Finite Group Databasebeta

← #372857

Subgroups — #372857

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34814C211yesmaximal, characteristic
2#21049C211yesmaximal, characteristic
3#13753C211yesmaximal, characteristic
4#11370C211yesmaximal, characteristic, complemented
5#11209C211yesmaximal, characteristic, complemented
6#11201C211yesmaximal, characteristic
7#11201C211yesmaximal, characteristic
8#10886C211yesmaximal, characteristic, complemented
9#10879C211yesmaximal, characteristic, complemented
10#10879C211yesmaximal, characteristic, complemented
11#10840C211yesmaximal, characteristic, complemented
12#10808C211yesmaximal, characteristic, complemented
13#10719C211yesmaximal, characteristic
14#10719C211yesmaximal, characteristic
15#10701C211yesmaximal, characteristic, complemented
16C23×C42C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2497C412yescomplemented
22#2496C412yescomplemented
23#2495C412yescomplemented
24#2494C412yescomplemented
25#2493—22no—
26#2480—22no—
27#2401—22no—
28C4×#126—22no—
29#2296C2211yescharacteristic
30#2296C2211yescharacteristic
31#2285C2211yescharacteristic
32#2280C2211yescharacteristic
33#2280C2211yescharacteristic
34C4×#117C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×#376C2211yescharacteristic
38C2×#376C2211yescharacteristic
39#1924—22no—
40#1923—22no—
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1728C2211yescharacteristic
58#1728C2211yescharacteristic
59#1434—22no—
60C22⋊#95—22no—
61C4×#95C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C2×#126—22no—
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—41no—
82C22×C42—41no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#423—22no—
90#423—41no—
91C2×C4⋊C8—22no—
92#407—22no—
93#407—22no—
94#407—41no—
95#407—41no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C4×M16—22no—
99C4×M16—41no—
100C2×#97—21no—
101C2×C4×C8—21no—
102#401—22no—
103#400—22no—
104#400—22no—
105#400—22no—
106#397—22no—
107#397—22no—
108#397—22no—
109#396—22no—
110#389C2×C412yescomplemented
111#389C2×C412yescomplemented
112#389C2×C412yescomplemented
113#389C2×C412yescomplemented
114#388—22no—
115#388—22no—
116#388—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#385—22no—
121#385—22no—
122#384C2×C411yescharacteristic, complemented
123#384C2×C411yescharacteristic, complemented
124#383—22no—
125#382C2×C411yescharacteristic, complemented
126#382C2×C411yescharacteristic, complemented
127#381—21no—
128#381—21no—
129#380—22no—
130#380C2×C412yes—
131#380C2×C412yes—
132#380C2×C412yes—
133#380C2×C412yes—
134#379—21no—
135#379—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2×C412yescomplemented
141C4×C4⋊C4C2×C412yescomplemented
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4C2×C412yes—
148C4×C22⋊C4C2×C412yes—
149C4×C22⋊C4D412yes—
150C4×C22⋊C4D412yes—
151#376—21no—
152#376—22no—
153#376—22no—
154#376—41no—
155#376—41no—
156C2×#95—21no—
157C2×#95—22no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95D412yes—
162C43—21no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—22no—
167#339—22no—
168#339—22no—
169#339—22no—
170#339—22no—
171#339—22no—
172#339—22no—
173#339—22no—
174#339—22no—
175#339—41no—
176#339—41no—
177#339—41no—
178#339—41no—
179#336—21no—
180#336—22no—
181#336—21no—
182C25C2×D411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—24no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C22×C412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2411yescharacteristic
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16#4112yes—
233C2×M16#4112yes—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238#126—48no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C412yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C22⋊C412yes—
279C2×C22⋊C4C22⋊C412yes—
280C2×C22⋊C4C22⋊C412yes—
281C2×C22⋊C4C22⋊C412yes—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42C22×C412yes—
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42#4112yes—
