Finite Group Databasebeta

← #10763282

Subgroups — #10763282

Order 768.

1232 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5668C211yesmaximal, characteristic
2D3×#432C211yesmaximal, characteristic, complemented
3#78517C211yesmaximal, characteristic, complemented
4C3×#1527C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic, complemented
7#69860C211yesmaximal, characteristic, complemented
8#68349C214yesmaximal, complemented
9#68345C214yesmaximal, complemented
10#17671—31nomaximal, Sylow
11C22×#239C411yescharacteristic
12C22×#237C411yescharacteristic
13C2×C3×#124C2211yescharacteristic
14C2×#944C411yescharacteristic
15C2×#940C411yescharacteristic
16#5668—21no—
17#5668—21no—
18#5668—21no—
19#5668—21no—
20#5668C412yes—
21#5668C412yes—
22#5668C2212yes—
23#5668C2212yes—
24C2×#884—21no—
25C2×#884—21no—
26C2×#67⋊C4—21no—
27C2×#67⋊C4—21no—
28C2×#875C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C3×#432C2211yescharacteristic
31C3×C4×M16C2211yescharacteristic
32#5126—21no—
33#5118—21no—
34#4926C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4835—21no—
37D3×C4⋊C8—21no—
38D3×C22⋊C8—21no—
39#4719—21no—
40#4705C2212yescomplemented
41#4705C2212yescomplemented
42#4705C2212yescomplemented
43D3×#97C2212yescomplemented
44#4685—21no—
45#4685—21no—
46C3×#329C2214yescomplemented
47#4540C2214yescomplemented
48#4464C2214yescomplemented
49#4464C2214yescomplemented
50C2×#532—31no—
51#2920—31no—
52C2×#432—31no—
53#1536—34no—
54#1530—34no—
55#1527—31no—
56#1527—31no—
57#1527—31no—
58#1527D311yescharacteristic, complemented
59C22×C3×Q8C2×C411yescharacteristic
60C22×C3×D4C2×C411yescharacteristic
61C22×#67—21no—
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C2×#239—21no—
66C2×#239—21no—
67C2×#239—44no—
68C2×#239C2×C412yes—
69C2×#239C2×C412yes—
70C2×#237—21no—
71C2×#237—21no—
72C2×#237—44no—
73C2×#237C2×C412yes—
74C2×#237C2×C412yes—
75C22×D12—21no—
76C22×D12C2×C411yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2×C411yescharacteristic
79C22×C4×D3D411yescharacteristic
80C22×C4×D3D411yescharacteristic
81C22×Dic6C2×C411yescharacteristic
82C2×C3×M16—21no—
83C3×#124—21no—
84C3×#124—21no—
85C3×#124C2×C412yes—
86C3×#124C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90#944—21no—
91#944—21no—
92#944C2×C412yes—
93#944C2×C412yes—
94#940—21no—
95#940—21no—
96#940C2×C412yes—
97#940C2×C412yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3D411yescharacteristic
105C2×#208—21no—
106C2×D3×C8—22no—
107#884—22no—
108#884—22no—
109#884—22no—
110#884—22no—
111#884—22no—
112#884—22no—
113#884—22no—
114#884—22no—
115#67⋊C4—22no—
116#67⋊C4—22no—
117#67⋊C4—22no—
118#67⋊C4—22no—
119#67⋊C4—22no—
120#67⋊C4—22no—
121#67⋊C4—22no—
122#67⋊C4—22no—
123#875—21no—
124#875—21no—
125#875C2×C412yes—
126#875C2312yes—
127C42×D3C2×C411yescharacteristic
128C42×D3C2×C411yescharacteristic
129C42×D3D411yescharacteristic
130C42×D3D411yescharacteristic
131C42×D3D411yescharacteristic
132C42×D3D411yescharacteristic
133C42×D3C2311yescharacteristic
134C42×D3C2311yescharacteristic
135C3×C4⋊C8—21no—
136C3×C22⋊C8—21no—
137C3×#97C2312yes—
138C3×#97C2312yes—
139C3×C4×C8—21no—
140#830—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143(C2×D3)⋊C8—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147Dic3⋊C8—22no—
148C4⋊(C3⋊C8)—21no—
149#803C2312yes—
150#803C2312yes—
151C4×C3⋊C8—21no—
152C22×#41—31no—
153C22×#41—31no—
154#532—32no—
155#532—32no—
156#532—32no—
157#532—32no—
158#532—61no—
159#532—61no—
