4765C2214yes—
78#4764C2214yes—
79#4760C2214yes—
80#4714C2212yescomplemented
81#4714C2212yescomplemented
82#4713C2212yescomplemented
83#4703C2212yescomplemented
84#4701C2214yescomplemented
85#4701C2214yescomplemented
86#3506—31no—
87#3502—31no—
88#3377—34no—
89#3375—34no—
90#3371—34no—
91#3368—34no—
92#3136—32no—
93#3135—31no—
94#3083—32no—
95#3077—31no—
96#3043—32no—
97#3037—31no—
98#2961—31no—
99#2947—32no—
100C2×#440D311yescharacteristic, complemented
101C22×C3×Q8C2311yescharacteristic
102C22×C3×D4C2311yescharacteristic
103C23×C3×C4—21no—
104C2×#235—22no—
105C2×#235C2312yes—
106C2×C3×SD16—24no—
107C2×C3×SD16C2314yes—
108C22×C3×C8C2311yescharacteristic
109C22×C3×C8C2311yescharacteristic
110C3×C4×Q8—21no—
111C3×C4×Q8—21no—
112C3×C4×Q8C2312yes—
113C3×C4×Q8C2312yes—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4C2312yes—
117C3×C4×D4C2312yes—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C22⋊C4—21no—
122C2×C3×C42—21no—
123C2×C3×C42C2311yescharacteristic
124#953—24no—
125Q8×Dic3—22no—
126Q8×Dic3—22no—
127Q8×Dic3C2314yes—
128Q8×Dic3C2314yes—
129C2×#216C2314yes—
130#942—28no—
131C2×#215C2314yes—
132#940C2314yes—
133#938—22no—
134#938—22no—
135#936C2314yes—
136#930—21no—
137#930—21no—
138#930—22no—
139#929—24no—
140C4×#67—24no—
141C4×#67—24no—
142#926—24no—
143#926—22no—
144#926C2312yes—
145#926C2312yes—
146#924—21no—
147#924—21no—
148#924—22no—
149#924—24no—
150#924D412yes—
151#924D412yes—
152C2×#208C2312yes—
153#893C2314yes—
154#892—22no—
155D3×C4⋊C4—22no—
156#890—24no—
157#890—22no—
158#890—24no—
159#890C2314yes—
160#888—24no—
161#888—22no—
162#888—24no—
163#888C2314yes—
164#887C2314yes—
165#885—24no—
166#884—24no—
167#883—24no—
168#879—24no—
169#875C2312yes—
170#874C2311yescharacteristic
171C4×D12—21no—
172C4×D12—21no—
173#870—24no—
174#870D412yes—
175#870C2312yes—
176#869—22no—
177#869—22no—
178#869D411yescharacteristic
179#869D411yescharacteristic
180#869C2311yescharacteristic
181#869C2311yescharacteristic
182#869C2311yescharacteristic
183C4×Dic6—22no—
184C4×Dic6—21no—
185C4×Dic6—21no—
186C4×Dic6—22no—
187C4×Dic6—24no—
188C4×Dic6C2312yes—
189C4×Dic6C2312yes—
190C3×#107C2314yes—
191C3×#103C2314yes—
192C3×#102C2314yes—
193C3×#97C2312yes—
194C3×#97C2312yes—
195#832C2314yes—
196#832C2314yes—
197#821C2312yes—
198#821C2312yes—
199#821C2312yes—
200#821C2312yes—
201#816C2312yes—
202#816C2312yes—
203#816C2312yes—
204#816C2312yes—
205#815C2314yes—
206Dic3⋊C8C2314yes—
207Dic3⋊C8C2314yes—
208C8×Dic3C2314yes—
209#810C2314yes—
210#809C2314yes—
211#808C2314yes—
212#807C2314yes—
213C4⋊(C3⋊C8)C2312yes—
214C4⋊(C3⋊C8)C2312yes—
215C2×#137—32no—
216C22×SD16C2×D311yescharacteristic
217#559—34no—
218Q82—32no—
219Q82—61no—
