Finite Group Databasebeta

← #6470485

Subgroups — #6470485

Order 512.

2328 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic, complemented
2#34174C211yesmaximal, characteristic
3#32967C212yesmaximal, complemented
4#32964C212yesmaximal, complemented
5#29962C212yesmaximal, complemented
6#29959C212yesmaximal, complemented
7#25977C214yesmaximal, complemented
8#25975C214yesmaximal, complemented
9#25969C214yesmaximal, complemented
10#25967C214yesmaximal, complemented
11#21944C212yesmaximal, complemented
12#21935C211yesmaximal, characteristic, complemented
13#20601C212yesmaximal
14C23×#41C2211yescharacteristic
15C2×#519C2211yescharacteristic
16C2×#519C2211yescharacteristic
17C2×#479C2214yescomplemented
18C2×#477C2212yescomplemented
19C2×#475C2212yescomplemented
20C2×#417C2214yes—
21C2×#417C2214yes—
22C2×#407C2212yes—
23#2841—22no—
24#2659—22no—
25#2646—22no—
26#2629C2212yes—
27#2628C2212yes—
28#2442C2214yescomplemented
29#2440C2214yescomplemented
30#2437—22no—
31#2341—22no—
32#2341—22no—
33#2341C2214yes—
34#2341C2214yes—
35#2309—22no—
36#2309C2211yescharacteristic
37#2309C2211yescharacteristic
38#2280—22no—
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41C2×#380C2212yes—
42#2094C2212yescomplemented
43#2093C2212yescomplemented
44#2092C2214yescomplemented
45#2067C2214yescomplemented
46#2067C2214yescomplemented
47#2065C2214yescomplemented
48#2065C2214yescomplemented
49#2025C2212yescomplemented
50#2025C2212yescomplemented
51#2023C2214yescomplemented
52#1865C2214yescomplemented
53#1865C2214yescomplemented
54#1863C2214yescomplemented
55#1863C2214yescomplemented
56#1854C2214yes—
57#1854C2214yes—
58#1853C2212yescomplemented
59#1853C2212yescomplemented
60#1852C2212yescomplemented
61#1852C2212yescomplemented
62#1851C2214yescomplemented
63#1851C2214yescomplemented
64#1809C2214yescomplemented
65#1809C2214yescomplemented
66#1805C2212yescomplemented
67#1805C2212yescomplemented
68#1803C2214yescomplemented
69#1798C2212yes—
70#1798C2212yes—
71#1788C2214yes—
72#1788C2214yes—
73#1787C2214yes—
74#1786C2214yes—
75C22×#41—22no—
76C22×#41—24no—
77C22×#41—44no—
78C22×#41—44no—
79C22×#41Q818yescomplemented
80C22×#41C2314yes—
81C23×Q8C2311yescharacteristic
82C23×D4—21no—
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C22×M16C2314yes—
87C2×C4⋊Q8C2312yes—
88C2×#125C2312yes—
89C2×C22⋊Q8—24no—
90#519—44no—
91#519—44no—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2314yes—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107#479—48no—
108#477—44no—
109#475—44no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2314yes—
117#417—48no—
118#417—48no—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127#407—28no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131#400—22no—
132#400—24no—
133#399—22no—
134#398—24no—
135#398—24no—
136#398—24no—
137#396—24no—
138#395—22no—
139#395—24no—
140#393—22no—
141#390—21no—
142#390—22no—
143#390—21no—
144#390—24no—
145#390—24no—
146#390—24no—
147#390—24no—
148#390C2312yes—
149#390C2314yes—
150#390C2312yes—
151#389C2314yes—
152#389C2314yes—
153#385—24no—
154#385—24no—
155#384C2312yes—
156#382C2312yes—
157#381—21no—
158#381D412yes—
159#380—22no—
160#380—22no—
161#380—24no—
162#380—24no—
163#380—24no—
164#380D412yes—
165#380D412yes—
166#380C2312yes—
167#380C2312yes—
168#380C2314yes—
169#380C2314yes—
170#379—21no—
171#379—22no—
172#379—22no—
173#379D412yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4—24no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4C2314yes—
178C2×#95—22no—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2312yes—
185#336C2312yes—
186C25—21no—
187C25C2×D411yescharacteristic
188C2×#41—216no—
189C2×#41—28no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—44no—
193C2×#41—44no—
194C2×#41—432no—
195C2×#41—432no—
196C2×#41—48no—
197C2×#41—48no—
198C2×#41C2×Q8116yes—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—21no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8C2×Q814yes—
205C22×Q8C2412yes—
206C22×D4—28no—
207C22×D4—24no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4C2×Q814yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×Q814yes—
228C23×C4#4114yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C2×M16—28no—
232C2×M16—44no—
233C2×M16—44no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238#125—44no—
239C22⋊Q8—88no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—28no—
243C4×Q8—44no—
244C4×Q8—42no—
245C4×Q8—42no—
246C4×Q8—44no—
247C4×Q8—48no—
248C4×Q8—44no—
249#117—42no—
250#117—42no—
251#117—42no—
252#117—42no—
