Finite Group Databasebeta

← #6626288

Subgroups — #6626288

Order 512.

1793 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#32960C212yesmaximal, complemented
4#32957C211yesmaximal, characteristic, complemented
5#29303C212yesmaximal, complemented
6#29234C212yesmaximal
7#27950C211yesmaximal, characteristic, complemented
8#27678C211yesmaximal, characteristic
9#27316C214yesmaximal, complemented
10#26387C214yesmaximal, complemented
11#26047C212yesmaximal, complemented
12#24814C212yesmaximal, complemented
13#22971C212yesmaximal, complemented
14#22671C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21454C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#479C2214yes—
21C2×#477C2212yes—
22C2×#448C2211yescharacteristic
23C2×#440C2214yes—
24C2×C4×SD16C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2589—22no—
28#2563C2211yescharacteristic
29#2552C2211yescharacteristic
30#2509C2212yes—
31#2497—21no—
32#2487—21no—
33#2454C2212yes—
34#2450—22no—
35#2369C2211yescharacteristic
36Q8×C4⋊C4—22no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2214yes—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2212yes—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2211yescharacteristic
44#2345—21no—
45C4×#121C2211yescharacteristic
46C4×#121C2212yes—
47C2×#389—22no—
48C2×#389C2212yes—
49C2×#384—21no—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2212yes—
54#2094C2212yes—
55#2086C2214yes—
56#2085C2214yes—
57#2083C2214yes—
58#2067C2214yes—
59#2059C2214yes—
60#1990C2212yes—
61#1948C2212yes—
62#1944C2214yes—
63#1943C2214yes—
64#1939C2212yes—
65#1918C2214yes—
66#1917C2214yes—
67#1880C2212yes—
68#1880C2214yes—
69#1874C2212yes—
70#1874C2211yescharacteristic
71#1865C2214yes—
72#1865C2212yes—
73#1863C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1862C2214yes—
77#1853C2212yes—
78#1853C2211yescharacteristic
79#1852C2212yes—
80#1852C2212yes—
81#1852C2212yes—
82#1845C2212yes—
83#1805C2214yes—
84#1803C2212yes—
85#1802C2212yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1787C2212yes—
89#1787C2211yescharacteristic
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125C2312yes—
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—48no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
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#479—48no—
130#477—28no—
131#448Q818yescomplemented
132#440—48no—
133C4×SD16—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#400—22no—
155#399—22no—
156#398—22no—
157#396—22no—
158#393—22no—
159#391—21no—
160#391—22no—
161#390C2312yes—
162#390C2311yescharacteristic
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—24no—
170#389—24no—
171#389—24no—
172#389—24no—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#389C2311yescharacteristic
180#388—21no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—21no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—22no—
193#384—22no—
194#384—24no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—22no—
206#380C2311yescharacteristic
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2×Q814yes—
241C22×Q8C2412yes—
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16C2×Q818yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261#125—44no—
262C22⋊Q8—82no—
263#121—22no—
264#121C2×Q814yes—
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—
279C4×D4—44no—
280C4×D4—84no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
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—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C42—28no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42—48no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—44no—
355#107—44no—
356#106—24no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—24no—
361#103—28no—
362#103—28no—
363#103—24no—
364#103C2×Q818yes—
365#102—24no—
366#102—44no—
367C22⋊C8—22no—
368C22⋊C8—24no—
369C22⋊C8C2×Q814yes—
370#97—44no—
371C4×C8—24no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—216no—
394C2×Q8—216no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8C22×Q814yes—
403C2×Q8C2×#4114yes—
404C2×Q8H(5,𝔽2)14yes—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C22×Q812yes—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4#13714yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×Q812yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461SD16—216no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8C22×Q814yes—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C22×Q812yes—
508C4⋊C4C2×#4114yes—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—81no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C42—28no—
519C42—28no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—48no—
528C42—48no—
529C42—42no—
530C42—82no—
531C42C2×#4114yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23D4×Q812yes—
544C23C2×#13712yes—
545C23#57612yes—
546C23C23×D411yescharacteristic
547Q8—216no—
548Q8—24no—
549Q8—24no—
550Q8—24no—
551Q8—162no—
552Q8#55418yes—
553D4—24no—
554D4—48no—
555D4—42no—
556C2×C4—216no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
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—21no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—84no—
609C2×C4—88no—
610C2×C4—84no—
611C2×C4—164no—
612C2×C4#52512yes—
613C2×C4#52512yes—
614C2×C4D4×Q812yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4#57512yes—
618C2×C4C23×Q811yescharacteristic
619C2×C4C22×#4112yes—
620C2×C4C2×H(5,𝔽2)12yes—
621C2×C4C2×2−1+411yescharacteristic
622C8—24no—
623C8—42no—
624C8—44no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#338514yes—
639C22C2×#52511yescharacteristic
640C22C2×D4×Q811yescharacteristic
641C22#348411yescharacteristic
642C22C22×#13711yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57611yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—28no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—44no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—84no—
660C4—82no—
661C4—82no—
662C4#311714yes—
663C4C2×#55412yes—
664C4#349212yes—
665C2—24no—
666C2—82no—
667C2#3654611yescharacteristic
668C2C2×#311711yescharacteristic
669C2#6058411yescharacteristic
670C2C2×#338511yescharacteristic
671C2#6136911yescharacteristic
672C2#6158411yescharacteristic
673C2#6160511yescharacteristic
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
3. #32960C2semidirect
4. #32957C2semidirect
5. #29303C2semidirect
7. #27950C2semidirect
9. #27316C2semidirect
10. #26387C2semidirect
11. #26047C2semidirect
12. #24814C2semidirect
13. #22971C2semidirect
14. #22671C2semidirect
15. #22629C2semidirect
131. #448Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×Q8×C4⋊C42
#329602
#329572
#293032
#292342
#279502
#276782
#273162
#263872
#260472
#248142
#229712
#226712
#226292
#214542
C4×#4482