Finite Group Databasebeta

← #6493681

Subgroups — #6493681

Order 512.

1724 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#35308C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#33120C212yesmaximal, complemented
4#33119C212yesmaximal, complemented
5#29928C212yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#27980C211yesmaximal, characteristic, complemented
8#27976C211yesmaximal, characteristic, complemented
9#27914C211yesmaximal, characteristic, complemented
10#27910C211yesmaximal, characteristic, complemented
11#27902C212yesmaximal, complemented
12#27899C212yesmaximal, complemented
13#27586C212yesmaximal, complemented
14#27584C212yesmaximal, complemented
15#24014C212yesmaximal, complemented
16#24011C212yesmaximal
17#21991C212yesmaximal, complemented
18#21814C212yesmaximal, complemented
19#20720C211yesmaximal, characteristic, complemented
20C2×#537C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#484C2212yes—
25C2×#484C2212yes—
26C2×#483C2212yes—
27C2×#480C2212yes—
28C2×#433C2212yes—
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2850C2211yescharacteristic
32#2850C2211yescharacteristic
33#2621C2211yescharacteristic
34#2621C2211yescharacteristic
35#2620C2211yescharacteristic
36#2620C2211yescharacteristic
37#2429—22no—
38#2427—22no—
39#2425—22no—
40#2425—22no—
41#2355—22no—
42#2354—22no—
43#2352—22no—
44#2352—22no—
45#2317C2211yescharacteristic
46#2314C2212yes—
47#2314C2212yes—
48#2311C2212yes—
49#2297C2211yescharacteristic
50C2×#391C2211yescharacteristic
51#2285C2212yes—
52#2285C2212yes—
53#2285C2211yescharacteristic
54C2×#384C2211yescharacteristic
55#2274C2212yes—
56#2095C2212yes—
57#2095C2212yes—
58#2095C2212yes—
59#2094C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2071C2212yes—
64#2036C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2033C2214yes—
68#1983C2214yes—
69#1981C2214yes—
70#1960C2212yes—
71#1959C2212yes—
72#1958C2212yes—
73#1956C2212yes—
74#1949C2214yes—
75#1947C2214yes—
76#1928C2214yes—
77#1928C2214yes—
78#1927C2214yes—
79#1925C2214yes—
80#1875C2212yes—
81#1873C2212yes—
82#1873C2212yes—
83#1873C2212yes—
84#1854C2214yes—
85#1852C2214yes—
86#1847C2212yes—
87#1845C2212yes—
88#1836C2212yes—
89#1835C2212yes—
90#1817C2212yes—
91#1798C2212yes—
92#1797C2212yes—
93#1788C2214yes—
94#1787C2214yes—
95#1780C2212yes—
96#1750C2212yes—
97#1749C2211yescharacteristic
98C4×C22⋊C8C2211yescharacteristic
99C22×#41C2311yescharacteristic
100C22×#41C2311yescharacteristic
101C23×Q8C2311yescharacteristic
102#537—44no—
103#536—28no—
104#536—44no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#124—21no—
109C2×#124C2311yescharacteristic
110C2×#124C2311yescharacteristic
111C2×#123—21no—
112C2×#123—21no—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C22⋊Q8C2311yescharacteristic
118C2×C22⋊Q8C2311yescharacteristic
119C2×C22⋊Q8C2311yescharacteristic
120C2×C22⋊Q8C2311yescharacteristic
121C2×C22⋊Q8C2311yescharacteristic
122C2×#121—21no—
123C2×#121—21no—
124C2×#121—21no—
125C2×#121C2311yescharacteristic
126C2×#121C2311yescharacteristic
127C4×#41—28no—
128C2×C4×Q8D412yes—
129C2×C4×D4D412yes—
130C2×C4×D4D412yes—
131C2×C4×D4D412yes—
132C2×#117—22no—
133C2×#117—22no—
134C2×#117C2312yes—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—22no—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#484—44no—
146#484—44no—
147#483—44no—
148#480—44no—
149#433—28no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×#97C2312yes—
168C2×C4×C8C2312yes—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#396—22no—
174#396—22no—
175#394—22no—
176#394—22no—
177#391—22no—
178#391—22no—
179#391C2311yescharacteristic
180#391C2311yescharacteristic
181#391C2311yescharacteristic
182#391C2311yescharacteristic
183#391C2311yescharacteristic
184#391C2311yescharacteristic
185#390—22no—
186#390C2311yescharacteristic
187#390C2311yescharacteristic
188#389—22no—
189#389—22no—
190#389—22no—
191#388—24no—
192#388—24no—
193#388—22no—
194#388—22no—
195#387—24no—
196#387—24no—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#385—22no—
203#384—22no—
204#384C2312yes—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2312yes—
209#382—22no—
210#382—22no—
211#382C2312yes—
