Finite Group Databasebeta

← #6795680

Subgroups — #6795680

Order 512.

1655 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#31686C214yesmaximal, complemented
5#31078C214yesmaximal, complemented
6#29936C212yesmaximal, complemented
7#29286C211yesmaximal, characteristic, complemented
8#29208C211yesmaximal, characteristic, complemented
9#28827C212yesmaximal, complemented
10#27727C212yesmaximal, complemented
11#27644C212yesmaximal, complemented
12#26296C212yesmaximal, complemented
13#25010C212yesmaximal
14#24700C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21982C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#529—21no—
20C2×#480C2214yes—
21C2×#474C2211yescharacteristic
22C2×#474C2212yes—
23C2×#431C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2567—21no—
27#2553—22no—
28#2552—21no—
29#2552C2212yes—
30#2552C2211yescharacteristic
31#2547C2211yescharacteristic
32#2542C2211yescharacteristic
33#2537C2211yescharacteristic
34#2502—22no—
35#2496C2212yes—
36#2489—21no—
37#2440—22no—
38#2440C2214yes—
39#2427—21no—
40#2422C2212yes—
41#2395—22no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8—21no—
47C4×#121C2212yes—
48#2285C2212yes—
49C2×#384C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2099C2214yes—
53#2086C2214yes—
54#2083C2214yes—
55#2083C2214yes—
56#2075C2214yes—
57#2075C2214yes—
58#2075C2214yes—
59#2071C2214yes—
60#2057C2214yes—
61#2033C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1988C2212yes—
65#1988C2211yescharacteristic
66#1988C2212yes—
67#1954C2212yes—
68#1947C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1942C2214yes—
76#1934C2214yes—
77#1923C2214yes—
78#1918C2212yes—
79#1862C2212yes—
80#1852C2214yes—
81#1852C2214yes—
82#1845C2212yes—
83#1845C2214yes—
84#1842C2211yescharacteristic
85#1834C2214yes—
86#1823C2214yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101#529—48no—
102C2×#126—22no—
103C2×#125—21no—
104C2×#125—22no—
105C2×#124—21no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—24no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2314yes—
127C22×C42—21no—
128C22×C42C2312yes—
129#480—48no—
130#474—24no—
131#474—44no—
132#431—28no—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#399—22no—
153#398—22no—
154#398—24no—
155#398—24no—
156#397—22no—
157#397—22no—
158#397—22no—
159#395—24no—
160#394—22no—
161#394—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#390C2312yes—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#388—22no—
178#387—22no—
179#386—22no—
180#385—24no—
181#385—24no—
182#385—22no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2314yes—
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380C2314yes—
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×D4—22no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2411yescharacteristic
240C23×C4—24no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—44no—
257C4⋊Q8—82no—
258#126—48no—
259#125—44no—
260#125—48no—
261#124—44no—
262#123—48no—
263C22⋊Q8—48no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—416no—
267#121—24no—
268#121—42no—
269#121—44no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—41no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4D818yes—
284#117—48no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—42no—
355#107—44no—
356#107—44no—
357#107—44no—
358#107—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—42no—
363C4⋊C8#4114yes—
364#102—28no—
365#102—24no—
366#102—24no—
367C22⋊C8—28no—
368C22⋊C8—44no—
369C4×C8—28no—
370C4×C8—24no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D412yes—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4C2×D814yes—
409C2×D4H(5,𝔽2)12yes—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C2×D818yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8C2×#4114yes—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4C2×D814yes—
501C4⋊C4#13714yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4C2×D818yes—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—22no—
522C42—21no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—44no—
529C42C2×D814yes—
530C42C2×#4112yes—
531C422−1+412yes—
532C422−1+412yes—
533C422−1+412yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C22×D814yes—
548C23C23×D411yescharacteristic
549Q8—84no—
550Q8—84no—
551Q8—164no—
552D4—28no—
553D4—22no—
554D4—22no—
555D4—41no—
556D4—41no—
557D4—88no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—168no—
608C2×C4#53612yes—
609C2×C4#53612yes—
610C2×C4#54712yes—
611C2×C4#54712yes—
612C2×C4C22×D812yes—
613C2×C4C22×D814yes—
614C2×C4C22×D812yes—
615C2×C4C22×D812yes—
616C2×C4C2×#13714yes—
617C2×C4C2×#13712yes—
618C2×C4#57812yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58511yescharacteristic
625C8—24no—
626C8—44no—
627C8—42no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22C2×#53611yescharacteristic
643C22C2×#54711yescharacteristic
644C22#348811yescharacteristic
645C22C23×D811yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57811yescharacteristic
648C22#358511yescharacteristic
649C4—24no—
650C4—22no—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—41no—
656C4—81no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—84no—
663C4—82no—
664C4#322614yes—
665C4#333414yes—
666C4#351511yescharacteristic
667C4#352211yescharacteristic
668C4#354411yescharacteristic
669C4#355511yescharacteristic
670C2—44no—
671C2—82no—
672C2#3892111yescharacteristic
673C2C2×#322611yescharacteristic
674C2#6089311yescharacteristic
675C2C2×#333411yescharacteristic
676C2#6125411yescharacteristic
677C2#6194711yescharacteristic
678C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36180C2semidirect
3. #33195C2semidirect
4. #31686C2semidirect
5. #31078C2semidirect
6. #29936C2semidirect
7. #29286C2semidirect
8. #29208C2semidirect
9. #28827C2semidirect
10. #27727C2semidirect
11. #27644C2semidirect
12. #26296C2semidirect
14. #24700C2semidirect
15. #22795C2semidirect
16. #21982C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349322
#331952
#316862
#310782
#299362
#292862
#292082
#288272
#277272
#276442
#262962
#250102
#247002
#227952
#219822
C4×#4742