220#557—34no—
221#557—32no—
222#557—62no—
223#557—61no—
224#556—32no—
225#555—61no—
226#553—62no—
227#550—32no—
228#547—61no—
229#544—32no—
230#544—62no—
231#538—62no—
232#535—32no—
233#533—31no—
234#533—31no—
235#533—62no—
236#532—31no—
237#525—62no—
238#520—32no—
239#519—32no—
240C2×C4×Q8C2×D311yescharacteristic
241C2×C4×D4C2×D311yescharacteristic
242#491—32no—
243#490—34no—
244#490—32no—
245#489—34no—
246#489—32no—
247#489—32no—
248#488—34no—
249#486—34no—
250#479—34no—
251#478—34no—
252#478—34no—
253#477—34no—
254#476—32no—
255#474—32no—
256#474—34no—
257#464—34no—
258#464—34no—
259#464—34no—
260#462—32no—
261#460—34no—
262#460—34no—
263#460—32no—
264#446—34no—
265#446—34no—
266#441—34no—
267#440—34no—
268#440C2×D314yes—
269#440C2×D314yes—
270C4×Q16—34no—
271C4×SD16—34no—
272#428—32no—
273C2×#107C2×D311yescharacteristic
274#424—32no—
275#423—31no—
276#418—32no—
277#417—32no—
278#416—32no—
279#416—32no—
280C2×#103C2×D311yescharacteristic
281C2×#102C2×D311yescharacteristic
282#405—32no—
283C2×#97C2×D311yescharacteristic
284C24×C3—21no—
285C2×C3×Q8—21no—
286C2×C3×Q8—22no—
287C2×C3×Q8#4114yescomplemented
288C2×C3×Q8C2412yes—
289C2×C3×D4—21no—
290C2×C3×D4—22no—
291C2×C3×D4—22no—
292C2×C3×D4C2412yes—
293C22×C3×C4—21no—
294C22×C3×C4—21no—
295C22×C3×C4—21no—
296C22×C3×C4—22no—
297C22×C3×C4—22no—
298C22×C3×C4—82no—
299C22×C3×C4C2×D411yescharacteristic
300C22×C3×C4C2×D411yescharacteristic
301C22×C3×C4C2411yescharacteristic
302C2×#67—22no—
303C2×#67—24no—
304C2×#67—24no—
305#235—24no—
306#235—24no—
307#235—84no—
308C2×D12—21no—
309C2×D12—21no—
310C2×D12C2412yes—
311C2×C4×D3—22no—
312C2×C4×D3—24no—
313C2×Dic6—21no—
314C2×Dic6—21no—
315C2×Dic6—24no—
316C2×Dic6C2×D412yes—
317C2×Dic6C2×D412yes—
318C2×Dic6C2412yes—
319C3×SD16—28no—
320C2×C3×C8—22no—
321C2×C3×C8—22no—
322C2×C3×C8C2412yes—
323C2×C3×C8C2412yes—
324C3×C4⋊C4—22no—
325C3×C4⋊C4—22no—
326C3×C4⋊C4—21no—
327C3×C4⋊C4—21no—
328C3×C4⋊C4C2412yes—
329C3×C4⋊C4C2412yes—
330C3×C22⋊C4—22no—
331C3×C22⋊C4—22no—
332C3×C42—22no—
333C3×C42—22no—
334C3×C42C2×D411yescharacteristic
335C3×C42C2×D411yescharacteristic
336C3×C42C2411yescharacteristic
337C3×C42C2411yescharacteristic
338#217—22no—
339#217—24no—
340#217—24no—
341#217C2×D414yes—
342#216—28no—
343#215—28no—
344(C2×D3)⋊C4—22no—
345(C2×D3)⋊C4—22no—
346(C2×D3)⋊C4—24no—
347C4⋊Dic3—21no—
348C4⋊Dic3—21no—
349C4⋊Dic3—21no—
350C4⋊Dic3—21no—
351C4⋊Dic3—24no—
352C4⋊Dic3—24no—
353C4⋊Dic3—24no—
354C4⋊Dic3C2×D412yes—
355C4⋊Dic3C2×D412yes—
356C4⋊Dic3C2×D412yes—
357C4⋊Dic3C2×D412yes—
358C4⋊Dic3C2412yes—
359C4⋊Dic3C2412yes—
360Dic3⋊C4—22no—
361Dic3⋊C4—22no—
362Dic3⋊C4—24no—
363Dic3⋊C4—24no—
364Dic3⋊C4—24no—
365Dic3⋊C4C2×D414yes—
