Finite Group Databasebeta

← #6479558

Subgroups — #6479558

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2624C211yesmaximal, characteristic
2#34180C211yesmaximal, characteristic, complemented
3#33103C214yesmaximal, complemented
4#29917C214yesmaximal, complemented
5#27874C214yesmaximal, complemented
6#27868C214yesmaximal, complemented
7#27617C212yesmaximal, complemented
8#27614C212yesmaximal, complemented
9#27588C212yesmaximal, complemented
10#27582C212yesmaximal, complemented
11#21944C211yesmaximal, characteristic, complemented
12#21940C211yesmaximal, characteristic, complemented
13#21764C211yesmaximal, characteristic
14#21761C211yesmaximal, characteristic
15#20685C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2624—22no—
21#2624—21no—
22#2624—21no—
23#2624C2212yes—
24#2624C2214yes—
25#2624C2212yes—
26C2×#399—21no—
27C2×#397—21no—
28C2×#394—22no—
29#2347—21no—
30#2346—21no—
31#2345—21no—
32#2344—21no—
33#2343—22no—
34#2342—22no—
35#2317C2212yescomplemented
36#2314C2212yescomplemented
37#2314C2212yescomplemented
38C2×#390C2212yes—
39C2×#389C2212yes—
40#2285C2212yes—
41#2283C2212yes—
42#2280C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45#2274C2212yescomplemented
46#2101C2214yescomplemented
47#2100C2214yescomplemented
48#2093C2214yescomplemented
49#2092C2214yescomplemented
50#2085C2214yescomplemented
51#2083C2214yescomplemented
52#2073C2214yescomplemented
53#2071C2214yescomplemented
54#1926C2214yescomplemented
55#1925C2214yescomplemented
56#1925C2214yescomplemented
57#1925C2214yescomplemented
58#1921C2214yescomplemented
59#1921C2214yescomplemented
60#1919C2214yescomplemented
61#1918C2214yescomplemented
62#1895C2212yes—
63#1895C2212yes—
64#1893C2214yes—
65#1875C2214yes—
66#1874C2214yes—
67#1854C2214yescomplemented
68#1853C2212yes—
69#1852C2212yes—
70#1852C2212yes—
71#1852C2212yes—
72#1852C2214yescomplemented
73#1824C2211yescharacteristic
74#1824C2211yescharacteristic
75#1824C2212yes—
76#1818C2212yescomplemented
77#1818C2212yescomplemented
78#1817C2212yes—
79#1817C2212yes—
80#1809C2214yescomplemented
81#1808C2212yescomplemented
82#1808C2212yescomplemented
83#1786C2211yescharacteristic
84#1786C2211yescharacteristic
85#1786C2212yes—
86#1729C2214yescomplemented
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C23×C8C2312yes—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—22no—
101C2×#117D412yes—
102C2×#117D411yescharacteristic
103C2×#117D411yescharacteristic
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C4⋊C4—22no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4D412yes—
120C22×C42C2312yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#407—24no—
130C2×C22⋊C8—24no—
131C2×C22⋊C8C2311yescharacteristic
132C2×C22⋊C8C2311yescharacteristic
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136#399—22no—
137#399—24no—
138#399—22no—
139#397—22no—
140#397—24no—
141#397—22no—
142#394—24no—
143#394—24no—
144#394—24no—
145#394—24no—
146#390—22no—
147#390—22no—
148#390C2314yes—
149#390C2314yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#385—22no—
165#385—22no—
166#384C2312yes—
167#384C2312yes—
168#383—22no—
169#382—22no—
170#382C2312yes—
171#382C2312yes—
172#381—22no—
173#381—21no—
174#380—22no—
175#380—21no—
176#380—21no—
177#380—21no—
178#380—21no—
179#380C2311yescharacteristic
180#380C2311yescharacteristic
181#380C2312yes—
182#380C2312yes—
183#380C2312yes—
184#380C2311yescharacteristic
185#380C2311yescharacteristic
186#380C2312yes—
187#379—21no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200#376—22no—
201C2×#95—21no—
202C2×#95—22no—
203C2×#95—21no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25—21no—
217C25C2×D411yescharacteristic
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—44no—
224C22×D4—44no—
225C22×D4—84no—
226C22×D4C2412yes—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—44no—
231C23×C4C2×D412yes—
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4#4114yes—
237C23×C4#4114yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×M16—24no—
242C2×M16—44no—
