Finite Group Databasebeta

← #7534629

Subgroups — #7534629

Order 512.

1961 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#59823C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#31320C214yesmaximal, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28223C212yesmaximal, complemented
7#26234C212yesmaximal, complemented
8#25531C214yesmaximal, complemented
9#25385C212yesmaximal, complemented
10#25014C212yesmaximal, complemented
11#24967C212yesmaximal, complemented
12#24118C212yesmaximal, complemented
13#23173C211yesmaximal, characteristic, complemented
14#22703C212yesmaximal, complemented
15#22487C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#547C2212yes—
20C2×#532—21no—
21C2×#483C2214yes—
22C2×#476C2211yescharacteristic
23C2×#465C2214yes—
24C2×#461C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×C8×D4C2212yes—
27C2×C8×D4C2212yes—
28#2868C2211yescharacteristic
29#2857—21no—
30#2732—22no—
31#2552—21no—
32#2547C2212yes—
33#2542C2212yes—
34#2538—22no—
35#2538C2212yes—
36#2537C2212yes—
37#2537C2212yes—
38#2454—22no—
39#2432—22no—
40#2422—22no—
41#2422—22no—
42#2417C2214yes—
43#2366—21no—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46D4×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2077C2214yes—
52#2058C2214yes—
53#2035C2214yes—
54#2035C2214yes—
55#2035C2214yes—
56#2034C2214yes—
57#2024C2214yes—
58#2000C2214yes—
59#1990C2212yes—
60#1987C2211yescharacteristic
61#1981C2211yescharacteristic
62#1956C2212yes—
63#1954C2212yes—
64#1954C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1948C2211yescharacteristic
69#1927C2212yes—
70#1920C2212yes—
71#1892C2214yes—
72#1877C2212yes—
73#1874C2214yes—
74#1872C2214yes—
75#1869C2214yes—
76#1864C2212yes—
77#1853C2214yes—
78#1853C2214yes—
79#1844C2212yes—
80#1834C2212yes—
81#1824C2214yes—
82#1807C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87C4×#106C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#102C2212yes—
92C2×#340C2214yes—
93C22×#41—22no—
94C22×SD16C2312yes—
95C23×C8C2314yes—
96#554—28no—
97#554—44no—
98#547—44no—
99#547—44no—
100#532—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127D411yescharacteristic
107C2×#125—22no—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C42—21no—
132#483—48no—
133#476—24no—
134#465—48no—
135#461—24no—
136C4×SD16—44no—
137C8×D4—28no—
138C8×D4—44no—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#399—24no—
162#398—24no—
163#397—24no—
164#396—24no—
165#396—24no—
166#394—24no—
167#394—24no—
168#391—22no—
169#391—21no—
170#391C2312yes—
171#390—22no—
172#390—22no—
173#390D412yes—
174#390D412yes—
175#390D412yes—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389C2312yes—
184#386—24no—
185#385—24no—
186#385—24no—
187#385—24no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384D411yescharacteristic
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#380—22no—
197#380—22no—
198#380—24no—
199#380C2314yes—
200#380C2314yes—
201#380C2314yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C43C2311yescharacteristic
213#340C2312yes—
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#336C2312yes—
224#336C2312yes—
225C2×#41—48no—
226C2×#41—48no—
227C2×#41—84no—
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8C2412yes—
231C22×D4—24no—
232C22×D4C2×D412yes—
233C22×D4C2×D412yes—
234C22×D4C2×D412yes—
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2414yes—
242C2×SD16—24no—
243C2×SD16—24no—
244C2×SD16—42no—
245C22×C8—28no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8#4114yescomplemented
260#127SD1618yescomplemented
261#125—48no—
262#124—48no—
263#123—416no—
264C22⋊Q8—28no—
265C22⋊Q8—416no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270#121—28no—
271#121—44no—
272#121—48no—
273#121—48no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—82no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—48no—
290C4×D4—48no—
291C4×D4—48no—
292C4×D4—84no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4C2×D414yes—
335C2×C22⋊C4C2×D414yes—
336C2×C22⋊C4C2×D414yes—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#106—24no—
351#106—24no—
352#106—42no—
353#106—44no—
354#106—44no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8#4114yescomplemented
359#103—24no—
360#103—24no—
361#103—42no—
362#103—44no—
363#103—44no—
364#102—24no—
365#102—24no—
366#102—42no—
367C22⋊C8—28no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370C4×C8—24no—
371C4×C8—24no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382C24—21no—
383C24C22×D414yes—
384#41—832no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—84no—
411C2×D4C2×SD1618yes—
412C2×D4C2×SD1618yes—
413C2×D4C2×SD1618yes—
414C2×D4H(5,𝔽2)12yes—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D414yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—88no—
465C2×C8C2×#4114yes—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C2×#4114yes—
500C4⋊C4H(5,𝔽2)12yes—
501C22⋊C4—24no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—41no—
518C42—44no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42C2×SD1614yes—
524C42C2×#4112yes—
525C42C2×#4112yes—
526C42H(5,𝔽2)12yes—
527C422−1+412yes—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×SD1618yes—
542C23C23×D411yescharacteristic
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—84no—
548Q8—88no—
549D4—216no—
550D4—22no—
551D4—22no—
552D4—216no—
553D4—48no—
554D4—41no—
555D4—41no—
556D4—816no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—816no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—168no—
602C2×C4#54612yes—
603C2×C4#54612yes—
604C2×C4#54712yes—
605C2×C4#54712yes—
606C2×C4C22×SD1612yes—
607C2×C4C22×SD1612yes—
608C2×C4C22×SD1612yes—
609C2×C4#57712yes—
610C2×C4#57812yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C8—24no—
618C8—42no—
619C8—42no—
620C8—42no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22C2×#54611yescharacteristic
636C22C2×#54711yescharacteristic
637C22#348411yescharacteristic
638C22C23×SD1611yescharacteristic
639C22C2×#57711yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358511yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—41no—
649C4—82no—
650C4—81no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4#329814yes—
656C4#333514yes—
657C4#348311yescharacteristic
658C4#351911yescharacteristic
659C4#352211yescharacteristic
660C4#354811yescharacteristic
661C2—48no—
662C2—82no—
663C2#5903911yescharacteristic
664C2C2×#329811yescharacteristic
665C2#6120011yescharacteristic
666C2C2×#333511yescharacteristic
667C2#6128211yescharacteristic
668C2#6156011yescharacteristic
669C2#6166711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59823C2semidirect
3. #31320C2semidirect
4. #29206C2semidirect
5. #29141C2semidirect
6. #28223C2semidirect
7. #26234C2semidirect
8. #25531C2semidirect
9. #25385C2semidirect
10. #25014C2semidirect
11. #24967C2semidirect
12. #24118C2semidirect
13. #23173C2semidirect
14. #22703C2semidirect
15. #22487C2semidirect
16. C4×#476C2semidirect
17. C4×#461C2semidirect
259. C4⋊Q8#41semidirect
260. #127SD16semidirect
358. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#598232
#361272
#313202
#292062
#291412
#282232
#262342
#255312
#253852
#250142
#249672
#241182
#231732
#227032
#224872
C4×#4762
C4×#4612