Finite Group Databasebeta

← #10763283

Subgroups — #10763283

Order 768.

1232 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#532C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic, complemented
3C2×#4705C211yesmaximal, characteristic, complemented
4C3×#1530C211yesmaximal, characteristic, complemented
5#71798C211yesmaximal, characteristic, complemented
6#70806C211yesmaximal, characteristic, complemented
7#69798C211yesmaximal, characteristic, complemented
8#68349C214yesmaximal, 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
31C2×C3×#97C2211yescharacteristic, complemented
32C2×(C2×D3)⋊C8—21no—
33C2×Dic3⋊C8—21no—
34#4926C2211yescharacteristic, complemented
35C2×#803C2211yescharacteristic, complemented
36#4837—21no—
37#4833—21no—
38#4726—21no—
39#4718—21no—
40#4707C2212yes—
41#4705—21no—
42#4705—21no—
43#4705C2212yes—
44#4705C2212yescomplemented
45#4705C2212yescomplemented
46C3×#329C2214yescomplemented
47#4540C2214yescomplemented
48#4464C2214yescomplemented
49#4464C2214yescomplemented
50C2×#532—31no—
51#2920—31no—
52C2×#432—31no—
53#1536—34no—
54#1530—31no—
55#1530—31no—
56#1530—31no—
57#1530D311yescharacteristic, complemented
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
75C22×C3×C8—21no—
76C3×C4⋊Q8C2×C411yescharacteristic
77C3×#127C2×C411yescharacteristic
78C3×#124C2312yes—
79C3×#121—21no—
80C3×#121—21no—
81C3×C4×D4—21no—
82C3×C4×D4—21no—
83C2×C3×C42C2311yescharacteristic
84#947C2×C412yes—
85#944C2×C412yes—
86#940—21no—
87#940—21no—
88#940C2×C412yes—
89#939—22no—
90#939—22no—
91#938—21no—
92#938—21no—
93#936C2×C412yes—
94D4×Dic3—21no—
95D4×Dic3—21no—
96#930—21no—
97#930—21no—
98C4×#67—22no—
99C4×#67—22no—
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3D411yescharacteristic
103C22×C3⋊C8—21no—
104C2×ΓL(1,𝔽25)—22no—
105#895—21no—
106#895—21no—
107#893—21no—
108#893—21no—
109D3×C4⋊C4—21no—
110D3×C4⋊C4—21no—
111#888—21no—
112#888—21no—
113#885—22no—
114#885—22no—
115#884—22no—
116#884—22no—
117#67⋊C4—22no—
118#67⋊C4—22no—
119D3×C22⋊C4—22no—
120D3×C22⋊C4—22no—
121#875C2312yes—
122#874C2×C411yescharacteristic
123C4×D12—21no—
124C4×D12—21no—
125C42×D3C2×C411yescharacteristic
126C42×D3C2×C411yescharacteristic
127C42×D3D411yescharacteristic
128C42×D3D411yescharacteristic
129C42×D3D411yescharacteristic
130C42×D3D411yescharacteristic
131C42×D3C2311yescharacteristic
132C42×D3C2311yescharacteristic
133#869C2×C411yescharacteristic
134C3×C4⋊C8—21no—
135C3×C22⋊C8—21no—
136C3×#97—21no—
137C3×#97C2312yes—
138C3×#97C2312yes—
139#830—21no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146C8×Dic3—22no—
147C4⋊(C3⋊C8)—21no—
148#803—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—32no—
172#432—32no—
173#432—32no—
174#432—61no—
175#432—61no—
176#432C2×D311yescharacteristic, complemented
177#431—61no—
178#431—61no—
179#423—61no—
180C2×C4⋊C8—61no—
181#407—61no—
182C2×C22⋊C8—61no—
183C4×M16—32no—
184C2×#97—31no—
185C2×#97C2×D311yescharacteristic, complemented
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—
248C2×C3×C8—42no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C42C22×C411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256C3×C42C2411yescharacteristic
257#217—22no—
258#217—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—22no—
262(C2×D3)⋊C4—22no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267C4×Dic3C22⋊C411yescharacteristic
268C4×Dic3C22⋊C411yescharacteristic
269C4×Dic3C22⋊C411yescharacteristic
270C4×Dic3C22⋊C411yescharacteristic
271C4×Dic3C22⋊C411yescharacteristic
272C4×Dic3C22⋊C411yescharacteristic
273C4×Dic3C22×C411yescharacteristic
274C4×Dic3C22×C411yescharacteristic
275C4×Dic3C2×D411yescharacteristic
276C4×Dic3C2×D411yescharacteristic
