Finite Group Databasebeta

← #6556037

Subgroups — #6556037

Order 512.

1651 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34308C211yesmaximal, characteristic
3#30172C212yesmaximal, complemented
4#29868C211yesmaximal, characteristic, complemented
5#29302C212yesmaximal, complemented
6#29272C211yesmaximal, characteristic, complemented
7#28687C211yesmaximal, characteristic, complemented
8#27611C212yesmaximal, complemented
9#27477C214yesmaximal, complemented
10#27049C214yesmaximal, complemented
11#26302C212yesmaximal, complemented
12#25699C212yesmaximal
13#23774C211yesmaximal, characteristic, complemented
14#23761C212yesmaximal, complemented
15#23163C212yesmaximal, complemented
16#22969C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#547C2211yescharacteristic
19C2×#520—21no—
20C2×#498C2212yes—
21C2×#476C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2571—21no—
25#2563C2212yes—
26#2538—21no—
27#2497C2214yes—
28#2492—22no—
29#2439—22no—
30#2369C2212yes—
31#2365C2211yescharacteristic
32#2361C2212yes—
33#2360—21no—
34#2359—22no—
35#2355—21no—
36#2354—21no—
37D4×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39#2344C2212yes—
40#2339—21no—
41#2323—21no—
42#2322—22no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#121C2212yes—
46#2293C2211yescharacteristic
47C2×#389C2212yes—
48#2285C2212yes—
49C42×D4C2211yescharacteristic
50#2099C2212yes—
51#2099C2211yescharacteristic
52#2093C2211yescharacteristic
53#2086C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2084C2212yes—
57#2084C2212yes—
58#2076C2214yes—
59#2076C2214yes—
60#2071C2212yes—
61#2071C2212yes—
62#2067C2212yes—
63#2057C2212yes—
64#2057C2211yescharacteristic
65#2034C2214yes—
66#2034C2214yes—
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1988C2212yes—
70#1987C2212yes—
71#1954C2212yes—
72#1948C2214yes—
73#1947C2212yes—
74#1944C2214yes—
75#1944C2214yes—
76#1924C2214yes—
77#1920C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1862C2214yes—
82#1853C2214yes—
83#1853C2214yes—
84#1852C2214yes—
85#1831C2211yescharacteristic
86#1824C2214yes—
87#1817C2212yes—
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41—21no—
96#547—24no—
97#547—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—21no—
106C2×#121C2312yes—
107#520—48no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117—22no—
121C2×#117—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C42C2312yes—
127#498—28no—
128#476—28no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144#400—24no—
145#396—24no—
146#393—24no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—24no—
152#390—22no—
153#390C2311yescharacteristic
154#390C2311yescharacteristic
155#389—22no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2311yescharacteristic
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2314yes—
168#389C2312yes—
169#388—24no—
170#388—21no—
171#388—22no—
172#388—21no—
173#388—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#386—22no—
179#386—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—22no—
190#382—22no—
191#381—22no—
192#380—22no—
193#380—22no—
194#380—21no—
195#380C2314yes—
196#380C2314yes—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C2×#95—22no—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C2×#41—28no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—42no—
256C4⋊Q8—82no—
257C4⋊Q8—82no—
258C4⋊Q8#4114yes—
259#123—216no—
260C22⋊Q8—28no—
261C22⋊Q8—84no—
262#121—24no—
263#121—24no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—44no—
273C4×D4—48no—
274C4×D4—48no—
275C4×D4—44no—
276C4×D4#4114yes—
277#117—48no—
278#117—44no—
279#117—84no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—48no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—42no—
347#107—44no—
348#106—28no—
349#106—28no—
350#106—28no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353#102—28no—
354#102—24no—
355#102—42no—
356C22⋊C8—28no—
357C22⋊C8—44no—
358#97—44no—
359C4×C8—24no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C24C22×D412yes—
380#41—432no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×D4—28no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4#13714yes—
399C2×D4C2×#4112yes—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C82−1+414yes—
449C4⋊C4—28no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C2×#4112yes—
484C4⋊C4C2×#4114yes—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—44no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—82no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—41no—
506C42—44no—
507C42—44no—
508C42—42no—
509C42—44no—
510C42—84no—
511C42AGL(1,ℤ/8)14yes—
512C42C2×#4112yes—
513C42C2×#4112yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C2×#13714yes—
528C23C23×D411yescharacteristic
529Q8—48no—
530Q8—84no—
531Q8—164no—
532D4—28no—
533D4—24no—
534D4—416no—
535D4—42no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—48no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—164no—
587C2×C4—164no—
588C2×C4#52512yes—
589C2×C4#52512yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×#13712yes—
594C2×C4#57712yes—
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×2−1+412yes—
598C2×C4#58512yes—
599C8—24no—
600C8—44no—
601C8—42no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#52511yescharacteristic
617C22#348511yescharacteristic
618C22#349911yescharacteristic
619C22C22×AGL(1,ℤ/8)11yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57711yescharacteristic
622C22#358511yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—44no—
633C4—81no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4—84no—
639C4#311014yes—
640C4#350712yes—
641C4#351612yes—
642C2—44no—
643C2—82no—
644C2#3584611yescharacteristic
645C2C2×#311011yescharacteristic
646C2#6059311yescharacteristic
647C2#6172011yescharacteristic
648C2#6173511yescharacteristic
649C2#6247711yescharacteristic
650C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #30172C2semidirect
4. #29868C2semidirect
5. #29302C2semidirect
6. #29272C2semidirect
7. #28687C2semidirect
8. #27611C2semidirect
9. #27477C2semidirect
10. #27049C2semidirect
11. #26302C2semidirect
13. #23774C2semidirect
14. #23761C2semidirect
15. #23163C2semidirect
16. #22969C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343082
#301722
#298682
#293022
#292722
#286872
#276112
#274772
#270492
#263022
#256992
#237742
#237612
#231632
#229692
#228392