Finite Group Databasebeta

← #6989341

Subgroups — #6989341

Order 512.

1323 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36449C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic, complemented
3#32839C212yesmaximal, complemented
4#31360C214yesmaximal, complemented
5#30639C214yesmaximal
6#30377C212yesmaximal, complemented
7#30083C212yesmaximal, complemented
8#29922C211yesmaximal, characteristic, complemented
9#28790C212yesmaximal, complemented
10#28720C212yesmaximal, complemented
11#27972C211yesmaximal, characteristic, complemented
12#27768C212yesmaximal, complemented
13#27645C211yesmaximal, characteristic
14#26527C212yesmaximal, complemented
15#26107C212yesmaximal, complemented
16#23754C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic, complemented
18C2×#488C2214yes—
19C2×#481C2212yes—
20#2863C2212yes—
21#2859C2212yes—
22#2859C2212yes—
23#2857C2211yescharacteristic
24#2647—22no—
25#2578—22no—
26#2571—21no—
27#2571C2211yescharacteristic
28#2571C2212yes—
29#2552—21no—
30#2552C2212yes—
31#2547C2211yescharacteristic
32#2530—22no—
33#2495—21no—
34#2485—21no—
35#2442—22no—
36#2442C2214yes—
37#2427C2212yes—
38#2421C2212yes—
39#2365C2211yescharacteristic
40#2352—21no—
41D4×C4⋊C4C2211yescharacteristic
42#2345—21no—
43#2317—21no—
44#2316—21no—
45C4×#124C2212yes—
46C4×#121C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2099C2211yescharacteristic
52#2097C2214yes—
53#2096C2212yes—
54#2093C2211yescharacteristic
55#2088C2214yes—
56#2085C2214yes—
57#2084C2212yes—
58#2083C2211yescharacteristic
59#2075C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2071C2211yescharacteristic
64#2071C2212yes—
65#2066C2214yes—
66#2059C2214yes—
67#2042C2214yes—
68#2035C2214yes—
69#2033C2212yes—
70#2026C2214yes—
71#1991C2214yes—
72#1981C2212yes—
73#1956C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1941C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1923C2214yes—
80#1872C2214yes—
81#1864C2214yes—
82#1863C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1824C2212yes—
86#1805C2214yes—
87#1804C2214yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—22no—
97C2×#125C2314yes—
98C2×#124C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×#121C2311yescharacteristic
101C2×C4×Q8C2312yes—
102C2×C4×D4—21no—
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2312yes—
105C2×#117—21no—
106C2×#117D411yescharacteristic
107C2×#117D411yescharacteristic
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2312yes—
112C22×C42C2311yescharacteristic
113#488—48no—
114#481—44no—
115C2×#107C2312yes—
116C2×#107C2312yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2312yes—
127C2×#102C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2314yes—
132#400—24no—
133#400—22no—
134#400—24no—
135#400—22no—
136#398—24no—
137#398—24no—
138#397—22no—
139#397—22no—
140#397—22no—
141#394—22no—
142#391C2312yes—
143#391C2312yes—
144#391C2312yes—
145#390C2311yescharacteristic
146#389—22no—
147#389—22no—
148#389—21no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389C2312yes—
159#389C2314yes—
160#389C2311yescharacteristic
161#389C2314yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—21no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#386—21no—
173#386—22no—
174#385—24no—
175#385—22no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2311yescharacteristic
184#383—22no—
185#382—22no—
186#382—21no—
187#382—22no—
188#382—21no—
189#382C2314yes—
190#382C2312yes—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—24no—
198#380—22no—
199#380C2311yescharacteristic
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214#376D412yes—
215C43C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#336C2312yes—
226#336C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2314yes—
230C22×Q8C2412yes—
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8C2412yes—
240C22×C8C2414yes—
241C22×C8C2412yes—
242C4⋊Q8—24no—
243#125—44no—
244#125—44no—
245#125—84no—
246#124—24no—
247#124—42no—
248C22⋊Q8—44no—
249#121—22no—
250#121—41no—
251C4×Q8—44no—
252C4×D4—24no—
253C4×D4—44no—
254C4×D4—44no—
255#117—24no—
256#117—42no—
257#117—44no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—22no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—42no—
323#107—42no—
324#106—44no—
325#106—44no—
326C4⋊C8—24no—
327C4⋊C8—44no—
328#103—44no—
329#103—44no—
330#103—44no—
331#103—44no—
332#102—28no—
333#102—24no—
334#102—42no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337C4×C8—24no—
338C4×C8—44no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—22no—
343#95—22no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—21no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C24—21no—
363C24C22×D411yescharacteristic
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—81no—
374C2×D4H(5,𝔽2)12yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—41no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4#13514yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2511yescharacteristic
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—81no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—24no—
477C42—22no—
478C42—42no—
479C42—42no—
480C42—44no—
481C42—42no—
482C42—82no—
483C422−1+414yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×#13512yes—
498C23C23×D411yescharacteristic
499Q8—44no—
500Q8—44no—
501D4—24no—
502D4—42no—
503D4—84no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—84no—
554C2×C4—164no—
555C2×C4—164no—
556C2×C4#53612yes—
557C2×C4#53612yes—
558C2×C4C2×#13512yes—
559C2×C4C2×#13512yes—
560C2×C4#57512yes—
561C2×C4#57712yes—
562C2×C4C2×H(5,𝔽2)11yescharacteristic
563C2×C4C2×2−1+412yes—
564C2×C4#58512yes—
565C2×C4#58511yescharacteristic
566C8—42no—
567C8—44no—
568C8—42no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×#53611yescharacteristic
584C22#348811yescharacteristic
585C22#350011yescharacteristic
586C22C22×#13511yescharacteristic
587C22C2×#57511yescharacteristic
588C22C2×#57711yescharacteristic
589C22#358611yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—84no—
599C4—81no—
600C4—82no—
601C4—82no—
602C4—81no—
603C4—82no—
604C4—82no—
605C4—81no—
606C4#323214yes—
607C4#355912yes—
608C4#356512yes—
609C2—42no—
610C2—82no—
611C2#4294411yescharacteristic
612C2C2×#323211yescharacteristic
613C2#6087211yescharacteristic
614C2#6195011yescharacteristic
615C2#6198611yescharacteristic
616C2#6253211yescharacteristic
617C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35286C2semidirect
3. #32839C2semidirect
4. #31360C2semidirect
6. #30377C2semidirect
7. #30083C2semidirect
8. #29922C2semidirect
9. #28790C2semidirect
10. #28720C2semidirect
11. #27972C2semidirect
12. #27768C2semidirect
14. #26527C2semidirect
15. #26107C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#364492
#352862
#328392
#313602
#306392
#303772
#300832
#299222
#287902
#287202
#279722
#277682
#276452
#265272
#261072
#237542
#226392