Finite Group Databasebeta

← #10947997

Subgroups — #10947997

Order 768.

1252 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86076C211yesmaximal, characteristic
2#78688C212yesmaximal, complemented
3C2×C3×#354C211yesmaximal, characteristic, complemented
4C2×#4541C211yesmaximal, characteristic, complemented
5#70019C212yesmaximal, complemented
6#69159C212yesmaximal, complemented
7#69158C212yesmaximal, complemented
8#69157C212yesmaximal, complemented
9D3×#354C212yesmaximal, complemented
10#33361—31nomaximal, Sylow
11C2×#1007C411yescharacteristic, complemented
12C2×#1004C2211yescharacteristic
13C2×C3×#124C2211yescharacteristic
14#5816C411yescharacteristic, complemented
15#5805C2211yescharacteristic
16#5677—22no—
17#5676—22no—
18#5675C411yescharacteristic, complemented
19#5675C2211yescharacteristic
20#5674C411yescharacteristic, complemented
21#5674C2211yescharacteristic
22D3×#124C411yescharacteristic, complemented
23D3×#124C2211yescharacteristic
24#5668C411yescharacteristic, complemented
25#5668C2211yescharacteristic
26#5490—22no—
27#5490—22no—
28#5486—22no—
29#5477C411yescharacteristic, complemented
30C4×#235C411yescharacteristic, complemented
31C2×C3×#99C2212yes—
32C2×#834C2212yes—
33C2×#806—22no—
34D3×#99—22no—
35D3×#99C2214yes—
36#4740—22no—
37#4740C2214yes—
38C3×#354C2212yescomplemented
39C3×#354C2212yescomplemented
40#4541C2212yescomplemented
41#4541C2212yescomplemented
42#4474C2214yescomplemented
43#4473C2214yescomplemented
44#3500—31no—
45#2883—32no—
46#2129—32no—
47#2127—32no—
48#2127—32no—
49#2126—32no—
50C2×#354—32no—
51C2×#354—31no—
52C2×#354D311yescharacteristic, complemented
53C22×C3×Q8C2×C411yescharacteristic, complemented
54C22×C3×D4C2311yescharacteristic
55C22×#67D412yes—
56C22×#67D412yes—
57#1007—42no—
58#1007—42no—
59C2×#239—21no—
60C2×#239C2×C411yescharacteristic, complemented
61C2×#239C2×C411yescharacteristic, complemented
62C2×D3×Q8—21no—
63C2×D3×Q8C2×C411yescharacteristic, complemented
64C2×D3×Q8C2×C411yescharacteristic, complemented
65#1004—42no—
66#1004—42no—
67C2×#237C2×C411yescharacteristic
68C2×#237D412yes—
69C2×#237C2311yescharacteristic
70C2×D3×D4C2×C411yescharacteristic
71C2×D3×D4D412yes—
72C2×D3×D4C2311yescharacteristic
73C2×#235—21no—
74C2×#235—21no—
75C2×#235C2×C411yescharacteristic
76C2×#235C2×C411yescharacteristic
77C2×#235C2×C411yescharacteristic, complemented
78C3×#124—22no—
79C3×#124C2×C411yescharacteristic, complemented
80C3×#124C2×C411yescharacteristic, complemented
81C3×#124C2311yescharacteristic
82C3×#124C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42C2×C411yescharacteristic, complemented
86#953—22no—
87C2×#217—22no—
88#947C2×C411yescharacteristic, complemented
89#947C2×C411yescharacteristic, complemented
90#946—22no—
91#944C2×C411yescharacteristic, complemented
92#944C2×C411yescharacteristic, complemented
93#940C2×C411yescharacteristic, complemented
94#940C2311yescharacteristic
95#938—22no—
96#936C2×C411yescharacteristic, complemented
97#936C2311yescharacteristic
98#930—22no—
99#929—22no—
100C4×#67—21no—
101C4×#67—21no—
102C4×#67—44no—
103#926C2×C411yescharacteristic, complemented
104#885—22no—
105#885—22no—
