Finite Group Databasebeta

← #6434108

Subgroups — #6434108

Order 512.

1875 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35263C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3#29918C214yesmaximal, complemented
4#29917C214yesmaximal, complemented
5#29913C212yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#29909C212yesmaximal, complemented
8#29906C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23737C212yesmaximal, complemented
11#23660C212yesmaximal, complemented
12#23634C212yesmaximal, complemented
13#20909C211yesmaximal, characteristic
14#20908C211yesmaximal, characteristic
15#20846C211yesmaximal, characteristic, complemented
16C2×#1852C211yesmaximal, characteristic, complemented
17#20685C211yesmaximal, characteristic, complemented
18C2×#525C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#428C2212yes—
23C22×#107C2212yes—
24C2×#417C2212yes—
25C2×#416C2212yes—
26C2×#407C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28#2855C2212yes—
29#2854C2212yes—
30#2625C2212yes—
31#2624C2212yes—
32#2498—22no—
33#2487—21no—
34#2486—21no—
35#2485—21no—
36#2485—21no—
37#2412—22no—
38#2317—21no—
39#2317—21no—
40#2317—22no—
41#2317C2212yes—
42#2317C2212yes—
43#2317C2214yes—
44#2316C2211yescharacteristic
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48C2×#389—21no—
49C2×#389—21no—
50C2×#385—22no—
51C2×#384C2212yes—
52C2×#382C2212yes—
53C2×#380C2211yescharacteristic
54#2101C2214yescomplemented
55#2101C2214yescomplemented
56#2100C2214yescomplemented
57#2099C2214yescomplemented
58#2097C2214yescomplemented
59#2095C2214yescomplemented
60#2093C2214yescomplemented
61#2092C2214yescomplemented
62#2085C2214yescomplemented
63#2085C2214yescomplemented
64#2083C2214yescomplemented
65#2083C2214yescomplemented
66#2074C2214yescomplemented
67#2073C2214yescomplemented
68#2072C2214yescomplemented
69#2071C2214yescomplemented
70#1875C2212yes—
71#1874C2211yescharacteristic
72#1874C2211yescharacteristic
73#1874C2212yes—
74#1874C2212yes—
75#1873C2212yes—
76#1872C2211yescharacteristic
77#1872C2211yescharacteristic
78#1872C2212yes—
79#1872C2212yes—
80#1854C2214yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1852C2214yes—
85#1852C2212yes—
86#1818C2214yes—
87#1817C2212yes—
88#1817C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C24×C4—21no—
97C24×C4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#525—24no—
101#525—24no—
102C2×#123—21no—
103C2×#123—22no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121C2311yescharacteristic
107C2×#121C2311yescharacteristic
108C2×#120—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×#117—24no—
112C2×#117—24no—
113C2×#117—21no—
114C2×#117—21no—
115C2×#117—44no—
116C2×#117D411yescharacteristic
117C2×#117D411yescharacteristic
118C2×#117D412yes—
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—44no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4D412yes—
143C22×C42—21no—
144C22×C42C2312yes—
145#428—44no—
146C2×#107—44no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151#417—44no—
152#416—44no—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159#407—24no—
160C2×C22⋊C8—24no—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2312yes—
166#400—22no—
167#399—22no—
168#398—22no—
169#397—22no—
170#397—22no—
171#397—22no—
172#394—22no—
173#394—22no—
174#391C2312yes—
175#391C2312yes—
176#390C2312yes—
177#390C2312yes—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—24no—
187#389—24no—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#385—22no—
200#385—24no—
201#385—24no—
202#385—24no—
203#385—24no—
204#384—22no—
205#384—22no—
206#384C2314yes—
207#384C2314yes—
208#382—22no—
209#382—22no—
210#382C2314yes—
211#382C2314yes—
212#381—22no—
213#380—21no—
214#380—21no—
215#380—21no—
216#380—21no—
217#380—21no—
218#380—21no—
219#380—22no—
220#380—22no—
221#380C2312yes—
222#380C2312yes—
223C2×#95—21no—
224C2×#95—22no—
225C2×#95—22no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240C25C2×D411yescharacteristic
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—24no—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—44no—
261C23×C4—82no—
262C23×C4—82no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4#4114yes—
269C23×C4#4114yes—
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×M16—24no—
274C2×M16—24no—
275C2×M16—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2414yes—
282#123—44no—
283#123—84no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288#121—22no—
289#121—22no—
290#121—22no—
291#121—22no—
292#120—44no—
293C4×D4—44no—
294C4×D4—44no—
295#117—24no—
296#117—24no—
297#117—28no—
298#117—24no—
