Finite Group Databasebeta

← #535403

Subgroups — #535403

Order 512.

3050 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#2326C211yesmaximal, characteristic
4#14416C214yesmaximal, complemented
5#14393C214yesmaximal, complemented
6#14142C212yesmaximal, complemented
7#14039C214yesmaximal, complemented
8#14032C214yesmaximal, complemented
9#14029C212yesmaximal, complemented
10#14027C212yesmaximal
11#13942C211yesmaximal, characteristic, complemented
12#13941C211yesmaximal, characteristic
13#13854C211yesmaximal, characteristic, complemented
14#13820C212yesmaximal
15#13779C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2355—28no—
22D4×C4⋊C4—28no—
23#2326C4116yescomplemented
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2214yes—
46C2×#376C2211yescharacteristic
47#1445C2214yes—
48#1443C2214yes—
49#1442C2212yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2212yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×#124—24no—
83C2×#124—24no—
84C2×#124C2×C418yes—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8C2×C418yescomplemented
88C2×C4×D4—28no—
89C2×C4×D4—28no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—44no—
93C2×C4×D4C2×C418yes—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—48no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123#391—28no—
124#391—44no—
125#391C2×C418yescomplemented
126#390—28no—
127#390C2×C418yes—
128#389—28no—
129#388C2×C4132yescomplemented
130#387—216no—
131#387—48no—
132#385—216no—
133#385C2×C4132yescomplemented
134#384C2×C418yescomplemented
135#384D418yes—
136#382—28no—
137#382—24no—
138#380—28no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—44no—
144C4×C22⋊C4—48no—
145#376C2×C418yescomplemented
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C25C2412yes—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8C22×C414yes—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4C22×C414yes—
187C22×D4#4114yes—
188C22×D4#4114yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C22×C418yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215#124—232no—
216C4×Q8—232no—
217C4×D4—232no—
218C4×D4—432no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
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—24no—
238C2×C4⋊C4—432no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C4116yes—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C4116yes—
270C2×C22⋊C4C22×C4116yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C22×C418yes—
290C2×C42C22×C414yes—
291C2×C42C22×C414yes—
292C2×C42C22×C414yes—
293C2×C42C2×D414yes—
294C2×C42#4114yes—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C22×C418yes—
321#95C22×C418yes—
322#95C22×C4116yes—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24C23×C414yes—
332C24C2×#4114yes—
333C24C2×#4114yes—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4#117116yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—216no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C23×C414yes—
431C22×C4C23×C414yes—
432C22×C4C23×C414yes—
433C22×C4C23×C414yes—
434C22×C4C23×C414yes—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C22×D418yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+414yes—
450C22×C42−1+412yes—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C22⋊C4—232no—
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C42—28no—
471C42—216no—
472C42—28no—
473C42—48no—
474C42—48no—
475C42—88no—
476C42—88no—
477C42C4×D4116yes—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23C2×#11718yes—
501C23C24×C411yescharacteristic
502C23C23×D411yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58512yes—
512Q8—416no—
513D4—232no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—432no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—832no—
597C2×C4C2×#11714yes—
598C2×C4C2×C4×D414yes—
599C2×C4C2×C4×D414yes—
600C2×C4C2×C4×D414yes—
601C2×C4#52014yes—
602C2×C4#54614yes—
603C2×C4#54814yes—
604C2×C4#55214yes—
605C2×C4#55314yes—
606C22—28no—
607C22—216no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22C22×#11711yescharacteristic
618C22C22×C4×D411yescharacteristic
619C22C2×#52011yescharacteristic
620C22C4×H(5,𝔽2)12yes—
621C22C4×2−1+412yes—
622C22C2×#54611yescharacteristic
623C22C2×#54811yescharacteristic
624C22C2×#55211yescharacteristic
625C22C2×#55311yescharacteristic
626C22#348411yescharacteristic
627C22#348511yescharacteristic
628C22#348611yescharacteristic
629C22#348711yescharacteristic
630C22#350612yes—
631C22#350712yes—
632C22#351312yes—
633C22#351512yes—
634C22#352012yes—
635C22#352112yes—
636C22#355212yes—
637C22#355412yes—
638C22#355812yes—
639C22#356512yes—
640C4—24no—
641C4—24no—
642C4—44no—
643C4—44no—
644C4—48no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4#235218yes—
654C2—48no—
655C2C2×#235211yescharacteristic
656C2#3426012yes—
657C2#3426511yescharacteristic
658C2#3426611yescharacteristic
659C2#3442812yes—
660C2#4514011yescharacteristic
661C2#4514211yescharacteristic
662C2#4515312yes—
663C2#4515512yes—
664C2#4533511yescharacteristic
665C2#4533711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2355C2semidirect
4. #14416C2semidirect
5. #14393C2semidirect
6. #14142C2semidirect
7. #14039C2semidirect
8. #14032C2semidirect
9. #14029C2semidirect
11. #13942C2semidirect
13. #13854C2semidirect
15. #13779C2semidirect
23. #2326C4semidirect
87. C2×C4×Q8C2×C4semidirect
125. #391C2×C4semidirect
129. #388C2×C4semidirect
133. #385C2×C4semidirect
134. #384C2×C4semidirect
145. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×D4×C4⋊C42
C2×#23262
#144162
#143932
#141422
#140392
#140322
#140292
#140272
#139422
#139412
#138542
#138202
#137792