106#884—22no—
107#884—22no—
108#884—22no—
109#884—22no—
110#883—22no—
111#883—22no—
112#882—22no—
113#882—22no—
114#67⋊C4—22no—
115#67⋊C4—22no—
116#67⋊C4—44no—
117D3×C22⋊C4—22no—
118D3×C22⋊C4—22no—
119D3×C22⋊C4—24no—
120#878—22no—
121#878—22no—
122#877—24no—
123#875C2×C411yescharacteristic, complemented
124#875C2×C411yescharacteristic, complemented
125#874C2×C411yescharacteristic, complemented
126C4×D12—21no—
127C4×D12—22no—
128#872C2×C411yescharacteristic, complemented
129#872C2×C411yescharacteristic, complemented
130C42×D3C2×C411yescharacteristic, complemented
131C42×D3C2×C411yescharacteristic, complemented
132#869C2×C411yescharacteristic, complemented
133C4×Dic6—21no—
134C3×#99—22no—
135C3×#99C2314yes—
136#834—22no—
137#834C2314yes—
138#806—24no—
139#806—24no—
140C2×2−1+4—31no—
141C2×H(5,𝔽2)—31no—
142#548—62no—
143#546—62no—
144#546—62no—
145#546—62no—
146#536—31no—
147#536—31no—
148#536—31no—
149#535—31no—
150#535—31no—
151#535—31no—
152#532—31no—
153#532—31no—
154#532—31no—
155C2×#124—31no—
156C2×#124—31no—
157C2×#124C2×D311yescharacteristic
158#518—62no—
159C4×#41—31no—
160#410—34no—
161#410—62no—
162C2×#99—32no—
163C2×#99—34no—
164C2×#99—62no—
165C2×#99—62no—
166C2×#99C2×D312yes—
167#354—32no—
168#354—32no—
169#354—34no—
170#354C2×D312yescomplemented
171#354C2×D312yescomplemented
172#353—34no—
173C24×C3C2×D412yes—
174C23×D3C22⋊C412yes—
175C23×D3C2×D412yes—
176C2×C3×Q8—21no—
177C2×C3×Q8—41no—
178C2×C3×Q8—41no—
179C2×C3×Q8C22×C411yescharacteristic
180C2×C3×Q8C22×C411yescharacteristic
181C2×C3×D4—41no—
182C2×C3×D4—41no—
183C2×C3×D4C22×C411yescharacteristic
184C2×C3×D4C2×D412yes—
185C2×C3×D4C2411yescharacteristic
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—42no—
189C22×C3×C4C22×C411yescharacteristic
190C2×#67—24no—
191C2×#67—24no—
192C2×#67—21no—
193C2×#67—22no—
194C2×#67—22no—
195C2×#67—42no—
196C2×#67—42no—
197C2×#67C22⋊C412yes—
198C2×#67C22⋊C412yes—
199C2×#67C22⋊C411yescharacteristic
200C2×#67C22⋊C412yes—
201C2×#67C22⋊C411yescharacteristic
202C2×#67C2×D412yes—
203C2×#67C2×D412yes—
204C22×Dic3—42no—
205C22×Dic3C22⋊C412yes—
206C22×Dic3C2×D412yes—
207#239—42no—
208#239—42no—
209#239—44no—
210D3×Q8—44no—
211D3×Q8—42no—
212D3×Q8—42no—
213#237—44no—
214#237—42no—
215#237—42no—
216D3×D4—42no—
217D3×D4—42no—
218D3×D4—44no—
219#235—22no—
220#235—22no—
221#235—41no—
222#235—41no—
223#235—42no—
224#235—42no—
225#235—42no—
226#235—42no—
227#235—41no—
228#235—41no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—22no—
232C2×D12C22⋊C411yescharacteristic
233C2×D12C22×C411yescharacteristic
234C2×D12C22×C411yescharacteristic
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C4×D3—21no—
240C2×C4×D3—21no—
241C2×C4×D3—21no—
242C2×C4×D3—44no—
243C2×C4×D3C22⋊C412yes—
244C2×C4×D3C22×C411yescharacteristic
245C2×C4×D3C22×C411yescharacteristic
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6C22⋊C411yescharacteristic
249C2×Dic6C22×C411yescharacteristic
250C2×Dic6C22×C411yescharacteristic
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—42no—
257C3×C42—21no—
258C3×C42C22×C411yescharacteristic
259C3×C42C22×C411yescharacteristic
260#217—22no—
261#217—21no—
