Finite Group Databasebeta

← #11675300

Subgroups — #11675300

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#86258C211yesmaximal, characteristic, complemented
2#86223C211yesmaximal, characteristic
3C3×#3135C211yesmaximal, characteristic, complemented
4#83350C211yesmaximal, characteristic, complemented
5#83314C211yesmaximal, characteristic, complemented
6#81735C214yesmaximal, complemented
7#81709C214yesmaximal, complemented
8#81675C214yesmaximal, complemented
9#81673C214yesmaximal, complemented
10#78864C212yesmaximal, complemented
11#78827C212yesmaximal, complemented
12#78823C212yesmaximal, complemented
13#78821C212yesmaximal, complemented
14C2×#4710C211yesmaximal, characteristic, complemented
15#78480C211yesmaximal, characteristic, complemented
16#61334—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19#5830C2212yes—
20#5827C2212yes—
21C2×#951C2212yes—
22C22×#204C2211yescharacteristic
23#5732C2212yes—
24#5722C2212yescomplemented
25#5721C2212yescomplemented
26#5718C2212yes—
27#5685C2214yescomplemented
28#5681C2214yes—
29#5623—22no—
30#5622—22no—
31#5617—22no—
32#5606—22no—
33#5567—22no—
34#5567—21no—
35Q8×Dic6—21no—
36#5565—22no—
37#5563—21no—
38#5555—21no—
39#5519C2212yes—
40#5516C2212yes—
41C2×C4×D12C2211yescharacteristic
42#5470C2211yescharacteristic
43C2×C4×Dic6C2211yescharacteristic
44C3×#491C2212yescomplemented
45C3×#490C2212yescomplemented
46C3×#489C2212yescomplemented
47C3×#489C2212yescomplemented
48C3×#416C2212yes—
49C3×#416C2212yes—
50C2×C3×#97C2211yescharacteristic
51#5232C2212yes—
52C2×#821C2211yescharacteristic
53C2×#818C2211yescharacteristic
54C2×#816C2211yescharacteristic
55#5102C2212yes—
56#5101C2212yes—
57#5090C2212yescomplemented
58#5087C2212yes—
59#5084C2212yes—
60#5075C2212yescomplemented
61#5066C2214yes—
62#5054C2214yescomplemented
63#5001C2212yes—
64#4995C2212yes—
65#4924C2211yescharacteristic
66#4800C2214yes—
67#4798C2214yes—
68#4798C2214yes—
69#4796C2214yescomplemented
70#4794C2214yes—
71#4793C2214yescomplemented
72#4790C2214yescomplemented
73#4789C2214yescomplemented
74#4766C2214yescomplemented
75#4764C2214yes—
76#4762C2214yes—
77#4762C2214yes—
78#4760C2214yes—
79#4759C2214yescomplemented
80#4758C2214yescomplemented
81#4757C2214yescomplemented
82#4710C2214yes—
83#4710C2214yes—
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#3135D311yescharacteristic, complemented
94#3083—32no—
95#3077—31no—
96#3043—32no—
97#3037—31no—
98#2961—31no—
99#2947—32no—
100C2×#440—31no—
101C2×C3×#41C2312yes—
102C2×#237—22no—
103C22×D12C2311yescharacteristic
104C22×C4×D3—21no—
105C22×Dic6C2311yescharacteristic
106C22×C3×C8C2311yescharacteristic
107C22×C3×C8C2311yescharacteristic
108C3×C4⋊Q8C2311yescharacteristic
109C3×C4⋊Q8C2311yescharacteristic
110C3×C4⋊Q8C2311yescharacteristic
111C3×#127C2311yescharacteristic
112C3×#125C2312yes—
113C3×#124C2312yes—
114C3×C22⋊Q8—21no—
115C3×C22⋊Q8—21no—
116C3×#121—21no—
117C3×#121—21no—
118C3×C4×Q8—21no—
119C3×C4×Q8—21no—
120C3×C4×D4—21no—
