Finite Group Databasebeta

← #6852154

Subgroups — #6852154

Order 512.

1635 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36544C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32689C214yesmaximal, complemented
5#32580C214yesmaximal, complemented
6#29973C212yesmaximal, complemented
7#29303C212yesmaximal, complemented
8#29234C212yesmaximal
9#28682C212yesmaximal, complemented
10#26279C212yesmaximal, complemented
11#25839C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic, complemented
13#25477C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic, complemented
16#22082C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#500C2212yes—
22C2×#481C2211yescharacteristic
23C2×#476C2212yes—
24C2×#446C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2584C2212yes—
28#2580C2212yes—
29#2567—21no—
30#2567—21no—
31#2567—22no—
32#2567C2211yescharacteristic
33#2567C2212yes—
34#2567C2214yes—
35#2567C2212yes—
36#2538C2211yescharacteristic
37#2465—22no—
38#2449—22no—
39#2423—21no—
40#2409—21no—
41C2×#395—22no—
42#2360—21no—
43#2355C2211yescharacteristic
44D4×C4⋊C4—21no—
45C4×#121C2212yes—
46C4×#121C2211yescharacteristic
47C2×#391—21no—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#384—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yes—
53#2088C2214yes—
54#2086C2214yes—
55#2086C2214yes—
56#2085C2214yes—
57#2067C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2058C2212yes—
61#2057C2214yes—
62#2023C2212yes—
63#1990C2212yes—
64#1987C2212yes—
65#1987C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1947C2212yes—
69#1944C2214yes—
70#1944C2214yes—
71#1943C2214yes—
72#1936C2214yes—
73#1926C2212yes—
74#1921C2214yes—
75#1919C2214yes—
76#1919C2212yes—
77#1918C2214yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1862C2214yes—
81#1862C2214yes—
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1852C2212yes—
85#1852C2212yes—
86#1852C2212yes—
87#1824C2211yescharacteristic
88#1824C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97#554—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×#121C2311yescharacteristic
110C2×C4×Q8—44no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D412yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D412yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#500—44no—
134#481Q818yescomplemented
135#476—44no—
136#446—44no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#399—22no—
154#398—22no—
155#396—22no—
156#395—22no—
157#395—24no—
158#395—24no—
159#395—24no—
160#395—24no—
161#395—22no—
162#393—22no—
163#391—22no—
164#391—22no—
165#391—24no—
166#391—21no—
167#391—21no—
168#391—21no—
169#390C2312yes—
170#390C2311yescharacteristic
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2311yescharacteristic
186#389C2312yes—
187#388—22no—
188#388—21no—
189#387—22no—
190#387—22no—
191#387—22no—
192#386—22no—
193#386—21no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#384—22no—
199#384—22no—
200#384—24no—
201#384—21no—
202#384C2311yescharacteristic
203#382C2312yes—
204#380—21no—
205#380—21no—
206#380C2311yescharacteristic
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—24no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8—44no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×D4—21no—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268#125—44no—
269C22⋊Q8—48no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—82no—
273#121—22no—
274#121C2×Q814yes—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—42no—
280C4×Q8—42no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—84no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×Q814yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C42—24no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—44no—
355#107—44no—
356#106—24no—
357#106—44no—
358#106C2×Q818yescomplemented
359C4⋊C8—24no—
360C4⋊C8—44no—
361#102—24no—
362#102—44no—
363#102C2×Q818yescomplemented
364C22⋊C8—22no—
365C22⋊C8—24no—
366C22⋊C8C2×Q814yes—
367#97—44no—
368C4×C8—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386C24—21no—
387C24C22×D411yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—416no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8—816no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C22×Q812yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4#13714yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×Q812yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8C22×Q814yes—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4C22×Q814yes—
509C4⋊C4C22×Q812yes—
510C4⋊C42−1+414yes—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—82no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—42no—
529C42—44no—
530C42—82no—
531C42H(5,𝔽2)14yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23D4×Q812yes—
544C23C2×#13712yes—
545C23#57612yes—
546C23C23×D411yescharacteristic
547Q8—48no—
548Q8—44no—
549Q8—44no—
550Q8—88no—
551Q8—88no—
552Q8—162no—
553D4—24no—
554D4—48no—
555D4—42no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—84no—
609C2×C4—88no—
610C2×C4—84no—
611C2×C4—164no—
612C2×C4#53412yes—
613C2×C4#53412yes—
614C2×C4D4×Q812yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4#57512yes—
618C2×C4C23×Q811yescharacteristic
619C2×C4C2×H(5,𝔽2)12yes—
620C2×C4C2×2−1+412yes—
621C2×C4C2×2−1+411yescharacteristic
622C8—24no—
623C8—42no—
624C8—44no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#338514yes—
639C22C2×#53411yescharacteristic
640C22C2×D4×Q811yescharacteristic
641C22#348611yescharacteristic
642C22C22×#13711yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57611yescharacteristic
645C22#358511yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—81no—
658C4—84no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4#319614yes—
663C4#350312yes—
664C4#355212yes—
665C2—24no—
666C2—82no—
667C2#3959311yescharacteristic
668C2C2×#319611yescharacteristic
669C2#6075611yescharacteristic
670C2C2×#338511yescharacteristic
671C2#6136711yescharacteristic
672C2#6183011yescharacteristic
673C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. #33149C2semidirect
4. #32689C2semidirect
5. #32580C2semidirect
6. #29973C2semidirect
7. #29303C2semidirect
9. #28682C2semidirect
10. #26279C2semidirect
11. #25839C2semidirect
12. #25813C2semidirect
13. #25477C2semidirect
15. #22510C2semidirect
134. #481Q8semidirect
358. #106C2×Q8semidirect
363. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365442
C2×#25672
#331492
#326892
#325802
#299732
#293032
#292342
#286822
#262792
#258392
#258132
#254772
#242992
#225102
#220822
C4×#4812