Finite Group Databasebeta

← #6561788

Subgroups — #6561788

Order 512.

1723 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36410C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29921C211yesmaximal, characteristic, complemented
4#29910C212yesmaximal, complemented
5#27973C211yesmaximal, characteristic, complemented
6#27857C212yesmaximal, complemented
7#27647C211yesmaximal, characteristic, complemented
8#27584C212yesmaximal, complemented
9#26323C212yesmaximal, complemented
10#26203C214yesmaximal, complemented
11#24997C212yesmaximal, complemented
12#24868C214yesmaximal, complemented
13#23757C211yesmaximal, characteristic, complemented
14#23627C212yesmaximal, complemented
15#22708C212yesmaximal, complemented
16#22469C211yesmaximal, characteristic, complemented
17#22418C212yesmaximal, complemented
18C2×#550C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#483C2212yes—
21C2×C8×D4C2212yes—
22C2×#416C2214yes—
23#2856C2211yescharacteristic
24#2854C2212yes—
25#2582C2211yescharacteristic
26#2548C2211yescharacteristic
27#2476—22no—
28#2464—22no—
29#2445—21no—
30#2431—21no—
31#2414C2212yes—
32#2407C2212yes—
33#2367—21no—
34#2363—21no—
35#2355—21no—
36#2355C2212yes—
37#2354—21no—
38#2354C2211yescharacteristic
39#2354C2212yes—
40#2352—22no—
41#2352C2214yes—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2211yescharacteristic
45#2297C2212yes—
46#2285C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2101C2214yes—
52#2100C2211yescharacteristic
53#2095C2211yescharacteristic
54#2094C2212yes—
55#2092C2214yes—
56#2084C2212yes—
57#2083C2211yescharacteristic
58#2076C2212yes—
59#2074C2214yes—
60#2073C2214yes—
61#2072C2211yescharacteristic
62#2072C2212yes—
63#2071C2212yes—
64#2071C2212yes—
65#2071C2212yes—
66#2036C2212yes—
67#2034C2214yes—
68#1984C2212yes—
69#1982C2212yes—
70#1957C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1927C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1919C2214yes—
77#1893C2214yes—
78#1875C2214yes—
79#1874C2214yes—
80#1869C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1834C2212yes—
84#1823C2212yes—
85#1817C2212yes—
86#1807C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1788C2214yes—
90C4×#107C2212yes—
91C4×#103C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C22×#41C2312yes—
97C23×C8C2314yes—
98#550—28no—
99C2×#127—21no—
100C2×#127C2311yescharacteristic
101C2×#124—21no—
102C2×#123—22no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—22no—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C4×#41—28no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4D412yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—21no—
119C2×#117—22no—
120C2×#117D412yes—
121C2×#117D411yescharacteristic
122C2×#117C2314yes—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#483—44no—
130C8×D4—28no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136#416—48no—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2312yes—
148#398—22no—
149#394—22no—
150#394—22no—
151#394—22no—
152#392—22no—
153#392—22no—
154#391—22no—
155#391—21no—
156#391C2312yes—
157#391C2311yescharacteristic
158#390—22no—
159#390—21no—
160#390—21no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—21no—
168#389C2312yes—
169#389C2311yescharacteristic
170#388—22no—
171#388—24no—
172#388—24no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—24no—
177#387—24no—
178#387—21no—
179#386—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#385—21no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2311yescharacteristic
193#382—21no—
194#382—22no—
195#382C2314yes—
196#380—22no—
197#380—22no—
198#380—21no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202#380C2311yescharacteristic
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—24no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—28no—
237C2×#41—42no—
238C2×#41—42no—
239C2×#41—44no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2411yescharacteristic
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D412yes—
248C22×D4C2×D411yescharacteristic
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C22×C8—28no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264#127—24no—
265#127—44no—
266#124—44no—
267#123—84no—
268C22⋊Q8—22no—
269C22⋊Q8—41no—
270#121—28no—
271#121—44no—
272#121—48no—
273#121—44no—
274#121—82no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×D4—24no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—44no—
282C4×D4#4118yes—
283#117—24no—
284#117—28no—
285#117—44no—
286#117—44no—
287#117—44no—
288#117—44no—
289#117—84no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—28no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—42no—
354#107—42no—
355#106—44no—
356#106—44no—
357C4⋊C8—24no—
358#103—24no—
359#103—28no—
360#103—42no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—28no—
366C22⋊C8—42no—
367C22⋊C8—42no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—24no—
381#95—22no—
382#95—21no—
383#95—22no—
384C24—22no—
385C24C22×D412yes—
386C24C22×D411yescharacteristic
387#41—416no—
388#41—416no—
389#41—88no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8H(5,𝔽2)12yes—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4#12314yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4#13514yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4118yes—
464C22×C4C2511yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—88no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4#12314yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—81no—
518C22⋊C4—84no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—82no—
522C22⋊C4#12318yes—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—24no—
529C42—42no—
530C42—44no—
531C42—42no—
532C42—84no—
533C42—82no—
534C42—84no—
535C42C2×#4114yes—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—21no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C2×#12314yes—
556C23C2×#13512yes—
557C23C23×D411yescharacteristic
558Q8—24no—
559Q8—44no—
560Q8—42no—
561D4—28no—
562D4—48no—
563D4—44no—
564D4—44no—
565D4—88no—
566D4—84no—
567D4—162no—
568D4—164no—
569D4—164no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—88no—
620C2×C4—164no—
621C2×C4C2×#12314yes—
622C2×C4C2×#12312yes—
623C2×C4C2×#12312yes—
624C2×C4C2×#13512yes—
625C2×C4C2×#13512yes—
626C2×C4#57512yes—
627C2×C4#57712yes—
628C2×C4C22×#4112yes—
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4#58512yes—
631C2×C4#58511yescharacteristic
632C8—44no—
633C8—42no—
634C8—42no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—42no—
645C22—41no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C22×#12311yescharacteristic
657C22#348811yescharacteristic
658C22#350011yescharacteristic
659C22C22×#13511yescharacteristic
660C22C2×#57511yescharacteristic
661C22C2×#57711yescharacteristic
662C22#358511yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—84no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4#309114yes—
679C4#351312yes—
680C4#352112yes—
681C2—42no—
682C2—44no—
683C2—162no—
684C2#3597611yescharacteristic
685C2C2×#309111yescharacteristic
686C2#6052511yescharacteristic
687C2#6195311yescharacteristic
688C2#6198711yescharacteristic
689C2#6253111yescharacteristic
690C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36410C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #29921C2semidirect
4. #29910C2semidirect
5. #27973C2semidirect
6. #27857C2semidirect
7. #27647C2semidirect
8. #27584C2semidirect
9. #26323C2semidirect
10. #26203C2semidirect
11. #24997C2semidirect
12. #24868C2semidirect
13. #23757C2semidirect
14. #23627C2semidirect
15. #22708C2semidirect
16. #22469C2semidirect
17. #22418C2semidirect

Maximal subgroups

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

subgroupindex
#364102
#341882
#299212
#299102
#279732
#278572
#276472
#275842
#263232
#262032
#249972
#248682
#237572
#236272
#227082
#224692
#224182