Finite Group Databasebeta

← #6489653

Subgroups — #6489653

Order 512.

3199 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#3239C211yesmaximal, characteristic, complemented
3C2×#3114C211yesmaximal, characteristic
4#35111C211yesmaximal, characteristic, complemented
5#33009C212yesmaximal, complemented
6#33004C212yesmaximal
7#29643C212yesmaximal, complemented
8#29640C212yesmaximal, complemented
9#25940C214yesmaximal, complemented
10#25845C214yesmaximal, complemented
11#24343C214yesmaximal, complemented
12#24339C214yesmaximal, complemented
13#21841C211yesmaximal, characteristic, complemented
14#21822C211yesmaximal, characteristic, complemented
15#20739C211yesmaximal, characteristic, complemented
16C2×#554—28no—
17C2×#554—22no—
18C2×#554—22no—
19C2×#554—28no—
20C2×#554C2214yes—
21C2×#554C2214yes—
22C22×C4⋊Q8C2211yescharacteristic
23C2×#529C2211yescharacteristic
24C22×#125C2212yes—
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—22no—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2212yes—
29C22×C4×D4C2211yescharacteristic
30C23×C4⋊C4—21no—
31#3239—28no—
32C2×#501C2212yescomplemented
33C2×#489C2212yescomplemented
34#3114—28no—
35C2×#484C2212yes—
36C2×#481C2212yes—
37C2×#483C2211yescharacteristic
38C2×#480C2211yescharacteristic
39C2×#475C2212yescomplemented
40C2×#478C2212yes—
41C2×#474C2212yescomplemented
42C2×#463C2212yes—
43C2×#450C2211yescharacteristic
44C2×#447C2211yescharacteristic
45C2×#442C2212yes—
46C2×#440C2212yes—
47C2×#432C2211yescharacteristic
48C2×#432C2211yescharacteristic
49#2855C2212yes—
50#2854C2212yes—
51#2623—21no—
52#2614—21no—
53#2604—22no—
54#2599—22no—
55#2592C2212yescomplemented
56#2584C2212yescomplemented
57#2581C2212yescomplemented
58#2580C2212yescomplemented
59#2322—22no—
60#2283C2211yescharacteristic
61#2059C2218yescomplemented
62#2059C2218yescomplemented
63#2026C2218yescomplemented
64#2025C2218yescomplemented
65#1993C2214yescomplemented
66#1988C2214yescomplemented
67#1985C2214yescomplemented
68#1985C2214yescomplemented
69#1879C2218yes—
70#1877C2218yes—
71#1864C2218yes—
72#1862C2218yes—
73#1824C2212yes—
74#1824C2212yes—
75#1823C2212yes—
76#1817C2212yes—
77#1817C2212yes—
78#1804C2214yes—
79#1786C2212yes—
80#1762C2214yes—
81C23×Q8—21no—
82C23×Q8D411yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4—22no—
86C22×SD16C2312yes—
87C22×D8C2312yes—
88#554—232no—
89#554—232no—
90#554—216no—
91#554—216no—
92#554—216no—
93#554—216no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8D418yes—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100#529—24no—
101#529—24no—
102C2×#126—21no—
103C2×#125—28no—
104C2×#125—28no—
105C2×#125—24no—
106C2×#125C2311yescharacteristic
107C2×#125C2318yes—
108C2×#124C2311yescharacteristic
109C2×#123—22no—
110C2×#123—21no—
111C2×C22⋊Q8—216no—
112C2×C22⋊Q8—232no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8C2311yescharacteristic
118C2×#121C2311yescharacteristic
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8D412yes—
122C2×C4×Q8D412yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—28no—
127C2×C4×D4—21no—
128C2×C4×D4—28no—
129C2×C4×D4—24no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—24no—
133C2×C4×D4D418yes—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2318yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×#117—21no—
139C22×C4⋊C4—216no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—22no—
143C22×C4⋊C4—24no—
144C22×C4⋊C4—24no—
145C22×C4⋊C4D412yes—
146C22×C4⋊C4D411yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2312yes—
149C22×C22⋊C4—22no—
150C22×C22⋊C4—21no—
151C22×C42C2312yes—
152C22×C42C2311yescharacteristic
153#501—28no—
154#489—28no—
155#484—28no—
156#483—24no—
157#481—28no—
158#480—24no—
159#478—28no—
160#475—28no—
161#474—28no—
162#463—28no—
163#450—24no—
164#447—24no—
165#442—28no—
166#440—28no—
167#432—24no—
168#432—24no—
169C2×#107C2312yes—
170C2×#107C2314yes—
171C2×#106C2312yes—
172C2×#106C2314yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#102C2312yes—
178C2×#102C2312yes—
179C2×#102C2318yes—
180C2×#102C2312yes—
181C2×#102C2314yes—
182C2×C22⋊C8C2311yescharacteristic
183C2×C22⋊C8C2311yescharacteristic
184C2×C22⋊C8C2312yes—
185C2×#97C2312yes—
186C2×#97C2312yes—
187#400—22no—
188#399—22no—
189#398—22no—
190#397—22no—
191#395—22no—
192#395—22no—
193#393—22no—
194#393—22no—
195#391C2314yes—
196#390C2314yes—
197#389—24no—
198#389C2314yes—
199#389C2314yes—
200#388—24no—
201#388—24no—
202#387—24no—
203#387—24no—
204#386—22no—
205#385—22no—
206#385—24no—
207#383—21no—
208#382C2312yes—
209#381—22no—
210#381—21no—
211#380C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214#376—21no—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#336C2318yes—
219#336C2318yes—
220C25—21no—
221C22×Q8—24no—
222C22×Q8—28no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—44no—
226C22×Q8—84no—
227C22×Q8C2×D412yes—
228C22×Q8C2×D412yes—
229C22×Q8C2411yescharacteristic
230C22×D4—28no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4C2×D414yes—
235C22×D4C2411yescharacteristic
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—216no—
240C23×C4—28no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—28no—
244C23×C4—21no—
245C23×C4—48no—
