Finite Group Databasebeta

← #6564241

Subgroups — #6564241

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36459C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#30149C212yesmaximal, complemented
4#29938C211yesmaximal, characteristic, complemented
5#29518C212yesmaximal, complemented
6#29280C212yesmaximal, complemented
7#27470C214yesmaximal, complemented
8#27344C214yesmaximal
9#26447C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25452C212yesmaximal, complemented
12#24298C211yesmaximal, characteristic
13#23841C212yesmaximal, complemented
14#23754C211yesmaximal, characteristic
15#23163C212yesmaximal, complemented
16#23066C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519—21no—
20C2×#498C2212yes—
21C2×#474C2212yes—
22#2860C2212yes—
23#2857C2211yescharacteristic
24#2563C2211yescharacteristic
25#2560—22no—
26#2552C2211yescharacteristic
27#2532—22no—
28#2496—21no—
29#2495C2212yes—
30#2490C2212yes—
31#2485—21no—
32#2369C2211yescharacteristic
33#2365—21no—
34#2365—21no—
35#2365C2212yes—
36#2365C2212yes—
37#2354—21no—
38#2354C2211yescharacteristic
39#2347—21no—
40#2344—22no—
41#2344C2214yes—
42#2327—22no—
43#2310—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#125C2212yes—
47C4×#121C2211yescharacteristic
48#2285C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2211yescharacteristic
52#2093C2211yescharacteristic
53#2088C2214yes—
54#2085C2214yes—
55#2084C2212yes—
56#2083C2211yescharacteristic
57#2077C2214yes—
58#2075C2212yes—
59#2071C2211yescharacteristic
60#2065C2214yes—
61#2059C2214yes—
62#2058C2212yes—
63#2058C2212yes—
64#2057C2212yes—
65#2033C2212yes—
66#2023C2212yes—
67#1990C2212yes—
68#1989C2214yes—
69#1988C2212yes—
70#1987C2212yes—
71#1949C2214yes—
72#1948C2212yes—
73#1944C2214yes—
74#1939C2212yes—
75#1936C2214yes—
76#1926C2214yes—
77#1926C2214yes—
78#1924C2214yes—
79#1921C2214yes—
80#1921C2214yes—
81#1920C2214yes—
82#1919C2214yes—
83#1874C2212yes—
84#1853C2214yes—
85#1852C2212yes—
86#1831C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#126—22no—
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×#121C2311yescharacteristic
106#519—48no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×D4—21no—
112C2×C4×D4C2311yescharacteristic
113C2×#117—21no—
114C2×#117—21no—
115C2×#117—24no—
116C2×#117D411yescharacteristic
117C2×#117D411yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121#498—28no—
122#474—28no—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#398—22no—
140#397—22no—
141#397—22no—
142#394—22no—
143#393—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—21no—
149#391—22no—
150#390—22no—
151#390C2312yes—
152#390C2311yescharacteristic
153#390C2312yes—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—21no—
159#389C2311yescharacteristic
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—21no—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—24no—
178#386—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#383—22no—
184#382—21no—
185#382C2312yes—
186#381—24no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380C2311yescharacteristic
193#380C2312yes—
194#380C2312yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214#376D412yes—
215C43C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2314yes—
232C22×Q8—28no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—44no—
237C22×Q8—44no—
238C22×D4—21no—
239C22×D4C2411yescharacteristic
240C23×C4—22no—
241C23×C4—21no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8#4118yes—
254#126—84no—
255#125—24no—
256#125—42no—
257C22⋊Q8—82no—
258#121—22no—
259#121—41no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×D4—24no—
265C4×D4—44no—
266#117—24no—
267#117—416no—
268#117—42no—
269#117—44no—
270#117—44no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333#107—24no—
334#107—28no—
335#107—24no—
336#106—28no—
337#106—28no—
338#106—44no—
339C4⋊C8—24no—
340C4⋊C8—44no—
341#102—24no—
342#102—28no—
343#102—42no—
344C22⋊C8—24no—
345C22⋊C8—42no—
346#97—28no—
347C4×C8—24no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—21no—
370C24C22×D411yescharacteristic
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—44no—
374C2×Q8—432no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—81no—
380C2×Q8—81no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—81no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4AGL(1,ℤ/8)14yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2511yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C82−1+414yes—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—81no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4C2×#4118yes—
465C4⋊C42−1+412yes—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C42—22no—
480C42—22no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—42no—
488C42—42no—
489C42—44no—
490C42—48no—
491C42—48no—
492C42—44no—
493C42—42no—
494C42—82no—
495C42C2×#4114yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×AGL(1,ℤ/8)12yes—
510C23C23×D411yescharacteristic
511Q8—416no—
512Q8—416no—
513Q8—162no—
514D4—42no—
515D4—42no—
516D4—84no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—416no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—84no—
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4C2×AGL(1,ℤ/8)12yes—
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4#57512yes—
573C2×C4#57712yes—
574C2×C4C22×#4112yes—
575C2×C4C2×2−1+412yes—
576C2×C4C2×2−1+411yescharacteristic
577C2×C4#58511yescharacteristic
578C8—24no—
579C8—42no—
580C8—44no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C2×#52511yescharacteristic
596C22#348911yescharacteristic
597C22#349911yescharacteristic
598C22C22×AGL(1,ℤ/8)11yescharacteristic
599C22C2×#57511yescharacteristic
600C22C2×#57711yescharacteristic
601C22#358611yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—42no—
606C4—42no—
607C4—48no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—82no—
613C4—81no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4#311414yes—
619C4#351812yes—
620C4#352312yes—
621C2—42no—
622C2—161no—
623C2#3600711yescharacteristic
624C2C2×#311411yescharacteristic
625C2#6058211yescharacteristic
626C2#6210711yescharacteristic
627C2#6213511yescharacteristic
628C2#6247711yescharacteristic
629C2#6251111yescharacteristic
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
2. #34209C2semidirect
3. #30149C2semidirect
4. #29938C2semidirect
5. #29518C2semidirect
6. #29280C2semidirect
7. #27470C2semidirect
9. #26447C2semidirect
10. #25811C2semidirect
11. #25452C2semidirect
13. #23841C2semidirect
15. #23163C2semidirect
16. #23066C2semidirect
17. #22636C2semidirect

Maximal subgroups

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

subgroupindex
#364592
#342092
#301492
#299382
#295182
#292802
#274702
#273442
#264472
#258112
#254522
#242982
#238412
#237542
#231632
#230662
#226362