Finite Group Databasebeta

← #6458454

Subgroups — #6458454

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#35172C211yesmaximal, characteristic
3#27743C214yesmaximal, complemented
4#27739C214yesmaximal, complemented
5#27736C212yesmaximal, complemented
6#27735C212yesmaximal, complemented
7#27728C212yesmaximal, complemented
8#27727C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23632C212yesmaximal
11#21322C211yesmaximal, characteristic, complemented
12#21311C211yesmaximal, characteristic, complemented
13#21089C212yesmaximal
14#20763C212yesmaximal, complemented
15C2×#1802C211yesmaximal, characteristic
16#20441C211yesmaximal, characteristic, complemented
17C4×#418C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#418C2211yescharacteristic
22C2×#418C2211yescharacteristic
23C22×#102C2212yes—
24C2×#405C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2859—21no—
27#2859—21no—
28#2848—22no—
29#2647—22no—
30#2626—22no—
31#2580—22no—
32#2571C2212yescomplemented
33#2571C2212yescomplemented
34#2566C2214yes—
35#2563C2212yes—
36#2552C2212yescomplemented
37#2552C2212yescomplemented
38#2542C2212yes—
39#2538—21no—
40#2538—21no—
41#2442—22no—
42#2442—22no—
43#2317C2212yes—
44#2317C2212yes—
45#2316C2212yes—
46#2309C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2101C2214yescomplemented
51#2101C2214yescomplemented
52#2100C2214yescomplemented
53#2099C2214yescomplemented
54#2073C2214yes—
55#2073C2214yes—
56#2071C2214yes—
57#2071C2214yes—
58#1992C2214yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1989C2212yescomplemented
62#1989C2212yescomplemented
63#1988C2211yescharacteristic, complemented
64#1988C2211yescharacteristic, complemented
65#1987C2211yescharacteristic, complemented
66#1987C2211yescharacteristic, complemented
67#1936C2214yes—
68#1936C2214yes—
69#1934C2214yes—
70#1934C2214yes—
71#1836C2214yescomplemented
72#1836C2214yescomplemented
73#1835C2214yescomplemented
74#1834C2214yescomplemented
75#1802C2212yes—
76#1802C2212yes—
77#1802C2214yes—
78#1771C2214yescomplemented
79#1770C2212yescomplemented
80C8×C4⋊C4C2212yescomplemented
81#1762C2212yes—
82#1762C2212yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85#1736C2211yescharacteristic
86#1736C2212yes—
87#1736C2212yes—
88#1736C2211yescharacteristic
89#1735C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—22no—
103C2×#125—22no—
104C2×#125C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C42C2311yescharacteristic
127C2×C4⋊C8—24no—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131#418—24no—
132#418—24no—
133C2×#102—28no—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140#405—28no—
141C4×M16—24no—
142C2×#97C2314yes—
143C2×#97C2312yes—
144C2×C4×C8C2314yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#400—24no—
148#399—24no—
149#398—24no—
150#398—24no—
151#396—24no—
152#396—24no—
153#393—24no—
154#390—22no—
155#390—22no—
156#390C2314yes—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#385—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#384—21no—
175#384—21no—
176#384D411yescharacteristic
177#384D411yescharacteristic
178#384C2314yes—
179#384C2312yes—
180#384C2312yes—
181#384C2314yes—
182#382—22no—
183#382—22no—
184#382D412yes—
185#382C2314yes—
186#382C2314yes—
187#380—22no—
188#380—22no—
189#380—24no—
190#380D412yes—
191#380D412yes—
192#379—24no—
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201#376—21no—
202C43—21no—
203#340C2314yes—
204#340C2314yes—
205#340C2312yes—
206#340C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—44no—
239C23×C4—82no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16#4114yes—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261#125—24no—
262#125—24no—
263#125—84no—
264#125—84no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—84no—
268C22⋊Q8—84no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—44no—
276C4×D4—44no—
277#117—24no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—22no—
282#117—22no—
283#117—22no—
284#117—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4114yes—
342C2×C42#4114yes—
343C2×C42C2414yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C4⋊C8—24no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349#102—28no—
350#102—28no—
351#102—24no—
352#102—24no—
353#102—24no—
354#102—24no—
355#97—28no—
356#97—24no—
357C4×C8—28no—
358C4×C8—24no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—22no—
372C24—24no—
373C24—42no—
374C24—42no—
375C24—81no—
376C24—81no—
377C24—81no—
378C24—81no—
379C24C22×D411yescharacteristic
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×D4—28no—
387C2×D4—24no—
388C2×D4—28no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4C2×#4114yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—816no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4#12414yes—
425C22×C4#12414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C22×C4C2511yescharacteristic
440M16—28no—
441M16—44no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—48no—
454C2×C8C2×#4114yes—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C42—24no—
504C42—24no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42#12414yes—
513C42#12414yes—
514C42C2×#4112yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—48no—
528C23—42no—
529C23—42no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23C2×#12412yes—
537C23#54712yes—
538C23#54812yes—
539C23C23×D411yescharacteristic
540Q8—84no—
541Q8—84no—
542D4—24no—
543D4—24no—
544D4—42no—
545D4—42no—
546D4—48no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—88no—
586C2×C4—84no—
587C2×C4—88no—
588C2×C4—164no—
589C2×C4C2×#12412yes—
590C2×C4C2×#12412yes—
591C2×C4C2×#12412yes—
592C2×C4C2×#12412yes—
593C2×C4C2×#12412yes—
594C2×C4#52914yes—
595C2×C4#54114yes—
596C2×C4#54712yes—
597C2×C4#54812yes—
598C2×C4#55214yes—
599C2×C4#55214yes—
600C2×C4#57612yes—
601C2×C4#57712yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C8—24no—
609C8—42no—
610C8—42no—
611C8—42no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—816no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22#248914yes—
627C22C22×#12411yescharacteristic
628C22C2×#54711yescharacteristic
629C22C2×#54811yescharacteristic
630C22C2×#57611yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358511yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—44no—
638C4—44no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—84no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#248914yes—
648C4#313614yes—
649C4C2×#52911yescharacteristic
650C4C2×#54111yescharacteristic
651C4C2×#55211yescharacteristic
652C4C2×#55211yescharacteristic
653C2—24no—
654C2—84no—
655C2C2×#248911yescharacteristic
656C2C2×#313611yescharacteristic
657C2#6062111yescharacteristic
658C2#6127811yescharacteristic
659C2#6127911yescharacteristic
660C2#6133911yescharacteristic
661C2#6134011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #27743C2semidirect
4. #27739C2semidirect
5. #27736C2semidirect
6. #27735C2semidirect
7. #27728C2semidirect
8. #27727C2semidirect
9. #23749C2semidirect
11. #21322C2semidirect
12. #21311C2semidirect
14. #20763C2semidirect
16. #20441C2semidirect
32. #2571C22semidirect
33. #2571C22semidirect
36. #2552C22semidirect
37. #2552C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
58. #1992C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1989C22semidirect
62. #1989C22semidirect
63. #1988C22semidirect
64. #1988C22semidirect
65. #1987C22semidirect
66. #1987C22semidirect
71. #1836C22semidirect
72. #1836C22semidirect
73. #1835C22semidirect
74. #1834C22semidirect
78. #1771C22semidirect
79. #1770C22semidirect
80. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#352862
#351722
#277432
#277392
#277362
#277352
#277282
#277272
#237492
#236322
#213222
#213112
#210892
#207632
C2×#18022
#204412
C4×#4182