212#382C2311yescharacteristic
213#382C2311yescharacteristic
214#380—22no—
215#380—22no—
216#380—22no—
217#380—22no—
218#380—22no—
219#380C2314yes—
220#380C2314yes—
221C4×C4⋊C4—24no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4C2314yes—
226C4×C4⋊C4C2314yes—
227C4×C22⋊C4—24no—
228C4×C22⋊C4—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—42no—
241C2×#41—42no—
242C2×#41—42no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2×D411yescharacteristic
255C22×D4C2×D411yescharacteristic
256C22×D4C2×D411yescharacteristic
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273#124—24no—
274#124—24no—
275#124—44no—
276#123—44no—
277#123—44no—
278#123—48no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285C22⋊Q8—44no—
286C22⋊Q8—48no—
287#121—24no—
288#121—42no—
289#121—44no—
290#121—44no—
291#121—44no—
292C4×Q8—28no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296#117—24no—
297#117—24no—
298#117—44no—
299#117—42no—
300#117—84no—
301#117—84no—
302#117#4118yes—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2414yes—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—44no—
367#107—44no—
368#107—44no—
369#106—44no—
370C4⋊C8—28no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#102—44no—
378#102—44no—
379C22⋊C8—28no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382C22⋊C8—44no—
383#97—24no—
384C4×C8—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398C24—21no—
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C24C22×D411yescharacteristic
402#41—416no—
403#41—88no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—41no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—88no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—82no—
438C2×D4—82no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—84no—
473C22×C4—82no—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C4C2511yescharacteristic
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4#12318yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—82no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4#12318yes—
537C42—24no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—44no—
543C42—42no—
544C42—84no—
545C42—84no—
546C42C2×#4114yes—
547C42C2×#4114yes—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23C2×#12312yes—
567C23#53512yes—
568C23C23×D411yescharacteristic
569Q8—44no—
570Q8—44no—
571Q8—44no—
572Q8—44no—
573Q8—84no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—44no—
578D4—84no—
579D4—162no—
580D4—162no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—164no—
629C2×C4—162no—
630C2×C4C2×#12312yes—
631C2×C4C2×#12314yes—
632C2×C4C2×#12314yes—
633C2×C4#53512yes—
634C2×C4#57814yes—
635C2×C4C22×#4112yes—
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4#58512yes—
638C8—44no—
639C8—44no—
640C22—24no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—42no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—82no—
656C22—82no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22C22×#12311yescharacteristic
661C22C2×#53511yescharacteristic
662C22#344911yescharacteristic
663C22C2×#57812yes—
664C22#358512yes—
665C4—24no—
666C4—24no—
667C4—24no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—41no—
672C4—84no—
673C4—81no—
674C4—81no—
675C4—84no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4#349414yes—
682C2—42no—
683C2—42no—
684C2—81no—
685C2—161no—
686C2#3526211yescharacteristic
687C2#6052612yes—
688C2#6082111yescharacteristic
689C2#6082511yescharacteristic
690C2#6099312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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. #33120C2semidirect
4. #33119C2semidirect
5. #29928C2semidirect
6. #29923C2semidirect
7. #27980C2semidirect
8. #27976C2semidirect
9. #27914C2semidirect
10. #27910C2semidirect
11. #27902C2semidirect
12. #27899C2semidirect
13. #27586C2semidirect
14. #27584C2semidirect
15. #24014C2semidirect
17. #21991C2semidirect
18. #21814C2semidirect
19. #20720C2semidirect

Maximal subgroups

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

subgroupindex
#353082
#341882
#331202
#331192
#299282
#299232
#279802
#279762
#279142
#279102
#279022
#278992
#275862
#275842
#240142
#240112
#219912
#218142
#207202