160#532—61no—
161#532—61no—
162C2×C4⋊Q8—31no—
163C2×#127—31no—
164C2×#124—31no—
165C2×#124C2×D311yescharacteristic
166C2×#121—61no—
167C2×#121—61no—
168C2×C4×D4—61no—
169C2×C4×D4—61no—
170C22×C42—31no—
171#432—31no—
172#432—32no—
173#432—32no—
174#432—32no—
175#432—61no—
176#432—61no—
177#432C2×D311yescharacteristic
178#431—61no—
179#431—61no—
180#423—61no—
181C2×C4⋊C8—61no—
182#407—61no—
183C2×C22⋊C8—61no—
184C4×M16—31no—
185C4×M16C2×D311yescharacteristic
186C2×#97—32no—
187#329—34no—
188#329—34no—
189#329—34no—
190#329C2×D314yescomplemented
191C24×C3—21no—
192C23×D3—21no—
193C23×D3C22⋊C411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—42no—
196C2×C3×Q8C22×C412yes—
197C2×C3×D4—21no—
198C2×C3×D4—42no—
199C2×C3×D4C22×C412yes—
200C22×C3×C4—21no—
201C22×C3×C4C22×C411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C2×#67—22no—
205C2×#67—22no—
206C2×#67—22no—
207C2×#67—22no—
208C2×#67—22no—
209C2×#67—22no—
210C2×#67—44no—
211C2×#67—44no—
212C2×#67—44no—
213C22×Dic3—21no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—44no—
217C22×Dic3C22⋊C411yescharacteristic
218C22×Dic3C22⋊C411yescharacteristic
219C22×Dic3C22⋊C411yescharacteristic
220#239—44no—
221#239—44no—
222#239—44no—
223#239—44no—
224#237—44no—
225#237—44no—
226#237—44no—
227#237—44no—
228C2×D12—22no—
229C2×D12—22no—
230C2×D12—21no—
231C2×D12—44no—
232C2×D12—42no—
233C2×D12C22×C412yes—
234C2×C4×D3—24no—
235C2×C4×D3—22no—
236C2×C4×D3—22no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—44no—
240C2×C4×D3C22⋊C411yescharacteristic
241C2×C4×D3C22⋊C411yescharacteristic
242C2×C4×D3C22⋊C411yescharacteristic
243C2×C4×D3C22⋊C411yescharacteristic
244C2×C4×D3C22⋊C411yescharacteristic
245C2×C4×D3C22⋊C411yescharacteristic
246C2×C4×D3C22×C411yescharacteristic
247C2×C4×D3C22×C411yescharacteristic
248C2×C4×D3C2×D411yescharacteristic
249C2×C4×D3C2×D411yescharacteristic
250C2×C4×D3C2×D411yescharacteristic
251C2×C4×D3C2×D411yescharacteristic
252C2×Dic6—21no—
253C2×Dic6—42no—
254C2×Dic6C22×C412yes—
255C3×M16—42no—
256C2×C3×C8—22no—
257C2×C3×C8—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C42C22×C411yescharacteristic
263C3×C42C2×D411yescharacteristic
264C3×C42C2×D411yescharacteristic
265C3×C42C2411yescharacteristic
266(C2×D3)⋊C4—22no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—22no—
270Dic3⋊C4—22no—
271Dic3⋊C4—22no—
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274C4×Dic3C22⋊C411yescharacteristic
275C4×Dic3C22⋊C411yescharacteristic
276C4×Dic3C22⋊C411yescharacteristic
277C4×Dic3C22⋊C411yescharacteristic
278C4×Dic3C22⋊C411yescharacteristic
279C4×Dic3C22⋊C411yescharacteristic
280C4×Dic3C22×C411yescharacteristic
281C4×Dic3C22×C411yescharacteristic
282C4×Dic3C2×D411yescharacteristic
283C4×Dic3C2×D411yescharacteristic
284C4×Dic3C2×D411yescharacteristic
285C4×Dic3C2×D411yescharacteristic
286#208—42no—
287C2×C3⋊C8—22no—
288C2×C3⋊C8—22no—
289D3×C8—44no—
290C2×#41—32no—
291C2×#41—32no—
292C2×#41—32no—
293C2×#41—32no—
294C2×#41—61no—
295C2×#41—61no—
296C2×#41—61no—
297C2×#41—61no—
298C2×#41—124no—
299C2×#41—124no—
300C22×Q8—31no—
301C22×Q8C4×D311yescharacteristic
302C22×D4—31no—
303C22×D4—61no—
304C22×D4—61no—
305C22×D4—61no—
306C22×D4—61no—
307C22×D4C4×D311yescharacteristic
308C23×C4—31no—
309C23×C4—31no—
310C23×C4—31no—
311C23×C4—61no—
312C23×C4—61no—
313C2×M16—21no—
314C2×M16—61no—
315C22×C8—62no—
316C4⋊Q8—32no—
317C4⋊Q8—32no—
318C4⋊Q8—61no—
319C4⋊Q8—61no—
320#127—32no—
321#127—32no—
322#127—61no—
323#127—61no—
