Finite Group Databasebeta

← #6660367

Subgroups — #6660367

Order 512.

3086 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C212yesmaximal
2#32710C214yesmaximal, complemented
3#29895C214yesmaximal, complemented
4#29731C214yesmaximal, complemented
5#29597C214yesmaximal, complemented
6#27603C211yesmaximal, characteristic, complemented
7#24552C214yesmaximal, complemented
8#24380C214yesmaximal, complemented
9#24006C212yesmaximal, complemented
10#23921C211yesmaximal, characteristic, complemented
11C2×#1779C211yesmaximal, characteristic
12C25×C4C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2212yes—
16C2×#484C2214yescomplemented
17C2×#467C2214yescomplemented
18C2×#451C2214yescomplemented
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2214yes—
21#2848—22no—
22#2848—22no—
23#2848—22no—
24#2848—22no—
25#2848C2214yes—
26#2848C2214yes—
27#2848C2214yes—
28#2848C2214yes—
29C2×#385—22no—
30C2×#385—22no—
31C2×#385—22no—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#379—22no—
36#2076C2214yescomplemented
37#2076C2214yescomplemented
38#2076C2218yescomplemented
39#2036C2218yescomplemented
40#2036C2218yescomplemented
41#2036C2218yescomplemented
42#2036C2218yescomplemented
43#2001C2214yescomplemented
44#2001C2214yescomplemented
45#2001C2218yescomplemented
46#1892C2218yescomplemented
47#1872C2214yescomplemented
48#1872C2214yescomplemented
49#1852C2214yescomplemented
50#1852C2214yescomplemented
51#1824C2214yescomplemented
52#1808C2214yescomplemented
53#1787C2214yescomplemented
54#1779C2211yescharacteristic
55#1779C2214yes—
56#1779C2211yescharacteristic
57#1779C2212yes—
58C26D411yescharacteristic
59C23×Q8C2312yes—
60C24×C4—48no—
61C24×C4—44no—
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4D412yes—
65C24×C4D412yes—
66C24×C4D411yescharacteristic
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C22×Q16C2314yescomplemented
73C23×C8C2314yes—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—48no—
81C2×C22⋊Q8—48no—
82C2×C22⋊Q8—48no—
83C2×C22⋊Q8C2314yes—
84C2×C22⋊Q8C2314yes—
85C2×C22⋊Q8C2314yes—
86C2×C22⋊Q8C2314yes—
87C2×C22⋊Q8C2314yes—
88C2×C22⋊Q8C2314yes—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4—22no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—22no—
101#484—48no—
102#467—48no—
103#451—48no—
104C2×#107C2318yescomplemented
105C2×#107C2314yescomplemented
106C2×#103C2318yescomplemented
107C2×#103C2318yescomplemented
108C2×#103C2318yescomplemented
109C2×C22⋊C8—48no—
110C2×C22⋊C8—44no—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2318yes—
116C2×C22⋊C8C2312yes—
117#385—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#385—24no—
123#385—24no—
124#385—24no—
125#385—24no—
126#385—24no—
127#385—24no—
128#385—24no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380C2314yes—
136#380C2314yes—
137#380C2314yes—
138#380C2314yes—
139#380C2314yes—
140#380C2314yes—
141#379—24no—
142#379—24no—
143#379—24no—
144#379—24no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—22no—
148#340C2314yescomplemented
149#340C2314yescomplemented
150#340C2318yescomplemented
151#340C2318yescomplemented
152C25—42no—
153C25—44no—
154C25—42no—
155C25—44no—
156C25Q1618yescomplemented
157C25C2×D411yescharacteristic
158C25C2×D412yes—
159C25C2×D412yes—
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C22×Q8—22no—
163C22×Q8—28no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8C2414yes—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—464no—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C2×Q16—48no—
215C2×Q16—44no—
216C22×C8—28no—
217C22×C8—48no—
218C22×C8C2414yes—
219C22×C8C2414yes—
220C22⋊Q8—28no—
221C22⋊Q8—216no—
222C22⋊Q8—28no—
223C22⋊Q8—44no—
224C22⋊Q8—48no—
225C22⋊Q8—48no—
226C22⋊Q8—48no—
227C22⋊Q8—44no—
228C22⋊Q8—48no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231C22⋊Q8—48no—
232C22⋊Q8—48no—
233C22⋊Q8—48no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2414yes—
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—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270#107—44no—
271#107—48no—
272#103—48no—
273#103—48no—
274#103—48no—
275C22⋊C8—28no—
276C22⋊C8—28no—
