Finite Group Databasebeta

← #10763281

Subgroups — #10763281

Order 768.

1232 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#532C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic, complemented
3D3×#432C211yesmaximal, characteristic, complemented
4C3×#1536C211yesmaximal, characteristic, complemented
5#71799C211yesmaximal, characteristic, complemented
6#70811C211yesmaximal, characteristic, complemented
7#70811C211yesmaximal, characteristic, complemented
8#68345C214yesmaximal, complemented
9#68343C214yesmaximal, complemented
10#17671—31nomaximal, Sylow
11C2×D3×#41C412yes—
12C3×#532C2211yescharacteristic
13#5832C412yes—
14#5725C411yescharacteristic
15D3×C4⋊Q8C411yescharacteristic
16#5716C411yescharacteristic
17D3×#127C411yescharacteristic
18D3×#124C2212yes—
19#5668C2212yes—
20D3×#121—21no—
21D3×#121—21no—
22#5596—21no—
23#5596—21no—
24#5548—21no—
25#5548—21no—
26C4×D3×D4—21no—
27C4×D3×D4—21no—
28#5477C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C3×#432C2211yescharacteristic, complemented
31C3×#432C2211yescharacteristic, complemented
32#4926C2211yescharacteristic, complemented
33#4926C2211yescharacteristic, complemented
34#4837—21no—
35#4835—21no—
36#4833—21no—
37D3×C4⋊C8—21no—
38#4726—21no—
39D3×C22⋊C8—21no—
40#4719—21no—
41#4718—21no—
42#4707C2212yes—
43#4705C2212yes—
44#4705C2212yescomplemented
45D3×#97C2212yescomplemented
46C3×#329C2214yescomplemented
47#4540C2214yescomplemented
48#4464C2214yescomplemented
49#4464C2214yescomplemented
50C2×#532—31no—
51#2920—31no—
52C2×#432—31no—
53#1536—31no—
54#1536—31no—
55#1536—31no—
56#1536D311yescharacteristic, complemented
57#1530—34no—
58#1527—34no—
59C2×C3×#41C2×C412yes—
60D3×#41—48no—
61C2×#239C2×C412yes—
62C2×D3×Q8C2×C412yes—
63C2×#237—22no—
64C2×#237C2×C412yes—
65C2×D3×D4—22no—
66C2×D3×D4—21no—
67C2×D3×D4—21no—
68C2×D3×D4C2×C412yes—
69C2×#235—22no—
70C2×#235C2×C412yes—
71C22×C4×D3—22no—
72C22×C4×D3C2×C411yescharacteristic
73C22×C4×D3D411yescharacteristic
74C22×C4×D3D411yescharacteristic
75C3×C4⋊Q8C2×C411yescharacteristic
76C3×#127C2×C411yescharacteristic
77C3×#124C2312yes—
78C3×#121—21no—
79C3×#121—21no—
80C3×C4×D4—21no—
81C3×C4×D4—21no—
82C2×C3×C42C2311yescharacteristic
83#947C2×C412yes—
84#944C2×C412yes—
85#940—21no—
86#940—21no—
87#940C2×C412yes—
88#939—22no—
89#939—22no—
90#938—21no—
91#938—21no—
92#936C2×C412yes—
93D4×Dic3—21no—
94D4×Dic3—21no—
95#930—21no—
96#930—21no—
97C4×#67—22no—
98C4×#67—22no—
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×ΓL(1,𝔽25)—22no—
103C2×D3×C8—22no—
104#895—21no—
105#895—21no—
106#893—21no—
107#893—21no—
108D3×C4⋊C4—21no—
109D3×C4⋊C4—21no—
110#888—21no—
111#888—21no—
112#885—22no—
113#885—22no—
114#884—22no—
115#884—22no—
116#67⋊C4—22no—
117#67⋊C4—22no—
118D3×C22⋊C4—22no—
119D3×C22⋊C4—22no—
120#875C2312yes—
121#874C2×C411yescharacteristic
122C4×D12—21no—
123C4×D12—21no—
124C42×D3C2×C411yescharacteristic
125C42×D3C2×C411yescharacteristic
126C42×D3D411yescharacteristic
127C42×D3D411yescharacteristic
128C42×D3D411yescharacteristic
129C42×D3D411yescharacteristic
130C42×D3C2311yescharacteristic
131C42×D3C2311yescharacteristic
132#869C2×C411yescharacteristic
133C3×C4⋊C8—21no—
134C3×C4⋊C8—21no—
135C3×C22⋊C8—21no—
136C3×C22⋊C8—21no—
137C3×#97C2312yes—
138C3×#97C2312yes—
139#830—21no—
