Finite Group Databasebeta

← #6800513

Subgroups — #6800513

Order 512.

1649 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36544C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#33148C211yesmaximal, characteristic, complemented
4#32634C214yesmaximal, complemented
5#32604C214yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#29106C212yesmaximal, complemented
8#28638C212yesmaximal, complemented
9#28118C212yesmaximal
10#26317C212yesmaximal, complemented
11#25934C212yesmaximal, complemented
12#25856C211yesmaximal, characteristic, complemented
13#25293C212yesmaximal, complemented
14#24209C211yesmaximal, characteristic
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2212yes—
22C2×#480C2211yescharacteristic
23C2×#461C2212yes—
24C2×#448C2212yes—
25C2×C8×Q8C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2584C2212yes—
29#2580C2212yes—
30#2568—21no—
31#2568—21no—
32#2568—22no—
33#2568C2212yes—
34#2568C2214yes—
35#2568C2212yes—
36#2567C2211yescharacteristic
37#2538C2211yescharacteristic
38#2465—22no—
39#2449—22no—
40#2423—21no—
41#2409—21no—
42C2×#400—22no—
43#2360—21no—
44#2355C2211yescharacteristic
45D4×C4⋊C4—21no—
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391—21no—
49C2×#391C2212yes—
50C2×#391C2212yes—
51C2×#382—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2074C2214yes—
55#2068C2214yes—
56#2068C2214yes—
57#2066C2214yes—
58#2055C2214yes—
59#2043C2214yes—
60#2043C2212yes—
61#2026C2214yes—
62#2026C2214yes—
63#2013C2212yes—
64#1987C2212yes—
65#1984C2212yes—
66#1984C2214yes—
67#1981C2212yes—
68#1968C2214yes—
69#1956C2212yes—
70#1948C2212yes—
71#1933C2214yes—
72#1933C2214yes—
73#1933C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1925C2212yes—
77#1920C2214yes—
78#1918C2212yes—
79#1877C2214yes—
80#1877C2214yes—
81#1874C2211yescharacteristic
82#1853C2212yes—
83#1852C2211yescharacteristic
84#1823C2211yescharacteristic
85#1823C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89#1787C2212yes—
90#1787C2212yes—
91C4×#107C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8D411yescharacteristic
96C23×Q8C2312yes—
97C22×SD16C2314yes—
98#554—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×#121C2311yescharacteristic
111C2×C4×Q8—44no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D412yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D412yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#487—44no—
133#480—24no—
134#461—44no—
135#448—28no—
136C8×Q8—44no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#400—24no—
157#400—24no—
158#400—24no—
159#400—24no—
160#399—22no—
161#398—22no—
162#396—22no—
163#395—22no—
164#395—22no—
165#393—22no—
166#391—21no—
167#391—22no—
168#391—24no—
169#391—22no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—21no—
176#391C2314yes—
177#391C2314yes—
178#391C2314yes—
179#391C2314yes—
180#390C2312yes—
181#390C2311yescharacteristic
182#389—22no—
183#389—22no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2311yescharacteristic
190#388—22no—
191#388—21no—
192#387—22no—
193#387—22no—
194#387—22no—
195#386—22no—
196#386—21no—
197#385—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384C2311yescharacteristic
203#382—22no—
204#382—22no—
205#382—24no—
206#382C2312yes—
207#380—21no—
208#380—21no—
209#380C2311yescharacteristic
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95—22no—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2414yes—
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×SD16—28no—
262C2×SD16—44no—
263C2×SD16—44no—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273#125—44no—
274C22⋊Q8—48no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—82no—
278#121—22no—
279#121#4114yes—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×D4—24no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—84no—
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—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352#107—24no—
353#107—42no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—28no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#102—24no—
365#102—42no—
366C22⋊C8—24no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C4×C8—24no—
370C4×C8—44no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392C24—21no—
393C24C22×D411yescharacteristic
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—42no—
408C2×Q8—416no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—81no—
414C2×Q8—81no—
415C2×Q8—88no—
416C2×Q8—88no—
417C2×Q8#13714yes—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×#4112yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4#13514yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C42−1+412yes—
473C22×C4C2511yescharacteristic
474SD16—88no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4C2×#4112yes—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—81no—
525C22⋊C4—82no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—22no—
530C42—24no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—44no—
535C42—42no—
536C42—82no—
537C42H(5,𝔽2)14yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23#54812yes—
550C23C2×#13512yes—
551C23#57712yes—
552C23C23×D411yescharacteristic
553Q8—28no—
554Q8—48no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—88no—
561Q8—162no—
562D4—24no—
563D4—48no—
564D4—42no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—164no—
620C2×C4#53412yes—
621C2×C4#53412yes—
622C2×C4#54812yes—
623C2×C4C2×#13512yes—
624C2×C4C2×#13512yes—
625C2×C4C2×#13714yes—
626C2×C4C2×#13712yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+411yescharacteristic
630C2×C4#58512yes—
631C8—42no—
632C8—44no—
633C8—42no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#336014yes—
648C22C2×#53411yescharacteristic
649C22C2×#54811yescharacteristic
650C22#348711yescharacteristic
651C22C22×#13511yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57711yescharacteristic
654C22#358611yescharacteristic
655C4—22no—
656C4—24no—
657C4—22no—
658C4—22no—
659C4—42no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—82no—
666C4—82no—
667C4—81no—
668C4—82no—
669C4—82no—
670C4—84no—
671C4#319014yes—
672C4#351412yes—
673C4#356112yes—
674C2—24no—
675C2—82no—
676C2#3912311yescharacteristic
677C2C2×#319011yescharacteristic
678C2#6075811yescharacteristic
679C2C2×#336011yescharacteristic
680C2#6132211yescharacteristic
681C2#6188211yescharacteristic
682C2#6190611yescharacteristic
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×#2568C2semidirect
3. #33148C2semidirect
4. #32634C2semidirect
5. #32604C2semidirect
6. #29957C2semidirect
7. #29106C2semidirect
8. #28638C2semidirect
10. #26317C2semidirect
11. #25934C2semidirect
12. #25856C2semidirect
13. #25293C2semidirect
15. Q8×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#365442
C2×#25682
#331482
#326342
#326042
#299572
#291062
#286382
#281182
#263172
#259342
#258562
#252932
#242092
Q8×C22⋊C82
#220412
C4×#4802