Finite Group Databasebeta

← #803727

Subgroups — #803727

Order 512.

2738 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×#2423C212yesmaximal
2C2×#2360C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#17061C212yesmaximal, complemented
5#15347C214yesmaximal, complemented
6#15294C214yesmaximal
7#15195C211yesmaximal, characteristic
8#15194C211yesmaximal, characteristic, complemented
9#15145C211yesmaximal, characteristic, complemented
10#15135C212yesmaximal
11#15083C212yesmaximal, complemented
12#14225C212yesmaximal
13#14005C212yesmaximal, complemented
14#13935C211yesmaximal, characteristic, complemented
15#13931C212yesmaximal, complemented
16#13907C212yesmaximal
17C4×#397C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2423—216no—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#397C2211yescharacteristic
27#2360—28no—
28Q8×C4⋊C4—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#388C2214yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#385C2212yes—
42C2×#384C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53#1445C2214yes—
54#1445C2211yescharacteristic
55#1445C2212yes—
56#1445C2214yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2214yes—
61#1443C2212yes—
62#1442C2211yescharacteristic
63#1442C2212yes—
64#1442C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2212yes—
71#1439C2211yescharacteristic
72#1439C2212yes—
73#1439C2212yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8D418yes—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8D418yes—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131#399—216no—
132#398—216no—
133#398—48no—
134#397Q818yescomplemented
135#391—24no—
136#391—28no—
137#389—216no—
138#389—28no—
139#389—28no—
140#389—28no—
141#389—28no—
142#389—24no—
143#389—48no—
144#388—216no—
145#387—216no—
146#387—28no—
147#385—28no—
148#384—28no—
149#384—28no—
150#384—24no—
151#380—28no—
152#380—48no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—28no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C25C2411yescharacteristic
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8C2×D414yes—
193C22×Q8C2×D414yes—
194C22×Q8#4114yes—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4C2×D414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C22⋊Q8—232no—
222C4×Q8—232no—
223C4×Q8—432no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—416no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4#4118yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×Q818yes—
291C2×C22⋊C4C2×Q814yes—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—44no—
307C2×C42C2×D418yes—
308C2×C42C2×D414yes—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C2×Q818yes—
338#95C2×Q814yes—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24C22×D412yes—
348C24C22×Q812yes—
349C242−1+412yes—
350C24C2511yescharacteristic
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—216no—
354C2×Q8—216no—
355C2×Q8#121116yes—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×Q814yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C42−1+412yes—
457C22×C42−1+414yes—
458C4⋊C4—28no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—28no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4#123116yes—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—88no—
483C42—216no—
484C42—28no—
485C42—416no—
486C42—48no—
487C42—44no—
488C42—48no—
489C42—84no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23D4×Q814yes—
513C23C23×D411yescharacteristic
514C23C23×D411yescharacteristic
515C23C23×Q811yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)12yes—
519C23C2×2−1+412yes—
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23#58512yes—
523C23#58511yescharacteristic
524Q8—232no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—816no—
611C2×C4C2×#12118yes—
612C2×C4C2×#12114yes—
613C2×C4C2×#12318yes—
614C2×C4C2×#12314yes—
615C2×C4#53614yes—
616C2×C4#54718yes—
617C2×C4D4×Q814yes—
618C2×C4#55218yes—
619C22—24no—
620C22—28no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22C22×#12111yescharacteristic
631C22C22×#12311yescharacteristic
632C22C2×#53611yescharacteristic
633C22C2×#54712yes—
634C22C2×D4×Q811yescharacteristic
635C22C2×D4×Q811yescharacteristic
636C22C2×#55212yes—
637C22Q8×#4111yescharacteristic
638C22Q8×#4112yes—
639C22#348711yescharacteristic
640C22#348812yes—
641C22#348911yescharacteristic
642C22#349711yescharacteristic
643C22#349811yescharacteristic
644C22#350312yes—
645C22#351312yes—
646C22#351612yes—
647C22#351712yes—
648C22#352012yes—
649C22#352311yescharacteristic
650C22#354412yes—
651C22#356312yes—
652C22#356512yes—
653C4—28no—
654C4—24no—
655C4—28no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—84no—
668C4#248618yes—
669C2—44no—
670C2C2×#248611yescharacteristic
671C2#3598711yescharacteristic
672C2#3602911yescharacteristic
673C2Q8×#12111yescharacteristic
674C2Q8×#12311yescharacteristic
675C2#3961711yescharacteristic
676C2#4288811yescharacteristic
677C2#4501612yes—
678C2#4508012yes—
679C2#5911312yes—
680C2#5949912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×Q8×C4⋊C4C2semidirect
4. #17061C2semidirect
5. #15347C2semidirect
8. #15194C2semidirect
9. #15145C2semidirect
11. #15083C2semidirect
13. #14005C2semidirect
14. #13935C2semidirect
15. #13931C2semidirect
134. #397Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24232
C2×#23602
C2×Q8×C4⋊C42
#170612
#153472
#152942
#151952
#151942
#151452
#151352
#150832
#142252
#140052
#139352
#139312
#139072
C4×#3972