Finite Group Databasebeta

← #6493668

Subgroups — #6493668

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35327C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#33121C212yesmaximal, complemented
4#33120C212yesmaximal, complemented
5#29932C212yesmaximal, complemented
6#29931C212yesmaximal, complemented
7#27979C211yesmaximal, characteristic, complemented
8#27979C211yesmaximal, characteristic, complemented
9#27914C211yesmaximal, characteristic, complemented
10#27913C211yesmaximal, characteristic, complemented
11#27895C212yesmaximal, complemented
12#27894C212yesmaximal, complemented
13#27596C212yesmaximal, complemented
14#27594C212yesmaximal, complemented
15#24014C212yesmaximal, complemented
16#24013C212yesmaximal
17#21988C212yesmaximal, complemented
18#21808C212yesmaximal, complemented
19#20719C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#484C2212yescomplemented
25C2×#481C2212yes—
26C2×#483C2212yescomplemented
27C2×#480C2212yes—
28C2×#432C2212yes—
29#2850C2211yescharacteristic
30#2850C2211yescharacteristic
31#2850C2211yescharacteristic
32#2850C2211yescharacteristic
33#2620C2211yescharacteristic
34#2620C2211yescharacteristic
35#2620C2211yescharacteristic
36#2620C2211yescharacteristic
37#2545—22no—
38#2544—22no—
39#2534—22no—
40#2534—22no—
41#2355—22no—
42#2354—22no—
43#2352—22no—
44#2352—22no—
45#2314C2212yes—
46#2314C2212yescomplemented
47#2311C2212yes—
48#2310C2211yescharacteristic
49#2309C2211yescharacteristic
50#2285C2212yes—
51#2285C2211yescharacteristic
52#2285C2212yescomplemented
53#2285C2211yescharacteristic, complemented
54C2×#384C2211yescharacteristic, complemented
55#2274C2212yescomplemented
56#2096C2212yescomplemented
57#2095C2212yescomplemented
58#2094C2212yes—
59#2093C2212yes—
60#2074C2212yescomplemented
61#2074C2212yescomplemented
62#2073C2212yes—
63#2073C2212yes—
64#2036C2214yescomplemented
65#2035C2214yescomplemented
66#2034C2214yes—
67#2033C2214yes—
68#1983C2214yes—
69#1983C2214yes—
70#1959C2212yes—
71#1959C2212yes—
72#1959C2212yes—
73#1959C2212yes—
74#1948C2214yescomplemented
75#1947C2214yescomplemented
76#1928C2214yescomplemented
77#1927C2214yescomplemented
78#1926C2214yes—
79#1925C2214yes—
80#1875C2212yes—
81#1875C2212yes—
82#1873C2212yescomplemented
83#1873C2212yescomplemented
84#1854C2214yes—
85#1854C2214yes—
86#1847C2212yescomplemented
87#1847C2212yescomplemented
88#1836C2212yescomplemented
89#1836C2212yescomplemented
90#1818C2212yes—
91#1798C2212yescomplemented
92#1798C2212yescomplemented
93#1787C2214yes—
94#1786C2214yes—
95#1780C2212yes—
96#1749C2211yescharacteristic, complemented
97#1749C2211yescharacteristic, complemented
98#1749C2212yescomplemented
99C22×#41C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101C23×D4C2311yescharacteristic
102#536—44no—
103#535—44no—
104C2×C4⋊Q8—21no—
105C2×#127—21no—
106C2×#124—21no—
107C2×#124—21no—
108C2×#124—22no—
109C2×#124—28no—
110C2×#124C2311yescharacteristic
111C2×#124C2311yescharacteristic
112C2×#124C2311yescharacteristic
113C2×#124C2311yescharacteristic
114C2×#123—21no—
115C2×#123—21no—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C22⋊Q8C2311yescharacteristic
118C2×C22⋊Q8C2311yescharacteristic
119C2×C22⋊Q8C2311yescharacteristic
120C2×#121C2311yescharacteristic
121C2×#121C2311yescharacteristic
122C2×#121C2311yescharacteristic
123C2×#121C2311yescharacteristic
124C2×#120—21no—
125C2×#120—21no—
126C4×#41—28no—
127C2×C4×Q8D412yes—
128C2×C4×D4D412yes—
129C2×C4×D4D412yes—
130C2×C4×D4D412yes—
131C2×#117—22no—
132C2×#117—22no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—22no—
140C22×C4⋊C4C2312yes—
141C22×C22⋊C4—22no—
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#484—44no—
146#483—44no—
147#481—44no—
148#480—44no—
149#432—28no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×#97C2312yes—
168C2×#97C2312yes—
169#396—22no—
170#396—22no—
171#396—22no—
172#396—22no—
173#394—22no—
174#394—22no—
175#394—22no—
176#394—22no—
177#391—22no—
178#391—22no—
179#391C2311yescharacteristic
180#391C2311yescharacteristic
181#391C2311yescharacteristic
182#391C2311yescharacteristic
183#390—22no—
184#390—22no—
185#390C2311yescharacteristic
186#390C2311yescharacteristic
187#390C2311yescharacteristic
188#390C2311yescharacteristic
189#388—24no—
190#388—24no—
191#388—22no—
192#388—22no—
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#386—22no—
203#384—22no—
204#384C2312yes—
205#384C2312yes—
206#384C2312yes—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382—22no—
210#382—22no—
211#382C2312yes—
212#382C2311yescharacteristic
213#382C2311yescharacteristic
214#381—22no—
215#380—22no—
216#380—22no—
217#380—22no—
218#380—22no—
219#380C2314yes—
220#380C2314yes—
221#379—22no—
222C4×C4⋊C4—24no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4C2314yes—
