Finite Group Databasebeta

← #6848668

Subgroups — #6848668

Order 512.

1689 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32207C214yesmaximal, complemented
5#31496C214yesmaximal, complemented
6#29692C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#28597C212yesmaximal
9#28280C211yesmaximal, characteristic, complemented
10#27627C212yesmaximal, complemented
11#26447C212yesmaximal, complemented
12#25341C212yesmaximal, complemented
13#25279C212yesmaximal, complemented
14#24836C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21887C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#529C2212yes—
20C2×#486C2212yes—
21C2×#477C2211yescharacteristic
22C2×#475C2212yes—
23C2×#476C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2568—21no—
29#2554—22no—
30#2552C2211yescharacteristic
31#2548C2211yescharacteristic
32#2547—21no—
33#2547C2212yes—
34#2541C2211yescharacteristic
35#2538C2211yescharacteristic
36#2512—21no—
37#2499—22no—
38#2496—22no—
39#2496C2214yes—
40#2490C2212yes—
41#2445—21no—
42#2426C2212yes—
43#2392—22no—
44#2365—21no—
45#2365C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127—21no—
48C4×#127C2211yescharacteristic
49C4×#121C2212yes—
50#2293—21no—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2088C2214yes—
55#2077C2214yes—
56#2071C2214yes—
57#2065C2214yes—
58#2065C2214yes—
59#2059C2214yes—
60#2042C2214yes—
61#2035C2214yes—
62#2033C2214yes—
63#2033C2214yes—
64#1990C2212yes—
65#1989C2214yes—
66#1988C2212yes—
67#1987C2211yescharacteristic
68#1983C2214yes—
69#1982C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1926C2214yes—
76#1921C2214yes—
77#1921C2212yes—
78#1893C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1863C2212yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1807C2214yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127—21no—
102C2×#127C2311yescharacteristic
103#529—28no—
104#529—44no—
105C2×#126—24no—
106C2×#126—22no—
107C2×#125C2312yes—
108C2×#124—22no—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121C2312yes—
114C2×#121C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117—22no—
130C22×C42C2312yes—
131#486—44no—
132#477—24no—
133#476—44no—
134#475—44no—
135C8×D4—44no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8—44no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#400—24no—
157#398—22no—
158#397—24no—
159#397—24no—
160#397—22no—
161#396—22no—
162#396—22no—
163#394—22no—
164#394—22no—
165#394—22no—
166#392—22no—
167#391—22no—
168#391—22no—
169#391—21no—
170#391—22no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—22no—
176#390—22no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389C2312yes—
186#389C2314yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—24no—
192#387—24no—
193#386—22no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2314yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262#127—22no—
263#127—22no—
264#127—44no—
265#126—48no—
266#126—416no—
267#125—24no—
268#125—42no—
269#124—24no—
270#124—42no—
271#124—48no—
272#123—48no—
273C22⋊Q8—24no—
274C22⋊Q8—42no—
275C22⋊Q8—48no—
276#121—24no—
277#121—24no—
278#121—42no—
279#121—42no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—41no—
285C4×Q8—44no—
286C4×Q8—48no—
287C4×Q8#4114yes—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—48no—
295C4×D4—44no—
296C4×D4—82no—
297C4×D4—84no—
298#117—48no—
299#117—48no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—24no—
366#107—24no—
367#106—44no—
368#106—44no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373C4⋊C8—44no—
374C4⋊C8—42no—
375#103—24no—
376#103—42no—
377#103—44no—
378#102—44no—
379#102—44no—
380#102—44no—
381#102—44no—
382C22⋊C8—44no—
383C4×C8—24no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—24no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24C22×D412yes—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8C2×#4112yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4AGL(1,ℤ/8)14yes—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—88no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4C2×#4112yes—
511C4⋊C42−1+412yes—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42—42no—
539C42—48no—
540C42—42no—
541C42—44no—
542C42—42no—
543C42—42no—
544C42—81no—
545C42#13514yes—
546C42C2×#4112yes—
547C42H(5,𝔽2)12yes—
548C42H(5,𝔽2)12yes—
549C422−1+412yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C2×AGL(1,ℤ/8)14yes—
564C23C23×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—84no—
570Q8—88no—
571D4—28no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—84no—
577D4—88no—
578D4—88no—
579D4—84no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
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—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—88no—
629C2×C4#53712yes—
630C2×C4#53712yes—
631C2×C4#54612yes—
632C2×C4#54612yes—
633C2×C4C2×#13512yes—
634C2×C4C2×#13512yes—
635C2×C4C2×#13512yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4#57712yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C2×C4#58511yescharacteristic
644C8—42no—
645C8—44no—
646C8—42no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22C2×#53711yescharacteristic
662C22C2×#54611yescharacteristic
663C22#348911yescharacteristic
664C22C22×#13511yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C2×#57711yescharacteristic
667C22#358611yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4—81no—
683C4#324714yes—
684C4#330414yes—
685C4#350711yescharacteristic
686C4#351111yescharacteristic
687C4#354611yescharacteristic
688C4#355011yescharacteristic
689C2—44no—
690C2—162no—
691C2#3946811yescharacteristic
692C2C2×#324711yescharacteristic
693C2#6092311yescharacteristic
694C2C2×#330411yescharacteristic
695C2#6116111yescharacteristic
696C2#6210111yescharacteristic
697C2#6212811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36188C2semidirect
3. #33196C2semidirect
4. #32207C2semidirect
5. #31496C2semidirect
6. #29692C2semidirect
7. #28716C2semidirect
9. #28280C2semidirect
10. #27627C2semidirect
11. #26447C2semidirect
12. #25341C2semidirect
13. #25279C2semidirect
14. #24836C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21887C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349342
#331962
#322072
#314962
#296922
#287162
#285972
#282802
#276272
#264472
#253412
#252792
#248362
D4×C4⋊C82
#218872
C4×#4772