Finite Group Databasebeta

← #6567793

Subgroups — #6567793

Order 512.

1513 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#29980C211yesmaximal, characteristic, complemented
4#29971C212yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#29219C211yesmaximal, characteristic, complemented
7#29157C212yesmaximal, complemented
8#27683C212yesmaximal, complemented
9#26511C212yesmaximal, complemented
10#26409C214yesmaximal, complemented
11#26111C212yesmaximal, complemented
12#25989C214yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23744C212yesmaximal, complemented
15#22707C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22489C212yesmaximal
18C2×#555C2211yescharacteristic
19C2×C8×D4C2212yes—
20#2868C2211yescharacteristic
21#2857C2212yes—
22#2571C2212yes—
23#2563—21no—
24#2538—21no—
25#2497—22no—
26#2492—22no—
27#2439C2214yes—
28#2369—21no—
29#2369C2211yescharacteristic
30#2367—22no—
31#2363C2212yes—
32#2361—21no—
33#2355—21no—
34#2354C2211yescharacteristic
35#2354C2212yes—
36D4×C4⋊C4C2211yescharacteristic
37#2347—21no—
38#2347C2212yes—
39#2337—22no—
40#2324—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#121C2212yes—
44#2296C2212yes—
45#2293C2211yescharacteristic
46#2285C2212yes—
47#2274—21no—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2101C2214yes—
51#2100C2212yes—
52#2100C2211yescharacteristic
53#2099C2212yes—
54#2099C2212yes—
55#2099C2212yes—
56#2094C2211yescharacteristic
57#2093C2212yes—
58#2088C2214yes—
59#2088C2212yes—
60#2087C2214yes—
61#2065C2212yes—
62#2058C2211yescharacteristic
63#2057C2212yes—
64#2057C2212yes—
65#2023C2212yes—
66#2023C2211yescharacteristic
67#1990C2212yes—
68#1988C2214yes—
69#1987C2212yes—
70#1949C2214yes—
71#1947C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1921C2214yes—
75#1918C2214yes—
76#1893C2214yes—
77#1875C2214yes—
78#1862C2214yes—
79#1854C2214yes—
80#1854C2214yes—
81#1850C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1825C2214yes—
85#1808C2214yes—
86#1804C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89C4×#106C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92#1735C2214yes—
93#1735C2214yes—
94C2×#336C2212yes—
95C23×C8C2314yes—
96#555—24no—
97#555Q818yescomplemented
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#124—22no—
104C2×C22⋊Q8—22no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×D4—22no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—21no—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2314yes—
120C22×C4⋊C4—21no—
121C22×C42C2312yes—
122C8×D4—28no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×#102C2312yes—
133C2×C22⋊C8C2314yes—
134C2×C4×C8C2312yes—
135#400—24no—
136#396—24no—
137#393—24no—
138#391—21no—
139#391—21no—
140#391—22no—
141#391—22no—
142#391—24no—
143#390—22no—
144#390—21no—
145#390—22no—
146#390C2311yescharacteristic
147#390C2311yescharacteristic
148#389—22no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2311yescharacteristic
156#389C2311yescharacteristic
157#389C2314yes—
158#389C2312yes—
159#389C2312yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—24no—
164#388—21no—
165#388—22no—
166#388—21no—
167#388—22no—
168#387—24no—
169#386—22no—
170#386—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#384—21no—
175#384—21no—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2312yes—
179#384C2312yes—
180#383—22no—
181#382—22no—
182#382C2314yes—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—21no—
187#380C2312yes—
188#380C2314yes—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C22⋊C4—22no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×D4—22no—
235C22×D4C2×D411yescharacteristic
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—22no—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C22×C8—28no—
243C22×C8—44no—
