Finite Group Databasebeta

← #72253102

Subgroups — #72253102

Order 1536.

2115 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#73104C212yesmaximal, complemented
2#11711494C214yesmaximal, complemented
3C23×C8×A4C211yesmaximal, characteristic
4#493333—31nomaximal, Sylow
5C23×C4×A4C411yescharacteristic
6C22×C8×A4—24no—
7C22×C8×A4C412yes—
8C22×C8×A4C411yescharacteristic
9C22×C8×A4C2212yes—
10C22×C8×A4C2211yescharacteristic
11#73104—24no—
12C2×A4⋊C16C2214yescomplemented
13#71963—41nomaximal
14C25×C8D311yescharacteristic
15C2×#2112—32no—
16C2×#2112—31no—
17C2×#2112—31no—
18C2×#2112—32no—
19#12679—34no—
20#12679—34no—
21C24×A4C811yescharacteristic
22C22×C4×A4—22no—
23C22×C4×A4—22no—
24C22×C4×A4C811yescharacteristic
25C22×C4×A4C812yes—
26C22×C4×A4C2×C412yes—
27C22×C4×A4C2×C411yescharacteristic
28C23×C3×C8—41no—
29C2×C8×A4—22no—
30C2×C8×A4—24no—
31C2×C8×A4—24no—
32C2×C8×A4C2×C412yes—
33C2×C8×A4C2×C411yescharacteristic
34C2×C8×A4D414yes—
35C2×C8×A4C2311yescharacteristic
36C2×#812—42no—
37A4⋊C16—24no—
38#4549—44no—
39C25×C4Dic311yescharacteristic
40C24×C8—24no—
41C24×C8—31no—
42C24×C8—31no—
43C24×C8—31no—
44C24×C8—32no—
45C24×C8—31no—
46C24×C8—32no—
47C24×C8—64no—
48C24×C8Dic311yescharacteristic
49C24×C8Dic312yes—
50C24×C8C2×D312yes—
51C24×C8C2×D311yescharacteristic
52C22×M32—32no—
53C22×M32—32no—
54#2112—64no—
55#2112—68no—
56#2112—61no—
57#2112—61no—
58#2112—62no—
59#2112—61no—
60#2112—62no—
61#2112—64no—
62#2112—61no—
63C2×C22⋊C16—38no—
64C2×C22⋊C16—34no—
65C2×C22⋊C16—34no—
66C2×C22⋊C16—34no—
67C2×C22⋊C16—34no—
68C23×A4—22no—
69C23×A4—24no—
70C23×A4C2×C811yescharacteristic
71C23×A4C2×C812yes—
72C23×C3×C4—41no—
73C2×C4×A4—22no—
74C2×C4×A4—24no—
75C2×C4×A4—22no—
76C2×C4×A4—22no—
77C2×C4×A4C22⋊C412yes—
78C2×C4×A4C2×C812yes—
79C2×C4×A4C2×C811yescharacteristic
80C2×C4×A4M1612yes—
81C2×C4×A4C22×C411yescharacteristic
82C22×C3×C8—42no—
83C22×C3×C8—41no—
84C22×C3×C8—41no—
85C22×C3×C8—42no—
86C22×C3×C8—84no—
87C8×A4—22no—
88C8×A4C22⋊C412yes—
89C8×A4C2×D412yes—
90#812—84no—
91C2×C3⋊C16—44no—
92C26C3⋊C811yescharacteristic
93C24×C4—22no—
94C24×C4—22no—
95C24×C4—31no—
96C24×C4—31no—
97C24×C4—31no—
98C24×C4—32no—
99C24×C4—31no—
100C24×C4—32no—
101C24×C4—62no—
102C24×C4—62no—
103C24×C4C3⋊C811yescharacteristic
104C24×C4C3⋊C812yes—
105C24×C4C2×Dic312yes—
106C24×C4C2×Dic311yescharacteristic
107C23×C8—22no—
108C23×C8—24no—
109C23×C8—24no—
110C23×C8—31no—
111C23×C8—31no—
112C23×C8—32no—
113C23×C8—34no—
114C23×C8—32no—
115C23×C8—31no—
116C23×C8—31no—
117C23×C8—31no—
118C23×C8—38no—
119C23×C8—31no—
120C23×C8—31no—
121C23×C8—32no—
122C23×C8—32no—
123C23×C8—31no—
124C23×C8—32no—
125C23×C8—32no—
126C23×C8—31no—
127C23×C8—32no—
128C23×C8—32no—
129C23×C8—32no—
130C23×C8—32no—
131C23×C8—32no—
132C23×C8—64no—
133C23×C8—64no—
134C23×C8—61no—
135C23×C8—61no—
136C23×C8—62no—
137C23×C8—64no—
138C23×C8—64no—
139C23×C8—61no—
140C23×C8—62no—
141C23×C8—62no—
142C23×C8—68no—
143C23×C8—64no—
144C23×C8—68no—
145C23×C8—64no—
146C23×C8—68no—
147C23×C8—64no—
148C23×C8—64no—
149C23×C8—68no—
150C23×C8—64no—
151C23×C8C2×Dic312yes—
152C23×C8C2×Dic311yescharacteristic
153C23×C8#6714yes—
154C23×C8S411yescharacteristic
155C23×C8C22×D311yescharacteristic
156C2×M32—62no—
