Finite Group Databasebeta

← #398478

Subgroups — #398478

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1944C212yesmaximal, complemented
4C2×#1749C211yesmaximal, characteristic
5#14697C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11207C211yesmaximal, characteristic, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10881C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2212yes—
19#2857—28no—
20C2×#391—21no—
21C2×#389—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#385—21no—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2057—24no—
32#1944—28no—
33#1749—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C22⋊#95—21no—
47C23×Q8—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—44no—
59C2×C22⋊Q8—44no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—24no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—42no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8Q818yescomplemented
92C2×#97—24no—
93C2×#97—24no—
94C2×#97—42no—
95#391—28no—
96#389—28no—
97#389—24no—
98#389—22no—
99#389—24no—
100#389—22no—
101#389D418yes—
102#385—44no—
103#384—24no—
104#380—28no—
105#380—24no—
106#380—24no—
107#380—22no—
108#380—22no—
109#380—42no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#336—22no—
127#336D414yes—
128C25C2×D411yescharacteristic
129C22×Q8—24no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—41no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q814yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C2×D414yes—
183C22×C8C2×Q814yes—
184C22×C8C2×Q814yes—
185C22×C8#4114yes—
186C4⋊Q8—48no—
187C4⋊Q8—48no—
188C22⋊Q8—432no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4#4112yes—
236C2×C4⋊C4#4112yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—48no—
256C2×C42—48no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42C2×D414yes—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262#107—48no—
263#106—216no—
264C22⋊C8—216no—
265#97—48no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24C4⋊Q812yes—
289C24AGL(1,ℤ/8)12yes—
290C24#13712yes—
291C24C22×D411yescharacteristic
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—21no—
310C22×C4—21no—
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—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4#12012yes—
398C22×C4#12012yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4C4⋊Q812yes—
402C22×C4AGL(1,ℤ/8)12yes—
403C22×C4AGL(1,ℤ/8)12yes—
404C22×C4#13712yes—
405C22×C4#13712yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×Q812yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+411yescharacteristic
415C22×C42−1+411yescharacteristic
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8C22⋊Q818yescomplemented
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C22⋊C4—48no—
455C22⋊C4—416no—
456C22⋊C4—84no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—44no—
462C42—44no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#50114yes—
498C23C2×#12011yescharacteristic
499C23C2×#12111yescharacteristic
500C23#52511yescharacteristic
501C23C2×C4⋊Q811yescharacteristic
502C23#53211yescharacteristic
503C23#53411yescharacteristic
504C23#53611yescharacteristic
505C23C2×AGL(1,ℤ/8)11yescharacteristic
506C23C2×AGL(1,ℤ/8)11yescharacteristic
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×#13711yescharacteristic
509C23C2×#13711yescharacteristic
510C23C2×#13711yescharacteristic
511C23#57511yescharacteristic
512C23#57511yescharacteristic
513Q8—816no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4#46914yes—
596C2×C4#47014yes—
597C2×C4C2×C22⋊Q812yes—
598C2×C4#52512yes—
599C2×C4#53212yes—
600C2×C4#53312yes—
601C2×C4#53912yes—
602C2×C4#54212yes—
603C2×C4D4212yes—
604C2×C4#54612yes—
605C2×C4#54711yescharacteristic
606C2×C4#54811yescharacteristic
607C2×C4D4×Q812yes—
608C2×C4D4×Q812yes—
609C2×C4#55211yescharacteristic
610C2×C4#55412yes—
611C2×C4#55412yes—
612C2×C4#55611yescharacteristic
613C8—28no—
614C8—44no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22#238811yescharacteristic
631C22#241011yescharacteristic
632C22#241511yescharacteristic
633C22#243411yescharacteristic
634C22#258311yescharacteristic
635C22#266311yescharacteristic
636C22#285711yescharacteristic
637C22#300311yescharacteristic
638C22#300411yescharacteristic
639C22#300611yescharacteristic
640C22#300711yescharacteristic
641C22C2×#46911yescharacteristic
642C22C2×#47011yescharacteristic
643C22#305511yescharacteristic
644C22#305611yescharacteristic
645C22#311011yescharacteristic
646C22#311211yescharacteristic
647C22#311511yescharacteristic
648C22#311611yescharacteristic
649C22C2×#50111yescharacteristic
650C22#316211yescharacteristic
651C22#316511yescharacteristic
652C22#316511yescharacteristic
653C22#318011yescharacteristic
654C22#318111yescharacteristic
655C22#318211yescharacteristic
656C22#318211yescharacteristic
657C22#319211yescharacteristic
658C22#319411yescharacteristic
659C22#319511yescharacteristic
660C22#319611yescharacteristic
661C22#323511yescharacteristic
662C22#323711yescharacteristic
663C22#323811yescharacteristic
664C22#323911yescharacteristic
665C4—24no—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—84no—
672C4—84no—
673C4—82no—
674C4—161no—
675C4—161no—
676C4#244912yes—
677C4#245212yes—
678C4#246512yes—
679C4#255612yes—
680C2—44no—
681C2#1476311yescharacteristic
682C2#2357411yescharacteristic
683C2#2357611yescharacteristic
684C2#2438411yescharacteristic
685C2#2438611yescharacteristic
686C2#2470911yescharacteristic
687C2#2471111yescharacteristic
688C2#2581611yescharacteristic
689C2#2581811yescharacteristic
690C2#2930211yescharacteristic
691C2#2930411yescharacteristic
692C2#3036411yescharacteristic
693C2#3036611yescharacteristic
694C2#3321211yescharacteristic
695C2#3321711yescharacteristic
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
2. C2×#2057C2semidirect
3. C2×#1944C2semidirect
6. #11384C2semidirect
7. #11323C2semidirect
8. #11207C2semidirect
11. #10881C2semidirect
91. C2×C22⋊C8Q8semidirect
429. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20572
C2×#19442
C2×#17492
#146972
#113842
#113232
#112072
#110962
#110952
#108812