Finite Group Databasebeta

← #6567803

Subgroups — #6567803

Order 512.

1659 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#29981C211yesmaximal, characteristic, complemented
4#29969C212yesmaximal, complemented
5#29272C211yesmaximal, characteristic, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29162C212yesmaximal, complemented
8#27683C212yesmaximal, complemented
9#26513C212yesmaximal, complemented
10#26426C214yesmaximal, complemented
11#26111C212yesmaximal, complemented
12#26006C214yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14#23747C212yesmaximal, complemented
15#22713C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22500C212yesmaximal
18C2×#547C2211yescharacteristic
19C2×#520—21no—
20C2×#436C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571C2212yes—
24#2563—21no—
25#2538—21no—
26#2497—22no—
27#2492—22no—
28#2439C2214yes—
29#2369—21no—
30#2365C2211yescharacteristic
31#2361C2212yes—
32#2360—21no—
33#2359—22no—
34#2355—21no—
35#2354C2212yes—
36D4×C4⋊C4C2211yescharacteristic
37D4×C4⋊C4C2211yescharacteristic
38#2344C2212yes—
39#2339—21no—
40#2323—21no—
41#2322—22no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#121C2212yes—
45#2293C2211yescharacteristic
46C2×#389C2212yes—
47#2285C2212yes—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2101C2214yes—
51#2100C2212yes—
52#2100C2212yes—
53#2100C2212yes—
54#2099C2212yes—
55#2099C2211yescharacteristic
56#2094C2212yes—
57#2093C2211yescharacteristic
58#2087C2214yes—
59#2087C2214yes—
60#2086C2212yes—
61#2067C2212yes—
62#2058C2212yes—
63#2057C2211yescharacteristic
64#2057C2212yes—
65#2023C2212yes—
66#2023C2211yescharacteristic
67#1990C2212yes—
68#1988C2212yes—
69#1987C2214yes—
70#1949C2214yes—
71#1948C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1919C2214yes—
75#1918C2214yes—
76#1895C2214yes—
77#1875C2214yes—
78#1862C2214yes—
79#1854C2214yes—
80#1852C2214yes—
81#1851C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1824C2214yes—
85#1809C2214yes—
86#1804C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89C4×#107C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41—21no—
96C22×M16C2314yes—
97#547—24no—
98#547Q818yescomplemented
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—21no—
107C2×#121C2312yes—
108#520—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—21no—
122C2×#117C2314yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128#436—28no—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C4×C8C2312yes—
141#400—24no—
142#396—24no—
143#393—24no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391—22no—
148#391—24no—
149#390—22no—
150#390C2311yescharacteristic
151#390C2311yescharacteristic
152#389—22no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2312yes—
160#389C2312yes—
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#388—24no—
167#388—21no—
168#388—22no—
169#388—21no—
170#388—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#386—22no—
176#386—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#382—22no—
187#382C2314yes—
188#381—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—21no—
193#380C2312yes—
194#380C2314yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C22⋊C4—21no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212C2×#95—22no—
213C43C2311yescharacteristic
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232C2×#41—28no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×D4—22no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×M16—28no—
249C2×M16—44no—
250C2×M16—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8C2×Q814yes—
260#123—216no—
261C22⋊Q8—28no—
262C22⋊Q8—84no—
263#121—24no—
264#121C2×Q818yes—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—42no—
271C4×D4—44no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4—44no—
275C4×D4C2×Q814yes—
276C4×D4C2×Q814yes—
277C4×D4#4114yes—
278#117—48no—
279#117—44no—
280#117—44no—
281#117—44no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×Q818yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—42no—
348#107—42no—
349#107—44no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353C4⋊C8—42no—
354#102—42no—
355#102—42no—
356#102—44no—
357C22⋊C8—28no—
358C4×C8—24no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—24no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D412yes—
379#41—432no—
380C2×Q8—24no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×D4—28no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4C22⋊Q814yes—
398C2×D4C22×Q812yes—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×Q814yes—
437C22×C4C2×#4114yes—
438C22×C4C2511yescharacteristic
439M16—88no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C4⋊C4—28no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4C22⋊Q814yes—
483C4⋊C4C22×Q812yes—
484C4⋊C4C22×Q814yes—
485C22⋊C4—24no—
486C22⋊C4—28no—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—81no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4C22⋊Q818yes—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—41no—
506C42—44no—
507C42—44no—
508C42—42no—
509C42—44no—
510C42—84no—
511C42AGL(1,ℤ/8)14yes—
512C42C22×Q812yes—
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×C22⋊Q814yes—
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—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—164no—
586C2×C4—164no—
587C2×C4C2×C22⋊Q812yes—
588C2×C4C2×C22⋊Q814yes—
589C2×C4C2×C22⋊Q812yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4#57512yes—
594C2×C4#57812yes—
595C2×C4C23×Q811yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4#58512yes—
598C2×C4#58512yes—
599C8—42no—
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—
616C22C22×C22⋊Q811yescharacteristic
617C22#348511yescharacteristic
618C22#350011yescharacteristic
619C22C22×AGL(1,ℤ/8)11yescharacteristic
620C22C2×#57511yescharacteristic
621C22C2×#57811yescharacteristic
622C22#358511yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—41no—
633C4—81no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—84no—
639C4#307714yes—
640C4#350412yes—
641C4#352112yes—
642C2—44no—
643C2—82no—
644C2#3603111yescharacteristic
645C2C2×#307711yescharacteristic
646C2#6049811yescharacteristic
647C2#6172011yescharacteristic
648C2#6173011yescharacteristic
649C2#6253811yescharacteristic
650C2#6259011yescharacteristic
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. #29981C2semidirect
4. #29969C2semidirect
5. #29272C2semidirect
6. #29218C2semidirect
7. #29162C2semidirect
8. #27683C2semidirect
9. #26513C2semidirect
10. #26426C2semidirect
11. #26111C2semidirect
12. #26006C2semidirect
13. #23774C2semidirect
14. #23747C2semidirect
15. #22713C2semidirect
16. #22700C2semidirect
98. #547Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
#343082
#299812
#299692
#292722
#292182
#291622
#276832
#265132
#264262
#261112
#260062
#237742
#237472
#227132
#227002
#225002