Finite Group Databasebeta

← #764166

Subgroups — #764166

Order 512.

2690 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2580C211yesmaximal, characteristic
4C2×#2331C211yesmaximal, characteristic, complemented
5#15786C214yesmaximal, complemented
6#15769C214yesmaximal, complemented
7#15499C214yesmaximal, complemented
8#15322C214yesmaximal, complemented
9#15225C212yesmaximal, complemented
10#15141C212yesmaximal
11#14150C211yesmaximal, characteristic, complemented
12#14141C212yesmaximal
13#14006C212yesmaximal, complemented
14#13850C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2580—28no—
21#2563—28no—
22#2546—28no—
23C2×#399C2212yes—
24C2×#398C2214yes—
25C2×#396C2214yes—
26C2×#393C2212yes—
27C2×#393C2214yes—
28#2331—28no—
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C2×#385C2214yes—
43C2×#382C2212yes—
44C2×#382C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51#1446C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2214yes—
75C4⋊#95C2211yescharacteristic
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—48no—
87C2×C4⋊Q8D418yes—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123#399—28no—
124#398—216no—
125#396—48no—
126#393—216no—
127#393—28no—
128#391—28no—
129#391—216no—
130#390—28no—
131#390—28no—
132#390—24no—
133#390Q818yescomplemented
134#389—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—28no—
139#388—48no—
140#387—216no—
141#385—216no—
142#382—28no—
143#382D418yes—
144#380—28no—
145#380—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149#376—24no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25C2412yes—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—42no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8C2×D418yes—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4C2×Q814yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—232no—
214C4⋊Q8—432no—
215C4×Q8—416no—
216C4×D4—416no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×Q814yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C2×Q8116yes—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42C2×Q814yes—
300C2×C42#4114yes—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24C22×Q814yes—
338C24C2511yescharacteristic
339C2×Q8—28no—
340C2×Q8—216no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—88no—
348C2×Q8—88no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
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—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C22×D418yes—
431C22×C4C22×D418yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q812yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+412yes—
448C22×C42−1+412yes—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—28no—
472C42—48no—
473C42—48no—
474C42—44no—
475C42—48no—
476C42—88no—
477C42#121116yes—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23C23×D411yescharacteristic
502C23C23×D411yescharacteristic
503C23C23×Q811yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)12yes—
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510C23#58512yes—
511C23#58512yes—
512Q8—416no—
513Q8—816no—
514D4—416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
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—42no—
572C2×C4—42no—
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—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—816no—
595C2×C4C2×#12114yes—
596C2×C4C2×#12114yes—
597C2×C4C2×#12114yes—
598C2×C4#53314yes—
599C2×C4#54114yes—
600C2×C4#54714yes—
601C2×C4#54814yes—
602C2×C4D4×Q814yes—
603C2×C4D4×Q814yes—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22—82no—
613C22—82no—
614C22—84no—
615C22C22×#12111yescharacteristic
616C22C2×#53311yescharacteristic
617C22C2×#54111yescharacteristic
618C22C2×#54711yescharacteristic
619C22C2×#54811yescharacteristic
620C22C2×D4×Q811yescharacteristic
621C22C2×D4×Q811yescharacteristic
622C22#348411yescharacteristic
623C22#348511yescharacteristic
624C22#348711yescharacteristic
625C22#348711yescharacteristic
626C22#348812yes—
627C22#349412yes—
628C22#350012yes—
629C22#350112yes—
630C22#350412yes—
631C22#351012yes—
632C22#351412yes—
633C22#352412yes—
634C22#354712yes—
635C22#355412yes—
636C22#355712yes—
637C22#356512yes—
638C4—24no—
639C4—24no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—48no—
644C4—44no—
645C4—48no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#242918yes—
652C2—84no—
653C2C2×#242911yescharacteristic
654C2#3551411yescharacteristic
655C2#3560212yes—
656C2#3567711yescharacteristic
657C2#3735812yes—
658C2#4493811yescharacteristic
659C2#4493912yes—
660C2#4513911yescharacteristic
661C2#4520712yes—
662C2#4527011yescharacteristic
663C2#4535711yescharacteristic
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×#2546C2semidirect
4. C2×#2331C2semidirect
5. #15786C2semidirect
6. #15769C2semidirect
7. #15499C2semidirect
8. #15322C2semidirect
9. #15225C2semidirect
11. #14150C2semidirect
13. #14006C2semidirect
14. #13850C2semidirect
133. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25462
C2×#25802
C2×#23312
#157862
#157692
#154992
#153222
#152252
#151412
#141502
#141412
#140062
#138502
#137852