Finite Group Databasebeta

← #6433945

Subgroups — #6433945

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic
3#30375C214yesmaximal, complemented
4#30374C214yesmaximal, complemented
5#30371C214yesmaximal, complemented
6#30370C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic, complemented
8#23732C211yesmaximal, characteristic, complemented
9#23642C212yesmaximal
10#23554C211yesmaximal, characteristic, complemented
11#23481C212yesmaximal
12#23455C211yesmaximal, characteristic, complemented
13#21159C212yesmaximal, complemented
14#21094C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#419C2212yes—
22C2×#419C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2629C2212yes—
26#2629C2212yes—
27#2628C2212yes—
28#2628C2212yes—
29#2627—21no—
30#2627—21no—
31#2626—21no—
32#2626—21no—
33#2625—22no—
34#2625—22no—
35#2625C2214yes—
36#2625C2214yes—
37#2396—22no—
38#2396—22no—
39C2×#400—21no—
40C2×#398—21no—
41C2×#397—21no—
42C2×#397—21no—
43#2296C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48#2285C2211yescharacteristic
49#2280C2212yes—
50C2×#380C2211yescharacteristic
51#2101C2214yescomplemented
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2097C2214yescomplemented
55#2096C2212yescomplemented
56#2095C2212yescomplemented
57#2077C2214yes—
58#2077C2214yes—
59#2076C2214yes—
60#2075C2214yes—
61#2059C2214yescomplemented
62#2058C2212yescomplemented
63#2057C2212yescomplemented
64#2036C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2035C2214yes—
68#2026C2214yescomplemented
69#2024C2212yescomplemented
70#2024C2212yescomplemented
71#1944C2214yescomplemented
72#1943C2214yescomplemented
73#1942C2214yescomplemented
74#1941C2214yescomplemented
75#1935C2214yescomplemented
76#1935C2214yescomplemented
77#1933C2214yescomplemented
78#1933C2214yescomplemented
79#1750C2214yes—
80#1750C2214yes—
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×M16C2312yes—
90C22×M16C2312yes—
91#533—44no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125C2312yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×#117—44no—
103C2×#117D412yes—
104C2×#117D412yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4D411yescharacteristic
120C22×C22⋊C4D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#428—28no—
124#428—44no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129#419—28no—
130#419—44no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135#407—28no—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C4×M16—44no—
139C2×#97C2314yes—
140C2×C4×C8C2314yes—
141#400—24no—
142#400—24no—
143#400—22no—
144#400—22no—
145#399—22no—
146#399—22no—
147#398—24no—
148#398—24no—
149#398—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#397—24no—
154#397—24no—
155#397—24no—
156#397—24no—
157#396—22no—
158#396—22no—
159#396—22no—
160#396—22no—
161#395—22no—
162#395—22no—
163#393—22no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391C2312yes—
169#391C2312yes—
170#391C2314yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#385—22no—
182#385—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382C2312yes—
186#381—22no—
187#380—22no—
188#380—21no—
189#380—21no—
190#380C2312yes—
191#380C2314yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#379—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2312yes—
220#336C2312yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—24no—
226C22×Q8—24no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8C2412yes—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C2×M16—24no—
250C2×M16—24no—
251C2×M16#4118yes—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256#125—42no—
257#125—42no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C22⋊Q8—48no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C4×Q8—44no—
264C4×Q8—44no—
265#117—24no—
266#117—24no—
267#117—24no—
268#117—24no—
269#117—44no—
270#117—44no—
271#117—42no—
272#117—42no—
273#117—42no—
274#117—42no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340#107—28no—
341#107—44no—
342#106—28no—
343#106—44no—
344#103—28no—
345#103—28no—
346#103—44no—
347#103—44no—
348C22⋊C8—28no—
349C22⋊C8—28no—
350#97—44no—
351C4×C8—44no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—41no—
376C24—41no—
377C24—41no—
378C24—41no—
379C24C22×D412yes—
380C24C22×D411yescharacteristic
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×Q8—81no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4114yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C22×C4C2511yescharacteristic
455M16—216no—
456M16—48no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8C2×#4118yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—84no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—44no—
518C42—44no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23#53212yes—
538C23#53412yes—
539C23#53612yes—
540C23C23×D411yescharacteristic
541Q8—44no—
542Q8—44no—
543Q8—84no—
544Q8—84no—
545Q8—84no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4#53212yes—
606C2×C4#53412yes—
607C2×C4#53612yes—
608C2×C4#53918yes—
609C2×C4#54218yes—
610C2×C4C22×#4112yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C8—28no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—88no—
630C22#266314yes—
631C22C2×#53211yescharacteristic
632C22C2×#53411yescharacteristic
633C22C2×#53611yescharacteristic
634C22#358511yescharacteristic
635C22#358611yescharacteristic
636C22#358611yescharacteristic
637C22#358611yescharacteristic
638C4—42no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4#266314yes—
654C4C2×#53912yes—
655C4C2×#54212yes—
656C2—24no—
657C2—82no—
658C2C2×#266311yescharacteristic
659C2#6070311yescharacteristic
660C2#6070511yescharacteristic
661C2#6075411yescharacteristic
662C2#6075511yescharacteristic
663C2#6086911yescharacteristic
664C2#6087011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30375C2semidirect
4. #30374C2semidirect
5. #30371C2semidirect
6. #30370C2semidirect
7. #23747C2semidirect
8. #23732C2semidirect
10. #23554C2semidirect
12. #23455C2semidirect
13. #21159C2semidirect
14. #21094C2semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
54. #2097C22semidirect
55. #2096C22semidirect
56. #2095C22semidirect
61. #2059C22semidirect
62. #2058C22semidirect
63. #2057C22semidirect
68. #2026C22semidirect
69. #2024C22semidirect
70. #2024C22semidirect
71. #1944C22semidirect
72. #1943C22semidirect
73. #1942C22semidirect
74. #1941C22semidirect
75. #1935C22semidirect
76. #1935C22semidirect
77. #1933C22semidirect
78. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#26252
#303752
#303742
#303712
#303702
#237472
#237322
#236422
#235542
#234812
#234552
#211592
#210942
#203772