324#124—21no—
325#124—21no—
326#124—32no—
327#124—32no—
328#124—61no—
329#124—61no—
330#124C4×D312yes—
331#124C22×D312yes—
332#121—62no—
333#121—62no—
334#121—62no—
335#121—62no—
336#121—62no—
337#121—62no—
338#121—62no—
339#121—62no—
340C4×D4—62no—
341C4×D4—62no—
342C4×D4—62no—
343C4×D4—62no—
344C4×D4—62no—
345C4×D4—62no—
346C4×D4—62no—
347C4×D4—62no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—61no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42C22×D311yescharacteristic
366C4⋊C8—21no—
367C4⋊C8—62no—
368C4⋊C8—62no—
369C4⋊C8—62no—
370C4⋊C8—61no—
371C22⋊C8—21no—
372C22⋊C8—62no—
373C22⋊C8—62no—
374C22⋊C8—62no—
375C22⋊C8—61no—
376#97—32no—
377#97—32no—
378#97—62no—
379#97C22×D312yes—
380#97C22×D312yes—
381C4×C8—21no—
382C4×C8—61no—
383C23×C3—22no—
384C23×C3—42no—
385C23×C3—42no—
386C23×C3—42no—
387C23×C3C2×C22⋊C411yescharacteristic
388C22×D3—22no—
389C22×D3—22no—
390C22×D3—22no—
391C22×D3—41no—
392C22×D3—42no—
393C22×D3—42no—
394C22×D3—42no—
395C22×D3C2×C22⋊C411yescharacteristic
396C22×D3C2×C22⋊C411yescharacteristic
397C3×Q8—42no—
398C3×Q8—42no—
399C3×D4—42no—
400C3×D4—42no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—41no—
405C2×C3×C4—42no—
406C2×C3×C4C2×C22⋊C411yescharacteristic
407C2×C3×C4C2×C22⋊C411yescharacteristic
408C2×C3×C4C2×C22⋊C411yescharacteristic
409C2×C3×C4C23×C411yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412#67—44no—
413#67—44no—
414#67—44no—
415#67—44no—
416#67—44no—
417#67—44no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—44no—
423C2×Dic3—42no—
424C2×Dic3—44no—
425C2×Dic3C2×C22⋊C411yescharacteristic
426C2×Dic3C2×C22⋊C411yescharacteristic
427C2×Dic3C2×C22⋊C411yescharacteristic
428C2×Dic3C2×C22⋊C411yescharacteristic
429C2×Dic3C2×C22⋊C411yescharacteristic
430C2×Dic3C2×C22⋊C411yescharacteristic
431D12—42no—
432D12—44no—
433D12—44no—
434D12—42no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—22no—
439C4×D3—22no—
440C4×D3—41no—
441C4×D3—41no—
442C4×D3—41no—
443C4×D3—41no—
444C4×D3—41no—
445C4×D3—41no—
446C4×D3—44no—
447C4×D3—44no—
448C4×D3#10014yes—
449Dic6—42no—
450Dic6—42no—
451C3×C8—42no—
452C3×C8—42no—
453C3⋊C8—42no—
454C3⋊C8—42no—
455C24—21no—
456C24—31no—
457C24—61no—
458#41—124no—
459#41—124no—
460#41—124no—
461#41—124no—
462#41—124no—
463#41—124no—
464#41—124no—
465#41—124no—
466C2×Q8—21no—
467C2×Q8—32no—
468C2×Q8—42no—
469C2×Q8—61no—
470C2×Q8—122no—
471C2×Q8C2×C4×D312yes—
472C2×D4—21no—
473C2×D4—32no—
474C2×D4—42no—
475C2×D4—62no—
476C2×D4—62no—
477C2×D4—62no—
478C2×D4—62no—
479C2×D4—62no—
480C2×D4—62no—
481C2×D4—62no—
482C2×D4—62no—
483C2×D4—61no—
484C2×D4—122no—
485C2×D4—124no—
486C2×D4—124no—
487C2×D4—124no—
488C2×D4—124no—
489C2×D4C2×C4×D312yes—
490C22×C4—21no—
491C22×C4—31no—
492C22×C4—31no—
493C22×C4—31no—
494C22×C4—31no—
495C22×C4—31no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—31no—
504C22×C4—31no—
505C22×C4—31no—
506C22×C4—64no—
507C22×C4—62no—
508C22×C4—62no—
509C22×C4—62no—
510C22×C4—62no—
511C22×C4—61no—
512C22×C4—122no—
513C22×C4—122no—
514C22×C4—124no—
515C22×C4—124no—
516C22×C4C2×C4×D311yescharacteristic
517C22×C4D3×D411yescharacteristic
518C22×C4D3×D411yescharacteristic
519M16—42no—
520M16—122no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—62no—
524C2×C8—62no—
525C2×C8—124no—
526C4⋊C4—62no—
527C4⋊C4—62no—
528C4⋊C4—62no—
529C4⋊C4—62no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—62no—
535C22⋊C4—62no—
536C22⋊C4—62no—
537C22⋊C4—62no—
538C42—31no—
539C42—31no—
