Finite Group Databasebeta

← #6993046

Subgroups — #6993046

Order 512.

1541 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#32801C212yesmaximal, complemented
4#31292C214yesmaximal
5#30530C214yesmaximal
6#30338C212yesmaximal, complemented
7#30063C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic
9#28748C212yesmaximal, complemented
10#28401C212yesmaximal, complemented
11#27769C212yesmaximal, complemented
12#25856C211yesmaximal, characteristic
13#25278C212yesmaximal, complemented
14#24632C212yesmaximal, complemented
15#24212C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#487C2214yes—
22C2×#477C2212yes—
23C2×#462C2212yes—
24C2×#451C2212yes—
25C2×C4×Q16C2212yes—
26#2868C2212yes—
27#2863C2212yes—
28#2862C2212yes—
29#2857C2211yescharacteristic
30#2660—22no—
31#2605—22no—
32#2567—21no—
33#2567—21no—
34#2567C2211yescharacteristic
35#2551C2211yescharacteristic
36#2546C2212yes—
37#2546C2212yes—
38#2509—21no—
39#2502—22no—
40#2497C2212yes—
41#2492C2212yes—
42#2465—22no—
43#2439—21no—
44#2434C2214yes—
45#2369C2211yescharacteristic
46#2365—21no—
47#2355C2211yescharacteristic
48#2347—21no—
49#2317—21no—
50#2317—21no—
51C4×C4⋊Q8C2212yes—
52C4×C22⋊Q8C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2095C2214yes—
55#2068C2214yes—
56#2057C2211yescharacteristic
57#2057C2212yes—
58#2056C2214yes—
59#2043C2212yes—
60#2043C2212yes—
61#2043C2211yescharacteristic
62#2043C2214yes—
63#2043C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2036C2212yes—
67#2036C2214yes—
68#2024C2211yescharacteristic
69#2024C2214yes—
70#2016C2211yescharacteristic
71#2016C2214yes—
72#2016C2214yes—
73#2016C2212yes—
74#2001C2212yes—
75#1981C2214yes—
76#1981C2214yes—
77#1981C2212yes—
78#1969C2214yes—
79#1956C2212yes—
80#1956C2214yes—
81#1939C2212yes—
82#1933C2214yes—
83#1928C2214yes—
84#1928C2214yes—
85#1923C2214yes—
86#1839C2212yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1787C2214yes—
90#1787C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2312yes—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C42C2311yescharacteristic
126#487—44no—
127#487—48no—
128#477—44no—
129#462—44no—
130#451—44no—
131C4×Q16—44no—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#400—22no—
155#398—24no—
156#398—24no—
157#396—22no—
158#395—24no—
159#395—24no—
160#395—22no—
161#395—22no—
162#393—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#391C2314yes—
172#391C2314yes—
173#391C2311yescharacteristic
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—21no—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#387—21no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—21no—
197#385—24no—
198#385—24no—
199#384—22no—
200#384—22no—
201#384—22no—
202#384—22no—
203#384C2312yes—
204#384C2311yescharacteristic
205#383—22no—
206#382—21no—
207#382—22no—
208#382—22no—
209#380—21no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226#376D412yes—
227C43C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235#336C2311yescharacteristic
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C22×Q8C2414yes—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×Q16—42no—
252C2×Q16—42no—
253C2×Q16—42no—
254C2×Q16—44no—
255C2×Q16—44no—
256C2×Q16—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275#125—44no—
276C22⋊Q8—22no—
277C22⋊Q8—41no—
278C22⋊Q8—42no—
279C22⋊Q8—42no—
280C22⋊Q8—44no—
281C22⋊Q8—82no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×Q8—84no—
287#117—24no—
288#117—44no—
289#117Q1618yescomplemented
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349#107—42no—
350#107—44no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353#103—28no—
354#103—24no—
355#103—42no—
356#103—44no—
357#103—44no—
358#103—42no—
359#103—42no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366C22⋊C8—42no—
367C22⋊C8—42no—
368C4×C8—24no—
369C4×C8—44no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24C22×D411yescharacteristic
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—81no—
421C2×Q8—81no—
422C2×Q8—88no—
423C2×Q8—88no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C2×Q1614yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456Q16—48no—
457Q16—88no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4C2×Q1614yes—
502C4⋊C4C2×Q1614yes—
503C4⋊C4#13714yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—81no—
509C22⋊C4—82no—
510C22⋊C4C2×Q1618yes—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—42no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—82no—
524C42—82no—
525C42C2×Q1618yes—
526C42H(5,𝔽2)14yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C22×Q1612yes—
534C23C23×D411yescharacteristic
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—44no—
539Q8—44no—
540Q8—42no—
541Q8—44no—
542Q8—42no—
543Q8—42no—
544Q8—44no—
545Q8—88no—
546Q8—84no—
547Q8—88no—
548Q8—162no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—88no—
599C2×C4#53512yes—
600C2×C4#53512yes—
601C2×C4C22×Q1612yes—
602C2×C4C22×Q1614yes—
603C2×C4C22×Q1614yes—
604C2×C4C22×Q1612yes—
605C2×C4C2×#13714yes—
606C2×C4C2×#13712yes—
607C2×C4#57712yes—
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4#58511yescharacteristic
611C2×C4#58512yes—
612C8—42no—
613C8—42no—
614C8—44no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22C2×#53511yescharacteristic
623C22#348811yescharacteristic
624C22#350011yescharacteristic
625C22C23×Q1611yescharacteristic
626C22C22×#13711yescharacteristic
627C22C2×#57711yescharacteristic
628C22#358611yescharacteristic
629C4—24no—
630C4—24no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4—82no—
645C4—81no—
646C4#320914yes—
647C4#354012yes—
648C4#354612yes—
649C2—42no—
650C2#4298711yescharacteristic
651C2C2×#320911yescharacteristic
652C2#6081311yescharacteristic
653C2#6195211yescharacteristic
654C2#6198011yescharacteristic
655C2#6253011yescharacteristic
656C2#6258911yescharacteristic
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. #32801C2semidirect
6. #30338C2semidirect
7. #30063C2semidirect
9. #28748C2semidirect
10. #28401C2semidirect
11. #27769C2semidirect
13. #25278C2semidirect
14. #24632C2semidirect
289. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#364802
#352812
#328012
#312922
#305302
#303382
#300632
#296912
#287482
#284012
#277692
#258562
#252782
#246322
#242122
#235712
#226392