299#117—24no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—24no—
307#117—24no—
308#117—42no—
309#117—42no—
310#117—44no—
311#117—44no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4—84no—
343C2×C4⋊C4—84no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C22⋊C4—28no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—82no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C42—24no—
378C2×C42—24no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—44no—
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387#107—44no—
388#107—44no—
389#107—44no—
390#106—44no—
391#103—42no—
392#103—42no—
393#103—44no—
394#102—42no—
395#102—42no—
396#102—44no—
397C22⋊C8—22no—
398C22⋊C8—22no—
399C22⋊C8—24no—
400C22⋊C8—24no—
401C22⋊C8—24no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—21no—
421C24—41no—
422C24—41no—
423C24#12314yes—
424C24C22×D411yescharacteristic
425C24C22×D411yescharacteristic
426C24C22×D411yescharacteristic
427#41—48no—
428#41—48no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—81no—
445C2×D4—81no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—216no—
453C22×C4—216no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—81no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—816no—
497C22×C4#12314yes—
498C22×C4#12314yes—
499C22×C4#12314yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)14yes—
514C22×C42−1+414yes—
515C22×C4C2511yescharacteristic
516M16—48no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—24no—
526C2×C8—48no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—84no—
557C4⋊C4—84no—
558C4⋊C4—84no—
559C4⋊C4—84no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—44no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C42—24no—
580C42—24no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—44no—
586C42—42no—
587C42—44no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—28no—
599C23—28no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—41no—
610C23—41no—
611C23—41no—
612C23—41no—
613C23—41no—
614C23—41no—
615C23—41no—
616C23C2×#12312yes—
617C23C2×#12312yes—
618C23C2×#12312yes—
619C23#53512yes—
620C23#53612yes—
621C23#57612yes—
622C23#57812yes—
623C23C23×D411yescharacteristic
624Q8—42no—
625Q8—42no—
626D4—42no—
627D4—42no—
628D4—84no—
629D4—84no—
630D4—84no—
631C2×C4—28no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—88no—
683C2×C4—88no—
684C2×C4—88no—
685C2×C4—84no—
686C2×C4—162no—
687C2×C4—162no—
688C2×C4C2×#12312yes—
689C2×C4C2×#12312yes—
690C2×C4C2×#12312yes—
691C2×C4#53512yes—
692C2×C4#53612yes—
693C2×C4#53918yes—
694C2×C4#53918yes—
695C2×C4C22×#4111yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C2×H(5,𝔽2)12yes—
698C2×C4C2×2−1+412yes—
699C8—42no—
700C8—42no—
701C8—44no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—416no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—81no—
723C22—81no—
724C22—81no—
725C22—81no—
726C22—81no—
727C22—81no—
728C22—81no—
729C22#262314yes—
730C22#309814yes—
731C22C22×#12311yescharacteristic
732C22C2×#53511yescharacteristic
733C22C2×#53611yescharacteristic
734C22C2×#57611yescharacteristic
735C22C2×#57811yescharacteristic
736C22#358511yescharacteristic
737C22#358611yescharacteristic
738C4—22no—
739C4—22no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—42no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—82no—
748C4—84no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#262314yes—
753C4C2×#53912yes—
754C4C2×#53912yes—
755C2—24no—
756C2—24no—
757C2—42no—
758C2—161no—
759C2C2×#262311yescharacteristic
760C2C2×#309811yescharacteristic
761C2#6051711yescharacteristic
762C2#6081111yescharacteristic
763C2#6081411yescharacteristic
764C2#6087111yescharacteristic
765C2#6087411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2317C2semidirect
3. #29918C2semidirect
4. #29917C2semidirect
5. #29913C2semidirect
6. #29912C2semidirect
7. #29909C2semidirect
8. #29906C2semidirect
9. #23749C2semidirect
10. #23737C2semidirect
11. #23660C2semidirect
12. #23634C2semidirect
15. #20846C2semidirect
16. C2×#1852C2semidirect
17. #20685C2semidirect
54. #2101C22semidirect
55. #2101C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
58. #2097C22semidirect
59. #2095C22semidirect
60. #2093C22semidirect
61. #2092C22semidirect
62. #2085C22semidirect
63. #2085C22semidirect
64. #2083C22semidirect
65. #2083C22semidirect
66. #2074C22semidirect
67. #2073C22semidirect
68. #2072C22semidirect
69. #2071C22semidirect

Maximal subgroups

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

subgroupindex
#352632
C2×#23172
#299182
#299172
#299132
#299122
#299092
#299062
#237492
#237372
#236602
#236342
#209092
#209082
#208462
C2×#18522
#206852