Finite Group Databasebeta

← #6497419

Subgroups — #6497419

Order 512.

1537 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#35201C211yesmaximal, characteristic
3#33294C212yesmaximal, complemented
4#33282C212yesmaximal, complemented
5#30235C212yesmaximal, complemented
6#30216C212yesmaximal, complemented
7#27815C212yesmaximal, complemented
8#27813C212yesmaximal, complemented
9#27737C212yesmaximal, complemented
10#27735C212yesmaximal, complemented
11#27722C214yesmaximal, complemented
12#27720C214yesmaximal
13#21867C211yesmaximal, characteristic
14#21839C211yesmaximal, characteristic, complemented
15#21403C211yesmaximal, characteristic
16#21322C211yesmaximal, characteristic, complemented
17#20480C211yesmaximal, characteristic, complemented
18C2×#544—21no—
19C2×#533C2211yescharacteristic
20C2×#529C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2868C2212yes—
24#2861C2212yes—
25#2859C2212yes—
26#2859C2212yes—
27#2663—22no—
28#2660—22no—
29#2604—22no—
30#2568C2211yescharacteristic
31#2567C2211yescharacteristic
32#2563C2211yescharacteristic
33#2552—21no—
34#2552C2212yes—
35#2551—21no—
36#2551C2212yes—
37#2542C2211yescharacteristic
38#2500C2212yes—
39#2499C2212yes—
40#2440—22no—
41#2440C2214yes—
42#2415—21no—
43#2415—21no—
44#2323—21no—
45#2323—21no—
46#2317—21no—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2072C2214yes—
55#2071C2214yes—
56#2024C2214yes—
57#2023C2214yes—
58#1992C2212yes—
59#1992C2212yes—
60#1992C2212yes—
61#1992C2212yes—
62#1992C2211yescharacteristic
63#1992C2212yes—
64#1992C2211yescharacteristic
65#1990C2214yes—
66#1990C2214yes—
67#1990C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1987C2212yes—
71#1959C2211yescharacteristic
72#1959C2212yes—
73#1959C2211yescharacteristic
74#1936C2214yes—
75#1935C2214yes—
76#1934C2214yes—
77#1933C2214yes—
78#1895C2212yes—
79#1894C2212yes—
80#1874C2212yes—
81#1872C2212yes—
82#1836C2214yes—
83#1836C2214yes—
84#1825C2214yes—
85#1803C2212yes—
86#1802C2212yes—
87#1770C2212yes—
88C8×C4⋊C4C2212yes—
89C4×#103C2212yes—
90C4×#102C2212yes—
91#1753C2214yes—
92#1745C2212yes—
93#1736C2214yes—
94C4×#97C2212yes—
95C22×M16C2312yes—
96#544—48no—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105#529—24no—
106#529—24no—
107C2×#126—21no—
108C2×#125—22no—
109C2×#125—22no—
110C2×#125C2311yescharacteristic
111C2×#125C2312yes—
112C2×#124C2311yescharacteristic
113C2×#123—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8C2311yescharacteristic
120C2×#121C2311yescharacteristic
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4C2311yescharacteristic
126C2×#117—21no—
127C2×#117—21no—
128C2×#117D411yescharacteristic
129C2×#117D411yescharacteristic
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135#432—24no—
136C2×C4⋊C8C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2312yes—
144C4×M16—24no—
145C2×#97C2314yes—
146C2×#97C2311yescharacteristic
147C2×#97C2311yescharacteristic
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#399—22no—
153#398—24no—
154#398—24no—
155#397—22no—
156#397—22no—
157#395—22no—
158#395—24no—
159#395—24no—
160#394—22no—
161#394—22no—
162#393—22no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2314yes—
180#389C2314yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#385—24no—
190#384—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2312yes—
196#384C2314yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2311yescharacteristic
201#382—22no—
202#382—21no—
203#382C2312yes—
204#382C2312yes—
205#380—21no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217#376D412yes—
218C43C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—24no—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—42no—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—48no—
259C4⋊Q8—48no—
260C4⋊Q8—84no—
261#126—44no—
262#125—22no—
263#125—22no—
264#125—24no—
265#125—24no—
266#125—48no—
267#125—48no—
268#124—22no—
269#124—22no—
270#123—44no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—416no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278#121—22no—
279#121—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—44no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—44no—
292#117—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
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—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
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—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42#4114yes—
356C2×C42#4114yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C4⋊C8—24no—
362#103—24no—
363#103—44no—
364#103—42no—
365#102—24no—
366#102—44no—
367#102—42no—
368C22⋊C8—24no—
369#97—28no—
370#97—22no—
371#97—22no—
372#97—24no—
373C4×C8—24no—
374C4×C8—44no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24C22×D411yescharacteristic
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—48no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—84no—
405C2×Q8—84no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—81no—
412C2×D4—81no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4#12414yes—
434C22×C4#12414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2511yescharacteristic
447M16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C82−1+414yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4AGL(1,ℤ/8)14yes—
500C4⋊C4#13714yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C42—24no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—42no—
522C42—44no—
523C42#12414yes—
524C42#12414yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C2×#12412yes—
543C23C23×D411yescharacteristic
544Q8—42no—
545Q8—42no—
546Q8—84no—
547Q8—84no—
548Q8—168no—
549D4—42no—
550D4—42no—
551D4—84no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—164no—
600C2×C4C2×#12412yes—
601C2×C4C2×#12412yes—
602C2×C4C2×#12412yes—
603C2×C4C2×#12412yes—
604C2×C4C2×#12412yes—
605C2×C4C2×AGL(1,ℤ/8)14yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×#13714yes—
608C2×C4C2×#13712yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C2×2−1+412yes—
612C2×C4#58512yes—
613C8—24no—
614C8—44no—
615C8—42no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C22×#12411yescharacteristic
631C22#349911yescharacteristic
632C22#350011yescharacteristic
633C22C22×AGL(1,ℤ/8)11yescharacteristic
634C22C22×#13711yescharacteristic
635C22#358511yescharacteristic
636C22#358611yescharacteristic
637C4—24no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4—81no—
653C4#314114yes—
654C4#346112yes—
655C4#349512yes—
656C2—42no—
657C2—161no—
658C2#3531211yescharacteristic
659C2C2×#314111yescharacteristic
660C2#6062111yescharacteristic
661C2#6250911yescharacteristic
662C2#6251011yescharacteristic
663C2#6257611yescharacteristic
664C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35275C2semidirect
3. #33294C2semidirect
4. #33282C2semidirect
5. #30235C2semidirect
6. #30216C2semidirect
7. #27815C2semidirect
8. #27813C2semidirect
9. #27737C2semidirect
10. #27735C2semidirect
11. #27722C2semidirect
14. #21839C2semidirect
16. #21322C2semidirect
17. #20480C2semidirect

Maximal subgroups

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

subgroupindex
#352752
#352012
#332942
#332822
#302352
#302162
#278152
#278132
#277372
#277352
#277222
#277202
#218672
#218392
#214032
#213222
#204802