Finite Group Databasebeta

← #6461972

Subgroups — #6461972

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#29590C214yesmaximal, complemented
4#29586C214yesmaximal, complemented
5#29582C214yesmaximal, complemented
6#29576C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic
9#23656C211yesmaximal, characteristic, complemented
10#23628C211yesmaximal, characteristic
11#21156C211yesmaximal, characteristic, complemented
12#21155C211yesmaximal, characteristic, complemented
13#21091C211yesmaximal, characteristic
14#21091C211yesmaximal, characteristic
15#21038C211yesmaximal, characteristic, complemented
16#21037C211yesmaximal, characteristic, complemented
17#20879C211yesmaximal, characteristic
18#20879C211yesmaximal, characteristic
19#20368C211yesmaximal, characteristic, complemented
20C2×#518C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2864C2214yescomplemented
32#2647C2214yescomplemented
33#2570—22no—
34#2570—22no—
35#2569—24no—
36#2357—22no—
37#2354—22no—
38#2342—24no—
39#2319C2214yes—
40#2317C2211yescharacteristic
41#2317C2211yescharacteristic
42#2317C2211yescharacteristic
43#2316C2211yescharacteristic
44#2316C2211yescharacteristic
45#2309C2211yescharacteristic
46#2296C2211yescharacteristic
47#2293C2214yes—
48C2×#390C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#382C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2092C2212yes—
56#2092C2212yes—
57#2092C2212yes—
58#2092C2212yes—
59#2085C2212yescomplemented
60#2085C2212yescomplemented
61#2084C2212yescomplemented
62#2083C2212yescomplemented
63#2073C2214yes—
64#2071C2214yes—
65#1944C2214yescomplemented
66#1943C2214yescomplemented
67#1942C2214yescomplemented
68#1941C2214yescomplemented
69#1936C2214yes—
70#1936C2214yes—
71#1934C2214yes—
72#1934C2214yes—
73#1921C2214yescomplemented
74#1920C2214yescomplemented
75#1919C2214yescomplemented
76#1918C2214yescomplemented
77#1864C2214yes—
78#1864C2214yes—
79#1862C2214yes—
80#1862C2214yes—
81#1746C2212yescomplemented
82#1746C2212yescomplemented
83#1745C2212yescomplemented
84#1744C2212yescomplemented
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×D4C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102#518—24no—
103#518—24no—
104C2×C4×D4—22no—
105C2×C4×D4C2314yes—
106C2×#117—24no—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—22no—
117C22×C22⋊C4—22no—
118C22×C42C2311yescharacteristic
119#427—24no—
120#427—24no—
121C2×#107—24no—
122C2×#107C2312yes—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#106—24no—
127C2×#106C2312yes—
128C2×#106C2312yes—
129C2×#106C2312yes—
130C2×#106C2312yes—
131#418—24no—
132#418—24no—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141#405—24no—
142#405—24no—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147#400—24no—
148#400—24no—
149#398—24no—
150#398—24no—
151#390—21no—
152#390—21no—
153#390—22no—
154#390C2314yes—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#388—24no—
167#388—24no—
168#388—24no—
169#388—24no—
170#387—24no—
171#387—24no—
172#387—24no—
173#387—24no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384—21no—
181#384C2312yes—
182#382—21no—
183#382—21no—
184#382—21no—
185#382—22no—
186#382C2312yes—
187#382C2312yes—
188#382C2312yes—
189#381—24no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—24no—
194#379—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2314yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—24no—
203C4×C22⋊C4D414yes—
204#376C2312yes—
205C2×#95—22no—
206C43C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2311yescharacteristic
220#336C2311yescharacteristic
221#336C2311yescharacteristic
222#336C2311yescharacteristic
223#336C2312yes—
224#336C2312yes—
225#336C2312yes—
226#336C2312yes—
227C25—21no—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4#4114yes—
235C22×D4C2412yes—
236C23×C4—24no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D412yes—
241C23×C4C2411yescharacteristic
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8#4114yes—
