Finite Group Databasebeta

← #6434420

Subgroups — #6434420

Order 512.

1480 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#35266C211yesmaximal, characteristic
3#30392C214yesmaximal, complemented
4#30390C214yesmaximal, complemented
5#30384C212yesmaximal, complemented
6#30383C212yesmaximal, complemented
7#30379C212yesmaximal, complemented
8#30376C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23731C211yesmaximal, characteristic
11#23671C212yesmaximal, complemented
12#23658C211yesmaximal, characteristic, complemented
13C2×#2071C211yesmaximal, characteristic
14#23614C212yesmaximal
15#21160C212yesmaximal, complemented
16#21091C212yesmaximal
17#20430C211yesmaximal, characteristic, complemented
18C2×#428C2212yes—
19C2×#428C2212yes—
20C2×#424C2212yes—
21C2×#418C2212yes—
22C22×#102C2212yes—
23C2×#405C2212yes—
24#2860C2212yes—
25#2860C2212yes—
26#2859C2212yescomplemented
27#2859C2212yescomplemented
28#2647C2214yescomplemented
29#2644C2212yes—
30#2644C2212yes—
31#2571—21no—
32#2571—21no—
33#2552—21no—
34#2552—21no—
35#2513—21no—
36#2513—21no—
37#2512—21no—
38#2512—21no—
39#2480—22no—
40#2480—22no—
41#2442—22no—
42#2442—22no—
43#2317C2212yes—
44#2317C2211yescharacteristic
45#2316C2212yes—
46#2316C2211yescharacteristic
47#2309C2211yescharacteristic
48#2309C2211yescharacteristic
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2101C2212yescomplemented
55#2101C2212yescomplemented
56#2100C2211yescharacteristic, complemented
57#2100C2211yescharacteristic, complemented
58#2099C2211yescharacteristic, complemented
59#2099C2211yescharacteristic, complemented
60#2094C2212yes—
61#2093C2212yes—
62#2092C2214yes—
63#2088C2214yescomplemented
64#2088C2214yescomplemented
65#2087C2214yescomplemented
66#2086C2214yescomplemented
67#2084C2212yescomplemented
68#2084C2212yescomplemented
69#2083C2212yescomplemented
70#2083C2212yescomplemented
71#2071C2212yes—
72#2071C2212yes—
73#2071C2214yes—
74#2067C2214yes—
75#2067C2214yes—
76#2065C2214yes—
77#2065C2214yes—
78#1944C2214yescomplemented
79#1943C2214yescomplemented
80#1942C2214yescomplemented
81#1941C2214yescomplemented
82#1936C2214yes—
83#1936C2214yes—
84#1934C2214yes—
85#1934C2214yes—
86#1768C2214yescomplemented
87#1767C2212yescomplemented
88C4×C4⋊C8C2212yescomplemented
89#1736C2212yes—
90#1735C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×#126—22no—
99C2×#126—22no—
100C2×#125—22no—
101C2×#125—22no—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42C2311yescharacteristic
114#428—28no—
115#428—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120#424—28no—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2314yes—
123#418—28no—
124C2×#102—44no—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#405—44no—
130C2×#97C2314yes—
131C2×C4×C8C2314yes—
132#400—24no—
133#400—24no—
134#398—24no—
135#398—24no—
136#397—22no—
137#397—22no—
138#397—22no—
139#397—22no—
140#397—22no—
141#397—22no—
142#396—22no—
143#396—22no—
144#390—21no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2314yes—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#385—21no—
183#385—21no—
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#382—22no—
195#382—21no—
196#382—22no—
197#382—22no—
198#382C2312yes—
199#380—21no—
200#380—21no—
201#380—24no—
202#380—22no—
203#379—22no—
204#379—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215#376C2312yes—
216C43C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2312yes—
234C25—21no—
235C22×D4—21no—
236C22×D4—24no—
237C22×D4—22no—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—24no—
247C2×M16—24no—
248C2×M16#4118yes—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255#126—84no—
256#126—84no—
257#125—84no—
258#125—84no—
259#117—24no—
260#117—22no—
261#117—22no—
262#117—24no—
263#117—24no—
264#117—22no—
265#117—24no—
266#117—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—28no—
340#107—44no—
341#106—28no—
342#106—44no—
343C4⋊C8—28no—
344C4⋊C8—28no—
345#102—28no—
346#102—28no—
347#102—44no—
348#102—44no—
349#97—44no—
350C4×C8—44no—
351#95—22no—
352#95—22no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—24no—
358#95—24no—
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—22no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—81no—
374C24—81no—
375C24—81no—
376C24—81no—
377C24C22×D411yescharacteristic
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—48no—
386C2×D4—48no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4114yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+414yes—
419C22×C42−1+412yes—
420C22×C4C2511yescharacteristic
421M16—216no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—48no—
431C2×C8C2×#4118yes—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—24no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—42no—
501C42—42no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—84no—
514C23—84no—
515C23—84no—
516C23—162no—
517C23—162no—
518C23—162no—
519C23—162no—
520C23#53212yes—
521C23#53612yes—
522C23#53712yes—
523C23C23×D411yescharacteristic
524D4—44no—
525D4—44no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—162no—
576C2×C4—164no—
577C2×C4#53212yes—
578C2×C4#53612yes—
579C2×C4#53712yes—
580C2×C4#53918yes—
581C2×C4#54118yes—
582C2×C4#57612yes—
583C2×C4#57712yes—
584C2×C4C22×#4112yes—
585C2×C4C2×H(5,𝔽2)11yescharacteristic
586C2×C4C2×2−1+412yes—
587C2×C4C2×2−1+411yescharacteristic
588C8—28no—
589C8—42no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—168no—
605C22#266414yes—
606C22C2×#53211yescharacteristic
607C22C2×#53611yescharacteristic
608C22C2×#53711yescharacteristic
609C22C2×#57611yescharacteristic
610C22C2×#57711yescharacteristic
611C22#358511yescharacteristic
612C22#358611yescharacteristic
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—84no—
628C4—81no—
629C4#266414yes—
630C4C2×#53912yes—
631C4C2×#54112yes—
632C2—24no—
633C2—162no—
634C2C2×#266411yescharacteristic
635C2#6072211yescharacteristic
636C2#6072711yescharacteristic
637C2#6087111yescharacteristic
638C2#6087311yescharacteristic
639C2#6092211yescharacteristic
640C2#6092311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #30392C2semidirect
4. #30390C2semidirect
5. #30384C2semidirect
6. #30383C2semidirect
7. #30379C2semidirect
8. #30376C2semidirect
9. #23744C2semidirect
11. #23671C2semidirect
12. #23658C2semidirect
15. #21160C2semidirect
17. #20430C2semidirect
26. #2859C22semidirect
27. #2859C22semidirect
28. #2647C22semidirect
54. #2101C22semidirect
55. #2101C22semidirect
56. #2100C22semidirect
57. #2100C22semidirect
58. #2099C22semidirect
59. #2099C22semidirect
63. #2088C22semidirect
64. #2088C22semidirect
65. #2087C22semidirect
66. #2086C22semidirect
67. #2084C22semidirect
68. #2084C22semidirect
69. #2083C22semidirect
70. #2083C22semidirect
78. #1944C22semidirect
79. #1943C22semidirect
80. #1942C22semidirect
81. #1941C22semidirect
86. #1768C22semidirect
87. #1767C22semidirect
88. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#352862
#352662
#303922
#303902
#303842
#303832
#303792
#303762
#237442
#237312
#236712
#236582
C2×#20712
#236142
#211602
#210912
#204302