243C22×C8—24no—
244C22×C8—44no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248#121—24no—
249#121—24no—
250#121—24no—
251#121—24no—
252#121—44no—
253#121—44no—
254#117—28no—
255#117—24no—
256#117—24no—
257#117—22no—
258#117—22no—
259#117—22no—
260#117—22no—
261#117—24no—
262#117—24no—
263#117—44no—
264#117—84no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—84no—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—82no—
311C2×C22⋊C4—84no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4—84no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329#107—44no—
330#107—44no—
331#107—44no—
332#106—44no—
333#102—44no—
334#102—44no—
335#102—44no—
336#102—44no—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342#95—22no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—42no—
364C24—42no—
365C24—44no—
366C24—44no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24#12314yes—
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373C24C22×D411yescharacteristic
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—42no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—816no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4#12314yes—
416C22×C4#12314yes—
417C22×C4#12314yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2511yescharacteristic
432M16—84no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—84no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—82no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—24no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—44no—
490C42—82no—
491C42—82no—
492C42—82no—
493C42—82no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—84no—
516C23—88no—
517C23—88no—
518C23—84no—
519C23C2×#12312yes—
520C23C2×#12312yes—
521C23C2×#12312yes—
522C23#57612yes—
523C23#57812yes—
524C23C23×D411yescharacteristic
525D4—44no—
526D4—42no—
527D4—42no—
528D4—84no—
529D4—88no—
530D4—88no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
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—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—84no—
574C2×C4—88no—
575C2×C4—84no—
576C2×C4—164no—
577C2×C4C2×#12312yes—
578C2×C4C2×#12312yes—
579C2×C4C2×#12312yes—
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4#58514yes—
583C8—42no—
584C8—42no—
585C8—44no—
586C22—22no—
587C22—24no—
588C22—24no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—88no—
601C22—82no—
602C22—84no—
603C22—84no—
604C22—82no—
605C22—82no—
606C22—168no—
607C22#309814yes—
608C22C22×#12311yescharacteristic
609C22#344912yes—
610C22C2×#57611yescharacteristic
611C22C2×#57811yescharacteristic
612C22#358512yes—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—84no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4#349112yes—
627C4#349112yes—
628C2—24no—
629C2—42no—
630C2—42no—
631C2—162no—
632C2#3509711yescharacteristic
633C2C2×#309811yescharacteristic
634C2#6051811yescharacteristic
635C2#6097812yes—
636C2#6098112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34180C2semidirect
3. #33103C2semidirect
4. #29917C2semidirect
5. #27874C2semidirect
6. #27868C2semidirect
7. #27617C2semidirect
8. #27614C2semidirect
9. #27588C2semidirect
10. #27582C2semidirect
11. #21944C2semidirect
12. #21940C2semidirect
15. #20685C2semidirect
35. #2317C22semidirect
36. #2314C22semidirect
37. #2314C22semidirect
45. #2274C22semidirect
46. #2101C22semidirect
47. #2100C22semidirect
48. #2093C22semidirect
49. #2092C22semidirect
50. #2085C22semidirect
51. #2083C22semidirect
52. #2073C22semidirect
53. #2071C22semidirect
54. #1926C22semidirect
55. #1925C22semidirect
56. #1925C22semidirect
57. #1925C22semidirect
58. #1921C22semidirect
59. #1921C22semidirect
60. #1919C22semidirect
61. #1918C22semidirect
67. #1854C22semidirect
72. #1852C22semidirect
76. #1818C22semidirect
77. #1818C22semidirect
80. #1809C22semidirect
81. #1808C22semidirect
82. #1808C22semidirect
86. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26242
#341802
#331032
#299172
#278742
#278682
#276172
#276142
#275882
#275822
#219442
#219402
#217642
#217612
#206852