Finite Group Databasebeta

← #6472784

Subgroups — #6472784

Order 512.

3068 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3#32985C214yesmaximal, complemented
4#29895C212yesmaximal, complemented
5#29893C212yesmaximal, complemented
6#27602C214yesmaximal, complemented
7#24561C218yesmaximal, complemented
8#24010C214yesmaximal, complemented
9#21942C212yesmaximal, complemented
10#21765C212yesmaximal, complemented
11#20579C211yesmaximal, characteristic
12C25×C4C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×#117C2212yes—
16C2×#485C2218yescomplemented
17C2×#407C2212yes—
18C2×#407C2214yes—
19#2848—21no—
20#2848—21no—
21#2848—22no—
22#2848C2212yes—
23#2848C2212yes—
24#2848C2214yes—
25C2×#385—21no—
26C2×#385—22no—
27#2280—22no—
28#2280—22no—
29#2280C2214yes—
30#2280C2214yes—
31C2×#381—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#379—21no—
38C2×#379—21no—
39C2×C4×C22⋊C4—21no—
40C2×C4×C22⋊C4—21no—
41#2077C2218yescomplemented
42#2076C2218yescomplemented
43#2036C2218yescomplemented
44#2035C2218yescomplemented
45#1894C2218yescomplemented
46#1873C2218yescomplemented
47#1872C2214yescomplemented
48#1872C2214yescomplemented
49#1854C2218yescomplemented
50#1853C2214yescomplemented
51#1853C2214yescomplemented
52#1852C2214yescomplemented
53#1852C2214yescomplemented
54#1825C2218yescomplemented
55#1823C2214yescomplemented
56#1809C2218yescomplemented
57#1787C2214yescomplemented
58#1779C2212yes—
59#1779C2214yes—
60#1779C2212yes—
61C26D411yescharacteristic
62C23×Q8C2311yescharacteristic
63C24×C4—28no—
64C24×C4—48no—
65C24×C4D411yescharacteristic
66C24×C4D411yescharacteristic
67C24×C4D412yes—
68C24×C4D411yescharacteristic
69C24×C4D412yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C22×M16C2314yes—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—48no—
82C2×C22⋊Q8C2314yes—
83C2×C22⋊Q8C2312yes—
84C2×C22⋊Q8C2312yes—
85C2×C22⋊Q8C2314yes—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—48no—
89C2×#117C2314yes—
90C2×#117C2314yes—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—21no—
105C22×C42—22no—
106#485—416no—
107C2×#107C2318yescomplemented
108C2×#106C2318yescomplemented
109C2×#103C2318yescomplemented
110C2×#103C2318yescomplemented
111#407—28no—
112#407—48no—
113C2×C22⋊C8C2318yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2318yes—
117#385—24no—
118#385—22no—
119#385—24no—
120#385—24no—
121#385—24no—
122#385—22no—
123#385—24no—
124#381—24no—
125#381—24no—
126#380—22no—
127#380—22no—
128#380—21no—
129#380—21no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380C2312yes—
135#380C2314yes—
136#380C2314yes—
137#380C2314yes—
138#380C2314yes—
139#380C2314yes—
140#380C2314yes—
141#380C2312yes—
142#379—22no—
143#379—24no—
144#379—22no—
145#379—24no—
146#379—24no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C2×#95—21no—
152C2×#95—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95—21no—
156#340C2318yescomplemented
157#340C2318yescomplemented
158#340C2318yescomplemented
159#340C2318yescomplemented
160C25—24no—
161C25—24no—
162C25—44no—
163C25—42no—
164C25—44no—
165C25C2×D411yescharacteristic
166C25C2×D412yes—
167C25C2×D411yescharacteristic
168C25C2×D412yes—
169C25C2×D411yescharacteristic
170C22×Q8—21no—
171C22×Q8—24no—
172C22×Q8—44no—
173C22×Q8C2412yes—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—464no—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4#4118yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C2×M16—28no—
223C2×M16—44no—
224C2×M16—44no—
225C22×C8C2418yes—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—216no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231C22⋊Q8—48no—
232C22⋊Q8—44no—
233C22⋊Q8—44no—
234#117—216no—
235#117—48no—
236#117—48no—
237#117—48no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2414yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—48no—
281#107—48no—
282#106—48no—
283#103—48no—
284#103—48no—
285C22⋊C8—28no—
286C22⋊C8—28no—
