Finite Group Databasebeta

← #7533240

Subgroups — #7533240

Order 512.

1861 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59821C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#31411C214yesmaximal, complemented
4#29219C211yesmaximal, characteristic, complemented
5#28717C211yesmaximal, characteristic, complemented
6#28606C212yesmaximal, complemented
7#28150C212yesmaximal, complemented
8#26303C212yesmaximal, complemented
9#26220C214yesmaximal, complemented
10#25571C212yesmaximal, complemented
11#24971C212yesmaximal, complemented
12#24119C212yesmaximal, complemented
13#23176C211yesmaximal, characteristic, complemented
14#22504C212yesmaximal, complemented
15#22500C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#547C2212yes—
20C2×#532—21no—
21C2×#487C2212yes—
22C2×#474C2211yescharacteristic
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#436C2212yes—
26C2×#436C2212yes—
27#2868C2211yescharacteristic
28#2857—21no—
29#2727—22no—
30#2548—21no—
31#2547C2212yes—
32#2542C2212yes—
33#2538C2212yes—
34#2537C2212yes—
35#2537C2212yes—
36#2512—22no—
37#2448—22no—
38#2432—22no—
39#2422—22no—
40#2422—22no—
41#2415C2214yes—
42#2354C2212yes—
43Q8×C4⋊C4—21no—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C42×D4C2211yescharacteristic
50#2099C2214yes—
51#2077C2214yes—
52#2076C2214yes—
53#2072C2214yes—
54#2036C2214yes—
55#2036C2214yes—
56#2000C2214yes—
57#2000C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2211yescharacteristic
63#1981C2211yescharacteristic
64#1954C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1927C2212yes—
69#1918C2212yes—
70#1895C2214yes—
71#1894C2214yes—
72#1894C2214yes—
73#1877C2212yes—
74#1875C2214yes—
75#1872C2214yes—
76#1872C2214yes—
77#1870C2214yes—
78#1862C2212yes—
79#1854C2214yes—
80#1834C2212yes—
81#1825C2214yes—
82#1824C2214yes—
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—28no—
99#547—44no—
100#532—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
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#487—44no—
132#474—24no—
133#461—24no—
134C4×SD16—44no—
135#436—44no—
136#436—44no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2312yes—
157#399—24no—
158#397—24no—
159#397—24no—
160#396—24no—
161#394—24no—
162#394—24no—
163#394—24no—
164#391—22no—
165#391—21no—
166#391C2312yes—
167#390—22no—
168#390—22no—
169#390—22no—
170#390D412yes—
171#390D412yes—
172#390D412yes—
173#389—22no—
174#389—22no—
175#389—24no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389C2312yes—
181#388—24no—
182#388—24no—
183#385—24no—
184#385—24no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384D411yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#380—22no—
194#380—22no—
195#380C2314yes—
196#380C2314yes—
197#380C2314yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C43C2311yescharacteristic
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#336C2312yes—
223#336C2312yes—
224C2×#41—48no—
225C2×#41—48no—
226C2×#41—84no—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8C2412yes—
230C22×D4—24no—
231C22×D4C2×D412yes—
232C22×D4C2×D412yes—
233C22×D4C2×D412yes—
234C22×D4C2411yescharacteristic
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2414yes—
240C2×SD16—24no—
241C2×SD16—42no—
242C2×SD16—42no—
243C2×M16—44no—
244C2×M16—44no—
245C2×M16—44no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yescomplemented
258#127—24no—
259#126—416no—
260#124—48no—
261#124—48no—
262#123—416no—
263C22⋊Q8—28no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268#121—28no—
269#121—44no—
270#121—48no—
271#121—48no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—48no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—41no—
284C4×D4—41no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4#4114yes—
290#117—28no—
291#117—44no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4C2×D414yes—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—42no—
350#106—42no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355#103—24no—
356#103—42no—
357#103—42no—
358#103—44no—
359#103—44no—
360#102—24no—
361#102—42no—
362#102—42no—
363C22⋊C8—44no—
364C22⋊C8—44no—
365C22⋊C8—44no—
366C4×C8—24no—
367C4×C8—24no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378C24—21no—
379C24C22×D414yes—
380#41—832no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—84no—
410C2×D4C2×#4112yes—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—48no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—84no—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D414yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445SD16—48no—
446M16—88no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
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—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4C2×#4114yes—
492C4⋊C4C2×#4112yes—
493C22⋊C4—24no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—42no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—42no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—41no—
511C42—48no—
512C42—44no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42AGL(1,ℤ/8)14yes—
518C42C2×#4112yes—
519C42C2×#4112yes—
520C42H(5,𝔽2)12yes—
521C42H(5,𝔽2)12yes—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C23×D411yescharacteristic
537Q8—48no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—88no—
543D4—22no—
544D4—22no—
545D4—48no—
546D4—48no—
547D4—41no—
548D4—41no—
549D4—48no—
550D4—816no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—84no—
593C2×C4—816no—
594C2×C4—84no—
595C2×C4#54612yes—
596C2×C4#54612yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4#57712yes—
603C2×C4#57812yes—
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4#58511yescharacteristic
609C2×C4#58511yescharacteristic
610C8—42no—
611C8—42no—
612C8—42no—
613C8—42no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22C2×#54611yescharacteristic
629C22C2×#54611yescharacteristic
630C22#348511yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C2×#57711yescharacteristic
633C22C2×#57811yescharacteristic
634C22#358611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—44no—
640C4—42no—
641C4—44no—
642C4—82no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4—84no—
647C4—81no—
648C4#332414yes—
649C4#333014yes—
650C4#348011yescharacteristic
651C4#350611yescharacteristic
652C4#351711yescharacteristic
653C4#354611yescharacteristic
654C2—84no—
655C2—82no—
656C2#5903311yescharacteristic
657C2C2×#332411yescharacteristic
658C2C2×#333011yescharacteristic
659C2#6120211yescharacteristic
660C2#6121011yescharacteristic
661C2#6171711yescharacteristic
662C2#6178911yescharacteristic
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. #31411C2semidirect
4. #29219C2semidirect
5. #28717C2semidirect
6. #28606C2semidirect
7. #28150C2semidirect
8. #26303C2semidirect
9. #26220C2semidirect
10. #25571C2semidirect
11. #24971C2semidirect
12. #24119C2semidirect
13. #23176C2semidirect
14. #22504C2semidirect
15. #22500C2semidirect
16. C4×#474C2semidirect
17. C4×#461C2semidirect
257. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#598212
#361272
#314112
#292192
#287172
#286062
#281502
#263032
#262202
#255712
#249712
#241192
#231762
#225042
#225002
C4×#4742
C4×#4612