Finite Group Databasebeta

← #6989178

Subgroups — #6989178

Order 512.

1719 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36404C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic, complemented
3#32741C212yesmaximal, complemented
4#31779C214yesmaximal, complemented
5#31674C214yesmaximal, complemented
6#30358C212yesmaximal, complemented
7#29954C212yesmaximal, complemented
8#29923C211yesmaximal, characteristic, complemented
9#29107C212yesmaximal, complemented
10#28639C212yesmaximal, complemented
11#26301C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25882C212yesmaximal, complemented
14#25469C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic, complemented
16#23755C211yesmaximal, characteristic, complemented
17#22530C211yesmaximal, characteristic, complemented
18C2×#547C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#487C2212yes—
23C2×#474C2212yes—
24C2×C8×Q8C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2855C2212yes—
29#2621—22no—
30#2568C2211yescharacteristic
31#2547—21no—
32#2547C2212yes—
33#2546—21no—
34#2546C2212yes—
35#2538C2211yescharacteristic
36#2486—22no—
37#2486C2214yes—
38#2470—22no—
39#2457—22no—
40#2420—21no—
41#2415C2212yes—
42#2407—21no—
43#2407C2212yes—
44#2360—21no—
45#2354C2211yescharacteristic
46#2352—21no—
47#2310—21no—
48C4×#121C2211yescharacteristic
49#2297C2212yes—
50#2285C2211yescharacteristic
51#2100C2211yescharacteristic
52#2095C2212yes—
53#2094C2211yescharacteristic
54#2086C2214yes—
55#2084C2211yescharacteristic
56#2077C2212yes—
57#2077C2214yes—
58#2076C2214yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2067C2214yes—
62#2066C2214yes—
63#2066C2214yes—
64#2057C2212yes—
65#2036C2214yes—
66#2036C2214yes—
67#2036C2214yes—
68#2034C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1981C2212yes—
75#1954C2212yes—
76#1954C2214yes—
77#1947C2212yes—
78#1942C2214yes—
79#1934C2214yes—
80#1933C2214yes—
81#1933C2214yes—
82#1874C2212yes—
83#1872C2214yes—
84#1854C2214yes—
85#1852C2212yes—
86#1825C2214yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1788C2214yes—
90#1788C2214yes—
91C4×#106C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96C22×#41C2312yes—
97C23×Q8—21no—
98#547—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#124—22no—
108C2×#123—24no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×#121C2311yescharacteristic
117C4×#41—44no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117D411yescharacteristic
128C2×#117D412yes—
129C2×#117C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#487—44no—
136#474—44no—
137C8×Q8—44no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#400—22no—
158#398—24no—
159#398—24no—
160#397—24no—
161#397—24no—
162#397—22no—
163#396—22no—
164#394—22no—
165#394—22no—
166#392—22no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#390—21no—
176#390C2312yes—
177#390C2311yescharacteristic
178#389—22no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389C2311yescharacteristic
184#389C2314yes—
185#389C2314yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—21no—
190#387—22no—
191#387—22no—
192#386—22no—
193#386—22no—
194#386—21no—
195#386—22no—
196#385—22no—
197#385—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—21no—
202#382—21no—
203#381—22no—
204#381—24no—
205#380—21no—
206#380—21no—
207#380—22no—
208#380—22no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2314yes—
213#380C2311yescharacteristic
214#380C2314yes—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—42no—
239C2×#41—44no—
240C2×#41—42no—
241C2×#41—44no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D412yes—
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—48no—
273#124—48no—
274#123—48no—
275#123—416no—
276C22⋊Q8—44no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#121—22no—
282#121—41no—
283#121—44no—
284C4×Q8—24no—
285C4×Q8—42no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—44no—
292#117—44no—
293#117—42no—
294#117—44no—
295#117—44no—
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—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#107—44no—
358#106—42no—
359#106—42no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—42no—
368#102—44no—
369C22⋊C8—42no—
370C22⋊C8—44no—
371C22⋊C8—42no—
372C22⋊C8—44no—
373C4×C8—24no—
374C4×C8—44no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392C24—21no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—416no—
396#41—88no—
397#41—88no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—816no—
411C2×Q8AGL(1,ℤ/8)14yes—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—81no—
426C2×D4—82no—
427C2×D42−1+412yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4#13514yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2511yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—22no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—82no—
541C42H(5,𝔽2)14yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×#13512yes—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—44no—
566Q8—44no—
567Q8—88no—
568Q8—88no—
569D4—24no—
570D4—48no—
571D4—44no—
572D4—42no—
573D4—84no—
574D4—88no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4—84no—
627C2×C4#53412yes—
628C2×C4#53412yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13512yes—
631C2×C4C2×AGL(1,ℤ/8)14yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4#57812yes—
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58512yes—
637C2×C4#58511yescharacteristic
638C8—42no—
639C8—42no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22C2×#53411yescharacteristic
663C22#348911yescharacteristic
664C22#350011yescharacteristic
665C22C22×#13511yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C2×#57811yescharacteristic
668C22#358611yescharacteristic
669C4—22no—
670C4—24no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—41no—
675C4—42no—
676C4—81no—
677C4—84no—
678C4—81no—
679C4—82no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4—82no—
684C4#318814yes—
685C4#356112yes—
686C4#356212yes—
687C2—42no—
688C2—82no—
689C2—82no—
690C2#4294311yescharacteristic
691C2C2×#318811yescharacteristic
692C2#6075911yescharacteristic
693C2#6209711yescharacteristic
694C2#6214811yescharacteristic
695C2#6252811yescharacteristic
696C2#6259511yescharacteristic
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. #35316C2semidirect
3. #32741C2semidirect
4. #31779C2semidirect
5. #31674C2semidirect
6. #30358C2semidirect
7. #29954C2semidirect
8. #29923C2semidirect
9. #29107C2semidirect
10. #28639C2semidirect
11. #26301C2semidirect
12. #25902C2semidirect
13. #25882C2semidirect
14. #25469C2semidirect
15. #24298C2semidirect
16. #23755C2semidirect
17. #22530C2semidirect

Maximal subgroups

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

subgroupindex
#364042
#353162
#327412
#317792
#316742
#303582
#299542
#299232
#291072
#286392
#263012
#259022
#258822
#254692
#242982
#237552
#225302