Finite Group Databasebeta

← #6471602

Subgroups — #6471602

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#33010C212yesmaximal
4#32954C211yesmaximal, characteristic
5#32906C211yesmaximal, characteristic
6#29967C211yesmaximal, characteristic
7#29908C212yesmaximal
8#29757C211yesmaximal, characteristic
9#27569C214yesmaximal, complemented
10#27313C214yesmaximal, complemented
11#26389C214yesmaximal, complemented
12#26187C214yesmaximal, complemented
13#22085C212yesmaximal
14#21916C211yesmaximal, characteristic
15#21455C212yesmaximal
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#475C2212yes—
21C2×#462C2212yes—
22C2×#451C2212yes—
23C2×#441C2214yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yes—
32#2507C2212yes—
33#2451C2212yes—
34#2446C2212yes—
35Q8×C4⋊C4—22no—
36Q8×C4⋊C4—22no—
37Q8×C4⋊C4C2214yes—
38Q8×C4⋊C4C2214yes—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#389—21no—
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#384—21no—
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2099C2212yes—
51#2099C2214yes—
52#2096C2214yes—
53#2088C2214yes—
54#2084C2214yes—
55#2083C2212yes—
56#2072C2214yes—
57#2068C2214yes—
58#2058C2212yes—
59#2043C2212yes—
60#1943C2214yes—
61#1935C2214yes—
62#1918C2214yes—
63#1917C2214yes—
64#1917C2214yes—
65#1878C2214yes—
66#1878C2214yes—
67#1878C2214yes—
68#1878C2214yes—
69#1872C2212yes—
70#1872C2212yes—
71#1865C2214yes—
72#1865C2214yes—
73#1865C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1853C2212yes—
77#1853C2212yes—
78#1824C2212yes—
79#1817C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1787C2212yes—
85#1763C2214yes—
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C22×Q16C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—44no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2314yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#484—28no—
125#475—44no—
126#462—44no—
127#451—28no—
128#441—48no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×#97C2314yes—
143#400—22no—
144#400—22no—
145#400—22no—
146#398—22no—
147#398—22no—
148#398—22no—
149#398—22no—
150#395—22no—
151#391—21no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—24no—
165#389—24no—
166#389—24no—
167#389—24no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—22no—
186#384—24no—
187#384—24no—
188#384—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2314yes—
192#381—22no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380C2312yes—
197C4×C4⋊C4—24no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2314yes—
220#336C2314yes—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—28no—
241C2×Q16—28no—
242C2×Q16—28no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—44no—
250C4×Q8—216no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—44no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—48no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4#4114yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334#107—28no—
335#106—44no—
336#106—44no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339#103—28no—
340#103—28no—
341#103—28no—
342#103—28no—
343#103—44no—
344#103—44no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347#97—44no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24C22×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—216no—
372C2×Q8—24no—
373C2×Q8—216no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×Q8C2×#4114yes—
386C2×Q8H(5,𝔽2)14yes—
387C2×Q82−1+414yes—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4#13714yes—
427C22×C4#13714yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2511yescharacteristic
440Q16—416no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C4⋊C4—216no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4C2×#4114yes—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C42—216no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—44no—
498C42—44no—
499C42—48no—
500C42—82no—
501C42—82no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23#52512yes—
507C23C2×#13712yes—
508C23C2×#13712yes—
509C23C23×D411yescharacteristic
510Q8—216no—
511Q8—24no—
512Q8—24no—
513Q8—24no—
514Q8—44no—
515Q8—44no—
516Q8—162no—
517Q8#53918yes—
518C2×C4—216no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4#52512yes—
578C2×C4C2×#13714yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C2×H(5,𝔽2)12yes—
584C2×C4C2×2−1+412yes—
585C8—44no—
586C8—44no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22#311314yes—
594C22C2×#52511yescharacteristic
595C22#348411yescharacteristic
596C22#348611yescharacteristic
597C22C22×#13711yescharacteristic
598C22C22×#13711yescharacteristic
599C22#358511yescharacteristic
600C22#358611yescharacteristic
601C4—24no—
602C4—22no—
603C4—22no—
604C4—28no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—44no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4C2×#53912yes—
619C4#349212yes—
620C2—24no—
621C2#3498911yescharacteristic
622C2C2×#311311yescharacteristic
623C2#6056511yescharacteristic
624C2#6157411yescharacteristic
625C2#6158811yescharacteristic
626C2#6182911yescharacteristic
627C2#6183011yescharacteristic
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
9. #27569C2semidirect
10. #27313C2semidirect
11. #26389C2semidirect
12. #26187C2semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×Q8×C4⋊C42
#330102
#329542
#329062
#299672
#299082
#297572
#275692
#273132
#263892
#261872
#220852
#219162
#214552