Finite Group Databasebeta

← #6534715

Subgroups — #6534715

Order 512.

1629 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36272C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30078C212yesmaximal, complemented
4#29688C211yesmaximal, characteristic, complemented
5#29392C212yesmaximal, complemented
6#29313C212yesmaximal, complemented
7#27948C211yesmaximal, characteristic, complemented
8#27645C211yesmaximal, characteristic
9#27034C214yesmaximal, complemented
10#26551C214yesmaximal
11#26088C212yesmaximal, complemented
12#24678C212yesmaximal, complemented
13#23677C212yesmaximal, complemented
14#23568C211yesmaximal, characteristic
15#22964C212yesmaximal, complemented
16C4⋊C4×SD16C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C2×#533C2211yescharacteristic
20C2×#481C2212yes—
21C2×#475C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2214yes—
24#2859C2212yes—
25#2857C2211yescharacteristic
26#2611—22no—
27#2553—22no—
28#2552C2211yescharacteristic
29#2542C2211yescharacteristic
30#2496C2212yes—
31#2489—21no—
32#2422C2212yes—
33#2415—21no—
34#2365C2211yescharacteristic
35#2360—21no—
36#2360—22no—
37#2360C2214yes—
38#2355—21no—
39#2355C2212yes—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42D4×C4⋊C4C2211yescharacteristic
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48#2293C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2096C2214yes—
53#2084C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2072C2214yes—
57#2071C2212yes—
58#2066C2214yes—
59#2058C2211yescharacteristic
60#2058C2212yes—
61#2055C2214yes—
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2035C2212yes—
65#2035C2214yes—
66#2023C2211yescharacteristic
67#2013C2211yescharacteristic
68#2000C2212yes—
69#1990C2212yes—
70#1948C2214yes—
71#1948C2214yes—
72#1948C2212yes—
73#1941C2214yes—
74#1941C2214yes—
75#1939C2212yes—
76#1926C2214yes—
77#1923C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1865C2214yes—
81#1865C2214yes—
82#1864C2214yes—
83#1853C2214yes—
84#1844C2212yes—
85#1839C2212yes—
86#1824C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2212yes—
93C4×#103C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×SD16C2312yes—
96#552—48no—
97#533—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#125—21no—
103C2×#125—21no—
104C2×#125—22no—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117—22no—
122C2×#117D411yescharacteristic
123C2×#117D411yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129#481—44no—
130#475—28no—
131C4×SD16—28no—
132C4×SD16—48no—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#399—22no—
153#398—22no—
154#397—22no—
155#397—22no—
156#394—22no—
157#391—21no—
158#391—22no—
159#391—21no—
160#391C2314yes—
161#390C2312yes—
162#390C2311yescharacteristic
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—24no—
172#389C2314yes—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—21no—
179#388—22no—
180#388—22no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#386—21no—
186#385—24no—
187#385—22no—
188#385—22no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#382—22no—
199#380—21no—
200#380—21no—
201#380C2311yescharacteristic
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—24no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8C2412yes—
237C22×D4—21no—
238C22×D4C2411yescharacteristic
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×SD16—24no—
246C2×SD16—24no—
247C2×SD16—24no—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254#126—48no—
255#125—28no—
256#125—44no—
257#125—48no—
258#123—48no—
259C22⋊Q8—28no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265#121—22no—
266#121—41no—
267C4×Q8—28no—
268C4×Q8—24no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—84no—
273C4×Q8#4118yes—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—42no—
277C4×D4—44no—
278#117—28no—
279#117—42no—
280#117—48no—
281#117SD1618yescomplemented
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#106—24no—
351#106—28no—
352#106—44no—
353#106—44no—
354#106—44no—
355C4⋊C8—28no—
356C4⋊C8—24no—
357#103—24no—
358#103—28no—
359#103—24no—
360#103—42no—
361#103—44no—
362#102—28no—
363#102—24no—
364C22⋊C8—24no—
365C22⋊C8—42no—
366C4×C8—24no—
367C4×C8—44no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D411yescharacteristic
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8C2×#4114yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4H(5,𝔽2)12yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C2×SD1614yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2511yescharacteristic
443SD16—216no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C82−1+414yes—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4C2×SD1614yes—
490C4⋊C4C2×SD1614yes—
491C4⋊C4C2×#4114yes—
492C22⋊C4—28no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4C2×SD1618yes—
504C42—22no—
505C42—22no—
506C42—28no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—22no—
512C42—42no—
513C42—42no—
514C42—44no—
515C42—42no—
516C42—84no—
517C42—84no—
518C42C2×SD1618yes—
519C42C2×#4114yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C22×SD1612yes—
534C23C23×D411yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—48no—
538Q8—42no—
539Q8—42no—
540Q8—84no—
541D4—24no—
542D4—42no—
543D4—84no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4—164no—
594C2×C4#52512yes—
595C2×C4#52512yes—
596C2×C4C22×SD1614yes—
597C2×C4C22×SD1612yes—
598C2×C4C22×SD1614yes—
599C2×C4C22×SD1612yes—
600C2×C4#57512yes—
601C2×C4#57712yes—
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×2−1+412yes—
605C2×C4#58511yescharacteristic
606C8—24no—
607C8—44no—
608C8—42no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×#52511yescharacteristic
624C22#348811yescharacteristic
625C22#349911yescharacteristic
626C22C23×SD1611yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358611yescharacteristic
630C4—24no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—44no—
639C4—44no—
640C4—81no—
641C4—84no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4#310314yes—
647C4#349212yes—
648C4#350712yes—
649C2—42no—
650C2—82no—
651C2#3568611yescharacteristic
652C2C2×#310311yescharacteristic
653C2#6058511yescharacteristic
654C2#6194811yescharacteristic
655C2#6198611yescharacteristic
656C2#6246911yescharacteristic
657C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #30078C2semidirect
4. #29688C2semidirect
5. #29392C2semidirect
6. #29313C2semidirect
7. #27948C2semidirect
9. #27034C2semidirect
11. #26088C2semidirect
12. #24678C2semidirect
13. #23677C2semidirect
15. #22964C2semidirect
16. C4⋊C4×SD16C2semidirect
17. #22639C2semidirect
281. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362722
#342142
#300782
#296882
#293922
#293132
#279482
#276452
#270342
#265512
#260882
#246782
#236772
#235682
#229642
C4⋊C4×SD162
#226392