157C2×M32—62no—
158C2×M32—64no—
159C2×M32—64no—
160C2×M32—64no—
161C2×M32—64no—
162C2×M32—62no—
163C2×M32—62no—
164C22×C16—34no—
165C22×C16—34no—
166C22⋊C16—68no—
167C22⋊C16—64no—
168C22⋊C16—616no—
169C22⋊C16—64no—
170C22⋊C16—64no—
171C22⋊C16—64no—
172C22⋊C16—64no—
173C22⋊C16—64no—
174C24×C3—41no—
175C22×A4—28no—
176C22×A4—22no—
177C22×A4—22no—
178C22×A4—22no—
179C22×A4C22⋊C812yes—
180C22×A4M3214yescomplemented
181C22×A4C22×C811yescharacteristic
182C22×C3×C4—42no—
183C22×C3×C4—41no—
184C22×C3×C4—41no—
185C22×C3×C4—42no—
186C22×C3×C4—82no—
187C22×C3×C4—82no—
188C4×A4—22no—
189C4×A4C22⋊C812yes—
190C4×A4C2×C22⋊C411yescharacteristic
191C4×A4C2×M1611yescharacteristic
192C2×C3×C8—41no—
193C2×C3×C8—42no—
194C2×C3×C8—44no—
195C2×C3×C8—41no—
196C2×C3×C8—84no—
197C2×C3×C8—84no—
198C2×C3×C8—82no—
199C3⋊C16—84no—
200C25—24no—
201C25—22no—
202C25—31no—
203C25—32no—
204C25—31no—
205C25—62no—
206C25—64no—
207C25C2×C3⋊C812yes—
208C25C2×C3⋊C811yescharacteristic
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—32no—
216C23×C4—32no—
217C23×C4—32no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4—34no—
223C23×C4—34no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—31no—
229C23×C4—32no—
230C23×C4—32no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—32no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4—32no—
237C23×C4—62no—
238C23×C4—62no—
239C23×C4—62no—
240C23×C4—62no—
241C23×C4—61no—
242C23×C4—61no—
243C23×C4—62no—
244C23×C4—62no—
245C23×C4—64no—
246C23×C4—62no—
247C23×C4—61no—
248C23×C4—62no—
249C23×C4—62no—
250C23×C4—64no—
251C23×C4—68no—
252C23×C4—64no—
253C23×C4—64no—
254C23×C4—62no—
255C23×C4—64no—
256C23×C4—62no—
257C23×C4—64no—
258C23×C4—64no—
259C23×C4—62no—
260C23×C4—62no—
261C23×C4—64no—
262C23×C4—62no—
263C23×C4—62no—
264C23×C4—64no—
265C23×C4C2×C3⋊C812yes—
266C23×C4C2×C3⋊C811yescharacteristic
267C23×C4#20812yes—
268C23×C4#21712yes—
269C23×C4SL(2,ℤ/4)11yescharacteristic
270C23×C4C22×Dic311yescharacteristic
271C22×C8—24no—
272C22×C8—22no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—34no—
277C22×C8—34no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8—32no—
282C22×C8—31no—
283C22×C8—32no—
284C22×C8—34no—
285C22×C8—34no—
286C22×C8—34no—
287C22×C8—32no—
288C22×C8—38no—
289C22×C8—34no—
290C22×C8—38no—
291C22×C8—32no—
292C22×C8—31no—
293C22×C8—32no—
294C22×C8—32no—
295C22×C8—32no—
296C22×C8—32no—
297C22×C8—32no—
298C22×C8—32no—
299C22×C8—32no—
300C22×C8—64no—
301C22×C8—64no—
302C22×C8—64no—
303C22×C8—64no—
304C22×C8—68no—
305C22×C8—68no—
306C22×C8—64no—
307C22×C8—62no—
308C22×C8—62no—
309C22×C8—62no—
310C22×C8—62no—
311C22×C8—61no—
312C22×C8—62no—
313C22×C8—62no—
314C22×C8—62no—
315C22×C8—61no—
316C22×C8—62no—
317C22×C8—64no—
318C22×C8—68no—
319C22×C8—68no—
320C22×C8—68no—
321C22×C8—64no—
322C22×C8—64no—
323C22×C8—64no—
324C22×C8—64no—
325C22×C8—64no—
326C22×C8—64no—
327C22×C8—64no—
328C22×C8—64no—
329C22×C8—64no—
330C22×C8—64no—
331C22×C8—64no—
332C22×C8—616no—
333C22×C8—616no—
334C22×C8—64no—
335C22×C8—64no—
336C22×C8—62no—
337C22×C8#21712yes—
338C22×C8SL(2,ℤ/4)11yescharacteristic
339C22×C8SL(2,ℤ/4)12yes—
340C22×C8C2×#6712yes—
341C22×C8C2×S412yes—
342C22×C8C2×S411yescharacteristic
343M32—124no—