228C4×C4⋊C4C2314yes—
229C4×C22⋊C4—24no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—42no—
241C22×Q8—21no—
242C22×Q8—24no—
243C22×Q8—42no—
244C22×Q8—42no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—24no—
254C22×D4—42no—
255C22×D4—42no—
256C22×D4C2×D411yescharacteristic
257C22×D4C2×D411yescharacteristic
258C22×D4C2×D411yescharacteristic
259C22×D4C2411yescharacteristic
260C22×D4C2411yescharacteristic
261C23×C4—21no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C22×C8C2414yes—
270C22×C8C2414yes—
271C4⋊Q8—44no—
272#127—44no—
273#124—24no—
274#124—24no—
275#124—24no—
276#124—24no—
277#124—44no—
278#124—44no—
279#124—48no—
280#123—44no—
281#123—44no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—42no—
285C22⋊Q8—42no—
286#121—24no—
287#121—24no—
288#121—42no—
289#121—42no—
290#120—44no—
291#120—44no—
292C4×Q8—28no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296#117—24no—
297#117—24no—
298#117—24no—
299#117—44no—
300#117—42no—
301#117—42no—
302#117—84no—
303#117—84no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2414yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42#4118yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—44no—
369#107—44no—
370#106—44no—
371#106—44no—
372C4⋊C8—28no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#102—44no—
378#102—44no—
379#102—44no—
380#102—44no—
381C22⋊C8—28no—
382C22⋊C8—24no—
383C22⋊C8—24no—
384C22⋊C8—44no—
385#97—24no—
386#97—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398C24—21no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—82no—
404C24—82no—
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407C24C22×D411yescharacteristic
408#41—416no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—41no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—88no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—88no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4—82no—
448C2×D4—82no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—84no—
477C22×C4#12318yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4114yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
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—44no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—44no—
546C42—42no—
547C42—42no—
548C42—84no—
549C42—84no—
550C42#12318yes—
551C42C2×#4114yes—
552C42C2×#4114yes—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—84no—
577C23—88no—
578C23—164no—
579C23—162no—
580C23—162no—
581C23C2×#12312yes—
582C23#53512yes—
583C23C23×D411yescharacteristic
584Q8—44no—
585Q8—44no—
586Q8—44no—
587Q8—84no—
588D4—44no—
589D4—44no—
590D4—44no—
591D4—44no—
592D4—44no—
593D4—84no—
594D4—162no—
595D4—162no—
596D4—162no—
597D4—162no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—84no—
642C2×C4—164no—
643C2×C4C2×#12314yes—
644C2×C4C2×#12312yes—
645C2×C4C2×#12314yes—
646C2×C4#53512yes—
647C2×C4C22×#4112yes—
648C2×C4C2×H(5,𝔽2)12yes—
649C2×C4#58512yes—
650C8—44no—
651C8—44no—
652C22—24no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—168no—
678C22C22×#12311yescharacteristic
679C22C2×#53511yescharacteristic
680C22#344911yescharacteristic
681C22#358512yes—
682C22#358612yes—
683C4—24no—
684C4—24no—
685C4—22no—
686C4—22no—
687C4—42no—
688C4—42no—
689C4—41no—
690C4—84no—
691C4—81no—
692C4—84no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4#349414yes—
699C2—42no—
700C2—42no—
701C2—81no—
702C2—161no—
703C2—161no—
704C2#3526211yescharacteristic
705C2#6051712yes—
706C2#6082511yescharacteristic
707C2#6082611yescharacteristic
708C2#6101612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #33121C2semidirect
4. #33120C2semidirect
5. #29932C2semidirect
6. #29931C2semidirect
7. #27979C2semidirect
8. #27979C2semidirect
9. #27914C2semidirect
10. #27913C2semidirect
11. #27895C2semidirect
12. #27894C2semidirect
13. #27596C2semidirect
14. #27594C2semidirect
15. #24014C2semidirect
17. #21988C2semidirect
18. #21808C2semidirect
19. #20719C2semidirect
24. C2×#484C22semidirect
26. C2×#483C22semidirect
46. #2314C22semidirect
52. #2285C22semidirect
53. #2285C22semidirect
54. C2×#384C22semidirect
55. #2274C22semidirect
56. #2096C22semidirect
57. #2095C22semidirect
60. #2074C22semidirect
61. #2074C22semidirect
64. #2036C22semidirect
65. #2035C22semidirect
74. #1948C22semidirect
75. #1947C22semidirect
76. #1928C22semidirect
77. #1927C22semidirect
82. #1873C22semidirect
83. #1873C22semidirect
86. #1847C22semidirect
87. #1847C22semidirect
88. #1836C22semidirect
89. #1836C22semidirect
91. #1798C22semidirect
92. #1798C22semidirect
96. #1749C22semidirect
97. #1749C22semidirect
98. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#353272
#341882
#331212
#331202
#299322
#299312
#279792
#279142
#279132
#278952
#278942
#275962
#275942
#240142
#240132
#219882
#218082
#207192