Finite Group Databasebeta

← #6616996

Subgroups — #6616996

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#33094C212yesmaximal, complemented
4#32953C211yesmaximal, characteristic, complemented
5#27950C211yesmaximal, characteristic, complemented
6#27677C211yesmaximal, characteristic, complemented
7#27582C212yesmaximal, complemented
8#27486C214yesmaximal, complemented
9#27358C214yesmaximal, complemented
10#26086C212yesmaximal, complemented
11#24676C212yesmaximal, complemented
12#24500C212yesmaximal
13#23060C212yesmaximal, complemented
14#22807C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#447C211yesmaximal, characteristic
17#21358C212yesmaximal, complemented
18C2×#552—21no—
19C22×C4×D4C2211yescharacteristic
20C22×#117—21no—
21C2×#474C2212yes—
22C2×#447C2211yescharacteristic
23C2×C4×D8C2212yes—
24C2×#431C2212yes—
25#2857C2211yescharacteristic
26#2854C2212yes—
27#2552C2211yescharacteristic, complemented
28#2542C2211yescharacteristic, complemented
29#2519—22no—
30#2512C2212yescomplemented
31#2506C2212yescomplemented
32#2498—22no—
33#2489—21no—
34#2417—21no—
35#2365C2211yescharacteristic, complemented
36#2362—21no—
37#2354—21no—
38#2354—21no—
39#2354—22no—
40#2354C2212yes—
41#2354C2214yes—
42#2354C2212yes—
43#2354C2211yescharacteristic, complemented
44C4×#121C2211yescharacteristic
45C2×#390C2212yes—
46C2×#388—22no—
47#2285C2212yes—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2212yescomplemented
53#2093C2214yes—
54#2093C2212yescomplemented
55#2083C2214yescomplemented
56#2071C2214yes—
57#2067C2214yes—
58#2054C2214yes—
59#1990C2212yescomplemented
60#1970C2214yes—
61#1947C2212yescomplemented
62#1947C2214yescomplemented
63#1947C2214yescomplemented
64#1939C2212yescomplemented
65#1934C2214yes—
66#1925C2214yes—
67#1925C2214yes—
68#1925C2214yescomplemented
69#1925C2214yescomplemented
70#1918C2214yescomplemented
71#1918C2214yescomplemented
72#1916C2214yescomplemented
73#1879C2212yes—
74#1874C2212yes—
75#1874C2211yescharacteristic
76#1862C2214yes—
77#1862C2212yes—
78#1852C2212yes—
79#1852C2212yes—
80#1852C2212yes—
81#1852C2212yes—
82#1852C2211yescharacteristic
83#1845C2212yescomplemented
84#1817C2212yes—
85#1802C2212yes—
86#1802C2212yescomplemented
87#1797C2211yescharacteristic
88#1786C2212yes—
89#1786C2211yescharacteristic
90C4×#107C2214yescomplemented
91C4×#107C2214yescomplemented
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×D8C2312yes—
98#552—48no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#126—22no—
101C2×#125—21no—
102C2×#125—22no—
103C2×#123—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×#121C2311yescharacteristic
108C2×C4×Q8—21no—
109C2×C4×D4—28no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C2×#117—22no—
118C2×#117—22no—
119C2×#117—24no—
120C2×#117—21no—
121C2×#117—48no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C42—22no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#474—28no—
135#447—24no—
136C4×D8—28no—
137#431—28no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#397—22no—
162#396—22no—
163#391—21no—
164#391—21no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2314yes—
169#390C2312yes—
170#390C2311yescharacteristic
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—21no—
181#388—24no—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#386—22no—
190#386—21no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—24no—
204#380—21no—
205#380C2312yes—
206#380C2311yescharacteristic
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C25—21no—
236C22×Q8—21no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—42no—
243C22×D4C2411yescharacteristic
244C22×D4C2412yes—
245C23×C4—28no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×D8—24no—
259C2×D8—24no—
260C2×D8—42no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265#126—48no—
266#125—48no—
267#125—44no—
268#123—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272#121—22no—
273#121—22no—
274C4×Q8—44no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4#4118yes—
283#117—48no—
284#117—48no—
285#117—48no—
286#117—416no—
287#117—44no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4D818yescomplemented