344M32—124no—
345M32—124no—
346M32—124no—
347C2×C16—64no—
348C2×C16—64no—
349C2×C16—64no—
350C2×C16—64no—
351C2×C16—64no—
352C2×C16—64no—
353C23×C3—42no—
354C23×C3—41no—
355C23×C3—82no—
356C23×C3—84no—
357C2×A4—22no—
358C2×A4—24no—
359C2×A4C2×C22⋊C811yescharacteristic
360C2×A4C2×M3212yes—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—82no—
367C2×C3×C4—84no—
368C2×C3×C4—82no—
369C2×C3×C4—82no—
370C3×C8—42no—
371C3×C8—42no—
372C3×C8—82no—
373C24—28no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—31no—
378C24—32no—
379C24—32no—
380C24—31no—
381C24—32no—
382C24—34no—
383C24—34no—
384C24—38no—
385C24—32no—
386C24—31no—
387C24—62no—
388C24—62no—
389C24—64no—
390C24—64no—
391C24—62no—
392C24—62no—
393C24—62no—
394C24—61no—
395C24—64no—
396C24—62no—
397C24—64no—
398C24—64no—
399C24—64no—
400C24—64no—
401C24—64no—
402C24—64no—
403C24—616no—
404C24—68no—
405C24—616no—
406C24#81214yescomplemented
407C24#83012yes—
408C24A4⋊C811yescharacteristic
409C24C22×C3⋊C811yescharacteristic
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—32no—
426C22×C4—31no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—32no—
433C22×C4—31no—
434C22×C4—34no—
435C22×C4—38no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—31no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—64no—
455C22×C4—64no—
456C22×C4—64no—
457C22×C4—62no—
458C22×C4—64no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—61no—
462C22×C4—62no—
463C22×C4—61no—
464C22×C4—62no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—61no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—64no—
473C22×C4—64no—
474C22×C4—64no—
475C22×C4—64no—
476C22×C4—64no—
477C22×C4—64no—
478C22×C4—64no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—62no—
482C22×C4—64no—
483C22×C4—62no—
484C22×C4—64no—
485C22×C4—64no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—64no—
491C22×C4—64no—
492C22×C4—68no—
493C22×C4—68no—
494C22×C4—68no—
495C22×C4—68no—
496C22×C4—62no—
497C22×C4—62no—
498C22×C4—64no—
499C22×C4—62no—
500C22×C4#83012yes—
501C22×C4A4⋊C811yescharacteristic
502C22×C4A4⋊C812yes—
503C22×C4C2×#20811yescharacteristic
504C22×C4C2×#21711yescharacteristic
505C22×C4C2×SL(2,ℤ/4)12yes—
506C22×C4C2×SL(2,ℤ/4)11yescharacteristic
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—31no—
511C2×C8—31no—
512C2×C8—34no—
513C2×C8—32no—
514C2×C8—32no—
515C2×C8—34no—
516C2×C8—34no—
517C2×C8—32no—
518C2×C8—32no—
519C2×C8—32no—
520C2×C8—32no—
521C2×C8—34no—
522C2×C8—32no—
523C2×C8—64no—
524C2×C8—64no—
525C2×C8—62no—
526C2×C8—62no—
527C2×C8—64no—
528C2×C8—61no—
529C2×C8—64no—
530C2×C8—61no—
531C2×C8—64no—
532C2×C8—64no—
533C2×C8—68no—
534C2×C8—68no—
535C2×C8—64no—
536C2×C8—62no—
537C2×C8—64no—
538C2×C8—64no—
539C2×C8—62no—
540C2×C8C2×SL(2,ℤ/4)12yes—
541C2×C8C2×SL(2,ℤ/4)11yescharacteristic
542C2×C8GL(2,ℤ/4)14yes—
543C2×C8C22×S411yescharacteristic
544C16—124no—
545C16—124no—
546C22×C3—44no—
547C22×C3—42no—
548C22×C3—41no—
549C22×C3—82no—
550C22×C3—82no—
551C22×C3—82no—
552C22×C3—88no—
553A4#211211yescharacteristic, complemented
554C3×C4—41no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—82no—
558C23—24no—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—31no—
563C23—32no—