277C4×Dic3C2×D411yescharacteristic
278C4×Dic3C2×D411yescharacteristic
279C2×C3⋊C8—22no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8—42no—
282ΓL(1,𝔽25)—44no—
283C2×#41—32no—
284C2×#41—32no—
285C2×#41—32no—
286C2×#41—62no—
287C2×#41—62no—
288C2×#41—128no—
289C2×#41C4×D312yes—
290C22×Q8—32no—
291C22×D4—32no—
292C22×D4—61no—
293C22×D4—61no—
294C22×D4—62no—
295C23×C4—31no—
296C23×C4—31no—
297C23×C4—31no—
298C23×C4—62no—
299C2×M16—62no—
300C22×C8—21no—
301C22×C8—61no—
302C4⋊Q8—32no—
303C4⋊Q8—31no—
304C4⋊Q8—61no—
305C4⋊Q8—61no—
306C4⋊Q8C4×D311yescharacteristic
307#127—32no—
308#127—31no—
309#127—61no—
310#127—61no—
311#127C4×D311yescharacteristic
312#124—32no—
313#124—32no—
314#124—32no—
315#124—62no—
316#124—62no—
317#124C22×D312yes—
318#121—21no—
319#121—21no—
320#121—62no—
321#121—62no—
322#121—61no—
323#121—61no—
324#121—61no—
325#121—61no—
326#121—62no—
327#121—62no—
328#121—61no—
329#121—61no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—62no—
333C4×D4—62no—
334C4×D4—61no—
335C4×D4—61no—
336C4×D4—62no—
337C4×D4—62no—
338C4×D4—61no—
339C4×D4—61no—
340C4×D4—61no—
341C4×D4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C22⋊C4—62no—
345C2×C22⋊C4—62no—
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×C42—31no—
356C2×C42—31no—
357C2×C42C22×D311yescharacteristic
358C4⋊C8—21no—
359C4⋊C8—62no—
360C4⋊C8—62no—
361C4⋊C8—62no—
362C4⋊C8—61no—
363C22⋊C8—21no—
364C22⋊C8—62no—
365C22⋊C8—62no—
366C22⋊C8—62no—
367C22⋊C8—61no—
368#97—21no—
369#97—32no—
370#97—32no—
371#97—61no—
372#97C22×D312yes—
373#97C22×D312yes—
374C4×C8—62no—
375C23×C3—22no—
376C23×C3C2×C22⋊C411yescharacteristic
377C22×D3—22no—
378C22×D3—22no—
379C22×D3—22no—
380C22×D3—41no—
381C22×D3—44no—
382C22×D3—44no—
383C22×D3—44no—
384C22×D3C2×C22⋊C411yescharacteristic
385C22×D3C2×C22⋊C411yescharacteristic
386C3×Q8—42no—
387C3×D4—42no—
388C3×D4—42no—
389C3×D4—44no—
390C3×D4—42no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—41no—
395C2×C3×C4—44no—
396C2×C3×C4C2×C22⋊C411yescharacteristic
397C2×C3×C4C2×C22⋊C411yescharacteristic
398C2×C3×C4C2×C22⋊C411yescharacteristic
399C2×C3×C4C23×C411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402#67—44no—
403#67—44no—
404#67—44no—
405#67—44no—
406#67—44no—
407C2×Dic3—22no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—44no—
411C2×Dic3#10014yes—
412C2×Dic3C2×C22⋊C411yescharacteristic
413C2×Dic3C2×C22⋊C411yescharacteristic
414C2×Dic3C2×C22⋊C411yescharacteristic
415C2×Dic3C2×C22⋊C411yescharacteristic
416C2×Dic3C2×C22⋊C411yescharacteristic
417C2×Dic3C2×C22⋊C411yescharacteristic
418D12—42no—
419D12—42no—
420D12—44no—
421D12—42no—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—22no—
428C4×D3—22no—
429C4×D3—22no—
430C4×D3—41no—
431C4×D3—41no—
432C4×D3—41no—
433C4×D3—41no—
434C4×D3—44no—
435C4×D3—44no—
436C4×D3—44no—
437Dic6—44no—
438Dic6—42no—
439C3×C8—42no—
440C3×C8—42no—
441C3⋊C8—42no—
442C3⋊C8—42no—
443C24—31no—
444C24—62no—
445#41—44no—
446#41—124no—
447#41—128no—
448#41—128no—
449#41—124no—
450#41—124no—
451C2×Q8—32no—
452C2×Q8—62no—
453C2×Q8—124no—
454C2×Q8C2×C4×D312yes—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—32no—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—62no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—124no—
468C2×D4—124no—
469C2×D4—124no—
470C2×D4—128no—
471C2×D4C2×C4×D312yes—
472C22×C4—22no—
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—31no—
488C22×C4—64no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—62no—
493C22×C4—122no—
494C22×C4—122no—
495C22×C4—128no—
496C22×C4C2×C4×D311yescharacteristic
497C22×C4D3×D411yescharacteristic
498C22×C4D3×D411yescharacteristic
499M16—124no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—42no—
503C2×C8—62no—
