Finite Group Databasebeta

← #6376007

Subgroups — #6376007

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic, complemented
3#34058C211yesmaximal, characteristic
4#22152C212yesmaximal, complemented
5#22146C212yesmaximal, complemented
6#22116C212yesmaximal, complemented
7#22110C212yesmaximal, complemented
8#22087C214yesmaximal, complemented
9#22052C214yesmaximal
10#21389C211yesmaximal, characteristic, complemented
11#21364C211yesmaximal, characteristic, complemented
12#21291C214yesmaximal, complemented
13#21128C214yesmaximal, complemented
14#20732C211yesmaximal, characteristic
15#20470C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3110C4116yescomplemented
19C2×#484C2212yes—
20C2×#474C2212yes—
21C2×#461C2212yes—
22C2×#450C2212yes—
23C2×#440C2214yes—
24C2×#432C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2370—22no—
27#2362—22no—
28#2356—21no—
29#2351—21no—
30#2347—21no—
31#2346—21no—
32#2323C2214yes—
33#2320C2212yescomplemented
34#2319—22no—
35#2319C2212yes—
36#2319C2212yes—
37#2316C2212yes—
38#2312—21no—
39#2312—21no—
40#2309C2212yes—
41C4×C4⋊Q8C2212yescomplemented
42C4×C4⋊Q8C2212yescomplemented
43#2293—21no—
44#2293—21no—
45#2293C2212yes—
46#2293C2212yes—
47#2285C2211yescharacteristic
48#2280—22no—
49#2276C2212yescomplemented
50C4×#117C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#1988C2211yescharacteristic, complemented
53#1988C2211yescharacteristic, complemented
54#1981C2211yescharacteristic, complemented
55#1981C2211yescharacteristic, complemented
56#1973C2214yes—
57#1968C2214yes—
58#1948C2212yescomplemented
59#1947C2212yescomplemented
60#1943C2214yescomplemented
61#1942C2214yescomplemented
62#1936C2214yes—
63#1935C2214yescomplemented
64#1934C2214yes—
65#1933C2214yescomplemented
66#1926C2214yes—
67#1925C2214yes—
68#1921C2214yescomplemented
69#1918C2214yescomplemented
70#1880C2214yes—
71#1880C2214yes—
72#1873C2214yes—
73#1865C2214yescomplemented
74#1863C2214yescomplemented
75#1854C2214yes—
76#1823C2212yes—
77#1802C2212yes—
78#1802C2212yes—
79#1786C2212yes—
80#1776C2214yescomplemented
81#1767C2212yescomplemented
82C4×C4⋊C8C2212yescomplemented
83C4×#103C2214yescomplemented
84C4×#102C2214yes—
85#1750C2214yes—
86#1746C2212yescomplemented
87#1726C2212yescomplemented
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91#533—24no—
92#533C2×C418yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#125—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8—22no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2314yes—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4C2×C418yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117—21no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C42C2311yescharacteristic
117#484C2×C4116yes—
118#474C2×C4116yes—
119#461C2×C4116yes—
120#450C2×C4116yes—
121#440C2×C4132yes—
122#432C2×C418yes—
123#431—24no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2314yes—
139C2×#97C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142#391—22no—
143#391C2314yes—
144#390—22no—
145#390—22no—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#389—22no—
151#389—22no—
152#389—24no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#388—24no—
162#388—24no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#387—24no—
172#387—24no—
173#384C2312yes—
174#383—22no—
175#382—21no—
176#382—22no—
177#381—22no—
178#381—22no—
179#380—21no—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380C2312yes—
185#379—22no—
186#379—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—24no—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4D411yescharacteristic
195C4×C4⋊C4D411yescharacteristic
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2311yescharacteristic
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4D411yescharacteristic
206C4×C22⋊C4D411yescharacteristic
207C4×C22⋊C4D412yes—
208#376—21no—
209#376—22no—
210#376C2311yescharacteristic
211#376C2311yescharacteristic
212C43—21no—
213C43C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C22×Q8—21no—
224C22×Q8C2412yes—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—42no—
228C22×D4—42no—
229C22×D4C22×C414yes—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—28no—
234C23×C4—22no—
235C23×C4—82no—
236C23×C4—82no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×SD16—44no—
241C2×SD16—44no—
242C2×SD16C22×C4116yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8C22×C414yes—
249C4⋊Q8C22×C414yes—
250#125—28no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8C22×C418yes—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—44no—