564C23—31no—
565C23—34no—
566C23—32no—
567C23—34no—
568C23—34no—
569C23—34no—
570C23—34no—
571C23—34no—
572C23—32no—
573C23—32no—
574C23—32no—
575C23—31no—
576C23—62no—
577C23—62no—
578C23—64no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23—64no—
584C23—64no—
585C23—68no—
586C23—68no—
587C23—616no—
588C23—64no—
589C23—62no—
590C23—61no—
591C23—62no—
592C23—62no—
593C23—64no—
594C23—64no—
595C23—64no—
596C23—62no—
597C23—62no—
598C23—64no—
599C23—64no—
600C23—68no—
601C23—616no—
602C23—616no—
603C23—616no—
604C23—68no—
605C23—616no—
606C23—68no—
607C23—68no—
608C23—616no—
609C23—64no—
610C23—62no—
611C23C2×#81212yes—
612C23C2×#83011yescharacteristic
613C23C2×A4⋊C812yes—
614C23C2×A4⋊C811yescharacteristic
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—32no—
620C2×C4—32no—
621C2×C4—31no—
622C2×C4—31no—
623C2×C4—31no—
624C2×C4—31no—
625C2×C4—32no—
626C2×C4—32no—
627C2×C4—32no—
628C2×C4—32no—
629C2×C4—32no—
630C2×C4—31no—
631C2×C4—31no—
632C2×C4—32no—
633C2×C4—32no—
634C2×C4—31no—
635C2×C4—31no—
636C2×C4—34no—
637C2×C4—32no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—64no—
641C2×C4—62no—
642C2×C4—62no—
643C2×C4—64no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—62no—
647C2×C4—62no—
648C2×C4—62no—
649C2×C4—62no—
650C2×C4—64no—
651C2×C4—64no—
652C2×C4—68no—
653C2×C4—64no—
654C2×C4—64no—
655C2×C4—62no—
656C2×C4—62no—
657C2×C4—64no—
658C2×C4—62no—
659C2×C4—62no—
660C2×C4C2×A4⋊C812yes—
661C2×C4C2×A4⋊C811yescharacteristic
662C2×C4#540912yes—
663C2×C4#543212yes—
664C2×C4C22×SL(2,ℤ/4)11yescharacteristic
665C8—22no—
666C8—32no—
667C8—32no—
668C8—62no—
669C8#543212yes—
670C8C2×GL(2,ℤ/4)12yes—
671C2×C3—42no—
672C2×C3—41no—
673C2×C3—84no—
674C2×C3—82no—
675C22—28no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—31no—
680C22—32no—
681C22—32no—
682C22—32no—
683C22—31no—
684C22—31no—
685C22—34no—
686C22—38no—
687C22—34no—
688C22—32no—
689C22—64no—
690C22—62no—
691C22—64no—
692C22—64no—
693C22—64no—
694C22—64no—
695C22—62no—
696C22—616no—
697C22—68no—
698C22—61no—
699C22—64no—
700C22—62no—
701C22—64no—
702C22—64no—
703C22—64no—
704C22—616no—
705C22—62no—
706C22—62no—
707C22—62no—
708C22#7196311yescharacteristic, complemented
709C22#7310414yescomplemented
710C22#7314112yes—
711C22C22×A4⋊C811yescharacteristic
712C4—22no—
713C4—31no—
714C4—31no—
715C4—32no—
716C4—62no—
717C4#7314112yes—
718C4C2×#540911yescharacteristic
719C4C2×#543211yescharacteristic
720C3—41noSylow
721C2—24no—
722C2—22no—
723C2—31no—
724C2—32no—
725C2—31no—
726C2—64no—
727C2—62no—
728C2C2×#7314111yescharacteristic
729C2C2×#7310412yes—
subgroupG/H
center, FrattiniC2×C8C22×S4
commutatorC22×A4C22×C8
FittingC25×C8D3
socleC24C22×C3⋊C8
prime radicalC24×A4C8
cosocleC2×C8×A4C23
hypercenterC23×C8S4
hypocenterA4#2112
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C8 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. C2×#73104C2semidirect
2. #11711494C2semidirect
12. C2×A4⋊C16C22semidirect
180. C22×A4M32semidirect
406. C24#812semidirect
553. A4#2112semidirect
708. C22#71963semidirect
709. C22#73104semidirect

Maximal subgroups

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

subgroupindex
C2×#731042
#117114942
C23×C8×A42
#4933333
#719634