Finite Group Databasebeta

← #6855927

Subgroups — #6855927

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic, complemented
3#33109C211yesmaximal, characteristic, complemented
4#32244C214yesmaximal, complemented
5#30824C214yesmaximal, complemented
6#29761C212yesmaximal
7#28799C212yesmaximal, complemented
8#27972C211yesmaximal, characteristic, complemented
9#27776C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic
11#26462C212yesmaximal, complemented
12#26020C212yesmaximal, complemented
13#25908C212yesmaximal
14#24301C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22085C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#489C2214yes—
21C2×#484C2212yes—
22C2×#481C2211yescharacteristic
23C2×#475C2214yes—
24#2857C2211yescharacteristic
25#2850C2212yes—
26#2596—22no—
27#2571C2211yescharacteristic
28#2551—22no—
29#2551—21no—
30#2551—21no—
31#2551C2212yes—
32#2551C2214yes—
33#2551C2212yes—
34#2547C2211yescharacteristic
35#2514C2212yes—
36#2510—22no—
37#2494—21no—
38#2487—21no—
39#2448C2212yes—
40C2×#395—22no—
41#2365C2211yescharacteristic
42#2354C2211yescharacteristic
43#2352—21no—
44#2347—21no—
45C4×C22⋊Q8C2212yes—
46C4×#121C2211yescharacteristic
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yes—
53#2094C2212yes—
54#2085C2214yes—
55#2072C2214yes—
56#2067C2214yes—
57#2066C2214yes—
58#2058C2214yes—
59#2042C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#2023C2214yes—
63#1993C2214yes—
64#1988C2214yes—
65#1981C2212yes—
66#1956C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1943C2214yes—
70#1939C2212yes—
71#1935C2214yes—
72#1928C2214yes—
73#1926C2212yes—
74#1919C2212yes—
75#1918C2214yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1865C2214yes—
83#1853C2212yes—
84#1853C2211yescharacteristic
85#1834C2212yes—
86#1824C2211yescharacteristic
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—48no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105C2×C4⋊Q8C2314yes—
106C2×#124C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4C2311yescharacteristic
112C2×#117—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#489—48no—
126#484—28no—
127#481—24no—
128#475—48no—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2311yescharacteristic
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147#401—22no—
148#400—22no—
149#400—22no—
150#399—22no—
151#397—22no—
152#397—22no—
153#395—24no—
154#395—24no—
155#395—24no—
156#395—24no—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#390C2312yes—
161#390C2311yescharacteristic
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#389C2311yescharacteristic
175#388—22no—
176#388—22no—
177#388—22no—
178#388—21no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#386—21no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#383—22no—
194#382—21no—
195#382C2312yes—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380C2311yescharacteristic
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—84no—
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2411yescharacteristic
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—48no—
261C4⋊Q8—416no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264#124—44no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267#121—22no—
268#121—22no—
269C4×Q8—44no—
270C4×D4—24no—
271C4×D4—44no—
272#117—44no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—28no—
342#106—24no—
343#106—24no—
344#106—44no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—28no—
348#103—28no—
349#103—44no—
350#103—44no—
351#102—24no—
352#102—24no—
353#102—44no—
354C22⋊C8—22no—
355C22⋊C8—24no—
356C22⋊C8—22no—
357#97—44no—
358C4×C8—24no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—21no—
381C24C22×D411yescharacteristic
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×Q8—832no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4#13714yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4112yes—
444C22×C42−1+412yes—
445C22×C4C2511yescharacteristic
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4AGL(1,ℤ/8)14yes—
490C4⋊C4C2×#4114yes—
491C4⋊C4H(5,𝔽2)14yes—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—81no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—42no—
513C42—82no—
514C422−1+414yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23#54812yes—
527C23C2×#13712yes—
528C23#57712yes—
529C23C23×D411yescharacteristic
530Q8—44no—
531Q8—44no—
532Q8—816no—
533Q8—816no—
534D4—42no—
535D4—42no—
536D4—84no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4#53612yes—
592C2×C4#53612yes—
593C2×C4#54812yes—
594C2×C4C2×AGL(1,ℤ/8)14yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)12yes—
600C2×C4C2×2−1+412yes—
601C2×C4C2×2−1+411yescharacteristic
602C8—42no—
603C8—44no—
604C8—42no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22#336914yes—
619C22C2×#53611yescharacteristic
620C22C2×#54811yescharacteristic
621C22#348411yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57711yescharacteristic
625C22#358611yescharacteristic
626C4—22no—
627C4—24no—
628C4—22no—
629C4—22no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—41no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—84no—
641C4—81no—
642C4#323914yes—
643C4#351112yes—
644C4#355212yes—
645C2—24no—
646C2—161no—
647C2#3972011yescharacteristic
648C2C2×#323911yescharacteristic
649C2#6087211yescharacteristic
650C2C2×#336911yescharacteristic
651C2#6131711yescharacteristic
652C2#6158411yescharacteristic
653C2#6160011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2551C2semidirect
3. #33109C2semidirect
4. #32244C2semidirect
5. #30824C2semidirect
7. #28799C2semidirect
8. #27972C2semidirect
9. #27776C2semidirect
11. #26462C2semidirect
12. #26020C2semidirect
14. #24301C2semidirect
15. #22629C2semidirect
16. #22085C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25512
#331092
#322442
#308242
#297612
#287992
#279722
#277762
#276782
#264622
#260202
#259082
#243012
#226292
#220852
C4×#4812