Finite Group Databasebeta

← #6800588

Subgroups — #6800588

Order 512.

1583 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#36545C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#33147C211yesmaximal, characteristic
4#32641C214yesmaximal, complemented
5#32619C214yesmaximal, complemented
6#29957C212yesmaximal
7#29112C212yesmaximal
8#28646C212yesmaximal, complemented
9#28121C212yesmaximal
10#26317C212yesmaximal, complemented
11#25931C212yesmaximal, complemented
12#25856C211yesmaximal, characteristic
13#25302C212yesmaximal, complemented
14#24212C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic
16#22050C212yesmaximal
17C4×#484C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2212yes—
22C2×#484C2211yescharacteristic
23C2×#462C2212yes—
24C2×#451C2212yes—
25C2×#446C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2584C2212yes—
29#2582C2211yescharacteristic
30#2581C2212yes—
31#2568—21no—
32#2568—21no—
33#2568—22no—
34#2568C2212yes—
35#2568C2214yes—
36#2568C2212yes—
37#2546C2211yescharacteristic
38#2473—22no—
39#2462—22no—
40#2446—21no—
41#2434—21no—
42C2×#400—22no—
43#2365—21no—
44#2360—21no—
45#2355C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391—21no—
49C2×#391C2212yes—
50C2×#391C2212yes—
51C2×#382—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2074C2214yes—
55#2068C2214yes—
56#2068C2214yes—
57#2066C2214yes—
58#2056C2214yes—
59#2043C2214yes—
60#2043C2212yes—
61#2026C2214yes—
62#2026C2214yes—
63#2016C2212yes—
64#1988C2212yes—
65#1985C2214yes—
66#1984C2212yes—
67#1981C2212yes—
68#1971C2214yes—
69#1956C2212yes—
70#1948C2212yes—
71#1935C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1928C2212yes—
75#1928C2214yes—
76#1927C2214yes—
77#1921C2212yes—
78#1919C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1872C2211yescharacteristic
82#1853C2212yes—
83#1852C2211yescharacteristic
84#1824C2211yescharacteristic
85#1824C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89#1787C2212yes—
90#1787C2212yes—
91C4×#107C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98#557—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—22no—
108C2×#125C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8—21no—
112C2×C4×Q8—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D412yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#487—44no—
131#484—24no—
132#462—44no—
133#451—28no—
134#446—44no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#400—24no—
156#400—24no—
157#400—24no—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#395—22no—
163#395—22no—
164#391—22no—
165#391—22no—
166#391—24no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—21no—
175#391C2312yes—
176#391C2314yes—
177#391C2314yes—
178#391C2314yes—
179#391C2314yes—
180#391C2311yescharacteristic
181#389—21no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2311yescharacteristic
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—21no—
197#385—22no—
198#385—22no—
199#385—22no—
200#385—21no—
201#384C2311yescharacteristic
202#382—22no—
203#382—22no—
204#382—24no—
205#382C2312yes—
206#380—21no—
207#380C2311yescharacteristic
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95—22no—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×D412yes—
248C22×Q8C2414yes—
249C22×Q8C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×Q16—28no—
259C2×Q16—44no—
260C2×Q16—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#125—44no—
273#125—48no—
274C22⋊Q8—22no—
275C22⋊Q8—44no—
276C22⋊Q8#4114yes—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
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—44no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346#107—24no—
347#107—42no—
348#106—44no—
349C4⋊C8—24no—
350C4⋊C8—44no—
351#103—24no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—42no—
357#103—44no—
358#103—44no—
359#103—44no—
360C22⋊C8—24no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363#97—44no—
364C4×C8—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390C24C22×D411yescharacteristic
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—416no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—88no—
417C2×Q8—88no—
418C2×Q8C2×#4112yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4#13714yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463Q16—88no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4C2×#4112yes—
509C22⋊C4—24no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—24no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—44no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42—82no—
525C42H(5,𝔽2)14yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#54812yes—
531C23C2×#13712yes—
532C23#57812yes—
533C23C23×D411yescharacteristic
534Q8—24no—
535Q8—48no—
536Q8—44no—
537Q8—44no—
538Q8—42no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—44no—
543Q8—88no—
544Q8—84no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
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—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—88no—
602C2×C4#53412yes—
603C2×C4#53412yes—
604C2×C4#54812yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4#57512yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+411yescharacteristic
611C2×C4#58512yes—
612C8—42no—
613C8—44no—
614C8—42no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22#337414yes—
622C22C2×#53411yescharacteristic
623C22C2×#54811yescharacteristic
624C22#348711yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57811yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—84no—
646C4#319614yes—
647C4#351412yes—
648C4#356112yes—
649C2—24no—
650C2#3912311yescharacteristic
651C2C2×#319611yescharacteristic
652C2#6075911yescharacteristic
653C2C2×#337411yescharacteristic
654C2#6132311yescharacteristic
655C2#6189311yescharacteristic
656C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32641C2semidirect
5. #32619C2semidirect
8. #28646C2semidirect
10. #26317C2semidirect
11. #25931C2semidirect
13. #25302C2semidirect

Maximal subgroups

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

subgroupindex
#365452
C2×#25682
#331472
#326412
#326192
#299572
#291122
#286462
#281212
#263172
#259312
#258562
#253022
#242122
#225102
#220502
C4×#4842