262#217—22no—
263#217—24no—
264#217—42no—
265(C2×D3)⋊C4—21no—
266(C2×D3)⋊C4—21no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—22no—
270(C2×D3)⋊C4—22no—
271(C2×D3)⋊C4—44no—
272C4⋊Dic3—21no—
273C4⋊Dic3—22no—
274Dic3⋊C4—21no—
275Dic3⋊C4—21no—
276Dic3⋊C4—22no—
277Dic3⋊C4—22no—
278Dic3⋊C4—44no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—44no—
282C4×Dic3C22×C411yescharacteristic
283C4×Dic3C22×C411yescharacteristic
2842−1+4—122no—
2852−1+4—122no—
286H(5,𝔽2)—122no—
287H(5,𝔽2)—122no—
288C2×#41—31no—
289C2×#41—32no—
290C2×#41—31no—
291C2×#41—31no—
292C2×#41—31no—
293C2×#41—31no—
294C2×#41—31no—
295C2×#41—31no—
296C2×#41—61no—
297C2×#41—61no—
298C2×#41—61no—
299C22×Q8—31no—
300C22×Q8—31no—
301C22×Q8—61no—
302C22×Q8C4×D311yescharacteristic, complemented
303C22×D4—32no—
304C22×D4—32no—
305C22×D4—31no—
306C22×D4—32no—
307C22×D4—31no—
308C22×D4C22×D311yescharacteristic
309C4⋊Q8—31no—
310C4⋊Q8—31no—
311C4⋊Q8—31no—
312#127—31no—
313#127—31no—
314#127—31no—
315#124—22no—
316#124—31no—
317#124—31no—
318#124—31no—
319#124—31no—
320#124—31no—
321#124—31no—
322#124—62no—
323#124—62no—
324#124C4×D311yescharacteristic, complemented
325#124C4×D311yescharacteristic, complemented
326#124C22×D311yescharacteristic
327#124C22×D311yescharacteristic
328#123—62no—
329#123—62no—
330#123—62no—
331C22⋊Q8—62no—
332C22⋊Q8—62no—
333C22⋊Q8—62no—
334#121—62no—
335#121—62no—
336#121—62no—
337#121—62no—
338#121—62no—
339#121—62no—
340#120—62no—
341#120—62no—
342#120—62no—
343C4×Q8—61no—
344C4×D4—62no—
345C4×D4—61no—
346C4×D4—62no—
347C4×D4—62no—
348C4×D4—61no—
349C4×D4—61no—
350C4×D4—124no—
351C4×D4—124no—
352#117—31no—
353#117—31no—
354#117—31no—
355#117—64no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—62no—
359C2×C22⋊C4—62no—
360C2×C22⋊C4—64no—
361C2×C22⋊C4—62no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42C4×D311yescharacteristic, complemented
365#99—22no—
366#99—34no—
367#99—64no—
368#99—64no—
369#99—62no—
370#99C22×D314yes—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—82no—
374C23×C3C2×C22⋊C412yes—
375C23×C3C2×C22⋊C411yescharacteristic
376C23×C3C22×D412yes—
377C22×D3—24no—
378C22×D3—21no—
379C22×D3—42no—
380C22×D3—82no—
381C22×D3—82no—
382C22×D3C2×C22⋊C412yes—
383C22×D3C2×C22⋊C411yescharacteristic
384C22×D3C2×C22⋊C411yescharacteristic
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—42no—
388C3×D4—41no—
389C3×D4—42no—
390C3×D4—41no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—44no—
400C2×C3×C4—81no—
401C2×C3×C4—162no—
402C2×C3×C4C2×C22⋊C411yescharacteristic
403C2×C3×C4C23×C411yescharacteristic
404#67—21no—
405#67—21no—
406#67—22no—
407#67—22no—
408#67—24no—
409#67—24no—
410#67—42no—
411#67—42no—
412#67—41no—
413#67—41no—
414#67—82no—
415#67—82no—
416#67#9912yes—
417#67#9912yes—
418C2×Dic3—24no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—42no—
423C2×Dic3—44no—
424C2×Dic3—162no—
425C2×Dic3C2×C22⋊C411yescharacteristic
426C2×Dic3C2×C22⋊C412yes—
427C2×Dic3C2×C22⋊C411yescharacteristic
428D12—22no—
429D12—41no—
430D12—41no—
431D12—42no—
432D12—42no—
433D12—82no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—42no—
437C4×D3—42no—
438C4×D3—42no—
