Finite Group Databasebeta

← #410856

Subgroups — #410856

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3#14697C211yesmaximal, characteristic
4#11325C212yesmaximal, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11208C212yesmaximal, complemented
8#11207C211yesmaximal, characteristic, complemented
9#11207C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×#117C2212yes—
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#389C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#385—21no—
20#2285—28no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2212yes—
24C2×#380—21no—
25C2×#380—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4—21no—
30#1854—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C22⋊#95—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#117—24no—
51C2×#117—42no—
52C2×#117—42no—
53C2×#117—42no—
54C2×#117—42no—
55C2×#117D418yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—24no—
72C22×C42—24no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—42no—
82C2×#107—42no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#106—42no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8Q818yes—
89#389—22no—
90#389—22no—
91#389—24no—
92#385—44no—
93#384—24no—
94#384—24no—
95#382—24no—
96#382—24no—
97#380—28no—
98#380—28no—
99#380—22no—
100#380—24no—
101#380Q814yes—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—28no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—22no—
121#336—22no—
122#336—24no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D414yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×Q814yes—
157C23×C4C2×Q812yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C2×Q814yes—
173C22×C8C2×Q814yes—
174#117—48no—
175#117—48no—
176#117—48no—
177#117—48no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×Q812yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C42—216no—
236C2×C42—216no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—28no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42#4114yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254#107—48no—
255#106—48no—
256C22⋊C8—216no—
257#95—28no—
258#95—28no—
259#95—24no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—42no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24#12012yes—
280C24C22⋊Q812yes—
281C24C4⋊Q812yes—
282C24C22×D411yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4#12012yes—
378C22×C4C22⋊Q818yes—
379C22×C4C22⋊Q812yes—
380C22×C4#12312yes—
381C22×C4#12312yes—
382C22×C4#12312yes—
383C22×C4#12312yes—
384C22×C4C4⋊Q818yes—
385C22×C4C4⋊Q812yes—
386C22×C4#13512yes—
387C22×C4#13512yes—
388C22×C4AGL(1,ℤ/8)12yes—
389C22×C4#13712yes—
390C22×C4C22×D412yes—
391C22×C4C22×Q812yes—
392C22×C4C22×Q811yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C22⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—84no—
441C42—28no—
442C42—28no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—416no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23#39514yes—
483C23C2×#12011yescharacteristic
484C23C2×C22⋊Q811yescharacteristic
485C23C2×#12311yescharacteristic
486C23C2×#12311yescharacteristic
487C23C2×C4⋊Q811yescharacteristic
488C23#53511yescharacteristic
489C23#53511yescharacteristic
490C23C2×#13511yescharacteristic
491C23C2×#13511yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494C23#57511yescharacteristic
495C23#57511yescharacteristic
496C23#57511yescharacteristic
497C23#57511yescharacteristic
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—416no—
527C2×C4—416no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4#39514yes—
579C2×C4#48214yes—
580C2×C4#48514yes—
581C2×C4C2×C22⋊Q812yes—
582C2×C4#52512yes—
583C2×C4#52512yes—
584C2×C4C2×C4⋊Q812yes—
585C2×C4#53412yes—
586C2×C4#53412yes—
587C2×C4#55311yescharacteristic
588C2×C4#55311yescharacteristic
589C2×C4#55312yes—
590C2×C4#55412yes—
591C2×C4#55412yes—
592C2×C4#55412yes—
593C2×C4#55412yes—
594C2×C4#55411yescharacteristic
595C2×C4#55411yescharacteristic
596C2×C4#55412yes—
597C8—44no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22C2×#39511yescharacteristic
615C22#248711yescharacteristic
616C22#248711yescharacteristic
617C22#250211yescharacteristic
618C22#250211yescharacteristic
619C22#261311yescharacteristic
620C22#284911yescharacteristic
621C22#300211yescharacteristic
622C22#300211yescharacteristic
623C22#300611yescharacteristic
624C22#300711yescharacteristic
625C22#307611yescharacteristic
626C22#307611yescharacteristic
627C22#307711yescharacteristic
628C22#307811yescharacteristic
629C22C2×#48211yescharacteristic
630C22C2×#48511yescharacteristic
631C22#309111yescharacteristic
632C22#309111yescharacteristic
633C22#309311yescharacteristic
634C22#309311yescharacteristic
635C22#309411yescharacteristic
636C22#309511yescharacteristic
637C22#316011yescharacteristic
638C22#316211yescharacteristic
639C22#316511yescharacteristic
640C22#316611yescharacteristic
641C22#320711yescharacteristic
642C22#321011yescharacteristic
643C22#321211yescharacteristic
644C22#321211yescharacteristic
645C22#321411yescharacteristic
646C22#321611yescharacteristic
647C22#321911yescharacteristic
648C22#321911yescharacteristic
649C4—24no—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4—84no—
658C4—161no—
659C4—161no—
660C4#257912yes—
661C4#257912yes—
662C4#258312yes—
663C4#258312yes—
664C2—44no—
665C2#1509211yescharacteristic
666C2#2355111yescharacteristic
667C2#2355411yescharacteristic
668C2#2767911yescharacteristic
669C2#2768211yescharacteristic
670C2#2768311yescharacteristic
671C2#2768511yescharacteristic
672C2#2794911yescharacteristic
673C2#2795111yescharacteristic
674C2#2795411yescharacteristic
675C2#2795511yescharacteristic
676C2#2962811yescharacteristic
677C2#2962911yescharacteristic
678C2#3298111yescharacteristic
679C2#3298511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11325C2semidirect
5. #11324C2semidirect
6. #11323C2semidirect
7. #11208C2semidirect
8. #11207C2semidirect
9. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#18542
#146972
#113252
#113242
#113232
#112082
#112072
#107312
#107122