287C22⋊C8—216no—
288C22⋊C8—48no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—24no—
297C24—22no—
298C24—22no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—22no—
303C24—24no—
304C24—22no—
305C24—24no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—22no—
312C24—24no—
313C24—22no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—416no—
325C24—416no—
326C24—432no—
327C24—416no—
328C24—432no—
329C24#12314yes—
330C24#13714yes—
331C24C22×D411yescharacteristic
332C24C22×D412yes—
333C24C22×D412yes—
334C24C22×D411yescharacteristic
335C24C22×D411yescharacteristic
336C2×Q8—24no—
337C2×Q8—28no—
338C2×Q8—44no—
339C2×Q8—42no—
340C2×Q8—88no—
341C22×C4—232no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—416no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—432no—
388C22×C4—432no—
389C22×C4—416no—
390C22×C4—432no—
391C22×C4—432no—
392C22×C4—416no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4#12314yes—
409C22×C4#12314yes—
410C22×C4#12314yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4H(5,𝔽2)18yes—
420C22×C4C2511yescharacteristic
421M16—88no—
422C2×C8—28no—
423C2×C8—28no—
424C2×C8—28no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C42—48no—
452C42—48no—
453C23—22no—
454C23—22no—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—28no—
459C23—216no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—21no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—44no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—48no—
490C23—48no—
491C23—48no—
492C23—44no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—44no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—4128no—
501C23—48no—
502C23—48no—
503C23—44no—
504C23C2×#12312yes—
505C23C2×#12312yes—
506C23C2×#12312yes—
507C23#53414yes—
508C23C2×#13714yes—
509C23C2×#13712yes—
510C23#57514yes—
511C23#57512yes—
512C23C23×D411yescharacteristic
513Q8—48no—
514Q8—84no—
515C2×C4—216no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—216no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—88no—
555C2×C4—88no—
556C2×C4—88no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C2×#12312yes—
560C2×C4C2×#12312yes—
561C2×C4C2×#12312yes—
562C2×C4#53414yes—
563C2×C4C22×#4112yes—
564C2×C4C2×H(5,𝔽2)14yes—
565C8—48no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—24no—
581C22—24no—
582C22—22no—
583C22—416no—
584C22—432no—
585C22—416no—
586C22—416no—
587C22—432no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22#309314yes—
599C22#309514yes—
600C22C22×#12311yescharacteristic
601C22C2×#53412yes—
602C22C22×#13711yescharacteristic
603C22C2×#57511yescharacteristic
604C22C2×#57512yes—
605C4—22no—
606C4—24no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—84no—
611C4—84no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4#349014yes—
616C2—24no—
617C2—24no—
618C2—44no—
619C2—44no—
620C2—42no—
621C2#3499211yescharacteristic
622C2C2×#309311yescharacteristic
623C2C2×#309511yescharacteristic
624C2#6074112yes—
625C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32985C2semidirect
4. #29895C2semidirect
5. #29893C2semidirect
6. #27602C2semidirect
7. #24561C2semidirect
8. #24010C2semidirect
9. #21942C2semidirect
10. #21765C2semidirect
16. C2×#485C22semidirect
41. #2077C22semidirect
42. #2076C22semidirect
43. #2036C22semidirect
44. #2035C22semidirect
45. #1894C22semidirect
46. #1873C22semidirect
47. #1872C22semidirect
48. #1872C22semidirect
49. #1854C22semidirect
50. #1853C22semidirect
51. #1853C22semidirect
52. #1852C22semidirect
53. #1852C22semidirect
54. #1825C22semidirect
55. #1823C22semidirect
56. #1809C22semidirect
57. #1787C22semidirect
107. C2×#107C23semidirect
108. C2×#106C23semidirect
109. C2×#103C23semidirect
110. C2×#103C23semidirect
156. #340C23semidirect
157. #340C23semidirect
158. #340C23semidirect
159. #340C23semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#22802
#329852
#298952
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