121C3×C4×D4—21no—
122C3×#117C2312yes—
123C2×C3×C42C2311yescharacteristic
124#951—28no—
125#947C2314yes—
126Q8×Dic3—22no—
127Q8×Dic3C2314yes—
128#944—22no—
129#944—24no—
130C2×#216C2314yes—
131#940C2314yes—
132#939—24no—
133#938—22no—
134#937—24no—
135D4×Dic3—24no—
136D4×Dic3C2314yes—
137C2×#214C2314yes—
138#930—22no—
139#930—22no—
140C2×(C2×D3)⋊C4—21no—
141#926—22no—
142#926C2312yes—
143C2×C4⋊Dic3—22no—
144C2×C4⋊Dic3C2311yescharacteristic
145C2×C4⋊Dic3C2311yescharacteristic
146#924D412yes—
147#924D412yes—
148C2×Dic3⋊C4—21no—
149C2×C4×Dic3—21no—
150C2×#208C2312yes—
151C2×#204—24no—
152C2×#204C2314yes—
153#898—24no—
154#897—22no—
155#892—22no—
156#890C2314yes—
157#889—24no—
158#889—24no—
159#889—24no—
160#888—24no—
161#888—22no—
162#888C2314yes—
163#887—22no—
164#887—24no—
165#887C2314yes—
166#887C2314yes—
167#886—24no—
168#885—24no—
169#883—24no—
170#67⋊C4—24no—
171#878—24no—
172#877—24no—
173C4×D12—21no—
174C4×D12—21no—
175C4×D12C2312yes—
176C4×D12C2312yes—
177#870—22no—
178#870D412yes—
179#869—22no—
180#869D411yescharacteristic
181#869D411yescharacteristic
182C4×Dic6—21no—
183C4×Dic6—21no—
184C4×Dic6—22no—
185C4×Dic6—22no—
186C4×Dic6C2312yes—
187C4×Dic6C2312yes—
188C4×Dic6C2312yes—
189C4×Dic6C2312yes—
190C3×#103C2312yes—
191C3×#103C2312yes—
192C3×#103C2312yes—
193C3×#103C2312yes—
194C3×#102C2312yes—
195C3×#102C2312yes—
196C3×#102C2312yes—
197C3×#102C2312yes—
198C3×#97C2312yes—
199C3×#97C2312yes—
200#835C2314yes—
201#832C2314yes—
202#821C2314yes—
203#818C2314yes—
204#816C2314yes—
205#815C2314yes—
206Dic3⋊C8C2314yes—
207Dic3⋊C8C2314yes—
208C8×Dic3C2314yes—
209#809C2314yes—
210#809C2314yes—
211#808C2314yes—
212#807C2314yes—
213C4⋊(C3⋊C8)C2312yes—
214C4⋊(C3⋊C8)C2312yes—
215C2×#137—32no—
216C22×SD16—31no—
217#559—34no—
218Q82—32no—
219Q82—61no—
220#557—32no—
221#557—34no—
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—62no—
235#533C2×D311yescharacteristic
236#532C2×D311yescharacteristic
237#525—62no—
238#520—32no—
239#519—32no—
240C2×C4×Q8—31no—
241C2×C4×D4—31no—
242#491C2×D312yescomplemented
243#490—34no—
244#490C2×D312yescomplemented
245#489—34no—
246#489C2×D312yescomplemented
247#489C2×D312yescomplemented
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—32no—
262#460—34no—
263#460—34no—
264#446—34no—
265#446—34no—
266#441—34no—
267#440—34no—
268#440—34no—
269#440—34no—
270C4×Q16—34no—
271C4×SD16—34no—
272#428—32no—
273C2×#107—31no—
274#424—32no—
275#423—31no—
276#418—32no—
277#417—32no—
278#416C2×D312yes—
279#416C2×D312yes—
280C2×#103—31no—
281C2×#102—31no—
282#405—32no—
283C2×#97C2×D311yescharacteristic
284C23×D3—21no—
285C3×#41—24no—
286C3×#41—24no—
287C2×C3×Q8—21no—
288C2×C3×Q8—21no—
289C2×C3×Q8C2412yes—
290C2×C3×D4—22no—
291C2×C3×D4—21no—
292C2×C3×D4—21no—
293C2×C3×D4C2412yes—
294C22×C3×C4—22no—
295C22×C3×C4C2×D411yescharacteristic
296C22×C3×C4C2×D411yescharacteristic
297C22×C3×C4C2411yescharacteristic
298C2×#67—24no—
299C2×#67—24no—
300C22×Dic3—21no—
301C22×Dic3—21no—
