Finite Group Databasebeta

← #6446790

Subgroups — #6446790

Order 512.

1766 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#35181C211yesmaximal, characteristic, complemented
2#34183C211yesmaximal, characteristic
3#26121C214yesmaximal, complemented
4#26118C214yesmaximal, complemented
5#26113C212yesmaximal, complemented
6#26110C212yesmaximal, complemented
7#26108C212yesmaximal, complemented
8#26107C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23668C212yesmaximal, complemented
11C2×#1948C211yesmaximal, characteristic, complemented
12#21145C211yesmaximal, characteristic, complemented
13#20882C212yesmaximal
14#20764C211yesmaximal, characteristic, complemented
15#20652C212yesmaximal
16#20645C211yesmaximal, characteristic
17C4×#417C211yesmaximal, characteristic
18C2×#520C2212yes—
19C2×C4×#41C2211yescharacteristic
20C22×#106C2212yes—
21C2×#424C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C2×#417C2211yescharacteristic
24C2×#417C2211yescharacteristic
25C2×#416C2212yes—
26C2×C4×M16C2211yescharacteristic
27#2851—22no—
28#2646—22no—
29#2571—21no—
30#2571C2212yes—
31#2570—22no—
32#2570C2214yes—
33#2552—21no—
34#2552C2212yes—
35Q8×C4⋊C4C2212yes—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—22no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41#2317—22no—
42#2317C2212yes—
43#2317C2212yes—
44#2317C2212yes—
45#2285—21no—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384—21no—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2100C2214yes—
54#2099C2214yes—
55#2088C2214yes—
56#2088C2214yes—
57#2088C2214yes—
58#2088C2214yes—
59#1948C2212yes—
60#1948C2214yes—
61#1948C2212yes—
62#1940C2214yes—
63#1939C2212yes—
64#1939C2212yes—
65#1865C2214yes—
66#1865C2214yes—
67#1864C2214yes—
68#1864C2214yes—
69#1836C2214yes—
70#1834C2212yes—
71#1834C2212yes—
72#1805C2214yes—
73#1805C2212yes—
74#1804C2212yes—
75#1804C2214yes—
76#1803C2211yescharacteristic
77#1803C2214yes—
78#1803C2211yescharacteristic
79#1802C2214yes—
80#1802C2211yescharacteristic
81#1802C2211yescharacteristic
82#1763C2212yes—
83#1762C2212yes—
84C4×#103C2212yes—
85C4×#102C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93C22×#41C2311yescharacteristic
94C22×M16C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C23×C8C2312yes—
97C2×#125—22no—
98#520—28no—
99#520—44no—
100C4×#41—24no—
101C4×#41Q818yes—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—24no—
108C2×C4×D4C2314yes—
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×#117—22no—
112C2×#117—21no—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C2×#106—44no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130#424—44no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137#417—24no—
138#417—24no—
139#416—28no—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C4×M16—24no—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151#400—24no—
152#400—24no—
153#398—24no—
154#398—24no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389—24no—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#387—24no—
166#387—24no—
167#385—24no—
168#385—24no—
169#385—24no—
170#385—24no—
171#385—24no—
172#384—22no—
173#384—22no—
174#384—22no—
175#384—22no—
176#384—21no—
177#384—21no—
178#384D411yescharacteristic
179#384D411yescharacteristic
180#384C2314yes—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382—22no—
187#382—22no—
188#382D412yes—
189#382C2312yes—
190#382C2314yes—
191#380—21no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380D412yes—
196#380D412yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C2×#95—21no—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#336C2314yes—
223#336C2314yes—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41C2×Q814yes—
228C2×#41#4114yes—
229C2×#41#4114yes—
230C22×Q8C2411yescharacteristic
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256#125—84no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—28no—
260C4×Q8—44no—
261C4×Q8C2×Q814yes—