140#830—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#815—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146C8×Dic3—22no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)—21no—
149#803C2312yes—
150#803C2312yes—
151C22×#41—32no—
152#532—32no—
153#532—31no—
154#532—31no—
155#532—32no—
156#532—31no—
157#532—61no—
158#532—61no—
159#532—61no—
160#532—61no—
161#532C2×D311yescharacteristic
162C2×C4⋊Q8—31no—
163C2×#127—31no—
164C2×#124—32no—
165C2×#121—61no—
166C2×#121—61no—
167C2×C4×D4—61no—
168C2×C4×D4—61no—
169C22×C42—31no—
170#432—31no—
171#432—31no—
172#432—32no—
173#432—32no—
174#432—61no—
175#432—61no—
176#432C2×D311yescharacteristic, complemented
177#432C2×D311yescharacteristic, complemented
178#431—61no—
179#431—61no—
180#423—61no—
181C2×C4⋊C8—61no—
182#407—61no—
183C2×C22⋊C8—61no—
184C4×M16—32no—
185C2×#97—32no—
186#329—34no—
187#329—34no—
188#329—34no—
189#329C2×D314yescomplemented
190C23×D3—22no—
191C23×D3C22⋊C411yescharacteristic
192C3×#41—44no—
193C2×C3×Q8C22×C412yes—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4C22×C412yes—
198C22×C3×C4—22no—
199C22×C3×C4C22×C411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C2×#67—22no—
203C2×#67—22no—
204C2×#67—22no—
205C2×#67—22no—
206C2×#67—22no—
207C22×Dic3—22no—
208C22×Dic3C22⋊C411yescharacteristic
209C22×Dic3C22⋊C411yescharacteristic
210C22×Dic3C22⋊C411yescharacteristic
211#239—44no—
212D3×Q8—44no—
213#237—44no—
214#237—48no—
215D3×D4—44no—
216D3×D4—44no—
217D3×D4—44no—
218D3×D4—48no—
219#235—48no—
220#235—44no—
221C2×D12—22no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C22×C412yes—
225C2×C4×D3—24no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—48no—
232C2×C4×D3C22⋊C411yescharacteristic
233C2×C4×D3C22⋊C411yescharacteristic
234C2×C4×D3C22⋊C411yescharacteristic
235C2×C4×D3C22⋊C411yescharacteristic
236C2×C4×D3C22⋊C411yescharacteristic
237C2×C4×D3C22⋊C411yescharacteristic
238C2×C4×D3C22×C411yescharacteristic
239C2×C4×D3C22×C411yescharacteristic
240C2×C4×D3C2×D411yescharacteristic
241C2×C4×D3C2×D411yescharacteristic
242C2×C4×D3C2×D411yescharacteristic
243C2×C4×D3C2×D411yescharacteristic
244C2×Dic6—22no—
245C2×Dic6C22×C412yes—
246C2×C3×C8—22no—
247C2×C3×C8—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C42C22×C411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42C2411yescharacteristic
256#217—22no—
257#217—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—22no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266C4×Dic3C22⋊C411yescharacteristic
267C4×Dic3C22⋊C411yescharacteristic
268C4×Dic3C22⋊C411yescharacteristic
269C4×Dic3C22⋊C411yescharacteristic
270C4×Dic3C22⋊C411yescharacteristic
271C4×Dic3C22⋊C411yescharacteristic
272C4×Dic3C22×C411yescharacteristic
273C4×Dic3C22×C411yescharacteristic
274C4×Dic3C2×D411yescharacteristic
275C4×Dic3C2×D411yescharacteristic
276C4×Dic3C2×D411yescharacteristic
277C4×Dic3C2×D411yescharacteristic
278C2×C3⋊C8—22no—
279C2×C3⋊C8—22no—
280ΓL(1,𝔽25)—44no—
281D3×C8—44no—
282C2×#41—32no—
283C2×#41—32no—
284C2×#41—32no—
285C2×#41—62no—
286C2×#41—62no—
287C2×#41—128no—
288C2×#41C4×D312yes—
289C22×Q8—32no—
290C22×D4—32no—
291C22×D4—61no—
292C22×D4—61no—
293C22×D4—62no—
294C23×C4—31no—
295C23×C4—31no—
296C23×C4—31no—
297C23×C4—62no—
298C2×M16—62no—
299C22×C8—62no—
300C4⋊Q8—32no—
301C4⋊Q8—31no—
302C4⋊Q8—61no—
