Finite Group Databasebeta

← #6447977

Subgroups — #6447977

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic, complemented
3#25003C214yesmaximal, complemented
4#25002C214yesmaximal, complemented
5#24999C214yesmaximal, complemented
6#24998C214yesmaximal, complemented
7#23747C212yesmaximal, complemented
8#23628C212yesmaximal
9#21303C211yesmaximal, characteristic, complemented
10#21172C211yesmaximal, characteristic, complemented
11#21060C212yesmaximal
12#21034C212yesmaximal, complemented
13#20765C211yesmaximal, characteristic, complemented
14#20646C211yesmaximal, characteristic
15C4×#418C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#428C2212yes—
19C2×#423C2211yescharacteristic
20C2×#418C2211yescharacteristic
21C2×#418C2212yes—
22C2×#418C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2580—21no—
25#2580—21no—
26#2580—22no—
27#2580C2214yes—
28#2580C2212yes—
29#2580C2212yes—
30C2×#399—21no—
31C2×#393—21no—
32#2355C2212yescomplemented
33#2355C2212yescomplemented
34#2355C2212yescomplemented
35Q8×C4⋊C4C2212yescomplemented
36#2345—21no—
37#2345—22no—
38Q8×C22⋊C4—21no—
39#2295—22no—
40#2295—21no—
41C2×#390C2212yes—
42C2×#389C2212yes—
43C2×#385—22no—
44#2285C2211yescharacteristic
45#2285C2212yes—
46C2×#382C2211yescharacteristic
47C42×Q8—21no—
48C4×#117C2212yes—
49C4×#117C2211yescharacteristic
50#2101C2214yescomplemented
51#2101C2214yescomplemented
52#2100C2214yescomplemented
53#2099C2214yescomplemented
54#2073C2214yes—
55#2073C2214yes—
56#2071C2214yes—
57#2071C2214yes—
58#1985C2212yescomplemented
59#1985C2212yescomplemented
60#1984C2214yescomplemented
61#1949C2214yescomplemented
62#1949C2214yescomplemented
63#1926C2214yes—
64#1926C2214yes—
65#1925C2214yes—
66#1925C2214yes—
67#1921C2214yescomplemented
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1918C2214yescomplemented
71#1836C2214yescomplemented
72#1835C2212yescomplemented
73#1835C2212yescomplemented
74#1804C2212yes—
75#1804C2212yes—
76#1802C2214yes—
77#1762C2214yes—
78C4×#102C2214yes—
79#1736C2212yes—
80#1736C2212yes—
81#1736C2211yescharacteristic
82#1736C2212yes—
83#1736C2212yes—
84#1736C2211yescharacteristic
85#1728C2212yes—
86#1728C2212yes—
87C23×Q8D411yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×M16C2311yescharacteristic
91C22×M16C2311yescharacteristic
92C22×M16C2312yes—
93C2×C4×Q8—22no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8D411yescharacteristic
97C2×C4×Q8D412yes—
98C2×C4×Q8D411yescharacteristic
99C2×C4×Q8C2311yescharacteristic
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—22no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D412yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4—22no—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121#428—28no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124#423—24no—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127#418—28no—
128#418—24no—
129#418—24no—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C4×M16—24no—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137#399—22no—
138#399—24no—
139#399—22no—
140#393—22no—
141#393—24no—
142#393—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#390—22no—
147#390—22no—
148#390C2314yes—
149#390C2314yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#385—24no—
164#385—24no—
165#385—24no—
166#385—24no—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#382—21no—
171#382—21no—
172#382C2311yescharacteristic
173#382C2311yescharacteristic
174#382C2312yes—
175#382C2312yes—
176#382C2312yes—
177#382C2312yes—
178#380—22no—
179#380—21no—
180#380—22no—
181#380—21no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—21no—
199#376—22no—
200#376—21no—
201C2×#95—21no—
202C2×#95—22no—
203C2×#95—21no—
204C43—22no—
205C43—21no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25—21no—
221C22×Q8—24no—
222C22×Q8—44no—
223C22×Q8C2×D411yescharacteristic
224C22×Q8C2×D412yes—
225C22×Q8C2×D411yescharacteristic
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—44no—
239C23×C4—84no—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—24no—