302C22×Dic3—21no—
303C22×Dic3—24no—
304#237—84no—
305C2×D12—22no—
306C2×D12—22no—
307C2×D12—21no—
308C2×D12C2412yes—
309C2×C4×D3—22no—
310C2×C4×D3—22no—
311C2×C4×D3—82no—
312C2×Dic6—21no—
313C2×Dic6—22no—
314C2×Dic6—24no—
315C2×Dic6C2×D412yes—
316C2×Dic6C2×D412yes—
317C2×Dic6#4114yescomplemented
318C2×Dic6C2412yes—
319C2×C3×C8—22no—
320C2×C3×C8—22no—
321C2×C3×C8C2412yes—
322C2×C3×C8C2412yes—
323C3×C4⋊C4—21no—
324C3×C4⋊C4—22no—
325C3×C4⋊C4—22no—
326C3×C4⋊C4—21no—
327C3×C4⋊C4—21no—
328C3×C4⋊C4—21no—
329C3×C4⋊C4C2412yes—
330C3×C4⋊C4C2412yes—
331C3×C22⋊C4—22no—
332C3×C22⋊C4—22no—
333C3×C22⋊C4—22no—
334C3×C42—22no—
335C3×C42C2×D411yescharacteristic
336C3×C42C2×D411yescharacteristic
337C3×C42C2411yescharacteristic
338C3×C42C2411yescharacteristic
339#217—24no—
340#217C2×D414yes—
341#216—28no—
342#214—28no—
343(C2×D3)⋊C4—22no—
344(C2×D3)⋊C4—22no—
345(C2×D3)⋊C4—24no—
346(C2×D3)⋊C4—24no—
347C4⋊Dic3—21no—
348C4⋊Dic3—21no—
349C4⋊Dic3—24no—
350C4⋊Dic3—24no—
351C4⋊Dic3C2×D412yes—
352C4⋊Dic3C2×D412yes—
353C4⋊Dic3C2×D412yes—
354C4⋊Dic3C2×D412yes—
355C4⋊Dic3C2412yes—
356C4⋊Dic3C2412yes—
357Dic3⋊C4—22no—
358Dic3⋊C4—22no—
359Dic3⋊C4—24no—
360Dic3⋊C4—24no—
361Dic3⋊C4—24no—
362Dic3⋊C4—24no—
363Dic3⋊C4C2×D414yes—
364Dic3⋊C4C2×D414yes—
365C4×Dic3—22no—
366C4×Dic3—22no—
367C4×Dic3—24no—
368C4×Dic3—24no—
369C4×Dic3—24no—
370C4×Dic3C2414yes—
371#208—24no—
372C2×C3⋊C8C2414yes—
373#204—28no—
374C2×#41—62no—
375C2×#41C22×D312yes—
376C22×Q8—31no—
377C22×D4—31no—
378C23×C4—61no—
379#137—68no—
380C2×Q16—34no—
381C2×SD16—34no—
382C2×SD16—34no—
383C2×SD16—64no—
384C2×M16—32no—
385C22×C8C22×D311yescharacteristic
386C22×C8C22×D311yescharacteristic
387C4⋊Q8—34no—
388C4⋊Q8—31no—
389C4⋊Q8—31no—
390C4⋊Q8—62no—
391C4⋊Q8—62no—
392C4⋊Q8—64no—
393C4⋊Q8—64no—
394C4⋊Q8—62no—
395C4⋊Q8C22×D311yescharacteristic
396C4⋊Q8C22×D311yescharacteristic
397C4⋊Q8C22×D311yescharacteristic
398#127—34no—
399#127C22×D311yescharacteristic
400#126—64no—
401#125—34no—
402#125—32no—
403#125—62no—
404#125—64no—
405#125—64no—
406#125—64no—
407#125C22×D312yes—
408#124—34no—
409#124—64no—
410#124C22×D312yes—
411#123—64no—
412#123—64no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—32no—
416C22⋊Q8—32no—
417C22⋊Q8—64no—
418C22⋊Q8—62no—
419#121—21no—
420#121—21no—
421#121—62no—
422#121—62no—
423#121—62no—
424#121—64no—
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—32no—
434C4×Q8—32no—
435C4×Q8—62no—
436C4×Q8—62no—
437C4×Q8—64no—
438C4×Q8—62no—
439C4×Q8—62no—
440C4×Q8—61no—
441C4×Q8—61no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—34no—
445C4×D4—32no—
446C4×D4—32no—
447C4×D4—64no—
448C4×D4—64no—
449C4×D4—61no—
450C4×D4—61no—
451#117—32no—
452#117—64no—
453#117—62no—
454#117—62no—
455#117C22×D312yes—
456C2×C4⋊C4—31no—
457C2×C4⋊C4—31no—
458C2×C4⋊C4—62no—
459C2×C4⋊C4—61no—
460C2×C22⋊C4—61no—
461C2×C42—61no—
462C2×C42C22×D311yescharacteristic
463#107—34no—
464#107—34no—
465#106—34no—
466C4⋊C8—34no—
467C4⋊C8—34no—
468C4⋊C8—32no—
469C4⋊C8—32no—
