Finite Group Databasebeta

← #6556958

Subgroups — #6556958

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36453C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic, complemented
3#30367C212yesmaximal, complemented
4#30230C211yesmaximal, characteristic, complemented
5#29320C211yesmaximal, characteristic, complemented
6#29156C212yesmaximal, complemented
7#29048C214yesmaximal, complemented
8#28520C211yesmaximal, characteristic, complemented
9#27858C212yesmaximal, complemented
10#27847C214yesmaximal, complemented
11#25341C212yesmaximal, complemented
12#25338C212yesmaximal
13#23817C211yesmaximal, characteristic, complemented
14#23760C212yesmaximal, complemented
15#23068C212yesmaximal, complemented
16#22968C212yesmaximal, complemented
17#20558C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#475C2212yes—
20C2×#474C2212yes—
21C2×C8×D4C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2855C2212yes—
26#2576—22no—
27#2568—21no—
28#2547—21no—
29#2547C2212yes—
30#2538C2211yescharacteristic
31#2537C2211yescharacteristic
32#2499—22no—
33#2496—22no—
34#2496C2214yes—
35#2494—21no—
36#2494C2212yes—
37#2460—22no—
38#2427C2212yes—
39#2420—21no—
40#2365—21no—
41#2365C2212yes—
42#2361—21no—
43#2355C2211yescharacteristic
44D4×C4⋊C4C2211yescharacteristic
45#2345—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48C4×#121C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yes—
52#2100C2214yes—
53#2100C2212yes—
54#2100C2212yes—
55#2095C2212yes—
56#2094C2212yes—
57#2083C2212yes—
58#2077C2214yes—
59#2075C2214yes—
60#2072C2211yescharacteristic
61#2071C2211yescharacteristic
62#2058C2212yes—
63#2042C2212yes—
64#2035C2214yes—
65#2033C2214yes—
66#2024C2211yescharacteristic
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1959C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1944C2214yes—
74#1943C2214yes—
75#1935C2214yes—
76#1928C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1864C2214yes—
82#1847C2212yes—
83#1808C2214yes—
84#1807C2214yes—
85#1776C2214yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88#1762C2214yes—
89C4×#103C2212yes—
90C4×#102C2212yes—
91#1752C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—24no—
102C2×#125—21no—
103C2×#124—22no—
104C2×#124C2311yescharacteristic
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—22no—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C22×C4⋊C4—21no—
120C22×C42C2312yes—
121#475—28no—
122#474—28no—
123C8×D4—28no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2312yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#401—22no—
140#400—24no—
141#397—22no—
142#397—24no—
143#397—24no—
144#397—22no—
145#396—22no—
146#394—22no—
147#394—22no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391C2311yescharacteristic
153#391C2312yes—
154#390—21no—
155#390C2311yescharacteristic
156#389—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#389C2312yes—
166#388—22no—
167#388—21no—
168#388—21no—
169#388—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#386—22no—
177#386—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#382—21no—
188#382—22no—
189#382—21no—
190#382C2311yescharacteristic
191#381—22no—
192#380—22no—
193#380—22no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200#380C2314yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2411yescharacteristic
235C22×D4—22no—
236C22×D4C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—22no—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C22×C8—28no—
244C22×C8—44no—
245C22×C8—44no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254#126—416no—
255#125—82no—
256#124—22no—
257#124—41no—
258#124—48no—
259C22⋊Q8—48no—
260C22⋊Q8—44no—
261#121—24no—
262#121—42no—
263#121—44no—
264C4×Q8—24no—
265C4×Q8—44no—
266C4×Q8—48no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—42no—
271C4×D4—44no—
272C4×D4—84no—
273#117—48no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—44no—
336#106—28no—
337#106—44no—
338C4⋊C8#4118yes—
339#103—28no—
340#103—42no—
341#103—42no—
342#102—24no—
343#102—28no—
344#102—42no—
345C22⋊C8—28no—
346#97—42no—
347#97—44no—
348C4×C8—24no—
349#95—22no—
350#95—21no—
351#95—22no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24—21no—
371C24C22×D412yes—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—82no—
390C2×D4AGL(1,ℤ/8)14yes—
391C2×D42−1+412yes—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—88no—
435C2×C8C2×#4118yes—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
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—24no—
449C4⋊C4—22no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—82no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—82no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—21no—
489C42—21no—
490C42—24no—
491C42—22no—
492C42—42no—
493C42—42no—
494C42—48no—
495C42—42no—
496C42—44no—
497C42—82no—
498C42C2×#4114yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C2×AGL(1,ℤ/8)14yes—
513C23C23×D411yescharacteristic
514Q8—42no—
515Q8—42no—
516Q8—84no—
517Q8—88no—
518D4—28no—
519D4—24no—
520D4—42no—
521D4—88no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—164no—
573C2×C4#53212yes—
574C2×C4#53212yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4#57512yes—
577C2×C4#57712yes—
578C2×C4#57812yes—
579C2×C4C22×#4112yes—
580C2×C4C2×2−1+411yescharacteristic
581C2×C4#58512yes—
582C2×C4#58511yescharacteristic
583C8—28no—
584C8—42no—
585C8—42no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C2×#53211yescharacteristic
601C22#348511yescharacteristic
602C22#348911yescharacteristic
603C22C22×AGL(1,ℤ/8)11yescharacteristic
604C22C2×#57511yescharacteristic
605C22C2×#57711yescharacteristic
606C22C2×#57811yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—84no—
617C4—81no—
618C4—82no—
619C4—82no—
620C4—81no—
621C4—82no—
622C4—82no—
623C4#318014yes—
624C4#351612yes—
625C4#352112yes—
626C2—44no—
627C2—82no—
628C2#3585611yescharacteristic
629C2C2×#318011yescharacteristic
630C2#6072311yescharacteristic
631C2#6173711yescharacteristic
632C2#6173911yescharacteristic
633C2#6212811yescharacteristic
634C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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
2. #36188C2semidirect
3. #30367C2semidirect
4. #30230C2semidirect
5. #29320C2semidirect
6. #29156C2semidirect
7. #29048C2semidirect
8. #28520C2semidirect
9. #27858C2semidirect
10. #27847C2semidirect
11. #25341C2semidirect
13. #23817C2semidirect
14. #23760C2semidirect
15. #23068C2semidirect
16. #22968C2semidirect
17. #20558C2semidirect

Maximal subgroups

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

subgroupindex
#364532
#361882
#303672
#302302
#293202
#291562
#290482
#285202
#278582
#278472
#253412
#253382
#238172
#237602
#230682
#229682
#205582