Finite Group Databasebeta

← #6477692

Subgroups — #6477692

Order 512.

2292 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×#3187C211yesmaximal, characteristic
2C22×#432C211yesmaximal, characteristic, complemented
3C2×#2855C211yesmaximal, characteristic, complemented
4C2×#2854C211yesmaximal, characteristic
5#33134C214yesmaximal, complemented
6#29775C214yesmaximal, complemented
7#24609C214yesmaximal, complemented
8#24607C214yesmaximal, complemented
9#24509C214yesmaximal, complemented
10#24507C214yesmaximal, complemented
11#21940C211yesmaximal, characteristic, complemented
12#21759C211yesmaximal, characteristic, complemented
13#21758C211yesmaximal, characteristic
14C22×#123—21no—
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#3187—28no—
19C2×#481C2212yes—
20C2×#483C2212yes—
21C2×#465C2214yes—
22C2×#448C2212yes—
23C2×#450C2212yes—
24C2×#432—28no—
25C2×#432C2214yes—
26C2×#432C2214yescomplemented
27C22×C4⋊C8C2212yescomplemented
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2212yescomplemented
31#2855—22no—
32#2855—21no—
33#2855—21no—
34#2855C2214yes—
35#2855C2212yescomplemented
36#2855C2212yescomplemented
37#2854—21no—
38#2854—21no—
39#2854—22no—
40#2854C2214yes—
41#2854C2212yes—
42#2854C2212yes—
43C2×#391C2212yescomplemented
44C2×#390C2212yes—
45C2×#389C2212yes—
46C2×#389C2212yescomplemented
47C2×#388—22no—
48C2×#387—22no—
49C2×#386—21no—
50C2×#385—21no—
51C2×#381—21no—
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54#2088C2218yescomplemented
55#2068C2218yescomplemented
56#2067C2218yescomplemented
57#2055C2218yescomplemented
58#1973C2218yescomplemented
59#1943C2218yescomplemented
60#1936C2218yescomplemented
61#1935C2218yescomplemented
62#1893C2214yes—
63#1892C2214yescomplemented
64#1874C2214yes—
65#1872C2214yes—
66#1852C2214yes—
67#1852C2214yescomplemented
68#1824C2212yescomplemented
69#1823C2212yes—
70#1808C2214yescomplemented
71#1787C2212yescomplemented
72#1786C2212yes—
73C23×Q8C2311yescharacteristic
74C23×D4—21no—
75C23×D4C2311yescharacteristic
76C24×C4D411yescharacteristic
77C24×C4D411yescharacteristic
78C24×C4C2311yescharacteristic
79C22×SD16C2314yes—
80C23×C8C2312yes—
81C23×C8C2312yes—
82C2×#123—24no—
83C2×#123—22no—
84C2×#123—22no—
85C2×#123—48no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D412yes—
116C22×C42D412yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121#483—28no—
122#481—28no—
123#465—216no—
124#450—28no—
125#448—28no—
126#432—216no—
127#432—216no—
128C2×#107C2318yes—
129C2×#106C2314yes—
130C2×C4⋊C8—28no—
131C2×C4⋊C8C2318yes—
132C2×#103C2314yes—
133C2×#103C2318yes—
134C2×#102C2314yes—
135C2×#102C2318yes—
136C2×C22⋊C8—24no—
137C2×C22⋊C8D418yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×#97—28no—
143C2×#97C2318yes—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#391C2314yes—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#388—24no—
161#388—24no—
162#388—24no—
163#388—24no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#386—22no—
169#386—24no—
170#386—22no—
171#385—24no—
172#385—22no—
173#385—22no—
174#381—24no—
175#381—22no—
176#381—22no—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380C2312yes—
182#380C2312yes—
183#380C2312yes—
184#380C2312yes—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—22no—
188C2×#95—22no—
189C2×#95—22no—
190C2×#95—22no—
191#340C2318yes—
192#340C2318yes—
193#340C2318yes—
194C25—21no—
195C25C2×D411yescharacteristic
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×Q8C2412yes—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4C2412yes—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D414yes—
234C23×C4#4114yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×SD16—28no—
239C2×SD16—48no—
240C22×C8—24no—
241C22×C8—44no—
242C22×C8—44no—
243C22×C8C2×D418yes—
244C22×C8C2414yes—
245C22×C8C2414yes—
246#123—48no—
