Finite Group Databasebeta

← #6856255

Subgroups — #6856255

Order 512.

1557 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#32451C214yesmaximal, complemented
5#31659C214yesmaximal, complemented
6#29715C212yesmaximal
7#28417C212yesmaximal, complemented
8#27983C212yesmaximal
9#27737C212yesmaximal, complemented
10#27686C212yesmaximal
11#25902C211yesmaximal, characteristic
12#25323C212yesmaximal, complemented
13#24683C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic
16#22050C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#475C2212yes—
22C2×#462C2212yes—
23C2×#451C2211yescharacteristic
24C2×#451C2212yes—
25C2×C4×Q16C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2585—22no—
29#2569—22no—
30#2568—22no—
31#2568—21no—
32#2568—21no—
33#2568C2212yes—
34#2568C2212yes—
35#2568C2214yes—
36#2552C2211yescharacteristic
37#2546C2211yescharacteristic
38#2511C2212yes—
39#2509—21no—
40#2446C2212yes—
41#2442—21no—
42C2×#400—22no—
43#2370—21no—
44#2365C2211yescharacteristic
45#2355C2211yescharacteristic
46#2345—21no—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#391—21no—
50C2×#391—21no—
51C2×#391C2212yes—
52C2×#382C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2074C2214yes—
57#2074C2214yes—
58#2072C2214yes—
59#2058C2212yes—
60#2056C2214yes—
61#2056C2214yes—
62#2024C2214yes—
63#2024C2212yes—
64#2016C2214yes—
65#1990C2212yes—
66#1988C2212yes—
67#1971C2214yes—
68#1960C2214yes—
69#1949C2214yes—
70#1939C2212yes—
71#1935C2214yes—
72#1927C2214yes—
73#1917C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1878C2212yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1863C2212yes—
83#1853C2212yes—
84#1852C2211yescharacteristic
85#1836C2214yes—
86#1805C2214yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—44no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#487—44no—
126#475—44no—
127#462—44no—
128#451—24no—
129#451—28no—
130C4×Q16—44no—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—24no—
151#400—24no—
152#400—24no—
153#400—24no—
154#400—22no—
155#400—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—24no—
163#391—22no—
164#391—24no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#391C2312yes—
169#391C2312yes—
170#391C2314yes—
171#391C2312yes—
172#391C2312yes—
173#391C2311yescharacteristic
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389C2311yescharacteristic
183#389C2312yes—
184#387—22no—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—21no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2312yes—
196#382—21no—
197#382—22no—
198#382—22no—
199#382C2314yes—
200#382C2314yes—
201#380—21no—
202#380—22no—
203#380—22no—
204#380—21no—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8#4114yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×Q16—24no—
258C2×Q16—28no—
259C2×Q16—42no—
260C2×Q16—42no—
261C2×Q16—44no—
262C2×Q16—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273#125—44no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—22no—
277C22⋊Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—44no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344#107—42no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—44no—
348#103—24no—
349#103—28no—
350#103—42no—
351#103—42no—
352#103—42no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357C22⋊C8—22no—
358C22⋊C8—24no—
359C22⋊C8—22no—
360#97—44no—
361C4×C8—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24C22×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—816no—
415C2×Q8C2×#4114yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4#13714yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C42−1+412yes—
459C22×C4C2511yescharacteristic
460Q16—48no—
461Q16—88no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—44no—
519C42—42no—
520C42—42no—
521C42—82no—
522C42—82no—
523C42H(5,𝔽2)14yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#54812yes—
529C23C2×#13712yes—
530C23#57812yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—24no—
534Q8—48no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—44no—
539Q8—42no—
540Q8—44no—
541Q8—88no—
542Q8—88no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4#53512yes—
601C2×C4#53512yes—
602C2×C4#54812yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4#57512yes—
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4C2×2−1+411yescharacteristic
609C2×C4#58512yes—
610C8—42no—
611C8—44no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22#337414yes—
620C22C2×#53511yescharacteristic
621C22C2×#54811yescharacteristic
622C22#348511yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57511yescharacteristic
625C22C2×#57811yescharacteristic
626C22#358611yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—41no—
636C4—81no—
637C4—81no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4#321514yes—
645C4#350512yes—
646C4#354112yes—
647C2—24no—
648C2#3972211yescharacteristic
649C2C2×#321511yescharacteristic
650C2#6081411yescharacteristic
651C2C2×#337411yescharacteristic
652C2#6132011yescharacteristic
653C2#6172311yescharacteristic
654C2#6173011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32451C2semidirect
5. #31659C2semidirect
7. #28417C2semidirect
9. #27737C2semidirect
12. #25323C2semidirect
13. #24683C2semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#25682
#329062
#324512
#316592
#297152
#284172
#279832
#277372
#276862
#259022
#253232
#246832
#243012
#226292
#220502
C4×#4512