504C2×C8—62no—
505C2×C8—122no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—62no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—62no—
518C22⋊C4—62no—
519C22⋊C4—62no—
520C42—31no—
521C42—31no—
522C42—31no—
523C42—31no—
524C42—31no—
525C42—31no—
526C42—31no—
527C42—31no—
528C42—31no—
529C42—31no—
530C42—31no—
531C42—31no—
532C42C2×C4×D311yescharacteristic
533C42D3×D411yescharacteristic
534C42D3×D411yescharacteristic
535C42C23×D311yescharacteristic
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3C2×#10012yes—
542C22×C3C22×C22⋊C411yescharacteristic
543C2×D3—21no—
544C2×D3—21no—
545C2×D3—21no—
546C2×D3—21no—
547C2×D3—21no—
548C2×D3—44no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—88no—
553C2×D3#42112yes—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—42no—
560C3×C4#42112yes—
561Dic3—21no—
562Dic3—21no—
563Dic3—21no—
564Dic3—21no—
565Dic3—42no—
566Dic3C2×#10012yes—
567Dic3C2×#10012yes—
568C23—22no—
569C23—31no—
570C23—31no—
571C23—62no—
572C23—62no—
573C23—62no—
574C23—121no—
575C23—124no—
576C23—124no—
577C23—124no—
578C23D3×C22⋊C411yescharacteristic
579Q8—42no—
580Q8—122no—
581Q8—124no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4—42no—
586D4—122no—
587D4—122no—
588D4—124no—
589D4—124no—
590D4—124no—
591D4—124no—
592D4—124no—
593D4—124no—
594D4—122no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—31no—
599C2×C4—31no—
600C2×C4—34no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—31no—
604C2×C4—31no—
605C2×C4—41no—
606C2×C4—44no—
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—62no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—124no—
621C2×C4—124no—
622C2×C4—124no—
623C2×C4—124no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4D3×C22⋊C411yescharacteristic
627C2×C4D3×C22⋊C411yescharacteristic
628C2×C4D3×C22⋊C411yescharacteristic
629C2×C4C22×C4×D311yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4C2×D3×D411yescharacteristic
632C8—42no—
633C8—42no—
634C8—122no—
635C8—122no—
636C2×C3—22no—
637C2×C3—82no—
638C2×C3C22×#10011yescharacteristic
639C2×C3C2×#42111yescharacteristic
640C2×C3C2×#42111yescharacteristic
641D3—22no—
642D3—41no—
643D3—82no—
644C22—21no—
645C22—21no—
646C22—32no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—61no—
651C22—61no—
652C22—61no—
653C22—61no—
654C22—61no—
655C22—124no—
656C22—122no—
657C22—122no—
658C22—122no—
659C22—248no—
660C22D3×#10012yes—
661C22C2×D3×C22⋊C411yescharacteristic
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—32no—
668C4—32no—
669C4—42no—
670C4—61no—
671C4—61no—
672C4—61no—
673C4—61no—
674C4—122no—
675C4#482112yes—
676C3#1767111yescharacteristic, complemented, Sylow
677C2—22no—
678C2—62no—
679C2—82no—
680C2—121no—
681C2—242no—
682C2C2×D3×#10011yescharacteristic
683C2C2×#482111yescharacteristic
684C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1530C2
socleC22×C3C22×C22⋊C4
prime radicalD3×#532C2
cosocleC3×C42C24
hypercenter#1530D3
hypocenterC3#17671
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1530

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. C2×#4705C2semidirect
4. C3×#1530C2semidirect
5. #71798C2semidirect
6. #70806C2semidirect
7. #69798C2semidirect
8. #68349C2semidirect
9. #68343C2semidirect
30. C3×#432C22semidirect
31. C2×C3×#97C22semidirect
34. #4926C22semidirect
35. C2×#803C22semidirect
44. #4705C22semidirect
45. #4705C22semidirect
46. C3×#329C22semidirect
47. #4540C22semidirect
48. #4464C22semidirect
49. #4464C22semidirect
57. #1530D3semidirect
176. #432C2×D3semidirect
185. C2×#97C2×D3semidirect
189. #329C2×D3semidirect
676. 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
C2×#47052
C3×#15302
#717982
#708062
#697982
#683492
#683432
#176713