Finite Group Databasebeta

← #6792848

Subgroups — #6792848

Order 512.

1849 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36126C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic, complemented
4#31570C214yesmaximal, complemented
5#31454C214yesmaximal, complemented
6#29906C212yesmaximal, complemented
7#28789C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic, complemented
9#27747C212yesmaximal, complemented
10#27677C211yesmaximal, characteristic, complemented
11#26526C212yesmaximal, complemented
12#26106C212yesmaximal, complemented
13#24600C212yesmaximal, complemented
14#24060C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21859C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C22×#125—21no—
21C2×#480C2211yescharacteristic
22C2×#480C2212yes—
23C2×#431C2212yes—
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2552—21no—
27#2552—22no—
28#2552—21no—
29#2552C2214yes—
30#2552C2212yes—
31#2552C2212yes—
32#2551C2211yescharacteristic
33#2541C2211yescharacteristic
34#2491—21no—
35#2461—22no—
36#2452C2212yes—
37#2445C2212yes—
38#2430—22no—
39#2416—21no—
40C2×#398—22no—
41#2366C2211yescharacteristic
42#2361—21no—
43D4×C4⋊C4C2211yescharacteristic
44C4×#121C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2099C2212yes—
52#2099C2214yes—
53#2093C2212yes—
54#2093C2214yes—
55#2086C2214yes—
56#2083C2214yes—
57#2071C2214yes—
58#2071C2214yes—
59#2067C2214yes—
60#2057C2214yes—
61#2041C2214yes—
62#1990C2214yes—
63#1982C2212yes—
64#1957C2212yes—
65#1957C2214yes—
66#1947C2212yes—
67#1947C2212yes—
68#1942C2214yes—
69#1939C2212yes—
70#1934C2214yes—
71#1925C2212yes—
72#1925C2214yes—
73#1918C2212yes—
74#1916C2214yes—
75#1874C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1874C2212yes—
79#1874C2212yes—
80#1862C2214yes—
81#1862C2214yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1834C2212yes—
85#1823C2211yescharacteristic
86#1817C2212yes—
87#1802C2212yes—
88#1802C2214yes—
89#1802C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—22no—
96#547—48no—
97#532—28no—
98#532—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#127C2312yes—
103C2×#125—22no—
104C2×#125—22no—
105C2×#125—24no—
106C2×#125—48no—
107C2×#124—22no—
108C2×#123—22no—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×#121C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#480—24no—
139#480—44no—
140#431—44no—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#398—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#396—22no—
164#395—22no—
165#394—22no—
166#394—22no—
167#393—22no—
168#392—22no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2311yescharacteristic
176#389—21no—
177#389—22no—
178#389—24no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#389C2312yes—
185#389C2312yes—
186#389C2311yescharacteristic
187#388—22no—
188#388—22no—
189#388—21no—
190#386—22no—
191#386—21no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385—22no—
196#384—22no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384C2314yes—
201#384C2312yes—
202#384C2312yes—
203#384C2314yes—
204#384C2311yescharacteristic
205#380—21no—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C2×#95—21no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C2×#41—48no—
236C2×#41—48no—
237C2×#41—84no—
238C22×Q8—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C2412yes—
246C22×D4C2411yescharacteristic
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264#127—24no—
265#127—24no—
266#127—42no—
267#125—48no—
268#125—48no—
269#125—416no—
270#124—48no—
271#123—48no—
272#123—48no—
273C22⋊Q8—48no—
274#121—22no—
275#121—24no—
276#121—24no—
277#121—48no—
278#121—44no—
279#121—44no—
280#121—44no—
281#121—82no—
282#121#4114yes—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—48no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—44no—
290C4×D4—44no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—84no—
321C2×C4⋊C4D818yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—44no—
366#107—42no—
367#107—44no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370#102—24no—
371#102—44no—
372#102—44no—
373#102—44no—
374#102—42no—
375#102—44no—
376#102—44no—
377C22⋊C8—22no—
378C22⋊C8—22no—
379C22⋊C8—24no—
380C4×C8—24no—
381C4×C8—44no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—22no—
397C24C22×D411yescharacteristic
398#41—832no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—82no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4C2×#4112yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C2×D814yes—
466C22×C4C2×D818yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4C2511yescharacteristic
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—84no—
516C4⋊C4—82no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4C2×D814yes—
521C4⋊C4C2×D814yes—
522C4⋊C4C2×D818yes—
523C4⋊C4#13714yes—
524C4⋊C4C2×#4112yes—
525C22⋊C4—24no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—81no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42—44no—
550C42—42no—
551C422−1+414yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23#54612yes—
571C23C22×D812yes—
572C23#57712yes—
573C23C23×D411yescharacteristic
574Q8—88no—
575D4—24no—
576D4—48no—
577D4—44no—
578D4—42no—
579D4—44no—
580D4—88no—
581D4—88no—
582D4—88no—
583D4—84no—
584D4—162no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—216no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4#53612yes—
637C2×C4#53612yes—
638C2×C4#54612yes—
639C2×C4C22×D812yes—
640C2×C4C22×D814yes—
641C2×C4C22×D814yes—
642C2×C4C22×D812yes—
643C2×C4C2×#13714yes—
644C2×C4C2×#13712yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+412yes—
648C2×C4#58512yes—
649C8—42no—
650C8—42no—
651C8—44no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#329414yes—
673C22C2×#53611yescharacteristic
674C22C2×#54611yescharacteristic
675C22#348711yescharacteristic
676C22C23×D811yescharacteristic
677C22C22×#13711yescharacteristic
678C22C2×#57711yescharacteristic
679C22#358511yescharacteristic
680C4—22no—
681C4—24no—
682C4—24no—
683C4—22no—
684C4—24no—
685C4—22no—
686C4—42no—
687C4—41no—
688C4—42no—
689C4—81no—
690C4—82no—
691C4—82no—
692C4—81no—
693C4—82no—
694C4—84no—
695C4#322614yes—
696C4#351512yes—
697C4#355912yes—
698C2—24no—
699C2—82no—
700C2—162no—
701C2#3890711yescharacteristic
702C2C2×#322611yescharacteristic
703C2#6087311yescharacteristic
704C2C2×#329411yescharacteristic
705C2#6116211yescharacteristic
706C2#6188011yescharacteristic
707C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36126C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #33106C2semidirect
4. #31570C2semidirect
5. #31454C2semidirect
6. #29906C2semidirect
7. #28789C2semidirect
8. #27973C2semidirect
9. #27747C2semidirect
10. #27677C2semidirect
11. #26526C2semidirect
12. #26106C2semidirect
13. #24600C2semidirect
14. #24060C2semidirect
15. #22599C2semidirect
16. #21859C2semidirect
17. C4×#480C2semidirect

Maximal subgroups

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

subgroupindex
#361262
C2×#25522
#331062
#315702
#314542
#299062
#287892
#279732
#277472
#276772
#265262
#261062
#246002
#240602
#225992
#218592
C4×#4802