Finite Group Databasebeta

← #6376002

Subgroups — #6376002

Order 512.

2066 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
1C2×#3109C211yesmaximal, characteristic
2#34222C211yesmaximal, characteristic, complemented
3C2×#2319C211yesmaximal, characteristic, complemented
4#22158C214yesmaximal, complemented
5#22123C214yesmaximal, complemented
6#22082C212yesmaximal, complemented
7#22076C212yesmaximal, complemented
8#22046C212yesmaximal, complemented
9#22040C212yesmaximal, complemented
10#21386C211yesmaximal, characteristic, complemented
11#21363C211yesmaximal, characteristic, complemented
12#21290C214yesmaximal, complemented
13#21126C214yesmaximal, complemented
14#20733C211yesmaximal, characteristic, complemented
15#20469C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3109C4116yescomplemented
19C2×#481C2212yes—
20C2×#476C2212yes—
21C2×#459C2212yes—
22C2×#447C2212yes—
23C2×#442C2214yes—
24C2×#432C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2372—21no—
27#2372—21no—
28#2364—21no—
29#2364—21no—
30#2353—22no—
31#2339—22no—
32#2331C2212yescomplemented
33#2328C2212yescomplemented
34#2319—22no—
35#2319—21no—
36#2319—21no—
37#2319C2212yes—
38#2319C2214yes—
39#2319C2212yes—
40#2319C2212yescomplemented
41C4×#121C2212yes—
42C4×#121C2212yes—
43C2×#390C2212yes—
44C2×#389C2212yes—
45C2×#388—22no—
46#2283C2211yescharacteristic
47C2×#381—21no—
48C2×#380—21no—
49#2275C2212yescomplemented
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×#376C2211yescharacteristic
52#1988C2212yescomplemented
53#1982C2212yescomplemented
54#1972C2214yes—
55#1970C2214yes—
56#1949C2214yescomplemented
57#1944C2214yescomplemented
58#1941C2214yescomplemented
59#1936C2214yes—
60#1936C2214yescomplemented
61#1934C2214yes—
62#1934C2214yescomplemented
63#1926C2214yes—
64#1925C2214yes—
65#1920C2214yescomplemented
66#1919C2214yescomplemented
67#1879C2214yes—
68#1879C2214yes—
69#1874C2212yes—
70#1874C2212yes—
71#1864C2214yescomplemented
72#1862C2214yescomplemented
73#1853C2212yes—
74#1852C2212yes—
75#1824C2211yescharacteristic
76#1824C2211yescharacteristic
77#1804C2214yes—
78#1786C2211yescharacteristic
79#1786C2211yescharacteristic
80#1776C2214yescomplemented
81#1768C2214yescomplemented
82C4×#102C2214yes—
83C4×#102C2214yescomplemented
84#1749C2212yes—
85C4×C22⋊C8C2212yes—
86#1746C2212yescomplemented
87#1726C2212yescomplemented
88C22×#41—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×D8C2314yes—
92#532—24no—
93#532C2×C418yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124—21no—
97C2×#121—21no—
98C2×#121C2312yes—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4C2×C418yescomplemented
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C22×C4⋊C4—22no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42—21no—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#481C2×C4116yescomplemented
121#476C2×C4116yescomplemented
122#459C2×C4116yescomplemented
123#447C2×C4116yescomplemented
124#442C2×C4132yescomplemented
125#432C2×C418yescomplemented
126#431—24no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2312yes—
141C2×#97C2314yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#391—22no—
146#391—22no—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#390C2314yes—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#389C2312yes—
158#389C2312yes—
159#388—22no—
160#388—22no—
161#388—24no—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#383—21no—
176#382C2312yes—
177#381—21no—
178#381—22no—
179#381—24no—
180#381—22no—
181#381—22no—
182#380—22no—
183#380—24no—
184#380—22no—
185#380—22no—
186#380C2311yescharacteristic
187#380C2311yescharacteristic
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201#376—21no—
202#376D412yes—
203#376D411yescharacteristic
204#376D411yescharacteristic
205#376C2311yescharacteristic
206#376C2312yes—
207#376C2311yescharacteristic
208C2×#95—21no—
209C2×#95—22no—
210C2×#95—22no—
211C2×#95D412yes—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C2×#41—28no—
224C2×#41—82no—
225C2×#41—82no—
226C22×Q8—21no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—44no—
233C22×D4C22×C414yes—
234C22×D4C2412yes—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—28no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—84no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×D8—44no—
251C2×D8—44no—
252C2×D8C22×C4116yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—22no—
257C4⋊Q8C22×C414yes—
258#127—22no—
259#127C22×C414yes—
260#124—28no—
261#121—28no—
262#121—24no—
263#121C22×C418yes—
