Finite Group Databasebeta

← #6593943

Subgroups — #6593943

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#33003C212yesmaximal
4#32957C211yesmaximal, characteristic, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27644C211yesmaximal, characteristic
7#27616C212yesmaximal, complemented
8#27553C214yesmaximal, complemented
9#27295C214yesmaximal, complemented
10#26109C212yesmaximal, complemented
11#24814C212yesmaximal, complemented
12#24600C212yesmaximal
13#22965C212yesmaximal, complemented
14#22767C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#448C211yesmaximal, characteristic
17#20735C212yesmaximal, complemented
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#480C2212yes—
21C2×#477C2212yes—
22C2×#448C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×#431C2212yes—
25#2857C2211yescharacteristic
26#2854C2212yes—
27#2571C2211yescharacteristic
28#2563C2211yescharacteristic
29#2532—22no—
30#2513C2212yes—
31#2508C2212yes—
32#2495—21no—
33#2488—22no—
34#2486—21no—
35#2369C2211yescharacteristic
36#2354C2211yescharacteristic
37#2352—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—22no—
41Q8×C4⋊C4C2214yes—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2212yes—
44#2346—21no—
45C4×#121C2211yescharacteristic
46C2×#389—22no—
47C2×#389C2212yes—
48C2×#384—21no—
49C2×#384C2211yescharacteristic
50#2283C2212yes—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2099C2214yes—
55#2086C2214yes—
56#2085C2214yes—
57#2083C2212yes—
58#2071C2212yes—
59#2067C2214yes—
60#1990C2212yes—
61#1947C2212yes—
62#1942C2214yes—
63#1939C2212yes—
64#1934C2214yes—
65#1921C2214yes—
66#1920C2214yes—
67#1916C2214yes—
68#1880C2212yes—
69#1880C2214yes—
70#1880C2214yes—
71#1874C2211yescharacteristic
72#1865C2212yes—
73#1863C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1853C2211yescharacteristic
77#1852C2212yes—
78#1834C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1817C2212yes—
83#1805C2214yes—
84#1805C2214yes—
85#1803C2212yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89#1787C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×#103C2214yes—
92C4×#103C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×#121C2311yescharacteristic
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—28no—
107C2×C4×Q8—44no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—21no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#480—44no—
129#477—28no—
130#448—24no—
131C4×SD16—28no—
132#431—44no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#400—22no—
152#398—22no—
153#397—22no—
154#397—22no—
155#393—22no—
156#391—21no—
157#391—22no—
158#390C2312yes—
159#390C2311yescharacteristic
160#390C2312yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—24no—
168#389—24no—
169#389—24no—
170#389C2312yes—
171#389C2314yes—
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—22no—
182#387—22no—
183#386—22no—
184#386—21no—
185#384—22no—
186#384—22no—
187#384—24no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#383—22no—
194#382—21no—
195#382—22no—
196#382C2312yes—
197#382C2312yes—
198#381—22no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#380C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—24no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#376—22no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8#4114yes—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—24no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261#123—84no—
262C22⋊Q8—82no—
263#121—22no—
264#121—22no—
265C4×Q8—28no—
266C4×Q8—28no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—28no—
271C4×Q8—28no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×Q8—48no—
276C4×Q8#4118yes—
277C4×D4—24no—
278C4×D4—44no—
279#117—44no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—48no—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352#107—28no—
353#107—44no—
354#106—24no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—24no—
358#103—24no—
359#103—28no—
360#103—28no—
361#103—24no—
362#102—24no—
363#102—44no—
364C22⋊C8—22no—
365C22⋊C8—22no—
366C22⋊C8—24no—
367C4×C8—24no—
368C4×C8—44no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24C22×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—216no—
392C2×Q8—24no—
393C2×Q8—216no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8C2×#4114yes—
402C2×Q8C2×#4114yes—
403C2×Q82−1+414yes—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
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—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4#13514yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4C2511yescharacteristic
457SD16—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4#13714yes—
501C4⋊C4C2×#4114yes—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—81no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C42—28no—
513C42—28no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—48no—
524C42—48no—
525C42—42no—
526C42—44no—
527C42—82no—
528C42C2×#4114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23#54612yes—
541C23C2×#13512yes—
542C23#57612yes—
543C23C23×D411yescharacteristic
544Q8—216no—
545Q8—24no—
546Q8—24no—
547Q8—24no—
548Q8—162no—
549Q8#55218yes—
550D4—42no—
551D4—42no—
552D4—84no—
553C2×C4—216no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—88no—
606C2×C4—84no—
607C2×C4#52512yes—
608C2×C4#52512yes—
609C2×C4#54612yes—
610C2×C4C2×#13512yes—
611C2×C4C2×#13512yes—
612C2×C4C2×#13714yes—
613C2×C4C2×#13712yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4112yes—
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+412yes—
618C8—42no—
619C8—42no—
620C8—44no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22#330514yes—
635C22C2×#52511yescharacteristic
636C22C2×#54611yescharacteristic
637C22#348611yescharacteristic
638C22C22×#13511yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57611yescharacteristic
641C22#358611yescharacteristic
642C4—28no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—24no—
647C4—41no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4—81no—
658C4#310814yes—
659C4C2×#55212yes—
660C4#349212yes—
661C2—24no—
662C2—161no—
663C2#3616311yescharacteristic
664C2C2×#310811yescharacteristic
665C2#6058411yescharacteristic
666C2C2×#330511yescharacteristic
667C2#6115711yescharacteristic
668C2#6182211yescharacteristic
669C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
4. #32957C2semidirect
5. #27960C2semidirect
7. #27616C2semidirect
8. #27553C2semidirect
9. #27295C2semidirect
10. #26109C2semidirect
11. #24814C2semidirect
13. #22965C2semidirect
14. #22767C2semidirect
15. #22599C2semidirect
17. #20735C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×Q8×C4⋊C42
#330032
#329572
#279602
#276442
#276162
#275532
#272952
#261092
#248142
#246002
#229652
#227672
#225992
C4×#4482
#207352