Finite Group Databasebeta

← #6537762

Subgroups — #6537762

Order 512.

1665 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34321C211yesmaximal, characteristic
3#30106C212yesmaximal, complemented
4#29798C211yesmaximal, characteristic, complemented
5#29338C212yesmaximal, complemented
6#29277C211yesmaximal, characteristic, complemented
7#28748C211yesmaximal, characteristic, complemented
8#27420C214yesmaximal, complemented
9#26642C214yesmaximal, complemented
10#26233C212yesmaximal, complemented
11#25568C212yesmaximal
12#24065C212yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23574C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23002C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C2×#533C2212yes—
20C2×#487C2212yes—
21C2×#485C2214yes—
22C2×#461C2212yes—
23C2×#440C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552—21no—
27#2542C2212yes—
28#2538—21no—
29#2441—22no—
30#2429C2214yes—
31#2423—22no—
32#2369C2211yescharacteristic
33#2367—22no—
34#2365C2212yes—
35#2363C2212yes—
36#2361—21no—
37#2354—21no—
38#2354C2211yescharacteristic
39Q8×C4⋊C4—21no—
40D4×C4⋊C4C2211yescharacteristic
41#2347—21no—
42#2347C2212yes—
43#2337—22no—
44#2324—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48#2296C2212yes—
49#2293C2211yescharacteristic
50#2274—21no—
51C42×D4C2211yescharacteristic
52#2100C2212yes—
53#2100C2211yescharacteristic
54#2094C2211yescharacteristic
55#2088C2212yes—
56#2065C2212yes—
57#2058C2211yescharacteristic
58#2057C2212yes—
59#2043C2212yes—
60#2043C2212yes—
61#2043C2212yes—
62#2043C2212yes—
63#2036C2214yes—
64#2035C2214yes—
65#2023C2211yescharacteristic
66#2013C2212yes—
67#2013C2212yes—
68#2000C2214yes—
69#2000C2214yes—
70#1987C2212yes—
71#1981C2214yes—
72#1981C2212yes—
73#1956C2212yes—
74#1954C2212yes—
75#1947C2212yes—
76#1935C2214yes—
77#1935C2214yes—
78#1927C2214yes—
79#1924C2214yes—
80#1921C2214yes—
81#1877C2214yes—
82#1873C2214yes—
83#1873C2214yes—
84#1873C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1831C2211yescharacteristic
88#1817C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×SD16C2314yes—
96#555—24no—
97#555—24no—
98#533—48no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—22no—
105C2×#124—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2314yes—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2314yes—
115C2×C4×D4—22no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C42C2312yes—
128#487—28no—
129#485—48no—
130#461—28no—
131#440—48no—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#399—24no—
151#398—24no—
152#396—24no—
153#391—21no—
154#391—21no—
155#391—22no—
156#391C2314yes—
157#390—21no—
158#390—22no—
159#390—22no—
160#390C2311yescharacteristic
161#390C2311yescharacteristic
162#389—21no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389C2312yes—
170#389C2312yes—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—24no—
179#388—21no—
180#388—22no—
181#388—21no—
182#388—22no—
183#387—24no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#383—22no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2312yes—
234#336C2311yescharacteristic
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8C2414yes—
238C22×D4—22no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×SD16—28no—
247C2×SD16—44no—
248C2×SD16—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8#4114yes—
259#125—48no—
260#124—216no—
261C22⋊Q8—28no—
262C22⋊Q8—44no—
263C22⋊Q8—48no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266#121—28no—
267#121—24no—
268#121—24no—
269C4×Q8—28no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—48no—
273C4×Q8—82no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—42no—
279C4×D4—44no—
280C4×D4—84no—
281C4×D4#4114yes—
282#117—28no—
283#117—24no—
284#117—24no—
285#117—48no—
286#117—44no—
287#117—44no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#106—24no—
355#106—42no—
356C4⋊C8—24no—
357C4⋊C8—42no—
358#103—28no—
359#103—28no—
360#103—28no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#102—24no—
366#102—42no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369#97—44no—
370C4×C8—24no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D412yes—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×D4—28no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4C2×#4112yes—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445SD16—416no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—82no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4C2×#4112yes—
488C4⋊C4C2×#4114yes—
489C4⋊C4H(5,𝔽2)14yes—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—48no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—84no—
505C42—28no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42—41no—
514C42—48no—
515C42—42no—
516C42—44no—
517C42—82no—
518C42—82no—
519C42—82no—
520C42#13714yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C23×D411yescharacteristic
538Q8—48no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—44no—
543Q8—84no—
544D4—24no—
545D4—44no—
546D4—42no—
547D4—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—164no—
598C2×C4#52512yes—
599C2×C4#52512yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4#57512yes—
604C2×C4#57712yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4#58512yes—
609C8—44no—
610C8—42no—
611C8—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22C2×#52511yescharacteristic
627C22#348411yescharacteristic
628C22#350011yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57511yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358611yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—81no—
645C4—82no—
646C4—84no—
647C4—81no—
648C4—82no—
649C4#311614yes—
650C4#350612yes—
651C4#351712yes—
652C2—82no—
653C2—82no—
654C2#3570411yescharacteristic
655C2C2×#311611yescharacteristic
656C2#6059111yescharacteristic
657C2#6158411yescharacteristic
658C2#6160411yescharacteristic
659C2#6254011yescharacteristic
660C2#6258111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36134C2semidirect
3. #30106C2semidirect
4. #29798C2semidirect
5. #29338C2semidirect
6. #29277C2semidirect
7. #28748C2semidirect
8. #27420C2semidirect
9. #26642C2semidirect
10. #26233C2semidirect
12. #24065C2semidirect
13. #23775C2semidirect
14. #23574C2semidirect
15. #23221C2semidirect
16. #23002C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343212
#301062
#297982
#293382
#292772
#287482
#274202
#266422
#262332
#255682
#240652
#237752
#235742
#232212
#230022
#228392