Finite Group Databasebeta

← #6986804

Subgroups — #6986804

Order 512.

1627 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36492C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic, complemented
3#32789C212yesmaximal, complemented
4#32265C214yesmaximal, complemented
5#32004C214yesmaximal, complemented
6#30294C212yesmaximal, complemented
7#30172C212yesmaximal, complemented
8#29938C211yesmaximal, characteristic, complemented
9#29157C212yesmaximal, complemented
10#28705C212yesmaximal, complemented
11#26277C212yesmaximal, complemented
12#25839C212yesmaximal, complemented
13#25813C211yesmaximal, characteristic, complemented
14#25339C212yesmaximal, complemented
15#24299C211yesmaximal, characteristic, complemented
16#23754C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#500C2212yes—
22C2×#476C2212yes—
23C2×C8×D4C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2628—22no—
29#2568—21no—
30#2567C2212yes—
31#2563C2211yescharacteristic
32#2563C2212yes—
33#2562C2212yes—
34#2542—21no—
35#2537C2211yescharacteristic
36#2535C2212yes—
37#2487C2214yes—
38#2477—22no—
39#2463—22no—
40#2421—21no—
41#2409—22no—
42#2407—21no—
43#2361—21no—
44#2355C2211yescharacteristic
45#2352—21no—
46#2317—21no—
47#2311—21no—
48C4×#121C2211yescharacteristic
49C4×#121C2212yes—
50#2285C2211yescharacteristic
51#2101C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2093C2211yescharacteristic
55#2085C2214yes—
56#2084C2212yes—
57#2083C2211yescharacteristic
58#2077C2214yes—
59#2077C2214yes—
60#2076C2214yes—
61#2076C2214yes—
62#2075C2212yes—
63#2071C2211yescharacteristic
64#2059C2214yes—
65#2059C2214yes—
66#2058C2212yes—
67#2057C2212yes—
68#2057C2212yes—
69#2034C2214yes—
70#2033C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1987C2212yes—
74#1987C2212yes—
75#1984C2212yes—
76#1948C2212yes—
77#1919C2214yes—
78#1918C2214yes—
79#1893C2214yes—
80#1874C2212yes—
81#1869C2214yes—
82#1862C2214yes—
83#1862C2214yes—
84#1853C2212yes—
85#1852C2214yes—
86#1817C2212yes—
87#1808C2214yes—
88#1808C2214yes—
89#1807C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98#554—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×#121—21no—
112C2×#121C2311yescharacteristic
113C2×#121C2312yes—
114C4×#41—44no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D412yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117D411yescharacteristic
127C2×#117D412yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#500—44no—
134#476—44no—
135C8×D4—44no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—24no—
152#399—24no—
153#397—22no—
154#395—24no—
155#394—22no—
156#394—22no—
157#393—24no—
158#393—22no—
159#392—22no—
160#391—22no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391—21no—
165#390—22no—
166#390—21no—
167#390C2311yescharacteristic
168#389—22no—
169#389—22no—
170#389—21no—
171#389C2314yes—
172#389C2314yes—
173#389C2312yes—
174#389C2311yescharacteristic
175#388—22no—
176#388—22no—
177#388—22no—
178#388—21no—
179#387—22no—
180#387—22no—
181#387—22no—
182#386—24no—
183#386—22no—
184#386—21no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#384—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382—21no—
196#382—21no—
197#382C2312yes—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2311yescharacteristic
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207#380C2314yes—
208#380C2312yes—
209#380C2314yes—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×Q8C2×D411yescharacteristic
244C22×D4—22no—
245C22×D4—21no—
246C22×D4C2×D412yes—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C22×C8—28no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#125—44no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—82no—
274C22⋊Q8—84no—
275C22⋊Q8—84no—
276#121—22no—
277#121—24no—
278#121—42no—
279#121—41no—
280#121—44no—
281C4×Q8—24no—
282C4×Q8—42no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—84no—
291#117—44no—
292#117—42no—
293#117—44no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—24no—
355#107—42no—
356#107—44no—
357#107—44no—
358#106—44no—
359#106—44no—
360C4⋊C8—24no—
361#102—24no—
362#102—24no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—42no—
366C22⋊C8—44no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385C24—21no—
386C24C22×D411yescharacteristic
387C24C22×D412yes—
388#41—416no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8—816no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—82no—
415C2×D4—81no—
416C2×D4#13714yes—
417C2×D42−1+412yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4#13514yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—88no—
473C2×C82−1+414yes—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—24no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42—42no—
532C42—82no—
533C42H(5,𝔽2)14yes—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—21no—
544C23—22no—
545C23—21no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×#13512yes—
554C23C2×#13714yes—
555C23C23×D411yescharacteristic
556Q8—44no—
557Q8—88no—
558Q8—88no—
559Q8—162no—
560D4—28no—
561D4—24no—
562D4—48no—
563D4—42no—
564D4—84no—
565D4—88no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4#53412yes—
620C2×C4#53412yes—
621C2×C4C2×#13512yes—
622C2×C4C2×#13512yes—
623C2×C4C2×#13712yes—
624C2×C4#57612yes—
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4C2×2−1+411yescharacteristic
627C2×C4C2×2−1+412yes—
628C2×C4#58511yescharacteristic
629C8—24no—
630C8—42no—
631C8—44no—
632C22—22no—
633C22—24no—
634C22—22no—
635C22—24no—
636C22—22no—
637C22—24no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C2×#53411yescharacteristic
654C22#348911yescharacteristic
655C22#349911yescharacteristic
656C22C22×#13511yescharacteristic
657C22C22×#13711yescharacteristic
658C22C2×#57611yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—81no—
668C4—82no—
669C4—81no—
670C4—84no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4#319014yes—
676C4#355212yes—
677C4#356112yes—
678C2—42no—
679C2—44no—
680C2—82no—
681C2#4291211yescharacteristic
682C2C2×#319011yescharacteristic
683C2#6075611yescharacteristic
684C2#6209811yescharacteristic
685C2#6212711yescharacteristic
686C2#6247411yescharacteristic
687C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35279C2semidirect
3. #32789C2semidirect
4. #32265C2semidirect
5. #32004C2semidirect
6. #30294C2semidirect
7. #30172C2semidirect
8. #29938C2semidirect
9. #29157C2semidirect
10. #28705C2semidirect
11. #26277C2semidirect
12. #25839C2semidirect
13. #25813C2semidirect
14. #25339C2semidirect
15. #24299C2semidirect
16. #23754C2semidirect
17. #22469C2semidirect

Maximal subgroups

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

subgroupindex
#364922
#352792
#327892
#322652
#320042
#302942
#301722
#299382
#291572
#287052
#262772
#258392
#258132
#253392
#242992
#237542
#224692