Finite Group Databasebeta

← #6570678

Subgroups — #6570678

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36411C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29939C211yesmaximal, characteristic, complemented
4#29914C212yesmaximal, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27856C212yesmaximal, complemented
7#27677C211yesmaximal, characteristic, complemented
8#27615C212yesmaximal, complemented
9#26279C212yesmaximal, complemented
10#26203C214yesmaximal, complemented
11#24993C212yesmaximal, complemented
12#24940C214yesmaximal, complemented
13#23755C211yesmaximal, characteristic, complemented
14#23744C212yesmaximal, complemented
15#22707C212yesmaximal, complemented
16#22489C212yesmaximal, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×C8×D4C2212yes—
21#2857C2211yescharacteristic
22#2854C2212yes—
23#2567C2211yescharacteristic
24#2538C2211yescharacteristic
25#2485C2212yes—
26#2474—22no—
27#2461—22no—
28#2422—21no—
29#2414C2212yes—
30#2407—21no—
31#2362—21no—
32#2355—21no—
33#2355C2212yes—
34#2354—21no—
35#2354—21no—
36#2354C2211yescharacteristic
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40#2314—22no—
41#2311—21no—
42C4×#121C2211yescharacteristic
43#2296C2212yes—
44#2285C2211yescharacteristic
45#2285C2212yes—
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48#2274—21no—
49#2101C2214yes—
50#2101C2214yes—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2099C2212yes—
54#2099C2212yes—
55#2099C2212yes—
56#2094C2211yescharacteristic
57#2093C2212yes—
58#2085C2214yes—
59#2084C2211yescharacteristic
60#2083C2212yes—
61#2077C2212yes—
62#2076C2214yes—
63#2073C2214yes—
64#2071C2211yescharacteristic
65#2034C2212yes—
66#1990C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1925C2214yes—
72#1921C2214yes—
73#1919C2214yes—
74#1918C2214yes—
75#1893C2214yes—
76#1875C2214yes—
77#1874C2212yes—
78#1869C2214yes—
79#1854C2214yes—
80#1854C2214yes—
81#1852C2212yes—
82#1834C2212yes—
83#1825C2214yes—
84#1823C2214yes—
85#1817C2212yes—
86#1808C2214yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2212yes—
90C4×#102C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92#1735C2214yes—
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×C8C2314yes—
97#555—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#124—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×#121C2311yescharacteristic
106C4×#41—28no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4D412yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—22no—
117C2×#117—21no—
118C2×#117D412yes—
119C2×#117D411yescharacteristic
120C2×#117C2312yes—
121C2×#117C2312yes—
122C2×#117C2314yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C8×D4—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C4×C8C2312yes—
141#397—22no—
142#396—22no—
143#395—22no—
144#394—22no—
145#394—22no—
146#392—22no—
147#391—22no—
148#391—21no—
149#391—21no—
150#390—22no—
151#390—21no—
152#390—22no—
153#390C2311yescharacteristic
154#390C2312yes—
155#389—21no—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2311yescharacteristic
160#389C2312yes—
161#388—22no—
162#388—24no—
163#388—24no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—21no—
169#387—24no—
170#387—24no—
171#387—22no—
172#386—24no—
173#386—22no—
174#386—21no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2311yescharacteristic
185#382—21no—
186#382—22no—
187#382C2314yes—
188#380—22no—
189#380—22no—
190#380—21no—
191#380—21no—
192#380—22no—
193#380—21no—
194#380C2312yes—
195#380C2314yes—
196#380C2312yes—
197#380C2312yes—
198#380C2311yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—24no—
212C4×C22⋊C4—21no—
213#340C2314yes—
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2314yes—
231C2×#41—24no—
232C2×#41—24no—
233C2×#41—42no—
234C22×Q8—21no—
235C22×Q8C2×D411yescharacteristic
236C22×D4—21no—
237C22×D4—22no—
238C22×D4C2×D412yes—
239C22×D4C2×D411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C22×C8—28no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256#124—48no—
257#123—84no—
258C22⋊Q8—44no—
259C22⋊Q8—82no—
260#121—28no—
261#121—22no—
262#121—41no—
263#121—44no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×D4—24no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—44no—
271C4×D4#4118yes—
272#117—24no—
273#117—28no—
274#117—28no—
275#117—24no—
276#117—42no—
277#117—44no—
278#117—44no—
279#117—44no—
280#117—44no—
281#117—84no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—28no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—44no—
348#107—44no—
349#106—24no—
350#106—42no—
351C4⋊C8—24no—
352#102—42no—
353#102—44no—
354#102—42no—
355C22⋊C8—28no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358C4×C8—24no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—21no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374C24—21no—
375C24C22×D412yes—
376C24C22×D411yescharacteristic
377#41—416no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—42no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—81no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4#12314yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4#13514yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4118yes—
442C22×C4C2511yescharacteristic
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—88no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4#12314yes—
486C4⋊C42−1+412yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—82no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4#12318yes—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—24no—
511C42—44no—
512C42—44no—
513C42—42no—
514C42—44no—
515C42—84no—
516C42—84no—
517C42C2×#4114yes—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—21no—
523C23—22no—
524C23—21no—
525C23—22no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×#12314yes—
538C23C2×#13512yes—
539C23C23×D411yescharacteristic
540Q8—44no—
541Q8—162no—
542D4—28no—
543D4—48no—
544D4—42no—
545D4—42no—
546D4—84no—
547D4—164no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—164no—
599C2×C4C2×#12314yes—
600C2×C4C2×#12312yes—
601C2×C4C2×#12312yes—
602C2×C4C2×#13512yes—
603C2×C4C2×#13512yes—
604C2×C4#57512yes—
605C2×C4#57612yes—
606C2×C4C22×#4112yes—
607C2×C4C2×2−1+411yescharacteristic
608C2×C4#58512yes—
609C2×C4#58511yescharacteristic
610C8—42no—
611C8—44no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—24no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22C22×#12311yescharacteristic
635C22#348911yescharacteristic
636C22#350011yescharacteristic
637C22C22×#13511yescharacteristic
638C22C2×#57511yescharacteristic
639C22C2×#57611yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—81no—
650C4—84no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#309114yes—
657C4#351512yes—
658C4#352112yes—
659C2—42no—
660C2—44no—
661C2—161no—
662C2#3606711yescharacteristic
663C2C2×#309111yescharacteristic
664C2#6052111yescharacteristic
665C2#6209711yescharacteristic
666C2#6213611yescharacteristic
667C2#6252711yescharacteristic
668C2#6258111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36411C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #29939C2semidirect
4. #29914C2semidirect
5. #27960C2semidirect
6. #27856C2semidirect
7. #27677C2semidirect
8. #27615C2semidirect
9. #26279C2semidirect
10. #26203C2semidirect
11. #24993C2semidirect
12. #24940C2semidirect
13. #23755C2semidirect
14. #23744C2semidirect
15. #22707C2semidirect
16. #22489C2semidirect
17. #22469C2semidirect

Maximal subgroups

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

subgroupindex
#364112
#341882
#299392
#299142
#279602
#278562
#276772
#276152
#262792
#262032
#249932
#249402
#237552
#237442
#227072
#224892
#224692