470#103—34no—
471#103—34no—
472#103C22×D312yes—
473#103C22×D312yes—
474#103C22×D312yes—
475#103C22×D312yes—
476#102—34no—
477#102—34no—
478#102—34no—
479#102—34no—
480#102C22×D312yes—
481#102C22×D312yes—
482#102C22×D312yes—
483#102C22×D312yes—
484#97—34no—
485#97C22×D312yes—
486#97C22×D312yes—
487C4×C8—34no—
488C23×C3—22no—
489C23×C3C22×D411yescharacteristic
490C22×D3—22no—
491C22×D3—42no—
492C22×D3—81no—
493C22×D3—81no—
494C3×Q8—22no—
495C3×Q8—22no—
496C3×Q8—42no—
497C3×Q8—81no—
498C3×D4—22no—
499C3×D4—22no—
500C3×D4—42no—
501C3×D4—44no—
502C3×D4—81no—
503C2×C3×C4—22no—
504C2×C3×C4—22no—
505C2×C3×C4—22no—
506C2×C3×C4—41no—
507C2×C3×C4—41no—
508C2×C3×C4—44no—
509C2×C3×C4C22×D411yescharacteristic
510C2×C3×C4C22×D411yescharacteristic
511C2×C3×C4C22×D411yescharacteristic
512C2×C3×C4C22×D411yescharacteristic
513C2×C3×C4C22×D411yescharacteristic
514C2×C3×C4C2×#4112yes—
515C2×C3×C42−1+412yes—
516C2×C3×C4C2511yescharacteristic
517#67—84no—
518#67—84no—
519C2×Dic3—24no—
520C2×Dic3—22no—
521C2×Dic3—22no—
522C2×Dic3—22no—
523C2×Dic3—84no—
524C2×Dic3—81no—
525C2×Dic3—81no—
526C2×Dic3C22×D414yes—
527C2×Dic3C22×D414yes—
528D12—22no—
529D12—22no—
530D12—42no—
531C4×D3—44no—
532C4×D3—82no—
533Dic6—24no—
534Dic6—22no—
535Dic6—42no—
536Dic6—42no—
537Dic6—84no—
538Dic6C2×#4114yes—
539C3×C8—22no—
540C3×C8—22no—
541C3⋊C8—24no—
542C24—61no—
543#41—24no—
544#41—24no—
545#41—244no—
546C2×Q8—21no—
547C2×Q8—21no—
548C2×Q8—32no—
549C2×Q8—32no—
550C2×Q8—34no—
551C2×Q8—32no—
552C2×Q8—64no—
553C2×Q8—61no—
554C2×Q8—62no—
555C2×Q8C23×D312yes—
556C2×D4—22no—
557C2×D4—21no—
558C2×D4—21no—
559C2×D4—32no—
560C2×D4—64no—
561C2×D4—64no—
562C2×D4—61no—
563C2×D4—62no—
564C2×D4—62no—
565C2×D4C23×D312yes—
566C22×C4—22no—
567C22×C4—64no—
568C22×C4—61no—
569C22×C4—61no—
570C22×C4—61no—
571C22×C4—62no—
572C22×C4—62no—
573C22×C4—242no—
574C22×C4D3×D411yescharacteristic
575C22×C4D3×D411yescharacteristic
576C22×C4C23×D311yescharacteristic
577Q16—68no—
578SD16—68no—
579SD16—68no—
580M16—64no—
581C2×C8—22no—
582C2×C8—22no—
583C2×C8—34no—
584C2×C8C23×D312yes—
585C2×C8C23×D312yes—
586C4⋊C4—21no—
587C4⋊C4—21no—
588C4⋊C4—22no—
589C4⋊C4—21no—
590C4⋊C4—21no—
591C4⋊C4—22no—
592C4⋊C4—32no—
593C4⋊C4—32no—
594C4⋊C4—32no—
595C4⋊C4—34no—
596C4⋊C4—34no—
597C4⋊C4—32no—
598C4⋊C4—32no—
599C4⋊C4—32no—
600C4⋊C4—64no—
601C4⋊C4—64no—
602C4⋊C4—64no—
603C4⋊C4—64no—
604C4⋊C4—64no—
605C4⋊C4—64no—
606C4⋊C4—61no—
607C4⋊C4—61no—
608C4⋊C4—62no—
609C4⋊C4—62no—
610C4⋊C4C23×D312yes—
611C4⋊C4C23×D312yes—
612C22⋊C4—22no—
613C22⋊C4—22no—
614C22⋊C4—22no—
615C22⋊C4—34no—
616C22⋊C4—64no—
617C22⋊C4—64no—
618C22⋊C4—64no—
619C22⋊C4—62no—
620C22⋊C4—62no—
621C42—22no—
622C42—34no—
623C42—64no—
624C42—64no—
625C42—64no—
626C42—62no—
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#54812yes—
638C22×C3C23×D411yescharacteristic
639C2×D3—42no—
640C2×D3—42no—
641C2×D3—42no—
642C2×D3—84no—
643C3×C4—21no—
644C3×C4—21no—
645C3×C4—21no—
646C3×C4—21no—
647C3×C4—42no—
