Finite Group Databasebeta

← #1006136

Subgroups — #1006136

Order 512.

2894 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C212yesmaximal, complemented
2#17210C212yesmaximal, complemented
3#16824C212yesmaximal, complemented
4#16404C214yesmaximal, complemented
5#15439C214yesmaximal, complemented
6#15429C212yesmaximal, complemented
7#15195C211yesmaximal, characteristic, complemented
8#15187C212yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#14572C212yesmaximal
11#14130C211yesmaximal, characteristic, complemented
12#13907C212yesmaximal, complemented
13#13872C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2212yes—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2487—216no—
21C2×#400C2214yes—
22C2×#399C2214yes—
23C2×#396C2214yes—
24C2×#392C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×#380C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42#1445C2214yes—
43#1444C2211yescharacteristic
44#1444C2212yes—
45#1443C2214yes—
46#1442C2212yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2211yescharacteristic
62#1439C2212yes—
63#1439C2212yes—
64#1438C2211yescharacteristic
65#1438C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×C4×D4—44no—
86C2×C4×D4—42no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#400—216no—
121#399—216no—
122#396—48no—
123#392Q818yes—
124#389—28no—
125#389—28no—
126#389—44no—
127#386—216no—
128#386—48no—
129#386—48no—
130#385—216no—
131#385—44no—
132#384—42no—
133#380—28no—
134#380—28no—
135#380—48no—
136#380—44no—
137#380—44no—
138#380D418yescomplemented
139C4×C22⋊C4—42no—
140C4×C22⋊C4—44no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C25C2411yescharacteristic
167C25C2412yes—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—41no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—41no—
178C22×D4C2×Q818yes—
179C22×D4C2×Q814yes—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2×D414yes—
203C23×C4#4118yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C4×D4—432no—
218C4×D4—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—432no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D418yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×Q814yes—
291C2×C22⋊C4C2×Q818yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—42no—
305#95—42no—
306#95—41no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—41no—
311#95—41no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C2×D418yes—
323#95C2×Q814yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24C22×D412yes—
345C24C22×Q812yes—
346C24C22×Q814yes—
347C24C2×#4114yes—
348C24H(5,𝔽2)14yes—
349C242−1+412yes—
350C24C2511yescharacteristic
351C2×D4—216no—
352C2×D4—216no—
353C2×D4—216no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—44no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—84no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4C22×D412yes—
455C22×C4C22×D418yes—
456C22×C4C22×D412yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q812yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+412yes—
466C4⋊C4—216no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—88no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C42—48no—
492C42—44no—
493C42—44no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23D4×Q814yes—
551C23#55418yes—
552C23C23×D411yescharacteristic
553C23C23×Q811yescharacteristic
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×2−1+411yescharacteristic
560C23C2×2−1+411yescharacteristic
561C23#58512yes—
562C23#58511yescharacteristic
563D4—416no—
564D4—432no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—416no—
607C2×C4—432no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—832no—
642C2×C4—816no—
643C2×C4#53714yes—
644C2×C4#53918yes—
645C2×C4#54618yes—
646C2×C4#55114yes—
647C2×C4#56014yes—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—28no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22C2×#53711yescharacteristic
675C22C2×#53912yes—
676C22C2×#54612yes—
677C22C2×D4×Q811yescharacteristic
678C22C2×#55111yescharacteristic
679C22C2×#55412yes—
680C22#348812yes—
681C22#349412yes—
682C22#349512yes—
683C22#349611yescharacteristic
684C22#350111yescharacteristic
685C22#350112yes—
686C22#350311yescharacteristic
687C22#350612yes—
688C22C2×#56011yescharacteristic
689C22#353612yes—
690C22#353611yescharacteristic
691C22#353712yes—
692C22#353711yescharacteristic
693C22#355112yes—
694C22#355111yescharacteristic
695C22#355912yes—
696C22#356311yescharacteristic
697C4—48no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—82no—
705C4—84no—
706C4—81no—
707C4—84no—
708C4—84no—
709C4#269318yes—
710C2—48no—
711C2—44no—
712C2#3961411yescharacteristic
713C2#3963112yes—
714C2#4295211yescharacteristic
715C2#4295612yes—
716C2#4510512yes—
717C2#4525111yescharacteristic
718C2#4633011yescharacteristic
719C2C2×#269311yescharacteristic
720C2#5298811yescharacteristic
721C2#5901612yes—
722C2#5919911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2487C2semidirect
2. #17210C2semidirect
3. #16824C2semidirect
4. #16404C2semidirect
5. #15439C2semidirect
6. #15429C2semidirect
7. #15195C2semidirect
8. #15187C2semidirect
9. #15131C2semidirect
11. #14130C2semidirect
12. #13907C2semidirect
13. #13872C2semidirect
14. #13872C2semidirect
138. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#172102
#168242
#164042
#154392
#154292
#151952
#151872
#151312
#145722
#141302
#139072
#138722
C4×#3922