Finite Group Databasebeta

← #6470451

Subgroups — #6470451

Order 512.

2638 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2339C211yesmaximal, characteristic
3#32956C212yesmaximal, complemented
4#32953C212yesmaximal, complemented
5#29953C212yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#25978C218yesmaximal, complemented
8#25970C218yesmaximal, complemented
9#21942C212yesmaximal, complemented
10#21934C211yesmaximal, characteristic, complemented
11#20601C212yesmaximal
12C23×#41C2211yescharacteristic
13C2×#543—21no—
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#476C2214yescomplemented
17C2×#474C2214yescomplemented
18C2×#417C2214yes—
19C2×#417C2214yes—
20C2×#407C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2659—21no—
24#2659—21no—
25#2549—24no—
26#2440C2214yescomplemented
27#2440C2214yescomplemented
28#2339—24no—
29#2339C2218yes—
30#2315—21no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#386—21no—
34C2×#386—21no—
35C2×#381—21no—
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C22×#95—21no—
40#2096C2214yescomplemented
41#2095C2214yescomplemented
42#2068C2218yescomplemented
43#2066C2218yescomplemented
44#2024C2214yescomplemented
45#2024C2214yescomplemented
46#1864C2218yescomplemented
47#1862C2218yescomplemented
48#1853C2214yes—
49#1853C2214yescomplemented
50#1852C2214yes—
51#1852C2214yescomplemented
52#1851C2218yescomplemented
53#1809C2218yescomplemented
54#1802C2214yescomplemented
55#1802C2214yescomplemented
56#1797C2212yes—
57#1797C2212yes—
58#1788C2218yes—
59#1787C2214yes—
60#1786C2214yes—
61C22×#41—22no—
62C22×#41—24no—
63C22×#41—44no—
64C22×#41—44no—
65C22×#41Q818yescomplemented
66C22×#41C2314yes—
67C23×Q8C2311yescharacteristic
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C24×C4—21no—
72C24×C4—21no—
73C24×C4C2311yescharacteristic
74C22×M16C2314yes—
75#543—48no—
76C2×C4⋊Q8C2312yes—
77C2×C4⋊Q8C2312yes—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4—44no—
85C2×C4×D4—44no—
86C2×C4×D4C2314yes—
87C2×C4×D4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#476—48no—
101#474—48no—
102C2×#107C2318yes—
103C2×#106C2318yes—
104C2×C4⋊C8C2314yes—
105C2×C4⋊C8C2314yes—
106#417—48no—
107#417—48no—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×#102C2318yes—
111C2×#102C2318yes—
112#407—28no—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115#398—24no—
116#398—24no—
117#395—24no—
118#395—24no—
119#391—22no—
120#391—22no—
121#391—22no—
122#391—22no—
123#391C2314yes—
124#391C2314yes—
125#389—22no—
126#389—22no—
127#389C2314yes—
128#389C2314yes—
129#386—28no—
130#386—28no—
131#385—24no—
132#385—24no—
133#384C2312yes—
134#384C2312yes—
135#381—28no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—22no—
141#380C2314yes—
142#380C2314yes—
143#380C2314yes—
144C4×C22⋊C4D414yes—
145C4×C22⋊C4C2314yes—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—28no—
153C2×#95D411yescharacteristic
154C2×#95D411yescharacteristic
155#340C2318yes—
156#340C2318yes—
157#336C2314yes—
158#336C2314yes—
159C25—21no—
160C25C2×D411yescharacteristic
161C2×#41—216no—
162C2×#41—28no—
163C2×#41—28no—
164C2×#41—28no—
165C2×#41—44no—
166C2×#41—44no—
167C2×#41—432no—
168C2×#41—432no—
169C2×#41—48no—
170C2×#41—48no—
171C2×#41C2×Q8116yes—
172C22×Q8—21no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8C2×Q814yes—
179C22×Q8C2412yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4C2×Q814yes—
189C22×D4C2412yes—
190C23×C4—21no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—82no—
208C23×C4—82no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×Q814yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C2×M16—28no—
220C2×M16—44no—
221C2×M16—44no—
222C22×C8C2418yes—
223C4⋊Q8—44no—
224C4⋊Q8—44no—
225C22⋊Q8—416no—
226C22⋊Q8—416no—
227C22⋊Q8—48no—
228C22⋊Q8—48no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—48no—
234C4×D4—48no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—82no—
266C2×C22⋊C4C2×D414yes—
267C2×C22⋊C4C2×D414yes—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2412yes—
273C2×C42C2412yes—
274#107—48no—
275#106—48no—