648C3×C4—42no—
649C3×C4—42no—
650C3×C4#54812yes—
651C3×C4C22×#4111yescharacteristic
652C3×C4C2×2−1+411yescharacteristic
653Dic3—24no—
654Dic3—24no—
655Dic3—24no—
656Dic3—42no—
657Dic3—42no—
658C23—22no—
659C23—62no—
660C23—122no—
661C23—241no—
662C23—241no—
663C23C2×D3×D411yescharacteristic
664Q8—22no—
665Q8—22no—
666Q8—34no—
667Q8—42no—
668Q8—64no—
669Q8—62no—
670Q8—81no—
671Q8—122no—
672Q8—122no—
673Q8—244no—
674D4—22no—
675D4—22no—
676D4—42no—
677D4—44no—
678D4—62no—
679D4—62no—
680D4—81no—
681D4—122no—
682D4—244no—
683D4—244no—
684C2×C4—22no—
685C2×C4—22no—
686C2×C4—22no—
687C2×C4—34no—
688C2×C4—34no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—44no—
692C2×C4—64no—
693C2×C4—62no—
694C2×C4—62no—
695C2×C4—62no—
696C2×C4—124no—
697C2×C4—242no—
698C2×C4—244no—
699C2×C4—241no—
700C2×C4—241no—
701C2×C4C2×D3×D411yescharacteristic
702C2×C4C2×D3×D411yescharacteristic
703C2×C4C2×D3×D411yescharacteristic
704C2×C4C2×D3×D411yescharacteristic
705C2×C4C2×D3×D411yescharacteristic
706C2×C4D3×#4112yes—
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—122no—
724C22—122no—
725C22—122no—
726C22—244no—
727C22#567712yes—
728C22C22×D3×D411yescharacteristic
729C4—21no—
730C4—21no—
731C4—21no—
732C4—21no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—64no—
737C4—64no—
738C4—64no—
739C4—122no—
740C4—122no—
741C4#567712yes—
742C4C2×D3×#4111yescharacteristic
743C4C2×#101011yescharacteristic
744C3#6133411yescharacteristic, complemented, Sylow
745C2—22no—
746C2—82no—
747C2—242no—
748C2C2×#567711yescharacteristic
749C2#8720011yescharacteristic
750C2#8720111yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3135C2
socleC22×C3C23×D4
prime radical#86223C2
hypercenter#3135D3
hypocenterC3#61334
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
1. #86258C2semidirect
3. C3×#3135C2semidirect
4. #83350C2semidirect
5. #83314C2semidirect
6. #81735C2semidirect
7. #81709C2semidirect
8. #81675C2semidirect
9. #81673C2semidirect
10. #78864C2semidirect
11. #78827C2semidirect
12. #78823C2semidirect
13. #78821C2semidirect
14. C2×#4710C2semidirect
15. #78480C2semidirect
24. #5722C22semidirect
25. #5721C22semidirect
27. #5685C22semidirect
44. C3×#491C22semidirect
45. C3×#490C22semidirect
46. C3×#489C22semidirect
47. C3×#489C22semidirect
57. #5090C22semidirect
60. #5075C22semidirect
62. #5054C22semidirect
69. #4796C22semidirect
71. #4793C22semidirect
72. #4790C22semidirect
73. #4789C22semidirect
74. #4766C22semidirect
79. #4759C22semidirect
80. #4758C22semidirect
81. #4757C22semidirect
84. #4701C22semidirect
85. #4701C22semidirect
93. #3135D3semidirect
242. #491C2×D3semidirect
244. #490C2×D3semidirect
246. #489C2×D3semidirect
247. #489C2×D3semidirect
317. C2×Dic6#41semidirect
744. C3#61334semidirect

Maximal subgroups

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

subgroupindex
#862582
#862232
C3×#31352
#833502
#833142
#817352
#817092
#816752
#816732
#788642
#788272
#788232
#788212
C2×#47102
#784802
#613343