Finite Group Databasebeta

← #6856694

Subgroups — #6856694

Order 512.

1663 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic
4#32082C214yesmaximal, complemented
5#30878C214yesmaximal
6#29798C212yesmaximal
7#29255C212yesmaximal, complemented
8#28748C212yesmaximal, complemented
9#28748C211yesmaximal, characteristic, complemented
10#28400C212yesmaximal
11#28283C211yesmaximal, characteristic, complemented
12#27769C212yesmaximal, complemented
13#25647C212yesmaximal, complemented
14#24187C212yesmaximal
15#22821C211yesmaximal, characteristic, complemented
16#22146C212yesmaximal
17C4×#461C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#533—21no—
20C2×#487C2212yes—
21C2×#487C2214yes—
22C2×#475C2212yes—
23C2×#461C2211yescharacteristic
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×#461C2212yes—
27C2×C4×SD16C2212yes—
28#2868C2212yes—
29#2868C2211yescharacteristic
30#2556—22no—
31#2552—22no—
32#2552—21no—
33#2552—22no—
34#2552—21no—
35#2551C2211yescharacteristic
36#2546C2211yescharacteristic
37#2546C2214yes—
38#2546C2212yes—
39#2542C2211yescharacteristic
40#2538C2211yescharacteristic
41#2506C2212yes—
42#2492—21no—
43#2452C2212yes—
44#2429—21no—
45#2396—22no—
46Q8×C4⋊C4—21no—
47Q8×C4⋊C4—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8—21no—
53#2293C2212yes—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2096C2214yes—
57#2066C2214yes—
58#2043C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2043C2214yes—
63#2024C2214yes—
64#2013C2214yes—
65#2013C2214yes—
66#1984C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2211yescharacteristic
72#1981C2212yes—
73#1981C2212yes—
74#1968C2214yes—
75#1956C2214yes—
76#1956C2212yes—
77#1948C2211yescharacteristic
78#1948C2212yes—
79#1933C2214yes—
80#1927C2212yes—
81#1916C2214yes—
82#1880C2214yes—
83#1877C2212yes—
84#1865C2214yes—
85#1865C2214yes—
86#1865C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1803C2212yes—
90#1802C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97#557—44no—
98#557—44no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#125—22no—
111C2×#125—22no—
112C2×#125—24no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C22×C42—21no—
130#487—44no—
131#487—48no—
132#475—44no—
133#461—24no—
134#461—28no—
135#461—44no—
136#461—44no—
137C4×SD16—44no—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#102C2312yes—
158C2×C4×C8C2312yes—
159C2×C4×C8C2314yes—
160#400—22no—
161#399—22no—
162#398—22no—
163#398—22no—
164#398—24no—
165#398—24no—
166#398—24no—
167#396—22no—
168#396—22no—
169#395—22no—
170#393—22no—
171#391—21no—
172#391—21no—
173#391—22no—
174#391C2312yes—
175#391C2314yes—
176#390C2312yes—
177#390C2312yes—
178#389—24no—
179#389—24no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—21no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#387—22no—
197#385—22no—
198#384—21no—
199#384—22no—
200#384—22no—
201#384—21no—
202#384—22no—
203#384—21no—
204#384—21no—
205#384—22no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2312yes—
209#380—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C43C2312yes—
231C43C2311yescharacteristic
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2414yes—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C2×SD16—24no—
249C2×SD16—28no—
250C2×SD16—42no—
251C2×SD16—42no—
252C2×SD16—44no—
253C2×SD16—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—42no—
270C4⋊Q8—44no—
271C4⋊Q8—42no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275#125—416no—
276#125—48no—
277#125—48no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C4×Q8—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×Q8—48no—
291C4×Q8—44no—
292C4×Q8SD1618yescomplemented
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—41no—
298C4×D4#4114yes—
299#117—48no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#106—42no—
367#106—44no—
368C4⋊C8—22no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—22no—
372C4⋊C8—42no—
373#103—28no—
374#103—24no—
375#103—28no—
376#103—24no—
377#103—44no—
378#103—42no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385#102—42no—
386C4×C8—24no—
387C4×C8—44no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—82no—
425C2×Q8C2×SD1614yes—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4C2×#4112yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460SD16—48no—
461SD16—88no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—84no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4C2×SD1618yes—
512C4⋊C4C2×SD1614yes—
513C4⋊C4#13714yes—
514C4⋊C4C2×#4112yes—
515C4⋊C4H(5,𝔽2)12yes—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C42—24no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—22no—
533C42—41no—
534C42—42no—
535C42—48no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42C2×SD1614yes—
542C42C2×SD1618yes—
543C42C2×#4112yes—
544C42H(5,𝔽2)12yes—
545C42H(5,𝔽2)12yes—
546C422−1+412yes—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C23×D411yescharacteristic
555Q8—28no—
556Q8—42no—
557Q8—44no—
558Q8—44no—
559Q8—42no—
560Q8—42no—
561Q8—44no—
562Q8—42no—
563Q8—44no—
564Q8—88no—
565Q8—84no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4#53512yes—
616C2×C4#53512yes—
617C2×C4#54812yes—
618C2×C4#54812yes—
619C2×C4C22×SD1612yes—
620C2×C4C22×SD1614yes—
621C2×C4C22×SD1614yes—
622C2×C4C22×SD1612yes—
623C2×C4C22×SD1612yes—
624C2×C4C2×#13714yes—
625C2×C4C2×#13712yes—
626C2×C4#57812yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C2×C4#58511yescharacteristic
633C8—42no—
634C8—44no—
635C8—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C2×#53511yescharacteristic
644C22C2×#54811yescharacteristic
645C22#348811yescharacteristic
646C22C23×SD1611yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358611yescharacteristic
650C4—24no—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—24no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—82no—
661C4—84no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4#320614yes—
667C4#335914yes—
668C4#350611yescharacteristic
669C4#350811yescharacteristic
670C4#354811yescharacteristic
671C4#355411yescharacteristic
672C2—82no—
673C2#3972511yescharacteristic
674C2C2×#320611yescharacteristic
675C2#6083711yescharacteristic
676C2C2×#335911yescharacteristic
677C2#6132211yescharacteristic
678C2#6194811yescharacteristic
679C2#6198411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36593C2semidirect
4. #32082C2semidirect
7. #29255C2semidirect
8. #28748C2semidirect
9. #28748C2semidirect
11. #28283C2semidirect
12. #27769C2semidirect
13. #25647C2semidirect
15. #22821C2semidirect
292. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365932
#349292
#331622
#320822
#308782
#297982
#292552
#287482
#284002
#282832
#277692
#256472
#241872
#228212
#221462
C4×#4612