253#117—48no—
254#117—48no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2411yescharacteristic
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2411yescharacteristic
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—84no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×D414yes—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42C2412yes—
296C2×C42C2412yes—
297#107—44no—
298#107—44no—
299#106—44no—
300#106—44no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303C4⋊C8—44no—
304#103—28no—
305#103—28no—
306#103—28no—
307#103—28no—
308#102—44no—
309#102—44no—
310#102—44no—
311#102—44no—
312C22⋊C8—28no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—22no—
326C24—24no—
327C24—42no—
328C24—41no—
329C24—41no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—82no—
334C24—82no—
335C24C22⋊Q814yes—
336C24C22×D411yescharacteristic
337C24C22×D412yes—
338#41—232no—
339#41—416no—
340#41—416no—
341#41—416no—
342#41—464no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—216no—
346C2×Q8—216no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—84no—
352C2×Q8—82no—
353C2×Q8C22×Q814yes—
354C2×Q8H(5,𝔽2)14yes—
355C2×Q8H(5,𝔽2)14yes—
356C2×D4—216no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—432no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—48no—
368C2×D4C22×Q814yes—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—432no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4C22⋊Q814yes—
399C22×C4C22⋊Q818yes—
400C22×C4C22×D412yes—
401C22×C4C22×D414yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q812yes—
406C22×C4C22×Q814yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4114yes—
409C22×C4C2511yescharacteristic
410M16—88no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—82no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—84no—
451C42—24no—
452C42—22no—
453C42—22no—
454C42—44no—
455C42—48no—
456C42—44no—
457C42—42no—
458C42—42no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—44no—
471C23—48no—
472C23—48no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—88no—
478C23—84no—
479C23—82no—
480C23—82no—
481C23—88no—
482C23—84no—
483C23—84no—
484C23C2×C22⋊Q812yes—
485C23C2×C22⋊Q814yes—
486C23C23×D411yescharacteristic
487Q8—216no—
488Q8—24no—
489Q8—24no—
490Q8—24no—
491Q8—164no—
492Q8#54318yes—
493D4—28no—
494D4—432no—
495D4—48no—
496D4—44no—
497C2×C4—216no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—432no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—164no—
533C2×C4—164no—
534C2×C4C2×C22⋊Q814yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4C23×Q811yescharacteristic
537C2×C4C22×#4111yescharacteristic
538C2×C4C2×H(5,𝔽2)12yes—
539C2×C4C2×H(5,𝔽2)12yes—
540C8—44no—
541C8—44no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—48no—
555C22—44no—
556C22—88no—
557C22—816no—
558C22—84no—
559C22—82no—
560C22—82no—
561C22C22×C22⋊Q811yescharacteristic
562C22#348411yescharacteristic
563C22#348411yescharacteristic
564C22#358512yes—
565C22#358612yes—
566C4—28no—
567C4—24no—
568C4—22no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—84no—
574C4—81no—
575C4—84no—
576C4—81no—
577C4C2×#54312yes—
578C4#349012yes—
579C2—44no—
580C2—42no—
581C2—84no—
582C2#3498211yescharacteristic
583C2#6049312yes—
584C2#6165012yes—
585C2#6165111yescharacteristic
586C2#6165211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37477C2semidirect
3. #32967C2semidirect
4. #32964C2semidirect
5. #29962C2semidirect
6. #29959C2semidirect
7. #25977C2semidirect
8. #25975C2semidirect
9. #25969C2semidirect
10. #25967C2semidirect
11. #21944C2semidirect
12. #21935C2semidirect
17. C2×#479C22semidirect
18. C2×#477C22semidirect
19. C2×#475C22semidirect
28. #2442C22semidirect
29. #2440C22semidirect
42. #2094C22semidirect
43. #2093C22semidirect
44. #2092C22semidirect
45. #2067C22semidirect
46. #2067C22semidirect
47. #2065C22semidirect
48. #2065C22semidirect
49. #2025C22semidirect
50. #2025C22semidirect
51. #2023C22semidirect
52. #1865C22semidirect
53. #1865C22semidirect
54. #1863C22semidirect
55. #1863C22semidirect
58. #1853C22semidirect
59. #1853C22semidirect
60. #1852C22semidirect
61. #1852C22semidirect
62. #1851C22semidirect
63. #1851C22semidirect
64. #1809C22semidirect
65. #1809C22semidirect
66. #1805C22semidirect
67. #1805C22semidirect
68. #1803C22semidirect
79. C22×#41Q8semidirect

Maximal subgroups

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

subgroupindex
#374772
#341742
#329672
#329642
#299622
#299592
#259772
#259752
#259692
#259672
#219442
#219352
#206012