262C4×Q8C2×Q814yes—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—28no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—44no—
269C4×D4—416no—
270C4×D4C2×Q814yes—
271C4×D4C2×Q814yes—
272#117—22no—
273#117—24no—
274#117—24no—
275#117—28no—
276#117—24no—
277#117—24no—
278#117—48no—
279#117C2×Q814yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4C2×D414yes—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—28no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—82no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×Q814yes—
342C2×C42#4114yes—
343C2×C42C2414yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#106—44no—
347#106—44no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353#103—24no—
354#103—24no—
355#103—24no—
356#103—24no—
357#102—24no—
358#102—24no—
359#102—24no—
360#102—24no—
361#97—24no—
362C4×C8—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D414yes—
374#95C2×D414yes—
375C24—21no—
376C24C22×D411yescharacteristic
377#41—28no—
378#41—28no—
379#41—28no—
380#41#125116yescomplemented
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8C22×Q812yes—
385C2×Q8C2×#4112yes—
386C2×Q8C2×#4112yes—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4C22×Q812yes—
394C2×D4C2×#4112yes—
395C2×D4C2×#4112yes—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4C22⋊Q814yes—
430C22×C4C22⋊Q814yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×Q812yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444C22×C4C2511yescharacteristic
445M16—44no—
446M16—44no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—48no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4C22×Q812yes—
499C4⋊C4C22×Q812yes—
500C22⋊C4—24no—
501C22⋊C4—28no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—48no—
512C42—44no—
513C42—44no—
514C42—82no—
515C42C22⋊Q814yes—
516C42C22⋊Q814yes—
517C42C22×Q812yes—
518C42C22×Q812yes—
519C42C2×#4112yes—
520C42C2×#4112yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×C22⋊Q812yes—
533C23#54612yes—
534C23#54812yes—
535C23C23×D411yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8C2×#12514yes—
540D4—216no—
541D4—22no—
542D4—22no—
543D4—22no—
544D4C2×#12514yes—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—216no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—168no—
589C2×C4—164no—
590C2×C4C2×C22⋊Q812yes—
591C2×C4C2×C22⋊Q812yes—
592C2×C4C2×C22⋊Q812yes—
593C2×C4C2×C22⋊Q812yes—
594C2×C4C2×C22⋊Q812yes—
595C2×C4C2×#12514yes—
596C2×C4#54414yes—
597C2×C4#54612yes—
598C2×C4#54812yes—
599C2×C4#55214yes—
600C2×C4#55314yes—
601C2×C4#57712yes—
602C2×C4#57712yes—
603C2×C4C23×Q811yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C8—42no—
610C8—42no—
611C8—42no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22#244214yes—
627C22C22×C22⋊Q811yescharacteristic
628C22C2×#54611yescharacteristic
629C22C2×#54811yescharacteristic
630C22C2×#57711yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358511yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—84no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#244214yes—
648C4#308414yes—
649C4C22×#12511yescharacteristic
650C4C2×#54411yescharacteristic
651C4C2×#55211yescharacteristic
652C4C2×#55311yescharacteristic
653C2—24no—
654C2—44no—
655C2C2×#244211yescharacteristic
656C2C2×#308411yescharacteristic
657C2#6049211yescharacteristic
658C2#6120611yescharacteristic
659C2#6120911yescharacteristic
660C2#6134611yescharacteristic
661C2#6134711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #26121C2semidirect
4. #26118C2semidirect
5. #26113C2semidirect
6. #26110C2semidirect
7. #26108C2semidirect
8. #26107C2semidirect
9. #23748C2semidirect
10. #23668C2semidirect
11. C2×#1948C2semidirect
12. #21145C2semidirect
14. #20764C2semidirect
380. #41#125semidirect

Maximal subgroups

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

subgroupindex
#351812
#341832
#261212
#261182
#261132
#261102
#261082
#261072
#237482
#236682
C2×#19482
#211452
#208822
#207642
#206522
#206452
C4×#4172