Finite Group Databasebeta

← #6447978

Subgroups — #6447978

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#25008C214yesmaximal, complemented
4#25007C214yesmaximal, complemented
5#24996C212yesmaximal, complemented
6#24995C212yesmaximal, complemented
7#24993C212yesmaximal, complemented
8#24990C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23634C212yesmaximal
11#21302C211yesmaximal, characteristic, complemented
12C2×#1947C211yesmaximal, characteristic, complemented
13#21067C212yesmaximal
14#21037C212yesmaximal, complemented
15#20764C211yesmaximal, characteristic, complemented
16#20645C211yesmaximal, characteristic
17C4×#417C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C22×#107C2212yes—
21C22×C4⋊C8C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#417C2211yescharacteristic
24C2×#416C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2857—21no—
27#2857—21no—
28#2629—22no—
29#2582C2212yes—
30#2582C2212yes—
31#2538C2212yes—
32#2538C2212yes—
33#2417—22no—
34#2417—22no—
35#2370—22no—
36#2369—21no—
37#2365—21no—
38#2355C2212yes—
39#2355C2214yes—
40Q8×C4⋊C4C2212yes—
41#2329—22no—
42#2317C2212yes—
43#2316C2212yes—
44#2311C2212yes—
45C4×C22⋊Q8—22no—
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2274C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2074C2214yes—
55#2073C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#1985C2214yes—
59#1984C2212yes—
60#1984C2212yes—
61#1947C2212yes—
62#1947C2212yes—
63#1947C2214yes—
64#1928C2214yes—
65#1928C2214yes—
66#1925C2214yes—
67#1925C2214yes—
68#1921C2214yes—
69#1921C2214yes—
70#1918C2214yes—
71#1918C2214yes—
72#1836C2214yes—
73#1834C2212yes—
74#1834C2212yes—
75#1805C2212yes—
76#1804C2212yes—
77#1803C2211yescharacteristic
78#1803C2211yescharacteristic
79#1802C2211yescharacteristic
80#1802C2211yescharacteristic
81#1763C2212yes—
82#1762C2212yes—
83C4×#103C2212yes—
84C4×#102C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93C22×#41C2311yescharacteristic
94C22×M16C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C23×C8C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8D411yescharacteristic
101C2×#125D412yes—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8D412yes—
105C2×C22⋊Q8D412yes—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×#117—21no—
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C2×#107—28no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136#417—24no—
137#417—24no—
138#416—28no—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C4×M16—24no—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#398—24no—
151#398—24no—
152#396—24no—
153#396—24no—
154#391—22no—
155#391—22no—
156#391—24no—
157#391—22no—
158#391C2314yes—
159#390C2314yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#387—24no—
171#387—24no—
172#387—24no—
173#386—24no—
174#385—24no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2314yes—
180#384C2312yes—
181#384C2312yes—
182#382C2312yes—
183#382C2314yes—
184#380—21no—
185#380—21no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—24no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—24no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—24no—
204C2×#95—21no—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222#336C2314yes—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41#4114yes—
229C22×Q8C2×D412yes—
230C22×Q8C2×D412yes—
231C22×Q8C2411yescharacteristic
232C22×D4—21no—
233C22×D4C2411yescharacteristic
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
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—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258#125—28no—
259C22⋊Q8—28no—
260C22⋊Q8—28no—
261C22⋊Q8—84no—
262C22⋊Q8—84no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—84no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—44no—
277C4×D4—44no—
278#117—24no—
279#117—24no—
280#117—22no—
281#117—22no—
282#117—24no—
283#117—24no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4#4114yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4C2×D414yes—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—44no—
339C2×C42—82no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4114yes—
343C2×C42#4114yes—
344C2×C42C2414yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#107—28no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—24no—
352#103—24no—
353#103—24no—
354#103—24no—
355#103—24no—
356#102—24no—
357#102—24no—
358#102—24no—
359#102—24no—
360#97—24no—
361C4×C8—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374C24—21no—
375C24C22×D411yescharacteristic
376#41—28no—
377#41—28no—
378#41—44no—
379#41—44no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8C2×#4112yes—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4C2×#4112yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#12314yes—
432C22×C4#12314yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C4C2511yescharacteristic
447M16—44no—
448M16—44no—
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—22no—
460C2×C8—22no—
461C2×C8—48no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4C2×#4114yes—
499C22⋊C4—28no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—24no—
507C42—24no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—44no—
512C42—44no—
513C42—84no—
514C42—82no—
515C42—84no—
516C42#12314yes—
517C42#12314yes—
518C42C2×#4112yes—
519C42C2×#4112yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C2×#12312yes—
534C23#54812yes—
535C23#54812yes—
536C23C23×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—48no—
540Q8—41no—
541Q8—41no—
542Q8—84no—
543D4—22no—
544D4—22no—
545D4—48no—
546D4—41no—
547D4—41no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—164no—
592C2×C4C2×#12312yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4C2×#12312yes—
596C2×C4C2×#12312yes—
597C2×C4#52914yes—
598C2×C4#54114yes—
599C2×C4#54812yes—
600C2×C4#54812yes—
601C2×C4#55214yes—
602C2×C4#55614yes—
603C2×C4#57612yes—
604C2×C4#57812yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C8—42no—
612C8—42no—
613C8—42no—
614C8—42no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#242114yes—
629C22C22×#12311yescharacteristic
630C22C2×#54811yescharacteristic
631C22C2×#54811yescharacteristic
632C22C2×#57611yescharacteristic
633C22C2×#57811yescharacteristic
634C22#358511yescharacteristic
635C22#358611yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—81no—
644C4—81no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#242114yes—
650C4#309814yes—
651C4C2×#52911yescharacteristic
652C4C2×#54111yescharacteristic
653C4C2×#55211yescharacteristic
654C4C2×#55611yescharacteristic
655C2—24no—
656C2—82no—
657C2C2×#242111yescharacteristic
658C2C2×#309811yescharacteristic
659C2#6052011yescharacteristic
660C2#6133911yescharacteristic
661C2#6134211yescharacteristic
662C2#6134511yescharacteristic
663C2#6134811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34213C2semidirect
3. #25008C2semidirect
4. #25007C2semidirect
5. #24996C2semidirect
6. #24995C2semidirect
7. #24993C2semidirect
8. #24990C2semidirect
9. #23749C2semidirect
11. #21302C2semidirect
12. C2×#1947C2semidirect
14. #21037C2semidirect
15. #20764C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#342132
#250082
#250072
#249962
#249952
#249932
#249902
#237492
#236342
#213022
C2×#19472
#210672
#210372
#207642
#206452
C4×#4172