439C4×D3—42no—
440C4×D3—42no—
441C4×D3—42no—
442C4×D3—81no—
443C4×D3—81no—
444C4×D3—164no—
445Dic6—22no—
446Dic6—41no—
447Dic6—42no—
448Dic6—42no—
449Dic6—41no—
450C24—32no—
451C24—32no—
452C24D3×D412yes—
453#41—62no—
454#41—62no—
455#41—121no—
456#41—121no—
457#41—121no—
458#41—121no—
459#41—122no—
460#41—122no—
461#41—122no—
462#41—122no—
463#41—122no—
464#41—122no—
465#41—124no—
466#41—122no—
467#41—122no—
468#41—124no—
469C2×Q8—21no—
470C2×Q8—31no—
471C2×Q8—31no—
472C2×Q8—31no—
473C2×Q8—41no—
474C2×Q8—41no—
475C2×Q8—61no—
476C2×Q8—61no—
477C2×Q8—122no—
478C2×Q8—122no—
479C2×Q8—124no—
480C2×Q8C2×C4×D311yescharacteristic
481C2×Q8C2×C4×D311yescharacteristic
482C2×D4—31no—
483C2×D4—32no—
484C2×D4—32no—
485C2×D4—31no—
486C2×D4—32no—
487C2×D4—32no—
488C2×D4—32no—
489C2×D4—31no—
490C2×D4—31no—
491C2×D4—31no—
492C2×D4—41no—
493C2×D4—41no—
494C2×D4—61no—
495C2×D4—61no—
496C2×D4—62no—
497C2×D4—64no—
498C2×D4—64no—
499C2×D4—61no—
500C2×D4—62no—
501C2×D4—62no—
502C2×D4—122no—
503C2×D4—122no—
504C2×D4—124no—
505C2×D4—122no—
506C2×D4—122no—
507C2×D4C2×C4×D311yescharacteristic
508C2×D4D3×D412yes—
509C2×D4C23×D311yescharacteristic
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—32no—
513C22×C4—32no—
514C22×C4—31no—
515C22×C4—31no—
516C22×C4—32no—
517C22×C4—42no—
518C22×C4—61no—
519C22×C4—61no—
520C22×C4—61no—
521C22×C4—61no—
522C22×C4—61no—
523C22×C4—61no—
524C22×C4—61no—
525C22×C4—124no—
526C22×C4—122no—
527C22×C4C2×C4×D311yescharacteristic
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—62no—
533C4⋊C4—61no—
534C4⋊C4—124no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—24no—
540C22⋊C4—42no—
541C22⋊C4—62no—
542C22⋊C4—62no—
543C22⋊C4—62no—
544C22⋊C4—62no—
545C22⋊C4—61no—
546C22⋊C4—61no—
547C22⋊C4—62no—
548C22⋊C4—61no—
549C22⋊C4—62no—
550C22⋊C4—64no—
551C22⋊C4—124no—
552C22⋊C4—122no—
553C42—21no—
554C42—31no—
555C42—31no—
556C42—61no—
557C42—61no—
558C42—124no—
559C42C2×C4×D311yescharacteristic
560C42C2×C4×D311yescharacteristic
561C22×C3—22no—
562C22×C3—22no—
563C22×C3—41no—
564C22×C3—42no—
565C22×C3—42no—
566C22×C3—82no—
567C22×C3—82no—
568C22×C3C2×#9911yescharacteristic
569C22×C3C2×#9911yescharacteristic
570C22×C3C22×C22⋊C411yescharacteristic
571C2×D3—21no—
572C2×D3—22no—
573C2×D3—41no—
574C2×D3—41no—
575C2×D3—42no—
576C2×D3—42no—
577C2×D3—41no—
578C2×D3—84no—
579C2×D3—162no—
580C2×D3C2×#9912yes—
581C3×C4—21no—
582C3×C4—21no—
583C3×C4—41no—
584C3×C4—41no—
585C3×C4—41no—
586C3×C4—41no—
587C3×C4—84no—
588Dic3—21no—
589Dic3—22no—
590Dic3—41no—
591Dic3—41no—
592Dic3—41no—
593Dic3—84no—
594Dic3C2×#9912yes—
595C23—22no—
596C23—22no—
597C23—31no—
598C23—32no—
599C23—31no—
600C23—64no—
601C23—61no—
602C23—82no—
603C23—122no—
604C23—242no—
605C23—242no—
606C23D3×C22⋊C412yes—
607C23D3×C22⋊C411yescharacteristic
608C23C2×D3×D412yes—
609Q8—41no—
610Q8—41no—
611Q8—42no—
612Q8—62no—
613Q8—121no—
614Q8—121no—
615Q8—122no—
616Q8—122no—
617D4—32no—
618D4—32no—
619D4—41no—
620D4—42no—
621D4—41no—
622D4—62no—
623D4—64no—
624D4—64no—
625D4—62no—
626D4—62no—
627D4—61no—
628D4—61no—
629D4—121no—
630D4—121no—
631D4—122no—
632D4—122no—
633D4—122no—
634D4—122no—
635D4—121no—
