Finite Group Databasebeta

← #6971691

Subgroups — #6971691

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36416C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#32789C212yesmaximal, complemented
4#31355C214yesmaximal, complemented
5#31084C214yesmaximal, complemented
6#30305C211yesmaximal, characteristic, complemented
7#29971C212yesmaximal, complemented
8#29694C212yesmaximal, complemented
9#29208C211yesmaximal, characteristic, complemented
10#29121C211yesmaximal, characteristic, complemented
11#26111C212yesmaximal, complemented
12#26045C212yesmaximal, complemented
13#25013C212yesmaximal, complemented
14#24968C212yesmaximal, complemented
15#24062C212yesmaximal, complemented
16#23815C211yesmaximal, characteristic, complemented
17#22008C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#482C2214yes—
21C2×#478C2214yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2589—22no—
27#2587—22no—
28#2580—21no—
29#2571C2211yescharacteristic
30#2566—21no—
31#2552C2211yescharacteristic
32#2547—21no—
33#2537—21no—
34#2537C2212yes—
35#2441C2212yes—
36#2439C2212yes—
37#2432—22no—
38#2422—22no—
39#2422C2214yes—
40#2370—21no—
41#2355—21no—
42#2354C2211yescharacteristic
43#2354C2211yescharacteristic
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2296C2212yes—
49C42×D4C2211yescharacteristic
50#2101C2214yes—
51#2099C2211yescharacteristic
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2088C2214yes—
55#2087C2214yes—
56#2084C2214yes—
57#2077C2214yes—
58#2076C2214yes—
59#2071C2212yes—
60#2059C2214yes—
61#2058C2212yes—
62#2058C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2041C2211yescharacteristic
66#2041C2211yescharacteristic
67#2041C2212yes—
68#2034C2214yes—
69#2033C2214yes—
70#2023C2212yes—
71#1990C2212yes—
72#1954C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1934C2214yes—
77#1923C2214yes—
78#1875C2214yes—
79#1864C2214yes—
80#1862C2214yes—
81#1854C2214yes—
82#1845C2212yes—
83#1844C2212yes—
84#1834C2212yes—
85#1825C2214yes—
86#1817C2212yes—
87#1804C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94#1744C2211yescharacteristic
95C22×#41—22no—
96#547—48no—
97#532—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125—21no—
104C2×#125C2312yes—
105C2×#124—22no—
106C2×#123—24no—
107C2×C22⋊Q8—22no—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121C2314yes—
111C2×C4×Q8—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2314yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4—22no—
123C22×C42C2312yes—
124#482—48no—
125#478—48no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#399—22no—
144#398—22no—
145#397—24no—
146#396—22no—
147#396—22no—
148#394—24no—
149#394—24no—
150#393—22no—
151#391—21no—
152#391—22no—
153#391—22no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#385—24no—
184#385—24no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—21no—
195#382—21no—
196#382C2312yes—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2312yes—
232C2×#41—48no—
233C2×#41—48no—
234C2×#41—84no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—24no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—22no—
251C4⋊Q8—41no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255#127—24no—
256#125—44no—
257#125—44no—
258#124—48no—
259#123—416no—
260C22⋊Q8—48no—
261#121—28no—
262#121—44no—
263#121—48no—
264#121—48no—
265C4×Q8—44no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—44no—
273C4×D4—48no—
274C4×D4—44no—
275C4×D4—44no—
276#117—24no—
277#117—42no—
278#117—44no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—44no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—44no—
343#107—44no—
344#106—44no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—44no—
348#102—24no—
349#102—28no—
350#102—24no—
351#102—24no—
352#102—44no—
353#102—42no—
354C22⋊C8—44no—
355C22⋊C8—44no—
356C4×C8—24no—
357C4×C8—42no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—22no—
376C24C22×D412yes—
377#41—832no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—81no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—84no—
401C2×D4H(5,𝔽2)14yes—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—84no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C42−1+412yes—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C42—24no—
498C42—22no—
499C42—21no—
500C42—21no—
501C42—42no—
502C42—42no—
503C42—41no—
504C42—41no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—82no—
509C42#13514yes—
510C42AGL(1,ℤ/8)14yes—
511C422−1+414yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C23×D411yescharacteristic
527Q8—84no—
528Q8—88no—
529D4—24no—
530D4—24no—
531D4—48no—
532D4—48no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—816no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—84no—
584C2×C4—816no—
585C2×C4—88no—
586C2×C4#53612yes—
587C2×C4#53612yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4C2×#13512yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×H(5,𝔽2)12yes—
595C2×C4C2×2−1+411yescharacteristic
596C2×C4C2×2−1+412yes—
597C2×C4#58511yescharacteristic
598C8—42no—
599C8—44no—
600C8—42no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C2×#53611yescharacteristic
616C22#348411yescharacteristic
617C22#348911yescharacteristic
618C22C22×#13511yescharacteristic
619C22C22×AGL(1,ℤ/8)11yescharacteristic
620C22#358511yescharacteristic
621C22#358611yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4—84no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4#322814yes—
639C4#354412yes—
640C4#355912yes—
641C2—82no—
642C2—84no—
643C2#4248211yescharacteristic
644C2C2×#322811yescharacteristic
645C2#6088611yescharacteristic
646C2#6156511yescharacteristic
647C2#6158211yescharacteristic
648C2#6210211yescharacteristic
649C2#6210711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36416C2semidirect
3. #32789C2semidirect
4. #31355C2semidirect
5. #31084C2semidirect
6. #30305C2semidirect
7. #29971C2semidirect
8. #29694C2semidirect
9. #29208C2semidirect
10. #29121C2semidirect
11. #26111C2semidirect
12. #26045C2semidirect
13. #25013C2semidirect
14. #24968C2semidirect
15. #24062C2semidirect
16. #23815C2semidirect
17. #22008C2semidirect

Maximal subgroups

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

subgroupindex
#364162
#361272
#327892
#313552
#310842
#303052
#299712
#296942
#292082
#291212
#261112
#260452
#250132
#249682
#240622
#238152
#220082