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—48no—
342C2×C22⋊C4—84no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—48no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—28no—
362#107—44no—
363#107—44no—
364#107—44no—
365C4⋊C8—28no—
366C4⋊C8—24no—
367#102—24no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—44no—
372#102—42no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—22no—
376C22⋊C8#4114yes—
377C4×C8—28no—
378C4×C8—24no—
379#95—24no—
380#95—24no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—24no—
395C24—22no—
396C24—21no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C22×D411yescharacteristic
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×#4114yes—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—432no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4C2×D818yes—
461C22×C4C2×D814yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C42−1+412yes—
473C22×C4C2511yescharacteristic
474D8—48no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8C2×#4114yes—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4C2×D814yes—
514C4⋊C4C2×D814yes—
515C4⋊C4C2×D818yes—
516C4⋊C4C2×#4114yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—82no—
530C22⋊C4—81no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—48no—
543C42—48no—
544C42—44no—
545C42—44no—
546C42—42no—
547C42C2×#4114yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23#54712yes—
574C23C22×D812yes—
575C23#57812yes—
576C23C23×D411yescharacteristic
577Q8—84no—
578D4—24no—
579D4—24no—
580D4—48no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—84no—
586C2×C4—21no—
587C2×C4—216no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—416no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—416no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4#52512yes—
638C2×C4#52512yes—
639C2×C4#54712yes—
640C2×C4C22×D814yes—
641C2×C4C22×D814yes—
642C2×C4C22×D812yes—
643C2×C4C22×D812yes—
644C2×C4#57512yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4112yes—
647C2×C4C2×2−1+411yescharacteristic
648C2×C4#58512yes—
649C8—24no—
650C8—44no—
651C8—42no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—816no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22#333414yes—
674C22C2×#52511yescharacteristic
675C22C2×#54711yescharacteristic
676C22#348711yescharacteristic
677C22C23×D811yescharacteristic
678C22C2×#57511yescharacteristic
679C22C2×#57811yescharacteristic
680C22#358511yescharacteristic
681C4—24no—
682C4—24no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—42no—
688C4—44no—
689C4—41no—
690C4—81no—
691C4—81no—
692C4—82no—
693C4—81no—
694C4—82no—
695C4—84no—
696C4#310014yes—
697C4#347512yes—
698C4#349512yes—
699C2—24no—
700C2—84no—
701C2—161no—
702C2#3642711yescharacteristic
703C2C2×#310011yescharacteristic
704C2#6058911yescharacteristic
705C2C2×#333411yescharacteristic
706C2#6125611yescharacteristic
707C2#6188011yescharacteristic
708C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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
1. #36186C2semidirect
2. C2×#2354C2semidirect
3. #33094C2semidirect
4. #32953C2semidirect
5. #27950C2semidirect
6. #27677C2semidirect
7. #27582C2semidirect
8. #27486C2semidirect
9. #27358C2semidirect
10. #26086C2semidirect
11. #24676C2semidirect
13. #23060C2semidirect
14. #22807C2semidirect
15. #22599C2semidirect
17. #21358C2semidirect
27. #2552C22semidirect
28. #2542C22semidirect
30. #2512C22semidirect
31. #2506C22semidirect
35. #2365C22semidirect
43. #2354C22semidirect
52. #2099C22semidirect
54. #2093C22semidirect
55. #2083C22semidirect
59. #1990C22semidirect
61. #1947C22semidirect
62. #1947C22semidirect
63. #1947C22semidirect
64. #1939C22semidirect
68. #1925C22semidirect
69. #1925C22semidirect
70. #1918C22semidirect
71. #1918C22semidirect
72. #1916C22semidirect
83. #1845C22semidirect
86. #1802C22semidirect
90. C4×#107C22semidirect
91. C4×#107C22semidirect
92. C8×C4⋊C4C22semidirect
310. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#23542
#330942
#329532
#279502
#276772
#275822
#274862
#273582
#260862
#246762
#245002
#230602
#228072
#225992
C4×#4472
#213582