Finite Group Databasebeta

← #6434426

Subgroups — #6434426

Order 512.

1427 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#35267C211yesmaximal, characteristic
3#30390C214yesmaximal, complemented
4#30388C214yesmaximal, complemented
5#30385C212yesmaximal, complemented
6#30383C212yesmaximal, complemented
7#30378C212yesmaximal, complemented
8#30376C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic
11#23669C212yesmaximal, complemented
12C2×#2083C211yesmaximal, characteristic, complemented
13#23630C211yesmaximal, characteristic
14#23611C212yesmaximal
15#21156C212yesmaximal, complemented
16#21087C212yesmaximal
17#20430C211yesmaximal, characteristic, complemented
18C2×#427C2212yes—
19C22×#107C2212yes—
20C2×#424C2212yes—
21C2×#417C2212yes—
22C2×#416C2212yes—
23C2×#405C2212yes—
24#2863C2212yes—
25#2860C2212yes—
26#2859C2212yes—
27#2859C2212yes—
28#2647C2214yes—
29#2645C2212yes—
30#2644C2212yes—
31#2571—21no—
32#2571—21no—
33#2552—21no—
34#2552—21no—
35#2510—22no—
36#2497—21no—
37#2496—21no—
38#2495—21no—
39#2494—21no—
40#2481—22no—
41#2442—22no—
42#2442—22no—
43#2317C2212yes—
44#2317C2211yescharacteristic
45#2316C2212yes—
46#2316C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49#2297C2211yescharacteristic
50C2×#389C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2101C2214yes—
55#2100C2212yes—
56#2099C2212yes—
57#2097C2212yes—
58#2096C2211yescharacteristic
59#2095C2211yescharacteristic
60#2094C2211yescharacteristic
61#2093C2211yescharacteristic
62#2092C2212yes—
63#2088C2214yes—
64#2088C2214yes—
65#2087C2214yes—
66#2086C2214yes—
67#2083C2212yes—
68#2083C2212yes—
69#2083C2214yes—
70#2072C2212yes—
71#2072C2212yes—
72#2071C2212yes—
73#2071C2212yes—
74#2068C2214yes—
75#2068C2214yes—
76#2067C2214yes—
77#2067C2214yes—
78#1944C2214yes—
79#1944C2214yes—
80#1942C2214yes—
81#1942C2214yes—
82#1936C2214yes—
83#1935C2214yes—
84#1934C2214yes—
85#1933C2214yes—
86#1768C2214yes—
87#1767C2212yes—
88C4×C4⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×#126—22no—
99C2×#125—22no—
100C2×#125—22no—
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C42C2311yescharacteristic
112#427—28no—
113C2×#107—44no—
114C2×#107C2312yes—
115C2×#107C2312yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2312yes—
119C2×#106C2312yes—
120#424—28no—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2314yes—
123#417—44no—
124#416—28no—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2314yes—
128C2×#102C2312yes—
129C2×#102C2312yes—
130C2×#102C2314yes—
131#405—44no—
132C2×#97C2314yes—
133C2×C4×C8C2314yes—
134#401—22no—
135#400—24no—
136#400—24no—
137#400—22no—
138#400—22no—
139#400—22no—
140#398—24no—
141#398—24no—
142#397—22no—
143#397—22no—
144#397—22no—
145#396—22no—
146#391C2312yes—
147#391C2312yes—
148#391C2311yescharacteristic
149#391C2311yescharacteristic
150#390C2312yes—
151#390C2312yes—
152#390C2311yescharacteristic
153#390C2311yescharacteristic
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389C2314yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—21no—
181#386—21no—
182#386—22no—
183#386—22no—
184#385—24no—
185#385—22no—
186#384—21no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#383—22no—
195#382—22no—
196#382—21no—
197#382—22no—
198#382—22no—
199#382C2312yes—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—24no—
207#380—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376C2312yes—
219C2×#95—21no—
220C43C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—22no—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×M16—24no—
249C2×M16—24no—
250C2×M16—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8#4118yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257#126—84no—
258#125—84no—
259#125—84no—
260#117—24no—
261#117—22no—
262#117—22no—
263#117—24no—
264#117—24no—
265#117—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—24no—
332#107—24no—
333#107—44no—
334#107—44no—
335#106—24no—
336#106—24no—
337C4⋊C8—28no—
338C4⋊C8—28no—
339#103—24no—
340#103—24no—
341#103—44no—
342#102—24no—
343#102—24no—
344#102—44no—
345#97—44no—
346C4×C8—44no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
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—
370#95C2×D412yes—
371C24—21no—
372C24C22×D411yescharacteristic
373#41—48no—
374#41—48no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—82no—
409C22×C4—81no—
410C22×C4—82no—
411C22×C4—81no—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4114yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+414yes—
424C22×C42−1+412yes—
425C22×C4C2511yescharacteristic
426M16—48no—
427C2×C8—216no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8C2×#4114yes—
436C2×C8C2×#4114yes—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—82no—
480C22⋊C4—81no—
481C22⋊C4—82no—
482C22⋊C4—81no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—44no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23#53212yes—
510C23#53612yes—
511C23#53712yes—
512C23C23×D411yescharacteristic
513Q8—42no—
514Q8—42no—
515D4—42no—
516D4—42no—
517D4—84no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—162no—
569C2×C4—164no—
570C2×C4#53212yes—
571C2×C4#53612yes—
572C2×C4#53712yes—
573C2×C4#53918yes—
574C2×C4#54118yes—
575C2×C4#57612yes—
576C2×C4#57812yes—
577C2×C4C22×#4112yes—
578C2×C4C2×H(5,𝔽2)11yescharacteristic
579C2×C4C2×2−1+412yes—
580C2×C4C2×2−1+411yescharacteristic
581C8—24no—
582C8—24no—
583C8—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22#266414yes—
598C22C2×#53211yescharacteristic
599C22C2×#53611yescharacteristic
600C22C2×#53711yescharacteristic
601C22C2×#57611yescharacteristic
602C22C2×#57811yescharacteristic
603C22#358511yescharacteristic
604C22#358611yescharacteristic
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—81no—
614C4—81no—
615C4—82no—
616C4—81no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4#266414yes—
622C4C2×#53912yes—
623C4C2×#54112yes—
624C2—24no—
625C2—161no—
626C2C2×#266411yescharacteristic
627C2#6071811yescharacteristic
628C2#6072111yescharacteristic
629C2#6087111yescharacteristic
630C2#6087411yescharacteristic
631C2#6092211yescharacteristic
632C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. #35286C2semidirect
3. #30390C2semidirect
4. #30388C2semidirect
5. #30385C2semidirect
6. #30383C2semidirect
7. #30378C2semidirect
8. #30376C2semidirect
9. #23747C2semidirect
11. #23669C2semidirect
12. C2×#2083C2semidirect
15. #21156C2semidirect
17. #20430C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#352672
#303902
#303882
#303852
#303832
#303782
#303762
#237472
#237302
#236692
C2×#20832
#236302
#236112
#211562
#210872
#204302