Finite Group Databasebeta

← #6556039

Subgroups — #6556039

Order 512.

1483 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34321C211yesmaximal, characteristic
3#30165C212yesmaximal, complemented
4#29868C211yesmaximal, characteristic, complemented
5#29304C212yesmaximal, complemented
6#29277C211yesmaximal, characteristic, complemented
7#28689C211yesmaximal, characteristic, complemented
8#27612C212yesmaximal, complemented
9#27484C214yesmaximal, complemented
10#27056C214yesmaximal, complemented
11#26299C212yesmaximal, complemented
12#25704C212yesmaximal
13#23775C211yesmaximal, characteristic, complemented
14#23760C212yesmaximal, complemented
15#23162C212yesmaximal, complemented
16#22969C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C2×#500C2212yes—
20C2×#476C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563C2212yes—
25#2538—21no—
26#2497C2214yes—
27#2492—22no—
28#2439—22no—
29#2369C2211yescharacteristic
30#2369C2212yes—
31#2367—22no—
32#2363C2212yes—
33#2361—21no—
34#2355—21no—
35#2354—21no—
36#2354C2211yescharacteristic
37D4×C4⋊C4C2211yescharacteristic
38#2347—21no—
39#2347C2212yes—
40#2337—22no—
41#2324—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#121C2212yes—
45#2296C2212yes—
46#2293C2211yescharacteristic
47#2285C2212yes—
48#2274—21no—
49C42×D4C2211yescharacteristic
50#2100C2212yes—
51#2100C2211yescharacteristic
52#2094C2211yescharacteristic
53#2088C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2077C2214yes—
59#2075C2214yes—
60#2071C2212yes—
61#2071C2212yes—
62#2065C2212yes—
63#2058C2211yescharacteristic
64#2057C2212yes—
65#2033C2214yes—
66#2033C2214yes—
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1987C2212yes—
70#1987C2212yes—
71#1954C2212yes—
72#1947C2214yes—
73#1947C2212yes—
74#1944C2214yes—
75#1943C2214yes—
76#1924C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1918C2214yes—
81#1862C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1831C2211yescharacteristic
86#1825C2214yes—
87#1817C2212yes—
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95#555—24no—
96#555—24no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#124—22no—
103C2×C22⋊Q8—22no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—22no—
109C2×C4×D4D411yescharacteristic
110C2×C4×D4D411yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×#117—22no—
115C2×#117—21no—
116C2×#117—21no—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C22×C4⋊C4—21no—
120C22×C42C2312yes—
121#500—28no—
122#476—28no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#396—24no—
140#393—24no—
141#391—21no—
142#391—21no—
143#391—22no—
144#391—22no—
145#391—24no—
146#390—22no—
147#390—21no—
148#390—22no—
149#390C2311yescharacteristic
150#390C2311yescharacteristic
151#389—22no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#389C2311yescharacteristic
159#389C2311yescharacteristic
160#389C2314yes—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—24no—
167#388—21no—
168#388—22no—
169#388—21no—
170#388—22no—
171#387—24no—
172#386—22no—
173#386—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#383—22no—
184#382—22no—
185#382—22no—
186#380—22no—
187#380—22no—
188#380—21no—
189#380C2314yes—
190#380C2314yes—
191#380C2312yes—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C43C2311yescharacteristic
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×D4—22no—
236C22×D4C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—22no—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—22no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—42no—
250C4⋊Q8—82no—
251C4⋊Q8—82no—
252C4⋊Q8#4114yes—
253#124—216no—
254C22⋊Q8—84no—
255#121—28no—
256#121—24no—
257#121—24no—
258C4×Q8—44no—
259C4×Q8—82no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—42no—
265C4×D4—44no—
266C4×D4—84no—
267C4×D4#4114yes—
268#117—28no—
269#117—24no—
270#117—24no—
271#117—44no—
272#117—44no—
273#117—84no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—28no—
338#107—28no—
339#107—44no—
340#106—24no—
341#106—28no—
342#106—42no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345#102—28no—
346#102—24no—
347#102—42no—
348C22⋊C8—28no—
349C22⋊C8—44no—
350#97—44no—
351C4×C8—24no—
352#95—22no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—24no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24—21no—
371C24C22×D412yes—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×D4—28no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4C2×#4112yes—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2511yescharacteristic
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C82−1+414yes—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C2×#4112yes—
467C4⋊C4C2×#4114yes—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—84no—
481C22⋊C4—82no—
482C22⋊C4—84no—
483C42—28no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—22no—
489C42—42no—
490C42—42no—
491C42—41no—
492C42—42no—
493C42—44no—
494C42—82no—
495C42—84no—
496C42—82no—
497C42—82no—
498C42#13714yes—
499C42C2×#4112yes—
500C42C2×#4112yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C23×D411yescharacteristic
516Q8—84no—
517Q8—84no—
518Q8—164no—
519D4—24no—
520D4—44no—
521D4—42no—
522D4—88no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—84no—
572C2×C4—164no—
573C2×C4#52512yes—
574C2×C4#52512yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4#57512yes—
579C2×C4#57612yes—
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C2×2−1+412yes—
583C2×C4#58512yes—
584C8—24no—
585C8—44no—
586C8—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22C2×#52511yescharacteristic
602C22#348511yescharacteristic
603C22#349911yescharacteristic
604C22C22×#13711yescharacteristic
605C22C2×#57511yescharacteristic
606C22C2×#57611yescharacteristic
607C22#358511yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—44no—
618C4—81no—
619C4—81no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—84no—
624C4#311614yes—
625C4#350712yes—
626C4#351612yes—
627C2—82no—
628C2—82no—
629C2#3584611yescharacteristic
630C2C2×#311611yescharacteristic
631C2#6059011yescharacteristic
632C2#6171911yescharacteristic
633C2#6174011yescharacteristic
634C2#6248011yescharacteristic
635C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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. #30165C2semidirect
4. #29868C2semidirect
5. #29304C2semidirect
6. #29277C2semidirect
7. #28689C2semidirect
8. #27612C2semidirect
9. #27484C2semidirect
10. #27056C2semidirect
11. #26299C2semidirect
13. #23775C2semidirect
14. #23760C2semidirect
15. #23162C2semidirect
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
#343212
#301652
#298682
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