Finite Group Databasebeta

← #6627945

Subgroups — #6627945

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#32909C212yesmaximal
4#32906C211yesmaximal, characteristic, complemented
5#29336C212yesmaximal
6#29240C212yesmaximal, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27678C211yesmaximal, characteristic
9#27570C214yesmaximal, complemented
10#26185C214yesmaximal, complemented
11#26113C212yesmaximal, complemented
12#24729C212yesmaximal, complemented
13#23004C212yesmaximal, complemented
14#22759C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21455C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#464C2214yes—
21C2×#462C2212yes—
22C2×#451C2212yes—
23C2×#448C2211yescharacteristic
24C2×#441C2214yes—
25C2×C4×SD16C2212yes—
26#2857C2211yescharacteristic
27#2850C2212yes—
28#2596—22no—
29#2571C2211yescharacteristic
30#2547C2211yescharacteristic
31#2514C2212yes—
32#2510—22no—
33#2494—21no—
34#2487—21no—
35#2448C2212yes—
36#2365C2211yescharacteristic
37#2354C2211yescharacteristic
38#2352—21no—
39Q8×C4⋊C4—22no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2214yes—
44Q8×C4⋊C4C2212yes—
45#2347—21no—
46C4×C22⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48C2×#389—22no—
49C2×#389C2212yes—
50C2×#384—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2100C2212yes—
56#2094C2212yes—
57#2088C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2042C2214yes—
61#1956C2212yes—
62#1948C2212yes—
63#1939C2212yes—
64#1936C2214yes—
65#1935C2214yes—
66#1919C2214yes—
67#1917C2214yes—
68#1880C2212yes—
69#1880C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1874C2212yes—
74#1874C2211yescharacteristic
75#1872C2212yes—
76#1872C2212yes—
77#1872C2212yes—
78#1865C2214yes—
79#1865C2214yes—
80#1865C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1834C2212yes—
84#1805C2214yes—
85#1803C2212yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1787C2211yescharacteristic
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#124C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×#121C2311yescharacteristic
102C2×C4×Q8—28no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—48no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C42—21no—
126C22×C42—22no—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#464—48no—
130#462—28no—
131#451—28no—
132#448—24no—
133#441—48no—
134C4×SD16—28no—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#401—22no—
154#400—22no—
155#400—22no—
156#399—22no—
157#397—22no—
158#397—22no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#390C2312yes—
163#390C2311yescharacteristic
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—24no—
172#389—24no—
173#389—24no—
174#389C2312yes—
175#389C2314yes—
176#389C2314yes—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—22no—
181#388—21no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—22no—
188#386—22no—
189#386—21no—
190#385—22no—
191#384—22no—
192#384—22no—
193#384—24no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#383—22no—
200#382—21no—
201#382C2312yes—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8#4114yes—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—28no—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—24no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264#124—44no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267#121—22no—
268#121—22no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—28no—
275C4×Q8—28no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—48no—
281C4×Q8#4118yes—
282C4×D4—24no—
283C4×D4—44no—
284#117—44no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
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—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C42—28no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42—48no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#106—24no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#103—24no—
360#103—24no—
361#103—28no—
362#103—24no—
363#103—28no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—44no—
368C22⋊C8—22no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371#97—44no—
372C4×C8—24no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—216no—
397C2×Q8—216no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8C2×#4114yes—
407C2×Q8C2×#4114yes—
408C2×Q8H(5,𝔽2)14yes—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462Q16—416no—
463SD16—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4C2×#4114yes—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—28no—
520C42—28no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—24no—
526C42—42no—
527C42—42no—
528C42—48no—
529C42—48no—
530C42—44no—
531C42—42no—
532C42—82no—
533C42C2×#4114yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23#54812yes—
546C23C2×#13712yes—
547C23#57712yes—
548C23C23×D411yescharacteristic
549Q8—216no—
550Q8—24no—
551Q8—24no—
552Q8—24no—
553Q8—44no—
554Q8—44no—
555Q8#55318yes—
556D4—42no—
557D4—42no—
558D4—84no—
559C2×C4—216no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—84no—
614C2×C4#52512yes—
615C2×C4#52512yes—
616C2×C4#54812yes—
617C2×C4C2×#13712yes—
618C2×C4C2×#13712yes—
619C2×C4#57512yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4112yes—
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×2−1+411yescharacteristic
624C8—42no—
625C8—44no—
626C8—42no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22#336914yes—
641C22C2×#52511yescharacteristic
642C22C2×#54811yescharacteristic
643C22#348411yescharacteristic
644C22C22×#13711yescharacteristic
645C22C2×#57511yescharacteristic
646C22C2×#57711yescharacteristic
647C22#358611yescharacteristic
648C4—22no—
649C4—22no—
650C4—28no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—44no—
658C4—81no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4—84no—
663C4—82no—
664C4#311714yes—
665C4C2×#55312yes—
666C4#349212yes—
667C2—24no—
668C2—161no—
669C2#3658111yescharacteristic
670C2C2×#311711yescharacteristic
671C2#6058511yescharacteristic
672C2C2×#336911yescharacteristic
673C2#6132011yescharacteristic
674C2#6158411yescharacteristic
675C2#6159611yescharacteristic
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. #32906C2semidirect
6. #29240C2semidirect
7. #27972C2semidirect
9. #27570C2semidirect
10. #26185C2semidirect
11. #26113C2semidirect
12. #24729C2semidirect
13. #23004C2semidirect
14. #22759C2semidirect
15. #22629C2semidirect
16. #21455C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×Q8×C4⋊C42
#329092
#329062
#293362
#292402
#279722
#276782
#275702
#261852
#261132
#247292
#230042
#227592
#226292
#214552
C4×#4482