257C4×Q8—44no—
258C4×Q8—84no—
259C4×Q8C22×C418yes—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4C22×C418yes—
265C4×D4C22×C418yes—
266#117—28no—
267#117—42no—
268#117—42no—
269#117—42no—
270#117—42no—
271#117—84no—
272#117—84no—
273#117—82no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4C22×C414yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42C22×C414yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42#4114yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—44no—
334#107C22×C4116yes—
335#106—44no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8C22×C418yes—
339#103—44no—
340#103—44no—
341#103C22×C4116yes—
342#102—44no—
343#102—44no—
344#102C22×C4116yes—
345C22⋊C8—24no—
346C22⋊C8C22×C418yes—
347#97—44no—
348#97C22×C418yes—
349C4×C8—24no—
350C4×C8—44no—
351#95—22no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95C2×D411yescharacteristic
365#95C2×D412yes—
366#95C2×D411yescharacteristic
367#95C2×D412yes—
368#95C2×D411yescharacteristic
369#95C2×D411yescharacteristic
370C24—22no—
371C24—24no—
372C24—44no—
373C24—81no—
374C24—81no—
375C24—82no—
376C24C22×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8C23×C414yes—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4C23×C414yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—816no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4C4×D414yes—
417C22×C4C4×D414yes—
418C22×C4C23×C412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2511yescharacteristic
430SD16—416no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8C23×C418yes—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4C23×C414yes—
463C4⋊C4C23×C414yes—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C22⋊C4—82no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C42—28no—
480C42—24no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—42no—
487C42—42no—
488C42—44no—
489C42—48no—
490C42—42no—
491C42—42no—
492C42—82no—
493C42—82no—
494C42—84no—
495C42C4×D414yes—
496C42C4×D414yes—
497C42C23×C412yes—
498C42C23×C412yes—
499C42C2×#4112yes—
500C42C2×#4112yes—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—24no—
506C23—48no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—84no—
514C23—84no—
515C23—88no—
516C23—88no—
517C23C2×C4×D412yes—
518C23C23×D411yescharacteristic
519Q8—28no—
520Q8—42no—
521Q8—42no—
522Q8—48no—
523D4—24no—
524D4—24no—
525D4—48no—
526D4—42no—
527D4—42no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—88no—
569C2×C4—84no—
570C2×C4C2×C4×D412yes—
571C2×C4C2×C4×D412yes—
572C2×C4C2×C4×D412yes—
573C2×C4C2×C4×D412yes—
574C2×C4C2×C4×D412yes—
575C2×C4#57612yes—
576C2×C4#57712yes—
577C2×C4C24×C411yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4#58512yes—
580C2×C4#58512yes—
581C8—28no—
582C8—42no—
583C8—42no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—816no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C22×C4×D411yescharacteristic
598C22#344912yes—
599C22C2×#57611yescharacteristic
600C22C2×#57711yescharacteristic
601C22#358612yes—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—44no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—44no—
612C4—44no—
613C4—82no—
614C4—84no—
615C4—82no—
616C4#294614yes—
617C4#342812yes—
618C4#345511yescharacteristic
619C4#345511yescharacteristic
620C2—42no—
621C2—84no—
622C2#3406511yescharacteristic
623C2C2×#294611yescharacteristic
624C2#6016911yescharacteristic
625C2#6098812yes—
626C2#6099112yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×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
2. #34225C2semidirect
4. #22152C2semidirect
5. #22146C2semidirect
6. #22116C2semidirect
7. #22110C2semidirect
8. #22087C2semidirect
10. #21389C2semidirect
11. #21364C2semidirect
12. #21291C2semidirect
13. #21128C2semidirect
15. #20470C2semidirect
18. #3110C4semidirect
33. #2320C22semidirect
41. C4×C4⋊Q8C22semidirect
42. C4×C4⋊Q8C22semidirect
49. #2276C22semidirect
52. #1988C22semidirect
53. #1988C22semidirect
54. #1981C22semidirect
55. #1981C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1943C22semidirect
61. #1942C22semidirect
63. #1935C22semidirect
65. #1933C22semidirect
68. #1921C22semidirect
69. #1918C22semidirect
73. #1865C22semidirect
74. #1863C22semidirect
80. #1776C22semidirect
81. #1767C22semidirect
82. C4×C4⋊C8C22semidirect
83. C4×#103C22semidirect
86. #1746C22semidirect
87. #1726C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
#342252
#340582
#221522
#221462
#221162
#221102
#220872
#220522
#213892
#213642
#212912
#211282
#207322
#204702