303C4⋊Q8—61no—
304C4⋊Q8C4×D311yescharacteristic
305#127—32no—
306#127—31no—
307#127—61no—
308#127—61no—
309#127C4×D311yescharacteristic
310#124—32no—
311#124—32no—
312#124—32no—
313#124—62no—
314#124—62no—
315#124C22×D312yes—
316#121—21no—
317#121—21no—
318#121—62no—
319#121—62no—
320#121—61no—
321#121—61no—
322#121—61no—
323#121—61no—
324#121—62no—
325#121—62no—
326#121—61no—
327#121—61no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—62no—
331C4×D4—62no—
332C4×D4—61no—
333C4×D4—61no—
334C4×D4—62no—
335C4×D4—62no—
336C4×D4—61no—
337C4×D4—61no—
338C4×D4—61no—
339C4×D4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C22⋊C4—62no—
343C2×C22⋊C4—62no—
344C2×C42—31no—
345C2×C42—31no—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—31no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42C22×D311yescharacteristic
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—62no—
359C4⋊C8—62no—
360C4⋊C8—61no—
361C4⋊C8—61no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364C22⋊C8—62no—
365C22⋊C8—62no—
366C22⋊C8—61no—
367C22⋊C8—61no—
368#97—32no—
369#97—32no—
370#97—62no—
371#97C22×D312yes—
372#97C22×D312yes—
373C4×C8—62no—
374C23×C3—22no—
375C23×C3C2×C22⋊C411yescharacteristic
376C22×D3—22no—
377C22×D3—22no—
378C22×D3—22no—
379C22×D3—41no—
380C22×D3—44no—
381C22×D3—44no—
382C22×D3—44no—
383C22×D3C2×C22⋊C411yescharacteristic
384C22×D3C2×C22⋊C411yescharacteristic
385C3×Q8—42no—
386C3×D4—42no—
387C3×D4—42no—
388C3×D4—42no—
389C3×D4—44no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—44no—
395C2×C3×C4C2×C22⋊C411yescharacteristic
396C2×C3×C4C2×C22⋊C411yescharacteristic
397C2×C3×C4C2×C22⋊C411yescharacteristic
398C2×C3×C4C23×C411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401#67—44no—
402#67—44no—
403#67—44no—
404#67—44no—
405#67—44no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—41no—
410C2×Dic3—44no—
411C2×Dic3C2×C22⋊C411yescharacteristic
412C2×Dic3C2×C22⋊C411yescharacteristic
413C2×Dic3C2×C22⋊C411yescharacteristic
414C2×Dic3C2×C22⋊C411yescharacteristic
415C2×Dic3C2×C22⋊C411yescharacteristic
416C2×Dic3C2×C22⋊C411yescharacteristic
417D12—42no—
418D12—42no—
419D12—42no—
420D12—44no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—22no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—41no—
432C4×D3—44no—
433C4×D3—44no—
434C4×D3—44no—
435C4×D3#10014yes—
436Dic6—44no—
437Dic6—42no—
438C3×C8—42no—
439C3×C8—42no—
440C3⋊C8—42no—
441C3⋊C8—42no—
442C24—31no—
443C24—62no—
444#41—44no—
445#41—124no—
446#41—128no—
447#41—128no—
448#41—124no—
449#41—124no—
450C2×Q8—32no—
451C2×Q8—62no—
452C2×Q8—124no—
453C2×Q8C2×C4×D312yes—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—32no—
458C2×D4—62no—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—124no—
467C2×D4—124no—
468C2×D4—124no—
469C2×D4—128no—
470C2×D4C2×C4×D312yes—
471C22×C4—22no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—31no—
487C22×C4—64no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—128no—
495C22×C4C2×C4×D311yescharacteristic
496C22×C4D3×D411yescharacteristic
497C22×C4D3×D411yescharacteristic
498M16—124no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—62no—
502C2×C8—62no—
503C2×C8—124no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—62no—
518C42—31no—
519C42—31no—
520C42—31no—
521C42—31no—