366Dic3⋊C4C2×D414yes—
367C4×Dic3—22no—
368C4×Dic3—24no—
369C4×Dic3—24no—
370C4×Dic3—24no—
371C4×Dic3C2414yes—
372#208—24no—
373C2×C3⋊C8C2414yes—
374C2×#41—32no—
375C2×#41—62no—
376C22×Q8C22×D311yescharacteristic
377C22×D4C22×D311yescharacteristic
378C23×C4—21no—
379#137—68no—
380C2×Q16—34no—
381C2×SD16—24no—
382C2×SD16—34no—
383C2×SD16C22×D314yes—
384C2×M16—32no—
385C22×C8C22×D311yescharacteristic
386C22×C8C22×D311yescharacteristic
387C4⋊Q8—34no—
388C4⋊Q8—31no—
389C4⋊Q8—31no—
390C4⋊Q8—31no—
391C4⋊Q8—31no—
392C4⋊Q8—31no—
393C4⋊Q8—62no—
394C4⋊Q8—62no—
395C4⋊Q8—62no—
396C4⋊Q8—64no—
397C4⋊Q8—64no—
398#127—34no—
399#127—31no—
400#126—64no—
401#125—34no—
402#125—32no—
403#125—32no—
404#125—64no—
405#125—62no—
406#125—64no—
407#125—64no—
408#124—34no—
409#124—32no—
410#124—64no—
411#123—64no—
412#123—64no—
413C22⋊Q8—32no—
414C22⋊Q8—32no—
415C22⋊Q8—64no—
416C22⋊Q8—62no—
417C22⋊Q8—61no—
418C22⋊Q8—61no—
419#121—64no—
420#121—62no—
421#121—62no—
422#121—62no—
423#121—61no—
424#121—61no—
425#120—64no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8—32no—
429C4×Q8—32no—
430C4×Q8—34no—
431C4×Q8—34no—
432C4×Q8—34no—
433C4×Q8—64no—
434C4×Q8—62no—
435C4×Q8—62no—
436C4×Q8—62no—
437C4×Q8—62no—
438C4×Q8—61no—
439C4×Q8—61no—
440C4×Q8C22×D312yes—
441C4×Q8C22×D312yes—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—34no—
445C4×D4—64no—
446C4×D4—64no—
447C4×D4—61no—
448C4×D4—61no—
449C4×D4C22×D312yes—
450C4×D4C22×D312yes—
451#117—32no—
452#117—32no—
453#117—64no—
454#117—62no—
455#117—62no—
456C2×C4⋊C4—21no—
457C2×C4⋊C4—62no—
458C2×C4⋊C4C22×D311yescharacteristic
459C2×C4⋊C4C22×D311yescharacteristic
460C2×C22⋊C4—21no—
461C2×C42—21no—
462C2×C42C22×D311yescharacteristic
463#107—34no—
464#107C22×D314yes—
465#106—34no—
466C4⋊C8—34no—
467C4⋊C8—34no—
468C4⋊C8—32no—
469C4⋊C8—32no—
470#103—32no—
471#103—32no—
472#103—32no—
473#103—32no—
474#103—34no—
475#103C22×D314yes—
476#102—34no—
477#102—34no—
478#102—32no—
479#102—32no—
480#102—32no—
481#102—32no—
482#102—34no—
483#102C22×D314yes—
484#97—34no—
485#97C22×D312yes—
486#97C22×D312yes—
487C4×C8—34no—
488C23×C3—22no—
489C23×C3—42no—
490C23×C3—81no—
491C23×C3—81no—
492C23×C3C22×D411yescharacteristic
493C22×D3—22no—
494C3×Q8—24no—
495C3×Q8—22no—
496C3×Q8C2×#4114yes—
497C3×D4—22no—
498C3×D4—22no—
499C3×D4—42no—
500C2×C3×C4—24no—
501C2×C3×C4—22no—
502C2×C3×C4—22no—
503C2×C3×C4—22no—
504C2×C3×C4—41no—
505C2×C3×C4—41no—
506C2×C3×C4—44no—
507C2×C3×C4—82no—
508C2×C3×C4—81no—
509C2×C3×C4—81no—
510C2×C3×C4C22×D411yescharacteristic
511C2×C3×C4C22×D411yescharacteristic
512C2×C3×C4C22×D411yescharacteristic
513C2×C3×C4C22×D411yescharacteristic
514C2×C3×C4C22×D411yescharacteristic
515C2×C3×C4C2×#4112yes—
