Finite Group Databasebeta

← #6470449

Subgroups — #6470449

Order 512.

2530 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2341C211yesmaximal, characteristic
3#32957C212yesmaximal, complemented
4#32954C212yesmaximal, complemented
5#29952C212yesmaximal, complemented
6#29950C212yesmaximal, complemented
7#25975C218yesmaximal, complemented
8#25967C218yesmaximal, complemented
9#21942C212yesmaximal, complemented
10#21934C211yesmaximal, characteristic, complemented
11#20601C212yesmaximal
12C23×#41C2211yescharacteristic
13C2×#543—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#477C2214yescomplemented
17C2×#475C2214yescomplemented
18C2×#417C2214yes—
19C2×#417C2214yes—
20C2×#407C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2659—21no—
24#2659—21no—
25#2549—24no—
26#2440C2214yescomplemented
27#2440C2214yescomplemented
28#2341—24no—
29#2341C2218yes—
30#2315—21no—
31C2×#390—21no—
32C2×#390—21no—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#380—21no—
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×C4×C22⋊C4—21no—
39C2×C4×C22⋊C4C2211yescharacteristic
40#2094C2214yescomplemented
41#2093C2214yescomplemented
42#2067C2218yescomplemented
43#2065C2218yescomplemented
44#2023C2214yescomplemented
45#2023C2214yescomplemented
46#1865C2218yescomplemented
47#1863C2218yescomplemented
48#1853C2214yes—
49#1853C2214yescomplemented
50#1852C2214yes—
51#1852C2214yescomplemented
52#1851C2218yescomplemented
53#1809C2218yescomplemented
54#1803C2214yescomplemented
55#1803C2214yescomplemented
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×C4C2311yescharacteristic
72C22×M16C2314yes—
73#543—48no—
74C2×C4⋊Q8C2312yes—
75C2×C4⋊Q8C2312yes—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—44no—
83C2×C4×Q8—44no—
84C2×C4×Q8C2314yes—
85C2×C4×Q8C2314yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
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⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#477—48no—
105#475—48no—
106C2×#107C2318yes—
107C2×#106C2318yes—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2314yes—
110#417—48no—
111#417—48no—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#102C2318yes—
115C2×#102C2318yes—
116#407—28no—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2314yes—
119#398—24no—
120#398—24no—
121#395—24no—
122#395—24no—
123#391—22no—
124#391—22no—
125#390—28no—
126#390—28no—
127#390—22no—
128#390—22no—
129#390C2314yes—
130#390C2314yes—
131#389—22no—
132#389—22no—
133#389C2314yes—
134#389C2314yes—
135#385—24no—
136#385—24no—
137#384C2312yes—
138#384C2312yes—
139#380—22no—
140#380—22no—
141#380—22no—
142#380—22no—
143#380—22no—
144#380—28no—
145#380C2314yes—
146#380C2314yes—
147#380C2314yes—
148C4×C22⋊C4—28no—
149C4×C22⋊C4D414yes—
150C4×C22⋊C4C2314yes—
151C2×#95—21no—
152C2×#95—22no—
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—24no—
173C22×Q8—24no—
174C22×Q8—21no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8C2×Q814yes—
179C22×Q8C2412yes—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4—42no—
185C22×D4—42no—
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—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×Q814yes—
208C23×C4#4114yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C2×M16—28no—
212C2×M16—44no—
213C2×M16—44no—
214C22×C8C2418yes—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C22⋊Q8—416no—
218C22⋊Q8—416no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221C4×Q8—28no—
222C4×Q8—28no—
223C4×Q8—44no—
224C4×Q8—44no—
225C4×Q8—48no—
226C4×Q8—48no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4—82no—
266C2×C22⋊C4C2×D414yes—
267C2×C22⋊C4C2×D414yes—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2412yes—
277C2×C42C2412yes—
278#107—48no—
