Finite Group Databasebeta

← #6856695

Subgroups — #6856695

Order 512.

1603 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34941C211yesmaximal, characteristic, complemented
3#33162C211yesmaximal, characteristic, complemented
4#32090C214yesmaximal, complemented
5#30886C214yesmaximal, complemented
6#29795C212yesmaximal
7#29257C212yesmaximal, complemented
8#28752C212yesmaximal, complemented
9#28748C211yesmaximal, characteristic, complemented
10#28398C212yesmaximal, complemented
11#28280C211yesmaximal, characteristic, complemented
12#27768C212yesmaximal, complemented
13#25646C212yesmaximal, complemented
14#24185C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22146C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×#488C2214yes—
20C2×#487C2212yes—
21C2×#476C2212yes—
22C2×#461C2212yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#459C2211yescharacteristic
26C2×C4×D8C2212yes—
27#2868C2211yescharacteristic
28#2858C2212yes—
29#2574—22no—
30#2567C2211yescharacteristic
31#2563C2211yescharacteristic
32#2552C2211yescharacteristic
33#2547C2212yes—
34#2545C2214yes—
35#2538—21no—
36#2538—21no—
37#2538C2211yescharacteristic
38#2512—22no—
39#2512—22no—
40#2508C2212yes—
41#2497—21no—
42#2454C2212yes—
43#2423—21no—
44#2396—22no—
45#2354—21no—
46#2354—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2212yes—
49C4×#127C2211yescharacteristic
50C4×#125C2212yes—
51C4×C22⋊Q8—21no—
52#2293C2212yes—
53#2285—21no—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2097C2214yes—
57#2065C2214yes—
58#2043C2214yes—
59#2042C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#2042C2214yes—
63#2026C2214yes—
64#2012C2214yes—
65#2012C2214yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1968C2214yes—
75#1958C2214yes—
76#1956C2212yes—
77#1948C2212yes—
78#1947C2211yescharacteristic
79#1933C2214yes—
80#1925C2212yes—
81#1916C2214yes—
82#1880C2214yes—
83#1879C2212yes—
84#1865C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#556—44no—
98#556—44no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#126—24no—
105C2×#125C2312yes—
106C2×#124C2314yes—
107C2×#124C2314yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C42—21no—
124#488—48no—
125#487—44no—
126#476—44no—
127#461—28no—
128#461—44no—
129#461—44no—
130#459—24no—
131C4×D8—44no—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#397—24no—
159#396—22no—
160#396—24no—
161#396—24no—
162#396—22no—
163#395—22no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391C2314yes—
168#390—21no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—24no—
189#388—24no—
190#387—22no—
191#387—22no—
192#387—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—22no—
203#382C2312yes—
204#380—21no—
205#380—21no—
206#380—22no—
207#380—22no—
208#380—21no—
209#380—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C43C2312yes—
230C43C2311yescharacteristic
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2414yes—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—44no—
252C2×D8—24no—
253C2×D8—42no—
254C2×D8—42no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266#127—22no—
267#127—22no—
268#126—416no—
269#125—24no—
270#125—42no—
271#124—44no—
272#124—44no—
273#124—44no—
274#124—44no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277C4×Q8—24no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—42no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—82no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—41no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4#4114yes—
294#117—48no—
295#117—84no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—44no—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—42no—
361#106—44no—
362C4⋊C8—22no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365C4⋊C8—22no—
366C4⋊C8—42no—
367#103—28no—
368#103—28no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#102—24no—
375#102—24no—
376#102—42no—
377#102—42no—
378#102—44no—
379#102—44no—
380C4×C8—24no—
381C4×C8—44no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C2×#4112yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460SD16—88no—
461D8—48no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C2×#4112yes—
507C4⋊C4H(5,𝔽2)12yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C42—24no—
523C42—22no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—22no—
532C42—41no—
533C42—42no—
534C42—42no—
535C42—48no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42#13514yes—
541C42C2×#4112yes—
542C42H(5,𝔽2)12yes—
543C42H(5,𝔽2)12yes—
544C422−1+412yes—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C23×D411yescharacteristic
560Q8—44no—
561Q8—44no—
562Q8—44no—
563Q8—44no—
564Q8—44no—
565Q8—84no—
566D4—22no—
567D4—22no—
568D4—42no—
569D4—42no—
570D4—41no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—84no—
575D4—84no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4#53512yes—
623C2×C4#53512yes—
624C2×C4#54812yes—
625C2×C4#54812yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4C2×#13512yes—
629C2×C4#57512yes—
630C2×C4#57712yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58511yescharacteristic
637C8—42no—
638C8—44no—
639C8—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C2×#53511yescharacteristic
655C22C2×#54811yescharacteristic
656C22#348811yescharacteristic
657C22C22×#13511yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—81no—
675C4—82no—
676C4#321214yes—
677C4#336014yes—
678C4#350611yescharacteristic
679C4#350811yescharacteristic
680C4#354811yescharacteristic
681C4#355411yescharacteristic
682C2—82no—
683C2—162no—
684C2#3972511yescharacteristic
685C2C2×#321211yescharacteristic
686C2#6083611yescharacteristic
687C2C2×#336011yescharacteristic
688C2#6132311yescharacteristic
689C2#6195011yescharacteristic
690C2#6198711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. #34941C2semidirect
3. #33162C2semidirect
4. #32090C2semidirect
5. #30886C2semidirect
7. #29257C2semidirect
8. #28752C2semidirect
9. #28748C2semidirect
10. #28398C2semidirect
11. #28280C2semidirect
12. #27768C2semidirect
13. #25646C2semidirect
14. #24185C2semidirect
15. #22821C2semidirect
16. #22146C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#349412
#331622
#320902
#308862
#297952
#292572
#287522
#287482
#283982
#282802
#277682
#256462
#241852
#228212
#221462
C4×#4592