516C2×C3×C42−1+412yes—
517C2×C3×C4C2511yescharacteristic
518#67—44no—
519#67—84no—
520#67—84no—
521C2×Dic3—22no—
522C2×Dic3—22no—
523C2×Dic3—22no—
524C2×Dic3C22×D414yes—
525C2×Dic3C22×D414yes—
526D12—22no—
527D12—22no—
528D12—42no—
529D12—81no—
530C4×D3—44no—
531C4×D3—84no—
532Dic6—22no—
533Dic6—22no—
534Dic6—42no—
535Dic6—42no—
536Dic6—42no—
537Dic6—84no—
538Dic6—81no—
539C3×C8—22no—
540C3×C8—22no—
541C3⋊C8—24no—
542C24—21no—
543#41—64no—
544#41—64no—
545#41—244no—
546C2×Q8—21no—
547C2×Q8—22no—
548C2×Q8—32no—
549C2×Q8—32no—
550C2×Q8—32no—
551C2×Q8—64no—
552C2×Q8—61no—
553C2×Q8—61no—
554C2×Q8#23514yescomplemented
555C2×Q8C23×D312yes—
556C2×D4—22no—
557C2×D4—21no—
558C2×D4—22no—
559C2×D4—32no—
560C2×D4—64no—
561C2×D4—64no—
562C2×D4—62no—
563C2×D4—61no—
564C2×D4—61no—
565C2×D4C23×D312yes—
566C22×C4—21no—
567C22×C4—21no—
568C22×C4—21no—
569C22×C4—22no—
570C22×C4—22no—
571C22×C4—64no—
572C22×C4—62no—
573C22×C4—82no—
574C22×C4D3×D411yescharacteristic
575C22×C4D3×D411yescharacteristic
576C22×C4C23×D311yescharacteristic
577Q16—68no—
578SD16—28no—
579SD16—68no—
580M16—64no—
581C2×C8—22no—
582C2×C8—22no—
583C2×C8—34no—
584C2×C8C23×D312yes—
585C2×C8C23×D312yes—
586C4⋊C4—22no—
587C4⋊C4—22no—
588C4⋊C4—21no—
589C4⋊C4—21no—
590C4⋊C4—32no—
591C4⋊C4—32no—
592C4⋊C4—32no—
593C4⋊C4—34no—
594C4⋊C4—34no—
595C4⋊C4—32no—
596C4⋊C4—32no—
597C4⋊C4—32no—
598C4⋊C4—64no—
599C4⋊C4—64no—
600C4⋊C4—64no—
601C4⋊C4—64no—
602C4⋊C4—64no—
603C4⋊C4—64no—
604C4⋊C4—61no—
605C4⋊C4—61no—
606C4⋊C4—62no—
607C4⋊C4—61no—
608C4⋊C4—62no—
609C4⋊C4—61no—
610C4⋊C4C23×D312yes—
611C4⋊C4C23×D312yes—
612C22⋊C4—22no—
613C22⋊C4—22no—
614C22⋊C4—34no—
615C22⋊C4—64no—
616C22⋊C4—64no—
617C22⋊C4—64no—
618C22⋊C4—62no—
619C22⋊C4—62no—
620C22⋊C4—62no—
621C42—22no—
622C42—22no—
623C42—34no—
624C42—64no—
625C42—64no—
626C42—64no—
627C42—62no—
628C42D3×D411yescharacteristic
629C42D3×D411yescharacteristic
630C42C23×D311yescharacteristic
631C42C23×D311yescharacteristic
632C22×C3—21no—
633C22×C3—21no—
634C22×C3—42no—
635C22×C3—42no—
636C22×C3—42no—
637C22×C3—84no—
638C22×C3#54812yes—
639C22×C3C23×D411yescharacteristic
640C2×D3—42no—
641C2×D3—42no—
642C2×D3—42no—
643C3×C4—21no—
644C3×C4—21no—
645C3×C4—21no—
646C3×C4—21no—
647C3×C4—24no—
648C3×C4—42no—
649C3×C4—42no—
650C3×C4#54812yes—
651C3×C4C22×#4111yescharacteristic
652C3×C4C2×2−1+411yescharacteristic
653Dic3—24no—
654Dic3—24no—
655Dic3—42no—
656Dic3—42no—
657Dic3—42no—
658C23—22no—
659C23—42no—
660C23—62no—
661C23—81no—
662C23—81no—
663C23C2×D3×D411yescharacteristic
664Q8—22no—
665Q8—24no—
666Q8—62no—
667Q8—62no—
668Q8—122no—
669Q8—122no—
670Q8—122no—
671Q8—241no—
672Q8—244no—
673Q8C2×#23514yes—
674D4—22no—
675D4—22no—
676D4—42no—