328C2×C42#4112yes—
329C2×C42#4112yes—
330C2×C42#4112yes—
331C2×C42#4112yes—
332C2×C42#4112yes—
333C4⋊C8—44no—
334C22⋊C8—44no—
335C22⋊C8—44no—
336#97—42no—
337C4×C8—42no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95C22⋊C412yes—
356#95C22⋊C412yes—
357#95C22⋊C412yes—
358#95C22⋊C412yes—
359C24—21no—
360C24—21no—
361C24—24no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—42no—
367C24C2×C22⋊C412yes—
368C24C22×D411yescharacteristic
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C412yes—
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C42−1+411yescharacteristic
460C22×C42−1+411yescharacteristic
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16—41no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—48no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—48no—
546C42—44no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—82no—
562C42#12614yescomplemented
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23#42114yes—
584C23C22×C22⋊C411yescharacteristic
585C23#52511yescharacteristic
586C23#52511yescharacteristic
587C23#54611yescharacteristic
588C23#54611yescharacteristic
589C23#54711yescharacteristic
590C23#54711yescharacteristic
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—432no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4C4×#4111yescharacteristic
658C2×C4C4×#4111yescharacteristic
659C2×C4C4×#4111yescharacteristic
660C2×C4C4×#4111yescharacteristic
661C2×C4#52011yescharacteristic
662C2×C4#52011yescharacteristic
663C2×C4C2×#12611yescharacteristic
664C2×C4#52911yescharacteristic
665C2×C4#52911yescharacteristic
666C2×C4#52911yescharacteristic
667C2×C4#53211yescharacteristic
668C2×C4#53211yescharacteristic
669C2×C4#53811yescharacteristic
670C2×C4#53911yescharacteristic
671C2×C4#54111yescharacteristic
672C2×C4#54211yescharacteristic
673C2×C4#54611yescharacteristic
674C2×C4#54611yescharacteristic
675C2×C4#54711yescharacteristic
676C2×C4#54711yescharacteristic
677C2×C4#55111yescharacteristic
678C2×C4#55111yescharacteristic
679C2×C4#55211yescharacteristic
680C2×C4#55211yescharacteristic
681C2×C4#55311yescharacteristic
682C2×C4#55311yescharacteristic
683C2×C4#55611yescharacteristic
684C2×C4#55611yescharacteristic
685C8—42no—
686C8—42no—
687C8—81no—
688C8—81no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22#229711yescharacteristic
709C22#233911yescharacteristic
710C22#233911yescharacteristic
711C22#246611yescharacteristic
712C22#246811yescharacteristic
713C22#247811yescharacteristic
714C22#248011yescharacteristic
715C22C2×#42112yes—
716C22#290112yes—
717C4—44no—
718C4—44no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—82no—
725C4—82no—
726C4—81no—
727C4—81no—
728C4—81no—
729C4—81no—
730C4—82no—
731C4—82no—
732C4C4×#12611yescharacteristic
733C4#233411yescharacteristic
734C4#237311yescharacteristic
735C4#237311yescharacteristic
736C4#249611yescharacteristic
737C4#252111yescharacteristic
738C4#255311yescharacteristic
739C4#255911yescharacteristic
740C2—21no—
741C2—21no—
742C2—21no—
743C2—21no—
744C2—44no—
745C2#1410211yescharacteristic
746C2#2141711yescharacteristic
747C2#2141911yescharacteristic
748C2#2250612yes—
749C2#2250612yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11370C2semidirect
5. #11209C2semidirect
8. #10886C2semidirect
9. #10879C2semidirect
10. #10879C2semidirect
11. #10840C2semidirect
12. #10808C2semidirect
15. #10701C2semidirect
21. #2497C4semidirect
22. #2496C4semidirect
23. #2495C4semidirect
24. #2494C4semidirect
110. #389C2×C4semidirect
111. #389C2×C4semidirect
112. #389C2×C4semidirect
113. #389C2×C4semidirect
122. #384C2×C4semidirect
123. #384C2×C4semidirect
125. #382C2×C4semidirect
126. #382C2×C4semidirect
140. C4×C4⋊C4C2×C4semidirect
141. C4×C4⋊C4C2×C4semidirect
562. C42#126semidirect

Maximal subgroups

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

subgroupindex
#348142
#210492
#137532
#113702
#112092
#112012
#108862
#108792
#108402
#108082
#107192
#107012