Finite Group Databasebeta

← #6624355

Subgroups — #6624355

Order 512.

1825 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33147C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic
5#29109C212yesmaximal
6#28123C212yesmaximal
7#27961C211yesmaximal, characteristic, complemented
8#27648C211yesmaximal, characteristic, complemented
9#27445C214yesmaximal, complemented
10#27381C214yesmaximal, complemented
11#26020C212yesmaximal, complemented
12#24680C212yesmaximal, complemented
13#23067C212yesmaximal, complemented
14#22759C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21581C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#484C2212yes—
21C2×#476C2212yes—
22C2×#450C2211yescharacteristic
23C2×C4×SD16C2212yes—
24#2857C2211yescharacteristic
25#2857C2212yes—
26#2586—22no—
27#2567C2211yescharacteristic, complemented
28#2551C2211yescharacteristic, complemented
29#2549—22no—
30#2507—21no—
31#2507C2212yescomplemented
32#2451C2212yescomplemented
33#2440—21no—
34#2369—21no—
35#2369C2211yescharacteristic, complemented
36#2354—22no—
37#2354—21no—
38#2354—21no—
39#2354C2212yes—
40#2354C2214yes—
41#2354C2212yes—
42Q8×C4⋊C4C2211yescharacteristic, complemented
43#2345—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46C2×#390C2212yes—
47C2×#388—22no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yescomplemented
53#2083C2212yescomplemented
54#2072C2212yescomplemented
55#2071C2214yes—
56#2067C2214yes—
57#2055C2214yes—
58#2023C2214yes—
59#1990C2212yescomplemented
60#1988C2214yescomplemented
61#1948C2214yescomplemented
62#1947C2212yescomplemented
63#1939C2212yescomplemented
64#1936C2214yes—
65#1936C2214yes—
66#1927C2214yescomplemented
67#1925C2214yes—
68#1919C2214yescomplemented
69#1919C2214yescomplemented
70#1918C2214yescomplemented
71#1917C2214yescomplemented
72#1880C2214yes—
73#1877C2212yes—
74#1872C2211yescharacteristic
75#1864C2214yes—
76#1864C2212yes—
77#1853C2212yes—
78#1853C2211yescharacteristic
79#1834C2212yescomplemented
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1803C2212yescomplemented
84#1802C2212yes—
85#1797C2211yescharacteristic
86#1797C2212yes—
87#1786C2211yescharacteristic
88#1786C2212yes—
89#1776C2214yescomplemented
90#1770C2212yescomplemented
91#1763C2214yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2312yes—
106C2×C4×D4—28no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C2×#117—24no—
113C2×#117—22no—
114C2×#117—48no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42—22no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#484—28no—
128#476—28no—
129#450—24no—
130C4×SD16—28no—
131C2×#107C2314yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2311yescharacteristic
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#398—22no—
149#398—22no—
150#398—22no—
151#395—22no—
152#395—22no—
153#395—22no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2311yescharacteristic
160#390—22no—
161#390—22no—
162#390C2314yes—
163#390C2314yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—24no—
181#387—22no—
182#387—22no—
183#387—21no—
184#387—22no—
185#385—22no—
186#385—21no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#380—21no—
195#380—22no—
196#380—22no—
197#380—24no—
198#380—22no—
199#380C2312yes—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C25—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8C2411yescharacteristic
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—42no—
243C22×D4C2412yes—
244C23×C4—28no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—44no—
250C23×C4—44no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—42no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C22⋊Q8—22no—
266C22⋊Q8—84no—
267C22⋊Q8—82no—
268C22⋊Q8#4114yes—
269C4×Q8—24no—
270C4×Q8—44no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4#4118yes—
277#117—48no—
278#117—416no—
279#117—48no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—28no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—48no—
331C2×C22⋊C4—84no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—44no—
340C2×C42—48no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349#107—28no—
350#106—24no—
351#106—28no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#103—28no—
355#103—24no—
356#102—24no—
357#102—28no—
358#102—24no—
359#102—44no—
360#102—42no—
361C22⋊C8—22no—
362C22⋊C8—24no—
363C22⋊C8—22no—
364#97—44no—
365C4×C8—24no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—24no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24C22×D411yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—81no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—81no—
403C2×Q8C2×#4112yes—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×#4114yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—432no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462SD16—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4AGL(1,ℤ/8)14yes—
506C4⋊C4C2×#4112yes—
507C4⋊C4C2×#4114yes—
508C22⋊C4—28no—
509C22⋊C4—24no—
510C22⋊C4—28no—
511C22⋊C4—48no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—48no—
525C42—42no—
526C42—48no—
527C42—42no—
528C42—82no—
529C42—82no—
530C42C2×#4114yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—24no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23#54812yes—
550C23C2×#13712yes—
551C23#57712yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—42no—
555Q8—164no—
556Q8—162no—
557D4—24no—
558D4—24no—
559D4—48no—
560D4—42no—
561D4—42no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—416no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—416no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4#52512yes—
617C2×C4#52512yes—
618C2×C4#54812yes—
619C2×C4C2×AGL(1,ℤ/8)14yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4112yes—
625C2×C4C2×2−1+411yescharacteristic
626C2×C4#58512yes—
627C8—44no—
628C8—42no—
629C8—42no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—816no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22#336914yes—
645C22C2×#52511yescharacteristic
646C22C2×#54811yescharacteristic
647C22#348511yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C22×#13711yescharacteristic
650C22C2×#57711yescharacteristic
651C22#358511yescharacteristic
652C4—24no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—82no—
664C4—81no—
665C4—84no—
666C4—81no—
667C4—82no—
668C4#311014yes—
669C4#347512yes—
670C4#349512yes—
671C2—24no—
672C2—84no—
673C2#3651611yescharacteristic
674C2C2×#311011yescharacteristic
675C2#6058811yescharacteristic
676C2C2×#336911yescharacteristic
677C2#6132111yescharacteristic
678C2#6172111yescharacteristic
679C2#6172811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36564C2semidirect
3. #33147C2semidirect
7. #27961C2semidirect
8. #27648C2semidirect
9. #27445C2semidirect
10. #27381C2semidirect
11. #26020C2semidirect
12. #24680C2semidirect
13. #23067C2semidirect
14. #22759C2semidirect
15. #22629C2semidirect
16. #21581C2semidirect
27. #2567C22semidirect
28. #2551C22semidirect
31. #2507C22semidirect
32. #2451C22semidirect
35. #2369C22semidirect
42. Q8×C4⋊C4C22semidirect
52. #2100C22semidirect
53. #2083C22semidirect
54. #2072C22semidirect
59. #1990C22semidirect
60. #1988C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1939C22semidirect
66. #1927C22semidirect
68. #1919C22semidirect
69. #1919C22semidirect
70. #1918C22semidirect
71. #1917C22semidirect
79. #1834C22semidirect
83. #1803C22semidirect
89. #1776C22semidirect
90. #1770C22semidirect
91. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#23542
#331472
#329562
#291092
#281232
#279612
#276482
#274452
#273812
#260202
#246802
#230672
#227592
#226292
#215812
C4×#4502