Finite Group Databasebeta

← #988885

Subgroups — #988885

Order 512.

2978 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C212yesmaximal
2C2×#2440C212yesmaximal, complemented
3#17112C212yesmaximal, complemented
4#16877C212yesmaximal, complemented
5#16321C214yesmaximal, complemented
6#15951C212yesmaximal
7#15430C214yesmaximal, complemented
8#15183C212yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15128C211yesmaximal, characteristic, complemented
11#14535C212yesmaximal
12#13993C212yesmaximal, complemented
13#13910C211yesmaximal, characteristic, complemented
14#13880C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20#2487—216no—
21#2440—216no—
22C2×#400C2214yes—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#395C2214yes—
27C2×#395C2212yes—
28C2×#394C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48#1446C2212yes—
49#1445C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2211yescharacteristic
60#1441C2212yes—
61#1441C2214yes—
62#1441C2211yescharacteristic
63#1440C2212yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2211yescharacteristic
69#1437C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2314yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—42no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—48no—
85C2×C22⋊Q8—48no—
86C2×C4×D4—28no—
87C2×C4×D4—42no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#400—216no—
122#399—216no—
123#398—216no—
124#398—48no—
125#395—216no—
126#395—44no—
127#394Q818yescomplemented
128#389—28no—
129#389—28no—
130#389—28no—
131#389—28no—
132#386—216no—
133#385—216no—
134#385—216no—
135#385—48no—
136#385—48no—
137#384—28no—
138#384—42no—
139#384—44no—
140#380—28no—
141#380—28no—
142#380D4116yes—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—42no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—28no—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C25C2412yes—
173C22×Q8—28no—
174C22×Q8—24no—
175C22×Q8—44no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4C2×Q814yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D418yes—
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C4⋊Q8—416no—
212C22⋊Q8—464no—
213C4×D4—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—432no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×D4116yes—
285C2×C22⋊C4C2×Q818yes—
286C2×C22⋊C4C2×Q818yes—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42#4114yes—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—41no—
315#95—44no—
316#95—42no—
317#95—41no—
318#95—41no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95C2×D4116yes—
324#95C2×Q814yes—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24C22×D414yes—
338C24C22×Q814yes—
339C242−1+414yes—
340C24C2511yescharacteristic
341C2×Q8—416no—
342C2×Q8—48no—
343C2×Q8—88no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×D4—216no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—44no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4C22×D414yes—
435C22×C4C22×D418yes—
436C22×C4C22×D418yes—
437C22×C4C22×D414yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q812yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+412yes—
450C22×C42−1+414yes—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—416no—
456C4⋊C4—44no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C4⋊C4—84no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—88no—
476C42—28no—
477C42—48no—
478C42—44no—
479C42—44no—
480C42—48no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23D4×Q818yes—
516C23C23×D412yes—
517C23C23×Q811yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+412yes—
525C23C2×2−1+412yes—
526C23#58511yescharacteristic
527Q8—816no—
528D4—416no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—432no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—832no—
611C2×C4#53618yes—
612C2×C4#53914yes—
613C2×C4D4214yes—
614C2×C4#54718yes—
615C2×C4#55414yes—
616C2×C4#56314yes—
617C22—28no—
618C22—28no—
619C22—28no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22C2×#53612yes—
633C22C2×#53911yescharacteristic
634C22C2×D4211yescharacteristic
635C22C2×#54712yes—
636C22C2×D4×Q812yes—
637C22C2×#55411yescharacteristic
638C22#348412yes—
639C22#348912yes—
640C22#349212yes—
641C22#349812yes—
642C22#350111yescharacteristic
643C22#350312yes—
644C22#350311yescharacteristic
645C22#350611yescharacteristic
646C22C2×#56311yescharacteristic
647C22#353711yescharacteristic
648C22#353811yescharacteristic
649C22#354911yescharacteristic
650C22#355212yes—
651C22#355212yes—
652C22#355212yes—
653C22#355912yes—
654C22#355911yescharacteristic
655C4—24no—
656C4—42no—
657C4—48no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—48no—
664C4—44no—
665C4—81no—
666C4—84no—
667C4—84no—
668C4#268018yes—
669C2—48no—
670C2#3959212yes—
671C2#3963111yescharacteristic
672C2#4250211yescharacteristic
673C2#4291412yes—
674C2#4487011yescharacteristic
675C2#4525512yes—
676C2#4615211yescharacteristic
677C2C2×#268011yescharacteristic
678C2#5298711yescharacteristic
679C2#5899811yescharacteristic
680C2#5931412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2440C2semidirect
3. #17112C2semidirect
4. #16877C2semidirect
5. #16321C2semidirect
7. #15430C2semidirect
8. #15183C2semidirect
9. #15137C2semidirect
10. #15128C2semidirect
12. #13993C2semidirect
13. #13910C2semidirect
14. #13880C2semidirect
127. #394Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#24402
#171122
#168772
#163212
#159512
#154302
#151832
#151372
#151282
#145352
#139932
#139102
#138802
C4×#3942