276C4⋊C8—44no—
277C4⋊C8—44no—
278#103—48no—
279#103—48no—
280#102—216no—
281#102—216no—
282C22⋊C8—28no—
283C22⋊C8—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95C2×D414yes—
294#95C2×D414yes—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24C22⋊Q814yes—
310C24C22×D412yes—
311C24C22×D411yescharacteristic
312#41—232no—
313#41—416no—
314#41—416no—
315#41—416no—
316#41—464no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—48no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—48no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8C22×Q814yes—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—216no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—432no—
344C2×D4C22×Q814yes—
345C2×D4H(5,𝔽2)14yes—
346C2×D4H(5,𝔽2)14yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—216no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—432no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—816no—
398C22×C4—816no—
399C22×C4C22⋊Q814yes—
400C22×C4C22⋊Q818yes—
401C22×C4AGL(1,ℤ/8)14yes—
402C22×C4#13714yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q814yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4114yes—
413C22×C4C2511yescharacteristic
414M16—88no—
415C2×C8—28no—
416C2×C8—28no—
417C2×C8—28no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—84no—
441C42—24no—
442C42—24no—
443C42—42no—
444C42—42no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—216no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—28no—
460C23—44no—
461C23—44no—
462C23—48no—
463C23—48no—
464C23—48no—
465C23—48no—
466C23—48no—
467C23—42no—
468C23—42no—
469C23—48no—
470C23—42no—
471C23—42no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23C2×C22⋊Q814yes—
477C23C2×C22⋊Q812yes—
478C23C2×AGL(1,ℤ/8)12yes—
479C23C2×#13712yes—
480C23C23×D411yescharacteristic
481Q8—28no—
482Q8—48no—
483Q8—44no—
484Q8—164no—
485Q8—164no—
486D4—216no—
487D4—24no—
488D4—24no—
489D4—24no—
490D4—432no—
491D4#54318yescomplemented
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—432no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—816no—
532C2×C4—816no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—164no—
538C2×C4—164no—
539C2×C4C2×C22⋊Q814yes—
540C2×C4C2×C22⋊Q812yes—
541C2×C4C2×AGL(1,ℤ/8)14yes—
542C2×C4C2×#13714yes—
543C2×C4C23×Q811yescharacteristic
544C2×C4C22×#4111yescharacteristic
545C2×C4C2×H(5,𝔽2)12yes—
546C2×C4C2×H(5,𝔽2)12yes—
547C8—48no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—28no—
558C22—432no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22—44no—
569C22—88no—
570C22#307914yes—
571C22C22×C22⋊Q811yescharacteristic
572C22#348411yescharacteristic
573C22#348411yescharacteristic
574C22C22×AGL(1,ℤ/8)11yescharacteristic
575C22C22×#13711yescharacteristic
576C22#358511yescharacteristic
577C22#358611yescharacteristic
578C4—24no—
579C4—24no—
580C4—24no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—44no—
585C4—82no—
586C4—84no—
587C4—82no—
588C4—84no—
589C4C2×#54312yes—
590C4#349012yes—
591C2—24no—
592C2—44no—
593C2—48no—
594C2#3498211yescharacteristic
595C2C2×#307911yescharacteristic
596C2#6049211yescharacteristic
597C2#6156911yescharacteristic
598C2#6157011yescharacteristic
599C2#6158511yescharacteristic
600C2#6158611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35002C2semidirect
3. #32956C2semidirect
4. #32953C2semidirect
5. #29953C2semidirect
6. #29951C2semidirect
7. #25978C2semidirect
8. #25970C2semidirect
9. #21942C2semidirect
10. #21934C2semidirect
16. C2×#476C22semidirect
17. C2×#474C22semidirect
26. #2440C22semidirect
27. #2440C22semidirect
40. #2096C22semidirect
41. #2095C22semidirect
42. #2068C22semidirect
43. #2066C22semidirect
44. #2024C22semidirect
45. #2024C22semidirect
46. #1864C22semidirect
47. #1862C22semidirect
49. #1853C22semidirect
51. #1852C22semidirect
52. #1851C22semidirect
53. #1809C22semidirect
54. #1802C22semidirect
55. #1802C22semidirect
65. C22×#41Q8semidirect
491. D4#543semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23392
#329562
#329532
#299532
#299512
#259782
#259702
#219422
#219342
#206012