277C22⋊C8—48no—
278C22⋊C8—42no—
279C22⋊C8—44no—
280C22⋊C8—42no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—22no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—41no—
303C24—42no—
304C24—416no—
305C24—432no—
306C24—41no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—41no—
313C24—416no—
314C24—416no—
315C24—432no—
316C24C2×Q1614yes—
317C24C2×Q1618yes—
318C24C2×Q1618yes—
319C24C2×Q1614yes—
320C24C2×Q1614yes—
321C24C22×D411yescharacteristic
322C24C22×D412yes—
323C24C22×D411yescharacteristic
324C24C22×D411yescharacteristic
325C24C22×D412yes—
326C2×Q8—24no—
327C2×Q8—28no—
328C2×Q8—28no—
329C2×Q8—24no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—816no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
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—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—432no—
383C22×C4—416no—
384C22×C4—416no—
385C22×C4—432no—
386C22×C4—416no—
387C22×C4—432no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—432no—
391C22×C4—416no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4H(5,𝔽2)18yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C4C2511yescharacteristic
412Q16—88no—
413C2×C8—28no—
414C2×C8—28no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—88no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—216no—
450C23—216no—
451C23—216no—
452C23—216no—
453C23—28no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—44no—
469C23—48no—
470C23—48no—
471C23—48no—
472C23—48no—
473C23—44no—
474C23—48no—
475C23—44no—
476C23—48no—
477C23—44no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—44no—
484C23—48no—
485C23—44no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—44no—
490C23—4128no—
491C23—44no—
492C23—48no—
493C23#53412yes—
494C23#53414yes—
495C23C22×Q1612yes—
496C23C22×Q1614yes—
497C23C22×Q1612yes—
498C23C22×Q1612yes—
499C23C22×Q1614yes—
500C23#57512yes—
501C23#57514yes—
502C23C23×D411yescharacteristic
503Q8—48no—
504Q8—48no—
505Q8—88no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—216no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—88no—
547C2×C4—88no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4#53412yes—
551C2×C4#53414yes—
552C2×C4C2×H(5,𝔽2)12yes—
553C2×C4C2×H(5,𝔽2)14yes—
554C8—44no—
555C8—44no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—24no—
572C22—22no—
573C22—416no—
574C22—432no—
575C22—432no—
576C22—416no—
577C22—416no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22#318914yes—
589C22#318918yes—
590C22C2×#53411yescharacteristic
591C22C2×#53412yes—
592C22C23×Q1611yescharacteristic
593C22C2×#57512yes—
594C22C2×#57511yescharacteristic
595C4—22no—
596C4—24no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—84no—
601C4—84no—
602C4—84no—
603C4—84no—
604C4#353214yes—
605C2—28no—
606C2—24no—
607C2—44no—
608C2—42no—
609C2—42no—
610C2#3743611yescharacteristic
611C2C2×#318911yescharacteristic
612C2C2×#318912yes—
613C2#6074111yescharacteristic
614C2#6075212yes—
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
2. #32710C2semidirect
3. #29895C2semidirect
4. #29731C2semidirect
5. #29597C2semidirect
6. #27603C2semidirect
7. #24552C2semidirect
8. #24380C2semidirect
9. #24006C2semidirect
10. #23921C2semidirect
16. C2×#484C22semidirect
17. C2×#467C22semidirect
18. C2×#451C22semidirect
36. #2076C22semidirect
37. #2076C22semidirect
38. #2076C22semidirect
39. #2036C22semidirect
40. #2036C22semidirect
41. #2036C22semidirect
42. #2036C22semidirect
43. #2001C22semidirect
44. #2001C22semidirect
45. #2001C22semidirect
46. #1892C22semidirect
47. #1872C22semidirect
48. #1872C22semidirect
49. #1852C22semidirect
50. #1852C22semidirect
51. #1824C22semidirect
52. #1808C22semidirect
53. #1787C22semidirect
72. C22×Q16C23semidirect
104. C2×#107C23semidirect
105. C2×#107C23semidirect
106. C2×#103C23semidirect
107. C2×#103C23semidirect
108. C2×#103C23semidirect
148. #340C23semidirect
149. #340C23semidirect
150. #340C23semidirect
151. #340C23semidirect
156. C25Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
#327102
#298952
#297312
#295972
#276032
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