522C42—31no—
523C42—31no—
524C42—31no—
525C42—31no—
526C42—31no—
527C42—31no—
528C42—31no—
529C42—31no—
530C42C2×C4×D311yescharacteristic
531C42D3×D411yescharacteristic
532C42D3×D411yescharacteristic
533C42C23×D311yescharacteristic
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3C22×C22⋊C411yescharacteristic
541C2×D3—21no—
542C2×D3—21no—
543C2×D3—21no—
544C2×D3—21no—
545C2×D3—44no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—88no—
550C2×D3C2×#10012yes—
551C2×D3#42112yes—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—42no—
558C3×C4C2×#10012yes—
559Dic3—21no—
560Dic3—21no—
561Dic3—21no—
562Dic3—21no—
563Dic3—42no—
564Dic3C2×#10012yes—
565Dic3#42112yes—
566C23—22no—
567C23—31no—
568C23—31no—
569C23—62no—
570C23—62no—
571C23—62no—
572C23—121no—
573C23—124no—
574C23—124no—
575C23—124no—
576C23D3×C22⋊C411yescharacteristic
577Q8—42no—
578Q8—122no—
579Q8—124no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—44no—
584D4—122no—
585D4—122no—
586D4—122no—
587D4—124no—
588D4—124no—
589D4—124no—
590D4—124no—
591D4—124no—
592D4—124no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—34no—
598C2×C4—31no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—124no—
620C2×C4—124no—
621C2×C4—121no—
622C2×C4—124no—
623C2×C4—124no—
624C2×C4D3×C22⋊C411yescharacteristic
625C2×C4D3×C22⋊C411yescharacteristic
626C2×C4D3×C22⋊C411yescharacteristic
627C2×C4C22×C4×D311yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×D3×D411yescharacteristic
630C8—42no—
631C8—42no—
632C8—122no—
633C8—122no—
634C2×C3—41no—
635C2×C3—82no—
636C2×C3C22×#10011yescharacteristic
637C2×C3C2×#42111yescharacteristic
638C2×C3C2×#42111yescharacteristic
639D3—22no—
640D3—22no—
641D3—82no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—32no—
646C22—32no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—61no—
651C22—61no—
652C22—61no—
653C22—61no—
654C22—124no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22—248no—
659C22C2×D3×C22⋊C411yescharacteristic
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—32no—
666C4—32no—
667C4—42no—
668C4—61no—
669C4—61no—
670C4—61no—
671C4—61no—
672C4—122no—
673C4D3×#10012yes—
674C3#1767111yescharacteristic, complemented, Sylow
675C2—41no—
676C2—62no—
677C2—62no—
678C2—82no—
679C2—242no—
680C2C2×D3×#10011yescharacteristic
681C2D3×#42111yescharacteristic
682C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1536C2
socleC22×C3C22×C22⋊C4
prime radicalD3×#532C2
cosocleC3×C42C24
hypercenter#1536D3
hypocenterC3#17671
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1536

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. D3×#432C2semidirect
4. C3×#1536C2semidirect
5. #71799C2semidirect
6. #70811C2semidirect
7. #70811C2semidirect
8. #68345C2semidirect
9. #68343C2semidirect
30. C3×#432C22semidirect
31. C3×#432C22semidirect
32. #4926C22semidirect
33. #4926C22semidirect
44. #4705C22semidirect
45. D3×#97C22semidirect
46. C3×#329C22semidirect
47. #4540C22semidirect
48. #4464C22semidirect
49. #4464C22semidirect
56. #1536D3semidirect
176. #432C2×D3semidirect
177. #432C2×D3semidirect
189. #329C2×D3semidirect
674. C3#17671semidirect

Maximal subgroups

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

subgroupindex
D3×#5322
#793312
D3×#4322
C3×#15362
#717992
#708112
#683452
#683432
#176713