264C4×D4—28no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—84no—
273C4×D4—84no—
274C4×D4C22×C418yes—
275C4×D4C22×C418yes—
276C4×D4C22×C418yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—84no—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—84no—
311C2×C22⋊C4—84no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—44no—
318C2×C42—82no—
319C2×C42C22×C414yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4114yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—44no—
330#106—44no—
331#106C22×C4116yes—
332C4⋊C8—24no—
333C4⋊C8C22×C418yes—
334#102—44no—
335#102—44no—
336#102—44no—
337#102—44no—
338#102C22×C4116yes—
339#102C22×C4116yes—
340C22⋊C8—22no—
341C22⋊C8—22no—
342C22⋊C8C22×C418yes—
343#97—44no—
344#97C22×C418yes—
345C4×C8—24no—
346C4×C8—44no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—21no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95C2×D412yes—
361#95C2×D411yescharacteristic
362#95C2×D411yescharacteristic
363#95C2×D412yes—
364#95C2×D411yescharacteristic
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D411yescharacteristic
368#95C2×D412yes—
369#95C2×D412yes—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—44no—
374C24—44no—
375C24—42no—
376C24—42no—
377C24C22×D411yescharacteristic
378#41—816no—
379C2×Q8—24no—
380C2×Q8—81no—
381C2×Q8—81no—
382C2×D4—28no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—82no—
398C2×D4—81no—
399C2×D4—81no—
400C2×D4C23×C414yes—
401C2×D4C23×C414yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—816no—
436C22×C4C4×D414yes—
437C22×C4C4×D414yes—
438C22×C4C23×C412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2511yescharacteristic
450D8—416no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8C23×C418yes—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C23×C414yes—
472C4⋊C4C23×C414yes—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C42—24no—
490C42—24no—
491C42—42no—
492C42—42no—
493C42—82no—
494C42—82no—
495C42—82no—
496C42—82no—
497C42—82no—
498C42C4×D414yes—
499C42C4×D414yes—
500C42C23×C412yes—
501C42C23×C412yes—
502C42C2×#4112yes—
503C42C2×#4112yes—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—48no—
509C23—48no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—88no—
522C23—88no—
523C23C2×C4×D412yes—
524C23#57612yes—
525C23#57712yes—
526C23C23×D411yescharacteristic
527Q8—84no—
528D4—24no—
529D4—28no—
530D4—24no—
531D4—48no—
532D4—416no—
533D4—44no—
534D4—44no—
535D4—84no—
536D4—84no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—88no—
581C2×C4—164no—
582C2×C4—164no—
583C2×C4C2×C4×D412yes—
584C2×C4C2×C4×D412yes—
585C2×C4C2×C4×D412yes—
586C2×C4C2×C4×D412yes—
587C2×C4C2×C4×D412yes—
588C2×C4C24×C411yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4#58514yes—
591C8—28no—
592C8—42no—
593C8—42no—
594C22—22no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—816no—
605C22—82no—
606C22—82no—
607C22—84no—
608C22—82no—
609C22—82no—
610C22—84no—
611C22#294614yes—
612C22C22×C4×D411yescharacteristic
613C22#344912yes—
614C22C2×#57611yescharacteristic
615C22C2×#57711yescharacteristic
616C22#358512yes—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4#342812yes—
632C4#345512yes—
633C2—24no—
634C2—84no—
635C2—84no—
636C2#3406511yescharacteristic
637C2C2×#294611yescharacteristic
638C2#6016811yescharacteristic
639C2#6097812yes—
640C2#6098012yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34222C2semidirect
3. C2×#2319C2semidirect
4. #22158C2semidirect
5. #22123C2semidirect
6. #22082C2semidirect
7. #22076C2semidirect
8. #22046C2semidirect
9. #22040C2semidirect
10. #21386C2semidirect
11. #21363C2semidirect
12. #21290C2semidirect
13. #21126C2semidirect
14. #20733C2semidirect
15. #20469C2semidirect
18. #3109C4semidirect
32. #2331C22semidirect
33. #2328C22semidirect
40. #2319C22semidirect
49. #2275C22semidirect
52. #1988C22semidirect
53. #1982C22semidirect
56. #1949C22semidirect
57. #1944C22semidirect
58. #1941C22semidirect
60. #1936C22semidirect
62. #1934C22semidirect
65. #1920C22semidirect
66. #1919C22semidirect
71. #1864C22semidirect
72. #1862C22semidirect
80. #1776C22semidirect
81. #1768C22semidirect
83. C4×#102C22semidirect
86. #1746C22semidirect
87. #1726C22semidirect
105. C2×C4×D4C2×C4semidirect
120. #481C2×C4semidirect
121. #476C2×C4semidirect
122. #459C2×C4semidirect
123. #447C2×C4semidirect
124. #442C2×C4semidirect
125. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
#342222
C2×#23192
#221582
#221232
#220822
#220762
#220462
#220402
#213862
#213632
#212902
#211262
#207332
#204692