Finite Group Databasebeta

← #6434423

Subgroups — #6434423

Order 512.

1427 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#35267C211yesmaximal, characteristic
3#30395C214yesmaximal, complemented
4#30391C214yesmaximal, complemented
5#30385C212yesmaximal, complemented
6#30384C212yesmaximal, complemented
7#30377C212yesmaximal, complemented
8#30376C212yesmaximal, complemented
9C2×#2099C211yesmaximal, characteristic, complemented
10#23734C211yesmaximal, characteristic
11#23671C212yesmaximal, complemented
12#23657C211yesmaximal, characteristic, complemented
13#23627C211yesmaximal, characteristic
14#23616C212yesmaximal
15#21160C212yesmaximal, complemented
16#21092C212yesmaximal
17#20430C211yesmaximal, characteristic, complemented
18C2×#428C2212yes—
19C2×#428C2212yes—
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#2099C2212yes—
55#2099C2212yes—
56#2099C2214yes—
57#2096C2212yes—
58#2095C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2088C2214yes—
62#2088C2214yes—
63#2087C2214yes—
64#2086C2214yes—
65#2085C2214yes—
66#2084C2212yes—
67#2083C2212yes—
68#2074C2212yes—
69#2073C2212yes—
70#2072C2211yescharacteristic
71#2072C2211yescharacteristic
72#2071C2211yescharacteristic
73#2071C2211yescharacteristic
74#2068C2214yes—
75#2067C2214yes—
76#2066C2214yes—
77#2065C2214yes—
78#1944C2214yes—
79#1943C2214yes—
80#1942C2214yes—
81#1941C2214yes—
82#1936C2214yes—
83#1935C2214yes—
84#1934C2214yes—
85#1933C2214yes—
86#1768C2214yes—
87#1767C2212yes—
88C4×C4⋊C8C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
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#428—28no—
113#428—44no—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118#424—28no—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2314yes—
121#417—28no—
122#416—44no—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#102C2314yes—
127C2×#102C2312yes—
128C2×#102C2312yes—
129#405—44no—
130C2×#97C2314yes—
131C2×C4×C8C2314yes—
132#401—22no—
133#400—24no—
134#400—24no—
135#400—22no—
136#400—22no—
137#400—22no—
138#398—24no—
139#398—24no—
140#397—22no—
141#397—22no—
142#397—22no—
143#396—22no—
144#391C2312yes—
145#391C2312yes—
146#391C2311yescharacteristic
147#391C2311yescharacteristic
148#390C2312yes—
149#390C2312yes—
150#390C2311yescharacteristic
151#390C2311yescharacteristic
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#386—21no—
179#386—21no—
180#386—22no—
181#386—22no—
182#385—24no—
183#385—22no—
184#384—21no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#383—22no—
193#382—22no—
194#382—21no—
195#382—22no—
196#382—22no—
197#382C2312yes—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—24no—
205#380—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216#376C2312yes—
217C2×#95—21no—
218C43C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2312yes—
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#4118yes—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
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—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
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—28no—
332#107—44no—
333#106—28no—
334#106—44no—
335C4⋊C8—28no—
336C4⋊C8—28no—
337#103—28no—
338#103—42no—
339#103—42no—
340#102—28no—
341#102—42no—
342#102—42no—
343#97—44no—
344C4×C8—44no—
345#95—22no—
346#95—22no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—21no—
370C24C22×D411yescharacteristic
371#41—48no—
372#41—48no—
373C2×Q8—22no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—81no—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+414yes—
422C22×C42−1+412yes—
423C22×C4C2511yescharacteristic
424M16—216no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—48no—
434C2×C8C2×#4118yes—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—24no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—44no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23#53212yes—
508C23#53612yes—
509C23#53712yes—
510C23C23×D411yescharacteristic
511Q8—42no—
512Q8—42no—
513D4—42no—
514D4—42no—
515D4—84no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—164no—
567C2×C4—162no—
568C2×C4#53212yes—
569C2×C4#53612yes—
570C2×C4#53712yes—
571C2×C4#53918yes—
572C2×C4#54118yes—
573C2×C4#57612yes—
574C2×C4#57712yes—
575C2×C4C22×#4112yes—
576C2×C4C2×H(5,𝔽2)11yescharacteristic
577C2×C4C2×2−1+412yes—
578C2×C4C2×2−1+411yescharacteristic
579C8—28no—
580C8—42no—
581C8—42no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#266414yes—
596C22C2×#53211yescharacteristic
597C22C2×#53611yescharacteristic
598C22C2×#53711yescharacteristic
599C22C2×#57611yescharacteristic
600C22C2×#57711yescharacteristic
601C22#358611yescharacteristic
602C22#358611yescharacteristic
603C4—22no—
604C4—22no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—81no—
615C4—81no—
616C4—84no—
617C4—81no—
618C4—82no—
619C4#266414yes—
620C4C2×#53912yes—
621C4C2×#54112yes—
622C2—24no—
623C2—161no—
624C2C2×#266411yescharacteristic
625C2#6072211yescharacteristic
626C2#6072611yescharacteristic
627C2#6087111yescharacteristic
628C2#6087211yescharacteristic
629C2#6092811yescharacteristic
630C2#6093111yescharacteristic
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. #30395C2semidirect
4. #30391C2semidirect
5. #30385C2semidirect
6. #30384C2semidirect
7. #30377C2semidirect
8. #30376C2semidirect
9. C2×#2099C2semidirect
11. #23671C2semidirect
12. #23657C2semidirect
15. #21160C2semidirect
17. #20430C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#352672
#303952
#303912
#303852
#303842
#303772
#303762
C2×#20992
#237342
#236712
#236572
#236272
#236162
#211602
#210922
#204302