247#123—48no—
248#123—416no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255#121—24no—
256#121—24no—
257#121—24no—
258#121—24no—
259#121—44no—
260#121—44no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2414yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—48no—
323#106—28no—
324C4⋊C8—216no—
325#103—28no—
326#103—48no—
327#102—28no—
328#102—48no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8C2×D418yes—
332C22⋊C8C2×D418yes—
333#97—216no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346C24—22no—
347C24—42no—
348C24—42no—
349C24—44no—
350C24—44no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24#12114yes—
355C24AGL(1,ℤ/8)14yes—
356C24#13714yes—
357C24C22×D411yescharacteristic
358C24C22×D411yescharacteristic
359C24C22×D411yescharacteristic
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—28no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—816no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4#12114yes—
432C22×C4#12114yes—
433C22×C4#12114yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2511yescharacteristic
448SD16—88no—
449C2×C8—28no—
450C2×C8—24no—
451C2×C8—28no—
452C2×C8—48no—
453C2×C8—48no—
454C2×C8C22×D418yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—24no—
487C42—24no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—44no—
500C23—44no—
501C23—48no—
502C23—48no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—84no—
509C23—84no—
510C23—88no—
511C23—88no—
512C23—84no—
513C23#468116yes—
514C23C2×#12112yes—
515C23C2×#12112yes—
516C23C2×#12112yes—
517C23C2×AGL(1,ℤ/8)12yes—
518C23C2×AGL(1,ℤ/8)12yes—
519C23C2×AGL(1,ℤ/8)12yes—
520C23C2×#13712yes—
521C23C2×#13712yes—
522C23C2×#13712yes—
523C23C23×D411yescharacteristic
524Q8—44no—
525Q8—44no—
526Q8—84no—
527D4—44no—
528D4—44no—
529D4—84no—
530D4—88no—
531D4—88no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
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—22no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—84no—
577C2×C4—88no—
578C2×C4—88no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4C2×#12112yes—
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4#57514yes—
585C2×C4C23×D411yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4#58514yes—
588C8—28no—
589C8—44no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—416no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—84no—
607C22—84no—
608C22—168no—
609C22C2×#46814yes—
610C22C2×#46814yes—
611C22C2×#46814yes—
612C22C22×#12111yescharacteristic
613C22#344912yes—
614C22C22×AGL(1,ℤ/8)11yescharacteristic
615C22C22×#13711yescharacteristic
616C22C2×#57512yes—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—24no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4—84no—
629C4#348512yes—
630C4#349112yes—
631C2—24no—
632C2—24no—
633C2—24no—
634C2—162no—
635C2#3508911yescharacteristic
636C2C22×#46811yescharacteristic
637C2#6044411yescharacteristic
638C2#6095812yes—
639C2#6096012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#432C2semidirect
3. C2×#2855C2semidirect
5. #33134C2semidirect
6. #29775C2semidirect
7. #24609C2semidirect
8. #24607C2semidirect
9. #24509C2semidirect
10. #24507C2semidirect
11. #21940C2semidirect
12. #21759C2semidirect
26. C2×#432C22semidirect
27. C22×C4⋊C8C22semidirect
30. C22×#97C22semidirect
35. #2855C22semidirect
36. #2855C22semidirect
43. C2×#391C22semidirect
46. C2×#389C22semidirect
54. #2088C22semidirect
55. #2068C22semidirect
56. #2067C22semidirect
57. #2055C22semidirect
58. #1973C22semidirect
59. #1943C22semidirect
60. #1936C22semidirect
61. #1935C22semidirect
63. #1892C22semidirect
67. #1852C22semidirect
68. #1824C22semidirect
70. #1808C22semidirect
71. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
C22×#4322
C2×#28552
C2×#28542
#331342
#297752
#246092
#246072
#245092
#245072
#219402
#217592
#217582