246C23×C4—88no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×SD16—24no—
253C2×SD16—28no—
254C2×D8—24no—
255C2×D8—28no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8C2×D418yes—
264C4⋊Q8C2×D418yes—
265#126—44no—
266#125—232no—
267#125—216no—
268#125—216no—
269#125—22no—
270#125—22no—
271#124—22no—
272#124—22no—
273#123—48no—
274#123—44no—
275C22⋊Q8—232no—
276C22⋊Q8—232no—
277C22⋊Q8—264no—
278C22⋊Q8—264no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281#121—22no—
282#121—22no—
283C4×Q8—28no—
284C4×Q8—28no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×D4—232no—
292C4×D4—216no—
293C4×D4—216no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—44no—
301C4×D4C2×D418yes—
302C4×D4C2×D418yes—
303#117—44no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—232no—
306C2×C4⋊C4—232no—
307C2×C4⋊C4—216no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D418yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C22⋊C4—216no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42C2×D414yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D414yes—
357C2×C42C2×D414yes—
358C2×C42C2414yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—28no—
363#106—24no—
364#106—28no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367#103—24no—
368#103—28no—
369#102—24no—
370#102—24no—
371#102—216no—
372#102—24no—
373#102—28no—
374C22⋊C8—22no—
375C22⋊C8—22no—
376C22⋊C8—24no—
377#97—24no—
378#97—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387C24—28no—
388C24—24no—
389C24—22no—
390C24—21no—
391C24—44no—
392C24—44no—
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—24no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—416no—
404C2×Q8—84no—
405C2×Q8—84no—
406C2×Q8—816no—
407C2×D4—28no—
408C2×D4—216no—
409C2×D4—216no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4C22×D414yes—
419C2×D42−1+418yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—232no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—432no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—82no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—832no—
449C22×C4#12014yes—
450C22×C4#12014yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C2511yescharacteristic
461SD16—48no—
462D8—48no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—28no—
467C2×C8—24no—
468C2×C8—24no—
469C4⋊C4—232no—
470C4⋊C4—232no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—28no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—84no—
492C4⋊C4C22×D418yes—
493C4⋊C4C22×D414yes—
494C22⋊C4—232no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—48no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—84no—
505C42—28no—
506C42—24no—
507C42—24no—
508C42—42no—
509C42—82no—
510C42#12014yes—
511C42#12014yes—
512C42C22×D412yes—
513C42C22×D412yes—
514C42C22×D412yes—
515C42C22×D412yes—
516C23—22no—
517C23—22no—
518C23—216no—
519C23—24no—
520C23—28no—
521C23—48no—
522C23—432no—
523C23—44no—
524C23—48no—
525C23—44no—
526C23—42no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23C2×#12012yes—
531C23#57514yes—
532C23C23×D411yescharacteristic
533Q8—48no—
534Q8—48no—
535Q8—42no—
536Q8—42no—
537Q8—88no—
538Q8—88no—
539D4—216no—
540D4—24no—
541D4—28no—
542D4—42no—
543D4—42no—
544D4—84no—
545D4#53618yes—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—44no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—48no—
566C2×C4—416no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—832no—
582C2×C4C2×#12012yes—
583C2×C4C2×#12012yes—
584C2×C4C2×#12012yes—
585C2×C4C2×#12012yes—
586C2×C4C2×#12012yes—
587C2×C4C23×D411yescharacteristic
588C2×C4C23×D411yescharacteristic
589C2×C4C2×2−1+414yes—
590C8—42no—
591C8—42no—
592C8—44no—
593C22—28no—
594C22—22no—
595C22—22no—
596C22—28no—
597C22—432no—
598C22—416no—
599C22—44no—
600C22—44no—
601C22—84no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22#300514yes—
606C22C22×#12011yescharacteristic
607C22#348612yes—
608C22C2×#57512yes—
609C22#358512yes—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—48no—
619C4—84no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4C2×#53612yes—
624C4#348612yes—
625C2—24no—
626C2—48no—
627C2—161no—
628C2#3524211yescharacteristic
629C2C2×#300511yescharacteristic
630C2#6036711yescharacteristic
631C2#6183114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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
1. C22×#554C2semidirect
2. C2×#3239C2semidirect
4. #35111C2semidirect
5. #33009C2semidirect
7. #29643C2semidirect
8. #29640C2semidirect
9. #25940C2semidirect
10. #25845C2semidirect
11. #24343C2semidirect
12. #24339C2semidirect
13. #21841C2semidirect
14. #21822C2semidirect
15. #20739C2semidirect
32. C2×#501C22semidirect
33. C2×#489C22semidirect
39. C2×#475C22semidirect
41. C2×#474C22semidirect
55. #2592C22semidirect
56. #2584C22semidirect
57. #2581C22semidirect
58. #2580C22semidirect
61. #2059C22semidirect
62. #2059C22semidirect
63. #2026C22semidirect
64. #2025C22semidirect
65. #1993C22semidirect
66. #1988C22semidirect
67. #1985C22semidirect
68. #1985C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32392
C2×#31142
#351112
#330092
#330042
#296432
#296402
#259402
#258452
#243432
#243392
#218412
#218222
#207392