253C22×C8#4114yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C22×C8C2411yescharacteristic
259C22×C8C2411yescharacteristic
260C4×D4—216no—
261C4×D4—28no—
262C4×D4—28no—
263#117—22no—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—88no—
275#117#4114yes—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—84no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—24no—
324C2×C42C2×D412yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—24no—
332#107—24no—
333#107—24no—
334#107—24no—
335#106—24no—
336#106—24no—
337#106—24no—
338#106—24no—
339#102—28no—
340#102—28no—
341#102—24no—
342#102—24no—
343#102—24no—
344#102—24no—
345#97—24no—
346#97—24no—
347C4×C8—24no—
348C4×C8—24no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95C2×D414yes—
362#95C2×D414yes—
363C24—22no—
364C24—24no—
365C24—42no—
366C24—42no—
367C24—82no—
368C24—82no—
369C24C22×D411yescharacteristic
370C2×D4—24no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4C2×#4114yes—
383C22×C4—22no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C42−1+412yes—
413C22×C42−1+412yes—
414C22×C4C2511yescharacteristic
415M16—44no—
416M16—44no—
417C2×C8—28no—
418C2×C8—28no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8C2×#4112yes—
434C2×C8C2×#4112yes—
435C2×C8C2×#4112yes—
436C2×C8C2×#4112yes—
437C4⋊C4—24no—
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—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—84no—
465C4⋊C4C2×#4114yes—
466C22⋊C4—216no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—48no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C42—24no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—44no—
488C42—44no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—84no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—48no—
501C23—88no—
502C23—162no—
503C23—162no—
504C23—164no—
505C23#52512yes—
506C23#53212yes—
507C23#53612yes—
508C23C23×D411yescharacteristic
509D4—24no—
510D4—24no—
511D4—48no—
512D4—42no—
513D4—42no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—816no—
547C2×C4—162no—
548C2×C4—162no—
549C2×C4—162no—
550C2×C4#52512yes—
551C2×C4#53212yes—
552C2×C4#53612yes—
553C2×C4#55214yes—
554C2×C4#55214yes—
555C2×C4#55214yes—
556C2×C4#55214yes—
557C2×C4C22×#4111yescharacteristic
558C2×C4C22×#4111yescharacteristic
559C2×C4C22×#4111yescharacteristic
560C2×C4C22×#4111yescharacteristic
561C2×C4C2×2−1+411yescharacteristic
562C2×C4C2×2−1+411yescharacteristic
563C8—22no—
564C8—22no—
565C8—22no—
566C8—22no—
567C8—42no—
568C8—42no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—24no—
573C22—44no—
574C22—48no—
575C22—82no—
576C22—82no—
577C22—84no—
578C22—168no—
579C22#261114yes—
580C22C2×#52511yescharacteristic
581C22C2×#53211yescharacteristic
582C22C2×#53611yescharacteristic
583C22#358412yes—
584C22#358512yes—
585C4—42no—
586C4—42no—
587C4—44no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4#261114yes—
598C4C2×#55211yescharacteristic
599C4C2×#55211yescharacteristic
600C4C2×#55211yescharacteristic
601C4C2×#55211yescharacteristic
602C2—24no—
603C2—84no—
604C2C2×#261111yescharacteristic
605C2#6060112yes—
606C2#6071311yescharacteristic
607C2#6071911yescharacteristic
608C2#6087512yes—
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. #35339C2semidirect
3. #29590C2semidirect
4. #29586C2semidirect
5. #29582C2semidirect
6. #29576C2semidirect
7. #23746C2semidirect
9. #23656C2semidirect
11. #21156C2semidirect
12. #21155C2semidirect
15. #21038C2semidirect
16. #21037C2semidirect
19. #20368C2semidirect
31. #2864C22semidirect
32. #2647C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2083C22semidirect
65. #1944C22semidirect
66. #1943C22semidirect
67. #1942C22semidirect
68. #1941C22semidirect
73. #1921C22semidirect
74. #1920C22semidirect
75. #1919C22semidirect
76. #1918C22semidirect
81. #1746C22semidirect
82. #1746C22semidirect
83. #1745C22semidirect
84. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353392
#342552
#295902
#295862
#295822
#295762
#237462
C2×#20922
#236562
#236282
#211562
#211552
#210912
#210382
#210372
#208792
#203682