677D4—62no—
678D4—62no—
679D4—122no—
680D4—124no—
681D4—241no—
682D4—244no—
683D4—244no—
684C2×C4—22no—
685C2×C4—24no—
686C2×C4—22no—
687C2×C4—22no—
688C2×C4—34no—
689C2×C4—34no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—44no—
693C2×C4—62no—
694C2×C4—62no—
695C2×C4—62no—
696C2×C4—82no—
697C2×C4—81no—
698C2×C4—81no—
699C2×C4—124no—
700C2×C4—244no—
701C2×C4C2×#23512yes—
702C2×C4C2×D3×D411yescharacteristic
703C2×C4C2×D3×D411yescharacteristic
704C2×C4C2×D3×D411yescharacteristic
705C2×C4C2×D3×D411yescharacteristic
706C2×C4C2×D3×D411yescharacteristic
707C2×C4#101012yes—
708C2×C4C24×D311yescharacteristic
709C8—22no—
710C8—22no—
711C8—64no—
712C2×C3—22no—
713C2×C3—82no—
714C2×C3C2×#54811yescharacteristic
715C2×C3#358511yescharacteristic
716C2×C3#358611yescharacteristic
717D3—82no—
718C22—21no—
719C22—21no—
720C22—42no—
721C22—42no—
722C22—42no—
723C22—84no—
724C22—122no—
725C22—122no—
726C22—122no—
727C22#555212yes—
728C22C22×D3×D411yescharacteristic
729C4—21no—
730C4—21no—
731C4—21no—
732C4—21no—
733C4—24no—
734C4—42no—
735C4—42no—
736C4—64no—
737C4—64no—
738C4—122no—
739C4—122no—
740C4—122no—
741C4#555212yes—
742C4C22×#23511yescharacteristic
743C4C2×#101011yescharacteristic
744C3#6133411yescharacteristic, complemented, Sylow
745C2—22no—
746C2—82no—
747C2—242no—
748C2C2×#555211yescharacteristic
749C2#8719211yescharacteristic
750C2#8719711yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#440C2
socleC22×C3C23×D4
prime radical#85788C2
hypercenterC2×#440D3
hypocenterC3#61334
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #85803C2semidirect
3. C2×C3×#440C2semidirect
4. #83849C2semidirect
5. #82441C2semidirect
6. #82296C2semidirect
7. #79758C2semidirect
8. #79756C2semidirect
9. #79754C2semidirect
10. #79752C2semidirect
11. #79162C2semidirect
12. #78774C2semidirect
13. #78737C2semidirect
14. #78497C2semidirect
15. #78480C2semidirect
25. #5570C22semidirect
27. #5569C22semidirect
29. Q8×Dic6C22semidirect
59. #5031C22semidirect
60. #5028C22semidirect
61. #5027C22semidirect
68. #4891C22semidirect
69. #4890C22semidirect
70. #4877C22semidirect
71. #4875C22semidirect
72. #4817C22semidirect
73. #4790C22semidirect
74. #4789C22semidirect
75. #4788C22semidirect
80. #4714C22semidirect
81. #4714C22semidirect
82. #4713C22semidirect
83. #4703C22semidirect
84. #4701C22semidirect
85. #4701C22semidirect
100. C2×#440D3semidirect
287. C2×C3×Q8#41semidirect
554. C2×Q8#235semidirect
744. C3#61334semidirect

Maximal subgroups

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

subgroupindex
#858032
#857882
C2×C3×#4402
#838492
#824412
#822962
#797582
#797562
#797542
#797522
#791622
#787742
#787372
#784972
#784802
#613343