Finite Group Databasebeta

← #6573043

Subgroups — #6573043

Order 512.

1537 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30060C212yesmaximal, complemented
4#29664C211yesmaximal, characteristic
5#29371C212yesmaximal, complemented
6#28385C212yesmaximal, complemented
7#27905C211yesmaximal, characteristic
8#27680C211yesmaximal, characteristic
9#26949C214yesmaximal
10#26807C214yesmaximal
11#26042C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#23783C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic
15#23102C212yesmaximal, complemented
16#22814C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#483C2214yes—
20C2×#475C2214yes—
21C2×#462C2212yes—
22C2×#462C2214yes—
23C2×C4×Q16C2212yes—
24#2862C2212yes—
25#2857C2211yescharacteristic
26#2612—22no—
27#2574—22no—
28#2551C2211yescharacteristic
29#2546C2211yescharacteristic
30#2508—21no—
31#2492C2212yes—
32#2439—21no—
33#2429C2212yes—
34#2369—21no—
35#2365C2211yescharacteristic
36#2360—22no—
37#2360C2214yes—
38#2355—21no—
39#2355C2211yescharacteristic
40#2355C2212yes—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2212yes—
43#2346—21no—
44#2330—22no—
45#2317—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49#2295C2212yes—
50C2×#384C2211yescharacteristic
51#2274—21no—
52C4×#117C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2088C2214yes—
58#2068C2214yes—
59#2058C2214yes—
60#2057C2211yescharacteristic
61#2043C2211yescharacteristic
62#2043C2214yes—
63#2036C2212yes—
64#2024C2211yescharacteristic
65#2024C2212yes—
66#2016C2211yescharacteristic
67#2016C2212yes—
68#2001C2212yes—
69#2001C2214yes—
70#1971C2214yes—
71#1971C2214yes—
72#1956C2212yes—
73#1948C2212yes—
74#1939C2212yes—
75#1927C2214yes—
76#1927C2214yes—
77#1927C2214yes—
78#1924C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1872C2212yes—
82#1865C2214yes—
83#1853C2212yes—
84#1844C2212yes—
85#1839C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97#533—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×#117—21no—
116C2×#117D411yescharacteristic
117C2×#117D411yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122#483—48no—
123#475—48no—
124#462—28no—
125#462—48no—
126C4×Q16—28no—
127C2×#107C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#398—22no—
148#398—22no—
149#396—22no—
150#395—22no—
151#393—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2314yes—
157#391C2312yes—
158#391C2312yes—
159#391C2311yescharacteristic
160#391C2312yes—
161#391C2312yes—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—24no—
172#389—24no—
173#389—22no—
174#389C2314yes—
175#389C2311yescharacteristic
176#389C2312yes—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#385—21no—
187#385—24no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#382—22no—
195#382—21no—
196#382—22no—
197#382—22no—
198#380—21no—
199#380—21no—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—24no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376—22no—
222#376D412yes—
223C43C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2411yescharacteristic
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—24no—
244C2×Q16—28no—
245C2×Q16—24no—
246C2×Q16—42no—
247C2×Q16—44no—
248C2×Q16—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258#125—28no—
259C22⋊Q8—28no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—41no—
264C22⋊Q8—44no—
265C4×Q8—28no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—42no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—84no—
273C4×Q8#4118yes—
274#117—28no—
275#117—24no—
276#117—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—24no—
340#106—28no—
341#106—44no—
342#106—44no—
343C4⋊C8—28no—
344C4⋊C8—24no—
345#103—24no—
346#103—28no—
347#103—24no—
348#103—24no—
349#103—28no—
350#103—24no—
351#103—44no—
352#103—44no—
353#103—44no—
354C22⋊C8—24no—
355C22⋊C8—42no—
356#97—44no—
357C4×C8—24no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8C2×#4114yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—41no—
420C22×C4—44no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4#13714yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2511yescharacteristic
434Q16—48no—
435Q16—88no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8H(5,𝔽2)14yes—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—28no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4#13714yes—
481C4⋊C4C2×#4114yes—
482C4⋊C42−1+412yes—
483C22⋊C4—28no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C42—22no—
490C42—22no—
491C42—28no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—44no—
501C42—42no—
502C42—42no—
503C42—82no—
504C42—84no—
505C42—84no—
506C42C2×#4114yes—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23C2×#13712yes—
514C23C23×D411yescharacteristic
515Q8—24no—
516Q8—24no—
517Q8—48no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521Q8—44no—
522Q8—42no—
523Q8—44no—
524Q8—164no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—164no—
577C2×C4#52512yes—
578C2×C4#52512yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13714yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4#57712yes—
584C2×C4C22×#4112yes—
585C2×C4C2×H(5,𝔽2)12yes—
586C2×C4C2×2−1+411yescharacteristic
587C2×C4#58511yescharacteristic
588C8—24no—
589C8—44no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22C2×#52511yescharacteristic
599C22#348911yescharacteristic
600C22#349811yescharacteristic
601C22C22×#13711yescharacteristic
602C22C22×#13711yescharacteristic
603C22C2×#57711yescharacteristic
604C22#358611yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—24no—
609C4—42no—
610C4—44no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—44no—
616C4—81no—
617C4—84no—
618C4—82no—
619C4—81no—
620C4—81no—
621C4—82no—
622C4#311314yes—
623C4#349412yes—
624C4#351112yes—
625C2—42no—
626C2#3608711yescharacteristic
627C2C2×#311311yescharacteristic
628C2#6058511yescharacteristic
629C2#6211311yescharacteristic
630C2#6214711yescharacteristic
631C2#6242211yescharacteristic
632C2#6244811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #30060C2semidirect
5. #29371C2semidirect
6. #28385C2semidirect
11. #26042C2semidirect
12. #24732C2semidirect
13. #23783C2semidirect
15. #23102C2semidirect
16. #22814C2semidirect

Maximal subgroups

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

subgroupindex
#362882
#342142
#300602
#296642
#293712
#283852
#279052
#276802
#269492
#268072
#260422
#247322
#237832
#235712
#231022
#228142
#226432