Finite Group Databasebeta

← #6446770

Subgroups — #6446770

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2#34181C211yesmaximal, characteristic
3#26117C214yesmaximal, complemented
4#26115C214yesmaximal, complemented
5#26111C212yesmaximal, complemented
6#26110C212yesmaximal, complemented
7#26107C212yesmaximal, complemented
8#26106C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23671C212yesmaximal, complemented
11#21170C211yesmaximal, characteristic, complemented
12#21143C211yesmaximal, characteristic, complemented
13#20876C212yesmaximal
14#20760C211yesmaximal, characteristic, complemented
15#20650C212yesmaximal
16#20646C211yesmaximal, characteristic
17C2×C4×#102C211yesmaximal, characteristic
18C2×#518C2212yes—
19C22×C4×D4C2211yescharacteristic
20C2×#428C2212yes—
21C2×#424C2212yes—
22C2×#423C2211yescharacteristic
23C2×#418C2212yes—
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2851—22no—
28#2646—22no—
29#2571—21no—
30#2571C2212yescomplemented
31#2570—22no—
32#2570C2214yescomplemented
33#2552—21no—
34#2552C2212yescomplemented
35#2349—22no—
36#2349C2214yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41#2317C2212yes—
42#2317C2212yes—
43#2316C2212yes—
44#2315—22no—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#380—22no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2101C2214yescomplemented
52#2100C2214yescomplemented
53#2099C2214yescomplemented
54#2088C2214yescomplemented
55#2088C2214yescomplemented
56#2087C2214yescomplemented
57#2086C2214yescomplemented
58#1949C2214yescomplemented
59#1948C2212yescomplemented
60#1947C2212yescomplemented
61#1939C2212yescomplemented
62#1939C2212yescomplemented
63#1939C2212yescomplemented
64#1939C2212yescomplemented
65#1864C2214yes—
66#1864C2214yes—
67#1862C2214yes—
68#1862C2214yes—
69#1835C2212yescomplemented
70#1835C2212yescomplemented
71#1834C2211yescharacteristic, complemented
72#1834C2211yescharacteristic, complemented
73#1834C2211yescharacteristic, complemented
74#1834C2211yescharacteristic, complemented
75#1804C2214yes—
76#1804C2214yes—
77#1804C2214yes—
78#1802C2214yes—
79#1802C2214yes—
80#1802C2212yes—
81#1802C2212yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85C4×#102C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1735C2212yes—
89#1735C2212yes—
90C2×#340C2211yescharacteristic
91C2×#340C2211yescharacteristic
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2312yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C2×#125—22no—
99#518—28no—
100#518—44no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—24no—
106C2×C4×D4Q818yes—
107C2×C4×D4C2314yes—
108C2×C4×D4C2314yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—24no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123#428—44no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126#424—44no—
127#423—24no—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132#418—28no—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C4×C8—24no—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#400—24no—
147#400—24no—
148#398—24no—
149#398—24no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#387—24no—
161#387—24no—
162#385—24no—
163#385—24no—
164#385—24no—
165#385—24no—
166#385—24no—
167#385—24no—
168#384—22no—
169#384D411yescharacteristic
170#384D411yescharacteristic
171#384C2314yes—
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#382—22no—
177#382—22no—
178#382D412yes—
179#382C2312yes—
180#382C2314yes—
181#382C2312yes—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—22no—
187#380—22no—
188#380—24no—
189#380—24no—
190#380D412yes—
191#380D412yes—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C2×#95—21no—
203C2×#95—24no—
204#340C2312yes—
205#340C2312yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C22×D4—21no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4C2×Q814yes—
228C22×D4#4114yes—
229C22×D4#4114yes—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—28no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—82no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×M16—24no—
245C2×M16—24no—
246C2×M16—24no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8C2411yescharacteristic
254C22×C8C2411yescharacteristic
255C22×C8C2411yescharacteristic
256C22×C8C2411yescharacteristic
257C22×C8C2414yes—
258#125—84no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—48no—
268C4×D4—48no—
269C4×D4C2×Q818yes—
270C4×D4C2×Q818yes—