540C42—31no—
541C42—31no—
542C42—31no—
543C42—31no—
544C42—31no—
545C42—31no—
546C42—31no—
547C42—31no—
548C42—31no—
549C42—31no—
550C42C2×C4×D311yescharacteristic
551C42D3×D411yescharacteristic
552C42D3×D411yescharacteristic
553C42C23×D311yescharacteristic
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—42no—
557C22×C3—42no—
558C22×C3—42no—
559C22×C3—84no—
560C22×C3#42112yes—
561C22×C3C22×C22⋊C411yescharacteristic
562C2×D3—21no—
563C2×D3—21no—
564C2×D3—21no—
565C2×D3—21no—
566C2×D3—21no—
567C2×D3—44no—
568C2×D3—42no—
569C2×D3—42no—
570C2×D3—42no—
571C2×D3—84no—
572C2×D3C2×#10012yes—
573C3×C4—21no—
574C3×C4—21no—
575C3×C4—21no—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—42no—
579C3×C4C2×#10012yes—
580Dic3—21no—
581Dic3—21no—
582Dic3—21no—
583Dic3—21no—
584Dic3—42no—
585Dic3C2×#10012yes—
586Dic3#42112yes—
587C23—22no—
588C23—31no—
589C23—31no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—62no—
594C23—62no—
595C23—62no—
596C23—121no—
597C23—122no—
598C23—122no—
599C23—122no—
600C23D3×C22⋊C411yescharacteristic
601Q8—42no—
602Q8—42no—
603Q8—122no—
604Q8—122no—
605D4—42no—
606D4—42no—
607D4—122no—
608D4—124no—
609D4—124no—
610D4—124no—
611D4—124no—
612D4—124no—
613D4—124no—
614D4—124no—
615D4—124no—
616D4—122no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—31no—
621C2×C4—31no—
622C2×C4—34no—
623C2×C4—31no—
624C2×C4—31no—
625C2×C4—31no—
626C2×C4—31no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—124no—
645C2×C4—124no—
646C2×C4—124no—
647C2×C4—124no—
648C2×C4—122no—
649C2×C4D3×C22⋊C411yescharacteristic
650C2×C4D3×C22⋊C411yescharacteristic
651C2×C4D3×C22⋊C411yescharacteristic
652C2×C4C22×C4×D311yescharacteristic
653C2×C4C2×D3×D411yescharacteristic
654C2×C4C2×D3×D411yescharacteristic
655C8—42no—
656C8—42no—
657C8—122no—
658C8—122no—
659C2×C3—22no—
660C2×C3—82no—
661C2×C3C22×#10011yescharacteristic
662C2×C3C2×#42111yescharacteristic
663C2×C3C2×#42111yescharacteristic
664D3—22no—
665D3—41no—
666D3—82no—
667C22—21no—
668C22—21no—
669C22—32no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—61no—
674C22—61no—
675C22—61no—
676C22—61no—
677C22—61no—
678C22—84no—
679C22—124no—
680C22—122no—
681C22—122no—
682C22—122no—
683C22—244no—
684C22#482112yes—
685C22C2×D3×C22⋊C411yescharacteristic
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—32no—
692C4—32no—
693C4—42no—
694C4—61no—
695C4—61no—
696C4—61no—
697C4—61no—
698C4—122no—
699C4D3×#10012yes—
700C3#1767111yescharacteristic, complemented, Sylow
701C2—22no—
702C2—62no—
703C2—82no—
704C2—121no—
705C2—242no—
706C2C2×D3×#10011yescharacteristic
707C2C2×#482111yescharacteristic
708C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1527C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5668C2
cosocleC3×C42C24
hypercenter#1527D3
hypocenterC3#17671
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. D3×#432C2semidirect
3. #78517C2semidirect
4. C3×#1527C2semidirect
5. #71718C2semidirect
6. #70804C2semidirect
7. #69860C2semidirect
8. #68349C2semidirect
9. #68345C2semidirect
40. #4705C22semidirect
41. #4705C22semidirect
42. #4705C22semidirect
43. D3×#97C22semidirect
46. C3×#329C22semidirect
47. #4540C22semidirect
48. #4464C22semidirect
49. #4464C22semidirect
58. #1527D3semidirect
190. #329C2×D3semidirect
700. C3#17671semidirect

Maximal subgroups

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

subgroupindex
C2×#56682
D3×#4322
#785172
C3×#15272
#717182
#708042
#698602
#683492
#683452
#176713