Finite Group Databasebeta

← #6989173

Subgroups — #6989173

Order 512.

1643 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36404C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic, complemented
3#32745C212yesmaximal, complemented
4#31783C214yesmaximal, complemented
5#31675C214yesmaximal, complemented
6#30359C212yesmaximal, complemented
7#29955C212yesmaximal, complemented
8#29922C211yesmaximal, characteristic, complemented
9#29107C212yesmaximal, complemented
10#28638C212yesmaximal, complemented
11#26297C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25886C212yesmaximal, complemented
14#25470C212yesmaximal, complemented
15#24299C211yesmaximal, characteristic, complemented
16#23754C211yesmaximal, characteristic, complemented
17#22530C211yesmaximal, characteristic, complemented
18C2×#547C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#487C2212yes—
22C2×#476C2212yes—
23C2×C8×Q8C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2628—22no—
29#2568—21no—
30#2568C2211yescharacteristic
31#2567C2212yes—
32#2563C2212yes—
33#2542—21no—
34#2538C2211yescharacteristic
35#2487C2214yes—
36#2470—22no—
37#2457—22no—
38#2420—21no—
39#2415C2212yes—
40#2409—22no—
41#2407—21no—
42#2407C2212yes—
43#2360—21no—
44#2354C2211yescharacteristic
45#2352—21no—
46#2317—21no—
47#2311—21no—
48C4×#121C2211yescharacteristic
49#2297C2212yes—
50#2285C2211yescharacteristic
51#2099C2211yescharacteristic
52#2096C2212yes—
53#2093C2211yescharacteristic
54#2088C2214yes—
55#2083C2211yescharacteristic
56#2077C2214yes—
57#2075C2212yes—
58#2075C2214yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2068C2214yes—
62#2068C2214yes—
63#2065C2214yes—
64#2058C2212yes—
65#2035C2214yes—
66#2035C2214yes—
67#2035C2214yes—
68#2033C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1987C2212yes—
74#1981C2212yes—
75#1954C2212yes—
76#1954C2214yes—
77#1948C2212yes—
78#1941C2214yes—
79#1934C2214yes—
80#1933C2214yes—
81#1933C2214yes—
82#1874C2212yes—
83#1872C2214yes—
84#1854C2214yes—
85#1853C2212yes—
86#1825C2214yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1788C2214yes—
90#1788C2214yes—
91C4×#107C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96C22×#41C2312yes—
97C23×Q8—21no—
98#547—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121C2311yescharacteristic
113C4×#41—44no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4D412yes—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C2×#117D411yescharacteristic
124C2×#117D412yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#487—44no—
132#476—44no—
133C8×Q8—44no—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—24no—
154#400—22no—
155#399—24no—
156#397—22no—
157#396—22no—
158#395—24no—
159#394—22no—
160#394—22no—
161#393—24no—
162#392—22no—
163#391—22no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#390—22no—
171#390—21no—
172#390C2312yes—
173#390C2311yescharacteristic
174#389—22no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389C2314yes—
180#389C2314yes—
181#389C2311yescharacteristic
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—21no—
186#387—22no—
187#387—22no—
188#386—24no—
189#386—22no—
190#386—22no—
191#386—21no—
192#386—22no—
193#385—22no—
194#385—22no—
195#385—24no—
196#384—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—22no—
201#382—21no—
202#382—21no—
203#381—22no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380—21no—
208#380—21no—
209#380—21no—
210#380C2314yes—
211#380C2314yes—
212#380C2312yes—
213#380C2311yescharacteristic
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—42no—
238C2×#41—44no—
239C2×#41—42no—
240C2×#41—44no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—44no—
244C22×Q8—82no—
245C22×Q8—82no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D412yes—
252C22×D4C2411yescharacteristic
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#123—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—84no—
276C22⋊Q8—84no—
277#121—22no—
278#121—41no—
279#121—44no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—42no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288#117—44no—
289#117—42no—
290#117—44no—
291#117—44no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—42no—
353#107—42no—
354#106—44no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#102—24no—
363#102—42no—
364#102—44no—
365C22⋊C8—42no—
366C22⋊C8—44no—
367C22⋊C8—42no—
368C22⋊C8—44no—
369C4×C8—24no—
370C4×C8—44no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388C24—21no—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—416no—
392#41—88no—
393#41—88no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—816no—
407C2×Q8#13714yes—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D42−1+412yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4#13514yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—42no—
536C42—82no—
537C42H(5,𝔽2)14yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×#13512yes—
559C23C23×D411yescharacteristic
560Q8—28no—
561Q8—44no—
562Q8—44no—
563Q8—88no—
564Q8—88no—
565D4—24no—
566D4—48no—
567D4—44no—
568D4—42no—
569D4—84no—
570D4—88no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4#53412yes—
624C2×C4#53412yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13714yes—
628C2×C4C2×#13712yes—
629C2×C4#57712yes—
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4C2×2−1+411yescharacteristic
632C2×C4#58512yes—
633C2×C4#58511yescharacteristic
634C8—42no—
635C8—42no—
636C8—44no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22C2×#53411yescharacteristic
659C22#348911yescharacteristic
660C22#350011yescharacteristic
661C22C22×#13511yescharacteristic
662C22C22×#13711yescharacteristic
663C22C2×#57711yescharacteristic
664C22#358511yescharacteristic
665C4—22no—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—42no—
670C4—41no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4#319014yes—
681C4#356112yes—
682C4#356212yes—
683C2—42no—
684C2—82no—
685C2—82no—
686C2#4294311yescharacteristic
687C2C2×#319011yescharacteristic
688C2#6075711yescharacteristic
689C2#6209811yescharacteristic
690C2#6214711yescharacteristic
691C2#6253211yescharacteristic
692C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36404C2
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. #32745C2semidirect
4. #31783C2semidirect
5. #31675C2semidirect
6. #30359C2semidirect
7. #29955C2semidirect
8. #29922C2semidirect
9. #29107C2semidirect
10. #28638C2semidirect
11. #26297C2semidirect
12. #25902C2semidirect
13. #25886C2semidirect
14. #25470C2semidirect
15. #24299C2semidirect
16. #23754C2semidirect
17. #22530C2semidirect

Maximal subgroups

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

subgroupindex
#364042
#352792
#327452
#317832
#316752
#303592
#299552
#299222
#291072
#286382
#262972
#259022
#258862
#254702
#242992
#237542
#225302