271#117—24no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—84no—
300C2×C4⋊C4—84no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×Q814yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—416no—
323C2×C22⋊C4C2×D414yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—82no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×Q814yes—
332C2×C42#4114yes—
333C2×C42C2414yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—44no—
337#106—44no—
338C4⋊C8—24no—
339C4⋊C8—24no—
340C4⋊C8—44no—
341C4⋊C8—44no—
342#102—24no—
343#102—24no—
344#102—24no—
345#102—24no—
346#102—28no—
347#102—28no—
348C4×C8—22no—
349C4×C8—22no—
350C4×C8—22no—
351C4×C8—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95C2×D414yes—
365#95C2×D414yes—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—81no—
371C24—81no—
372C24—81no—
373C24—81no—
374C24C22×D411yescharacteristic
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4#125116yescomplemented
385C2×D4C22×Q814yes—
386C2×D4C2×#4114yes—
387C2×D4C2×#4114yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—816no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4C22⋊Q814yes—
422C22×C4C22⋊Q814yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×Q812yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C22×C4C2511yescharacteristic
437M16—48no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—24no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—44no—
454C2×C8—44no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4C22×Q814yes—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C42—24no—
500C42—24no—
501C42—44no—
502C42—44no—
503C42—44no—
504C42—82no—
505C42C22⋊Q814yes—
506C42C22⋊Q814yes—
507C42C22×Q812yes—
508C42C22×Q812yes—
509C42C2×#4112yes—
510C42C2×#4112yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—216no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23C2×C22⋊Q812yes—
530C23#54612yes—
531C23#54812yes—
532C23C23×D411yescharacteristic
533D4—24no—
534D4—216no—
535D4—24no—
536D4—24no—
537D4C2×#12518yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—416no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—88no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—168no—
579C2×C4—168no—
580C2×C4C2×C22⋊Q812yes—
581C2×C4C2×C22⋊Q812yes—
582C2×C4C2×C22⋊Q812yes—
583C2×C4C2×C22⋊Q812yes—
584C2×C4C2×C22⋊Q812yes—
585C2×C4C2×#12514yes—
586C2×C4#54414yes—
587C2×C4#54612yes—
588C2×C4#54812yes—
589C2×C4#55214yes—
590C2×C4#55314yes—
591C2×C4#57612yes—
592C2×C4#57712yes—
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C8—41no—
600C8—41no—
601C8—41no—
602C8—41no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—28no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—816no—
618C22#244214yes—
619C22C22×C22⋊Q811yescharacteristic
620C22C2×#54611yescharacteristic
621C22C2×#54811yescharacteristic
622C22C2×#57611yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358411yescharacteristic
625C22#358411yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4#244214yes—
639C4#308114yes—
640C4C22×#12511yescharacteristic
641C4C2×#54411yescharacteristic
642C4C2×#55211yescharacteristic
643C4C2×#55311yescharacteristic
644C2—24no—
645C2—48no—
646C2C2×#244211yescharacteristic
647C2C2×#308111yescharacteristic
648C2#6048911yescharacteristic
649C2#6121111yescharacteristic
650C2#6121211yescharacteristic
651C2#6132511yescharacteristic
652C2#6132611yescharacteristic
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. #35181C2semidirect
3. #26117C2semidirect
4. #26115C2semidirect
5. #26111C2semidirect
6. #26110C2semidirect
7. #26107C2semidirect
8. #26106C2semidirect
9. #23749C2semidirect
10. #23671C2semidirect
11. #21170C2semidirect
12. #21143C2semidirect
14. #20760C2semidirect
30. #2571C22semidirect
32. #2570C22semidirect
34. #2552C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
54. #2088C22semidirect
55. #2088C22semidirect
56. #2087C22semidirect
57. #2086C22semidirect
58. #1949C22semidirect
59. #1948C22semidirect
60. #1947C22semidirect
61. #1939C22semidirect
62. #1939C22semidirect
63. #1939C22semidirect
64. #1939C22semidirect
69. #1835C22semidirect
70. #1835C22semidirect
71. #1834C22semidirect
72. #1834C22semidirect
73. #1834C22semidirect
74. #1834C22semidirect
384. C2×D4#125semidirect

Maximal subgroups

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

subgroupindex
#351812
#341812
#261172
#261152
#261112
#261102
#261072
#261062
#237492
#236712
#211702
#211432
#208762
#207602
#206502
#206462
C2×C4×#1022