Finite Group Databasebeta

← #6570706

Subgroups — #6570706

Order 512.

1699 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#36405C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#29936C211yesmaximal, characteristic, complemented
4#29908C212yesmaximal, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27861C212yesmaximal, complemented
7#27680C211yesmaximal, characteristic, complemented
8#27615C212yesmaximal, complemented
9#26278C212yesmaximal, complemented
10#26216C214yesmaximal, complemented
11#24996C212yesmaximal, complemented
12#24949C214yesmaximal, complemented
13#23756C211yesmaximal, characteristic, complemented
14#23745C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22504C212yesmaximal, complemented
17#22473C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#436C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2582C2211yescharacteristic
25#2546C2211yescharacteristic
26#2485C2212yes—
27#2478—22no—
28#2468—22no—
29#2447—21no—
30#2432—21no—
31#2415C2212yes—
32#2365—21no—
33#2362—21no—
34#2354C2211yescharacteristic
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2212yes—
41#2317—21no—
42#2317—22no—
43C4×C22⋊Q8C2211yescharacteristic
44#2296C2212yes—
45#2285C2211yescharacteristic
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49#2100C2214yes—
50#2100C2214yes—
51#2099C2212yes—
52#2099C2212yes—
53#2099C2212yes—
54#2099C2211yescharacteristic
55#2099C2212yes—
56#2096C2212yes—
57#2096C2211yescharacteristic
58#2084C2214yes—
59#2084C2211yescharacteristic
60#2083C2212yes—
61#2077C2212yes—
62#2075C2214yes—
63#2072C2214yes—
64#2072C2211yescharacteristic
65#2035C2212yes—
66#1990C2212yes—
67#1988C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1928C2214yes—
72#1921C2214yes—
73#1919C2214yes—
74#1919C2214yes—
75#1894C2214yes—
76#1872C2212yes—
77#1872C2214yes—
78#1870C2214yes—
79#1854C2214yes—
80#1853C2212yes—
81#1853C2214yes—
82#1834C2212yes—
83#1825C2214yes—
84#1823C2214yes—
85#1817C2212yes—
86#1809C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2212yes—
90C4×#103C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2314yes—
97#556—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#126—22no—
101C2×#124—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2311yescharacteristic
105#520—48no—
106C4×#41—28no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D412yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#436—28no—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145#398—22no—
146#398—22no—
147#397—22no—
148#397—22no—
149#394—22no—
150#394—22no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391C2312yes—
155#391C2311yescharacteristic
156#390—21no—
157#389—22no—
158#389—24no—
159#389—24no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2311yescharacteristic
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—21no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—21no—
180#385—24no—
181#385—24no—
182#385—24no—
183#384—22no—
184#384—21no—
185#384—22no—
186#384—21no—
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2314yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#382—22no—
194#382—21no—
195#381—22no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—24no—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4C2×D412yes—
239C22×D4C2×D411yescharacteristic
240C23×C4—24no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C2×M16—28no—
249C2×M16—44no—
250C2×M16—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256#126—48no—
257#124—44no—
258C22⋊Q8—28no—
259C22⋊Q8—22no—
260C22⋊Q8—41no—
261C22⋊Q8—44no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—48no—
266C4×Q8—44no—
267C4×D4—24no—
268C4×D4—28no—
269C4×D4—28no—
270C4×D4—416no—
271C4×D4—48no—
272C4×D4#4118yes—
273#117—24no—
274#117—28no—
275#117—28no—
276#117—24no—
277#117—42no—
278#117—48no—
279#117—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—28no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—44no—
350#106—24no—
351#106—42no—
352#106—44no—
353C4⋊C8—24no—
354#103—42no—
355#103—42no—
356#103—44no—
357C22⋊C8—28no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360C4×C8—24no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380#41—232no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8#13714yes—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4#12314yes—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—84no—
428C22×C4—82no—
429C22×C4—84no—
430C22×C4AGL(1,ℤ/8)14yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4118yes—
440C22×C4C2511yescharacteristic
441M16—88no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4#12314yes—
490C4⋊C42−1+412yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—81no—
501C22⋊C4—82no—
502C22⋊C4#12318yes—
503C42—24no—
504C42—22no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—44no—
510C42—48no—
511C42—44no—
512C42—42no—
513C42—44no—
514C42—42no—
515C42C2×#4114yes—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23C2×#12314yes—
529C23C2×AGL(1,ℤ/8)12yes—
530C23C23×D411yescharacteristic
531Q8—28no—
532Q8—42no—
533Q8—42no—
534Q8—84no—
535D4—216no—
536D4—28no—
537C2×C4—216no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—164no—
589C2×C4—168no—
590C2×C4C2×#12314yes—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×#13714yes—
596C2×C4C2×#13712yes—
597C2×C4#57712yes—
598C2×C4C22×#4112yes—
599C2×C4C2×2−1+411yescharacteristic
600C2×C4#58512yes—
601C2×C4#58511yescharacteristic
602C8—42no—
603C8—44no—
604C8—42no—
605C22—22no—
606C22—24no—
607C22—22no—
608C22—24no—
609C22—22no—
610C22—24no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22C22×#12311yescharacteristic
620C22#348911yescharacteristic
621C22#350011yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57711yescharacteristic
625C22#358611yescharacteristic
626C4—22no—
627C4—24no—
628C4—22no—
629C4—22no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—84no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—81no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4#309214yes—
643C4#351512yes—
644C4#352112yes—
645C2—42no—
646C2—44no—
647C2#3606711yescharacteristic
648C2C2×#309211yescharacteristic
649C2#6052211yescharacteristic
650C2#6211111yescharacteristic
651C2#6214711yescharacteristic
652C2#6253911yescharacteristic
653C2#6258911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29936C2semidirect
4. #29908C2semidirect
5. #27960C2semidirect
6. #27861C2semidirect
7. #27680C2semidirect
8. #27615C2semidirect
9. #26278C2semidirect
10. #26216C2semidirect
11. #24996C2semidirect
12. #24949C2semidirect
13. #23756C2semidirect
14. #23745C2semidirect
15. #22715C2semidirect
16. #22504C2semidirect

Maximal subgroups

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

subgroupindex
#364052
#341832
#299362
#299082
#279602
#278612
#276802
#276152
#262782
#262162
#249962
#249492
#237562
#237452
#227152
#225042
#224732