636D4—121no—
637D4—242no—
638D4—242no—
639D4—242no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—31no—
647C2×C4—31no—
648C2×C4—32no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—62no—
653C2×C4—62no—
654C2×C4—64no—
655C2×C4—61no—
656C2×C4—61no—
657C2×C4—61no—
658C2×C4—81no—
659C2×C4—122no—
660C2×C4—122no—
661C2×C4—122no—
662C2×C4—122no—
663C2×C4—122no—
664C2×C4—122no—
665C2×C4—122no—
666C2×C4—124no—
667C2×C4—162no—
668C2×C4—241no—
669C2×C4—241no—
670C2×C4—484no—
671C2×C4—482no—
672C2×C4D3×C22⋊C411yescharacteristic
673C2×C4C22×C4×D311yescharacteristic
674C2×C3—21no—
675C2×C3—21no—
676C2×C3—21no—
677C2×C3—42no—
678C2×C3—42no—
679C2×C3C22×#9911yescharacteristic
680D3—22no—
681D3—42no—
682D3—41no—
683D3—81no—
684C22—22no—
685C22—22no—
686C22—32no—
687C22—41no—
688C22—42no—
689C22—42no—
690C22—61no—
691C22—62no—
692C22—82no—
693C22—82no—
694C22—121no—
695C22—121no—
696C22—122no—
697C22—122no—
698C22—121no—
699C22—244no—
700C22—482no—
701C22D3×#9911yescharacteristic
702C22D3×#9911yescharacteristic
703C22C2×D3×C22⋊C411yescharacteristic
704C4—21no—
705C4—21no—
706C4—32no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—61no—
712C4—62no—
713C4—84no—
714C4—121no—
715C4—121no—
716C4—121no—
717C4—244no—
718C3#3336111yescharacteristic, complemented, Sylow
719C2—21no—
720C2—21no—
721C2—21no—
722C2—42no—
723C2—42no—
724C2—62no—
725C2—122no—
726C2—121no—
727C2—241no—
728C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC2×C3×#354C2
socleC2×C3C22×#99
prime radical#86076C2
cosocleC2×C3×D4C24
hypercenterC2×#354D3
hypocenterC3#33361
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
2. #78688C2semidirect
3. C2×C3×#354C2semidirect
4. C2×#4541C2semidirect
5. #70019C2semidirect
6. #69159C2semidirect
7. #69158C2semidirect
8. #69157C2semidirect
9. D3×#354C2semidirect
11. C2×#1007C4semidirect
14. #5816C4semidirect
18. #5675C4semidirect
20. #5674C4semidirect
22. D3×#124C4semidirect
24. #5668C4semidirect
29. #5477C4semidirect
30. C4×#235C4semidirect
38. C3×#354C22semidirect
39. C3×#354C22semidirect
40. #4541C22semidirect
41. #4541C22semidirect
42. #4474C22semidirect
43. #4473C22semidirect
52. C2×#354D3semidirect
53. C22×C3×Q8C2×C4semidirect
60. C2×#239C2×C4semidirect
61. C2×#239C2×C4semidirect
63. C2×D3×Q8C2×C4semidirect
64. C2×D3×Q8C2×C4semidirect
77. C2×#235C2×C4semidirect
79. C3×#124C2×C4semidirect
80. C3×#124C2×C4semidirect
85. C2×C3×C42C2×C4semidirect
88. #947C2×C4semidirect
89. #947C2×C4semidirect
91. #944C2×C4semidirect
92. #944C2×C4semidirect
93. #940C2×C4semidirect
96. #936C2×C4semidirect
103. #926C2×C4semidirect
123. #875C2×C4semidirect
124. #875C2×C4semidirect
125. #874C2×C4semidirect
128. #872C2×C4semidirect
129. #872C2×C4semidirect
130. C42×D3C2×C4semidirect
131. C42×D3C2×C4semidirect
132. #869C2×C4semidirect
170. #354C2×D3semidirect
171. #354C2×D3semidirect
302. C22×Q8C4×D3semidirect
324. #124C4×D3semidirect
325. #124C4×D3semidirect
364. C2×C42C4×D3semidirect
718. C3#33361semidirect

Maximal subgroups

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

subgroupindex
#860762
#786882
C2×C3×#3542
C2×#45412
#700192
#691592
#691582
#691572
D3×#3542
#333613