279#106—48no—
280C4⋊C8—44no—
281C4⋊C8—44no—
282#103—216no—
283#103—216no—
284#102—48no—
285#102—48no—
286C22⋊C8—28no—
287C22⋊C8—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95C2×D414yes—
294#95C2×D414yes—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
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—24no—
319C2×Q8—216no—
320C2×Q8—216no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8C22×Q814yes—
332C2×Q8H(5,𝔽2)14yes—
333C2×Q8H(5,𝔽2)14yes—
334C2×D4—216no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—432no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4C22×Q814yes—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—216no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—432no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4C22⋊Q814yes—
392C22×C4C22⋊Q818yes—
393C22×C4AGL(1,ℤ/8)14yes—
394C22×C4#13714yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×Q812yes—
402C22×C4C22×Q814yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4114yes—
405C22×C4C2511yescharacteristic
406M16—88no—
407C2×C8—28no—
408C2×C8—28no—
409C2×C8—28no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—24no—
438C42—24no—
439C42—42no—
440C42—42no—
441C42—48no—
442C42—48no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—28no—
454C23—24no—
455C23—44no—
456C23—44no—
457C23—48no—
458C23—48no—
459C23—42no—
460C23—42no—
461C23—48no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23C2×C22⋊Q814yes—
475C23C2×C22⋊Q812yes—
476C23C2×AGL(1,ℤ/8)12yes—
477C23C2×#13712yes—
478C23C23×D411yescharacteristic
479Q8—216no—
480Q8—24no—
481Q8—24no—
482Q8—24no—
483Q8—164no—
484Q8—164no—
485Q8#54318yescomplemented
486D4—28no—
487D4—432no—
488D4—48no—
489D4—44no—
490C2×C4—216no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—432no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—164no—
534C2×C4—164no—
535C2×C4C2×C22⋊Q814yes—
536C2×C4C2×C22⋊Q812yes—
537C2×C4C2×AGL(1,ℤ/8)14yes—
538C2×C4C2×#13714yes—
539C2×C4C23×Q811yescharacteristic
540C2×C4C22×#4111yescharacteristic
541C2×C4C2×H(5,𝔽2)12yes—
542C2×C4C2×H(5,𝔽2)12yes—
543C8—48no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—88no—
561C22—816no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22#307914yes—
567C22C22×C22⋊Q811yescharacteristic
568C22#348411yescharacteristic
569C22#348411yescharacteristic
570C22C22×AGL(1,ℤ/8)11yescharacteristic
571C22C22×#13711yescharacteristic
572C22#358511yescharacteristic
573C22#358611yescharacteristic
574C4—24no—
575C4—24no—
576C4—28no—
577C4—24no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—84no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4C2×#54312yes—
586C4#349012yes—
587C2—24no—
588C2—44no—
589C2—84no—
590C2#3498211yescharacteristic
591C2C2×#307911yescharacteristic
592C2#6049211yescharacteristic
593C2#6157211yescharacteristic
594C2#6157311yescharacteristic
595C2#6158711yescharacteristic
596C2#6158811yescharacteristic
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. #32957C2semidirect
4. #32954C2semidirect
5. #29952C2semidirect
6. #29950C2semidirect
7. #25975C2semidirect
8. #25967C2semidirect
9. #21942C2semidirect
10. #21934C2semidirect
16. C2×#477C22semidirect
17. C2×#475C22semidirect
26. #2440C22semidirect
27. #2440C22semidirect
40. #2094C22semidirect
41. #2093C22semidirect
42. #2067C22semidirect
43. #2065C22semidirect
44. #2023C22semidirect
45. #2023C22semidirect
46. #1865C22semidirect
47. #1863C22semidirect
49. #1853C22semidirect
51. #1852C22semidirect
52. #1851C22semidirect
53. #1809C22semidirect
54. #1803C22semidirect
55. #1803C22semidirect
65. C22×#41Q8semidirect
485. Q8#543semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23412
#329572
#329542
#299522
#299502
#259752
#259672
#219422
#219342
#206012