Finite Group Databasebeta

← #7534672

Subgroups — #7534672

Order 512.

1899 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59821C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#31343C214yesmaximal, complemented
4#29208C211yesmaximal, characteristic, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28230C212yesmaximal, complemented
7#26236C212yesmaximal, complemented
8#25547C214yesmaximal, complemented
9#25393C212yesmaximal, complemented
10#25016C212yesmaximal, complemented
11#24968C212yesmaximal, complemented
12#24119C212yesmaximal, complemented
13#23176C211yesmaximal, characteristic, complemented
14#22713C212yesmaximal, complemented
15#22504C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#547C2212yes—
20C2×#532—21no—
21C2×#484C2214yes—
22C2×#474C2211yescharacteristic
23C2×#468C2214yes—
24C2×#461C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×#436C2212yes—
27C2×#436C2212yes—
28#2868C2211yescharacteristic
29#2857—21no—
30#2727—22no—
31#2548—21no—
32#2547C2212yes—
33#2542C2212yes—
34#2538C2212yes—
35#2537C2212yes—
36#2537C2212yes—
37#2512—22no—
38#2448—22no—
39#2432—22no—
40#2422—22no—
41#2422—22no—
42#2415C2214yes—
43#2354C2212yes—
44Q8×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46D4×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2076C2214yes—
52#2057C2214yes—
53#2036C2214yes—
54#2036C2214yes—
55#2036C2214yes—
56#2033C2214yes—
57#2024C2214yes—
58#2000C2214yes—
59#1990C2212yes—
60#1988C2211yescharacteristic
61#1981C2211yescharacteristic
62#1956C2212yes—
63#1954C2212yes—
64#1954C2212yes—
65#1948C2211yescharacteristic
66#1947C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1927C2212yes—
70#1918C2212yes—
71#1894C2214yes—
72#1877C2212yes—
73#1875C2214yes—
74#1872C2214yes—
75#1870C2214yes—
76#1862C2212yes—
77#1854C2214yes—
78#1852C2214yes—
79#1845C2212yes—
80#1834C2212yes—
81#1825C2214yes—
82#1809C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87C4×#107C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#102C2212yes—
92#1736C2214yes—
93C22×#41—22no—
94C22×SD16C2312yes—
95C22×M16C2314yes—
96#556—28no—
97#556—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×#126—24no—
108C2×#124—22no—
109C2×#124—22no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×#117C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C42—21no—
131#484—48no—
132#474—24no—
133#468—48no—
134#461—24no—
135C4×SD16—44no—
136#436—28no—
137#436—44no—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2312yes—
159C2×C4×C8C2312yes—
160#399—24no—
161#397—24no—
162#397—24no—
163#396—24no—
164#394—24no—
165#394—24no—
166#394—24no—
167#391—22no—
168#391—21no—
169#391C2312yes—
170#390—22no—
171#390—22no—
172#390—22no—
173#390D412yes—
174#390D412yes—
175#390D412yes—
176#389—22no—
177#389—22no—
178#389—24no—
179#389—22no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389C2312yes—
184#388—24no—
185#388—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#380C2314yes—
199#380C2314yes—
200#380C2314yes—
201C4×C4⋊C4—22no—
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#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
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—21no—
239C23×C4—21no—
240C23×C4C2414yes—
241C2×SD16—24no—
242C2×SD16—24no—
243C2×SD16—42no—
244C2×M16—28no—
245C2×M16—44no—
246C2×M16—44no—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—44no—
257C4⋊Q8—82no—
258C4⋊Q8#4114yescomplemented
259#127—24no—
260#126—416no—
261#124—48no—
262#124—48no—
263#123—416no—
264C22⋊Q8—28no—
265C22⋊Q8—44no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—28no—
270#121—44no—
271#121—48no—
272#121—48no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—48no—
276C4×Q8—44no—
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—
291#117—28no—
292#117—44no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—24no—
350#107—24no—
351#107—44no—
352#107—44no—
353#106—42no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8#4114yescomplemented
358#103—24no—
359#103—24no—
360#103—42no—
361#103—44no—
362#103—44no—
363#102—24no—
364#102—24no—
365#102—42no—
366C22⋊C8—28no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369C4×C8—24no—
370C4×C8—24no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381C24—21no—
382C24C22×D414yes—
383#41—832no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—84no—
413C2×D4H(5,𝔽2)12yes—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—84no—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D414yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448SD16—48no—
449M16—88no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
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×C8C2×#4114yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×#4114yes—
495C4⋊C4H(5,𝔽2)12yes—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—41no—
514C42—48no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42AGL(1,ℤ/8)14yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C42H(5,𝔽2)12yes—
524C422−1+412yes—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C23×D411yescharacteristic
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—88no—
546D4—22no—
547D4—22no—
548D4—48no—
549D4—48no—
550D4—48no—
551D4—41no—
552D4—41no—
553D4—816no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—816no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4#54612yes—
599C2×C4#54612yes—
600C2×C4#54712yes—
601C2×C4#54712yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4#57812yes—
606C2×C4#57812yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C8—24no—
614C8—42no—
615C8—42no—
616C8—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×#54611yescharacteristic
632C22C2×#54711yescharacteristic
633C22#348411yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C2×#57811yescharacteristic
636C22C2×#57811yescharacteristic
637C22#358511yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—41no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—84no—
649C4—82no—
650C4—81no—
651C4#333014yes—
652C4#335314yes—
653C4#348311yescharacteristic
654C4#351911yescharacteristic
655C4#352211yescharacteristic
656C4#354811yescharacteristic
657C2—84no—
658C2—82no—
659C2#5903911yescharacteristic
660C2C2×#333011yescharacteristic
661C2#6120311yescharacteristic
662C2C2×#335311yescharacteristic
663C2#6128211yescharacteristic
664C2#6158211yescharacteristic
665C2#6166511yescharacteristic
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. #59821C2semidirect
3. #31343C2semidirect
4. #29208C2semidirect
5. #29141C2semidirect
6. #28230C2semidirect
7. #26236C2semidirect
8. #25547C2semidirect
9. #25393C2semidirect
10. #25016C2semidirect
11. #24968C2semidirect
12. #24119C2semidirect
13. #23176C2semidirect
14. #22713C2semidirect
15. #22504C2semidirect
16. C4×#474C2semidirect
17. C4×#461C2semidirect
258. C4⋊Q8#41semidirect
357. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#598212
#361272
#313432
#292082
#291412
#282302
#262362
#255472
#253932
#250162
#249682
#241192
#231762
#227132
#225042
C4×#4742
C4×#4612