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—22no—
253C2×M16—22no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C4×Q8—24no—
258C4×Q8—28no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—48no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—44no—
272C4×D4—44no—
273#117—22no—
274#117—24no—
275#117—22no—
276#117—24no—
277#117—22no—
278#117#4114yes—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—84no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—84no—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—28no—
343#106—28no—
344C4⋊C8—24no—
345C4⋊C8—24no—
346#102—28no—
347#102—28no—
348#102—28no—
349#102—28no—
350#97—24no—
351C4×C8—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372C24—24no—
373C24—22no—
374C24—44no—
375C24—44no—
376C24—42no—
377C24—42no—
378C24C22×D411yescharacteristic
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—416no—
387C2×D4—28no—
388C2×D4—24no—
389C2×D4—28no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4C2×#4114yes—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—816no—
424C22×C4#12314yes—
425C22×C4#12314yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C42−1+412yes—
440C22×C42−1+412yes—
441C22×C4C2511yescharacteristic
442M16—48no—
443M16—44no—
444M16—44no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4C2×#4114yes—
483C22⋊C4—24no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—44no—
500C42—44no—
501C42—44no—
502C42—42no—
503C42—44no—
504C42—82no—
505C42—82no—
506C42—82no—
507C42#12314yes—
508C42#12314yes—
509C42C2×#4112yes—
510C42C2×#4112yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—416no—
517C23—44no—
518C23—42no—
519C23—48no—
520C23—42no—
521C23—44no—
522C23—84no—
523C23—84no—
524C23—88no—
525C23C2×#12312yes—
526C23#54812yes—
527C23#54812yes—
528C23C23×D411yescharacteristic
529Q8—28no—
530Q8—48no—
531Q8—44no—
532D4—24no—
533D4—24no—
534D4—48no—
535D4—42no—
536D4—42no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—84no—
575C2×C4—88no—
576C2×C4—84no—
577C2×C4C2×#12312yes—
578C2×C4C2×#12312yes—
579C2×C4C2×#12312yes—
580C2×C4C2×#12312yes—
581C2×C4C2×#12312yes—
582C2×C4#52914yes—
583C2×C4#54114yes—
584C2×C4#54812yes—
585C2×C4#54812yes—
586C2×C4#55214yes—
587C2×C4#55614yes—
588C2×C4#57514yes—
589C2×C4C22×#4111yescharacteristic
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×2−1+411yescharacteristic
594C2×C4C2×2−1+411yescharacteristic
595C8—44no—
596C8—42no—
597C8—42no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—48no—
603C22—44no—
604C22—816no—
605C22—82no—
606C22—82no—
607C22—84no—
608C22#242114yes—
609C22C22×#12311yescharacteristic
610C22C2×#54811yescharacteristic
611C22C2×#54811yescharacteristic
612C22C2×#57512yes—
613C22#358512yes—
614C4—22no—
615C4—22no—
616C4—24no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—44no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4#242114yes—
629C4C2×#52911yescharacteristic
630C4C2×#54111yescharacteristic
631C4C2×#55211yescharacteristic
632C4C2×#55611yescharacteristic
633C2—24no—
634C2—84no—
635C2C2×#242111yescharacteristic
636C2#6051111yescharacteristic
637C2#6051811yescharacteristic
638C2#6131812yes—
639C2#6132412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. Q8×#117C2semidirect
3. #25003C2semidirect
4. #25002C2semidirect
5. #24999C2semidirect
6. #24998C2semidirect
7. #23747C2semidirect
9. #21303C2semidirect
10. #21172C2semidirect
12. #21034C2semidirect
13. #20765C2semidirect
32. #2355C22semidirect
33. #2355C22semidirect
34. #2355C22semidirect
35. Q8×C4⋊C4C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
58. #1985C22semidirect
59. #1985C22semidirect
60. #1984C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
67. #1921C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
70. #1918C22semidirect
71. #1836C22semidirect
72. #1835C22semidirect
73. #1835C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
Q8×#1172
#250032
#250022
#249992
#249982
#237472
#236282
#213032
#211722
#210602
#210342
#207652
#206462
C4×#4182