244C22×C8—44no—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—22no—
249C4⋊Q8—42no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253C4⋊Q8C2×Q814yes—
254#124—216no—
255C22⋊Q8—84no—
256#121—28no—
257#121—24no—
258#121C2×Q818yes—
259C4×Q8—44no—
260C4×Q8—82no—
261C4×D4—24no—
262C4×D4—22no—
263C4×D4—42no—
264C4×D4—44no—
265C4×D4—84no—
266C4×D4C2×Q814yes—
267C4×D4C2×Q814yes—
268C4×D4#4114yes—
269#117—24no—
270#117—28no—
271#117—44no—
272#117—44no—
273#117—44no—
274#117—44no—
275#117C2×Q818yes—
276C2×C4⋊C4—21no—
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—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—44no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—44no—
340#107—44no—
341#106—42no—
342#106—42no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345C4⋊C8—42no—
346#102—42no—
347#102—42no—
348#102—44no—
349C22⋊C8—28no—
350C4×C8—24no—
351#95—22no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—24no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—21no—
370C24C22×D412yes—
371C2×Q8—24no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—81no—
377C2×Q8—81no—
378C2×Q8—82no—
379C2×Q8—82no—
380C2×D4—28no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4C22⋊Q814yes—
389C2×D4C22×Q812yes—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×Q814yes—
422C22×C4C2×#4114yes—
423C22×C4C2511yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—88no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—82no—
461C4⋊C4—81no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4C22⋊Q814yes—
466C4⋊C4C22×Q812yes—
467C4⋊C4C22×Q814yes—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—84no—
480C22⋊C4—82no—
481C22⋊C4—84no—
482C22⋊C4C22⋊Q818yes—
483C42—28no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—42no—
490C42—42no—
491C42—41no—
492C42—42no—
493C42—44no—
494C42—82no—
495C42—84no—
496C42—82no—
497C42—82no—
498C42#13514yes—
499C42C22×Q812yes—
500C42C2×#4112yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×C22⋊Q814yes—
515C23C23×D411yescharacteristic
516Q8—84no—
517Q8—84no—
518Q8—164no—
519D4—28no—
520D4—24no—
521D4—42no—
522D4—88no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—164no—
572C2×C4C2×C22⋊Q814yes—
573C2×C4C2×C22⋊Q812yes—
574C2×C4C2×C22⋊Q812yes—
575C2×C4C2×#13512yes—
576C2×C4C2×#13512yes—
577C2×C4C2×#13512yes—
578C2×C4#57512yes—
579C2×C4#57612yes—
580C2×C4C23×Q811yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4#58512yes—
583C2×C4#58512yes—
584C8—42no—
585C8—44no—
586C8—42no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22C22×C22⋊Q811yescharacteristic
602C22#348511yescharacteristic
603C22#350011yescharacteristic
604C22C22×#13511yescharacteristic
605C22C2×#57511yescharacteristic
606C22C2×#57611yescharacteristic
607C22#358611yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—24no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—44no—
617C4—41no—
618C4—81no—
619C4—82no—
620C4—82no—
621C4—81no—
622C4—84no—
623C4—82no—
624C4#307614yes—
625C4#350412yes—
626C4#352112yes—
627C2—44no—
628C2—82no—
629C2#3603111yescharacteristic
630C2C2×#307611yescharacteristic
631C2#6049411yescharacteristic
632C2#6170111yescharacteristic
633C2#6174011yescharacteristic
634C2#6252711yescharacteristic
635C2#6258111yescharacteristic
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. #29980C2semidirect
4. #29971C2semidirect
5. #29277C2semidirect
6. #29219C2semidirect
7. #29157C2semidirect
8. #27683C2semidirect
9. #26511C2semidirect
10. #26409C2semidirect
11. #26111C2semidirect
12. #25989C2semidirect
13. #23775C2semidirect
14. #23744C2semidirect
15. #22707C2semidirect
16. #22696C2semidirect
97. #555Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
#343212
#299802
#299712
#292772
#292192
#291572
#276832
#265112
#264092
#261112
#259892
#237752
#237442
#227072
#226962
#224892