Finite Group Databasebeta

← #6499901

Subgroups — #6499901

Order 512.

3414 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#3189C212yesmaximal, complemented
3C2×#2417C211yesmaximal, characteristic
4#25545C218yesmaximal, complemented
5#24437C218yesmaximal, complemented
6#24064C212yesmaximal, complemented
7#24006C212yesmaximal, complemented
8#23934C212yesmaximal, complemented
9#23921C212yesmaximal, complemented
10#21738C211yesmaximal, characteristic
11C2×#1872C212yesmaximal, complemented
12#3501—216no—
13C2×#554—24no—
14C2×#554C2214yes—
15C2×D4×Q8C2214yes—
16C2×#544—21no—
17C2×#543—21no—
18C2×#543—21no—
19C2×#533C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2211yescharacteristic
24C2×#518—21no—
25C22×C4×D4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27#3189—216no—
28C2×#484C2214yes—
29C2×#484C2214yescomplemented
30C2×#467C2218yescomplemented
31C2×#451C2214yes—
32C2×#451C2214yescomplemented
33C2×#451C2214yescomplemented
34C22×#103C2214yescomplemented
35C22×C22⋊C8C2212yes—
36#2417—24no—
37#2417C2218yes—
38C2×#398—21no—
39C2×#396—21no—
40C2×#391C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#386—21no—
43C2×#380—21no—
44C2×#380C2211yescharacteristic
45#2057C2214yescomplemented
46#2043C2214yescomplemented
47#2036C2218yescomplemented
48#2024C2214yescomplemented
49#2016C2214yescomplemented
50#2001C2218yescomplemented
51#1878C2218yescomplemented
52#1872C2214yes—
53#1872C2214yes—
54#1872C2214yescomplemented
55#1872C2218yescomplemented
56#1869C2218yes—
57#1863C2218yescomplemented
58#1852C2214yescomplemented
59#1817C2212yes—
60#1787C2214yescomplemented
61#1779C2212yes—
62C2×#340C2214yescomplemented
63C23×Q8C2311yescharacteristic
64C23×Q8C2312yes—
65C23×Q8C2312yes—
66C23×D4D411yescharacteristic
67C24×C4—21no—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C22×Q16C2318yes—
71C23×C8C2314yes—
72#554—264no—
73#554—216no—
74#554—216no—
75D4×Q8—216no—
76D4×Q8—216no—
77#544—216no—
78#543—216no—
79#543—216no—
80#533—216no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×#125—24no—
85C2×#125—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—44no—
101C2×C22⋊Q8D4116yes—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2314yes—
107#518—216no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—24no—
111C2×C4×D4—24no—
112C2×C4×D4—28no—
113C2×C4×D4D414yes—
114C2×C4×D4C2314yes—
115C2×#117—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—44no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4D411yescharacteristic
137C22×C22⋊C4D411yescharacteristic
138C22×C42C2311yescharacteristic
139#484—216no—
140#484—216no—
141#467—232no—
142#451—216no—
143#451—216no—
144#451—48no—
145C2×#107C2318yes—
146C2×#103—48no—
147C2×#103C2318yes—
148C2×#103C2318yes—
149C2×#103C2318yes—
150C2×#103C2318yes—
151C2×C22⋊C8—28no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156#398—28no—
157#396—28no—
158#391—22no—
159#391C2314yes—
160#389—22no—
161#389C2314yes—
162#386—28no—
163#380—22no—
164#380—28no—
165#380C2314yes—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170#340C2318yes—
171#340C2318yes—
172#336C2314yes—
173#336C2314yes—
174C25C2×D411yescharacteristic
175C25C2×D411yescharacteristic
176C22×Q8—28no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—21no—
182C22×Q8—24no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2×D418yes—
190C22×Q8C2412yes—
191C22×Q8C2414yes—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4Q1618yescomplemented
195C22×D4C2×D414yes—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—24no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—82no—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D418yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C2×Q16—216no—
224C2×Q16—48no—
225C2×Q16—48no—
226C22×C8—28no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8C2418yes—
230C4⋊Q8—232no—
231C4⋊Q8—28no—
232C4⋊Q8—28no—
233#125—216no—
234#125—232no—
235C22⋊Q8—216no—
236C22⋊Q8—216no—
237C22⋊Q8—216no—
238C22⋊Q8—232no—
239C22⋊Q8—232no—
240C22⋊Q8—232no—
241C22⋊Q8—232no—
242C22⋊Q8—28no—
243C22⋊Q8—216no—
244C22⋊Q8—216no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C22⋊Q8—44no—
252C22⋊Q8—48no—
253C22⋊Q8—48no—
254C22⋊Q8C2×D4116yes—
255C22⋊Q8C2×D4116yes—
256C4×Q8—216no—
257C4×Q8—28no—
258C4×Q8—28no—
259C4×D4—232no—
260C4×D4—232no—
261C4×D4—216no—
262C4×D4—28no—
263C4×D4—28no—
264#117—216no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—216no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D414yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321#107—216no—
322#103—216no—
323#103—216no—
324#103—216no—
325#103—48no—
326C22⋊C8—28no—
327C22⋊C8—28no—
328C22⋊C8—44no—
329C22⋊C8—44no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334C24—22no—
335C24—22no—
336C24—41no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24#12014yes—
345C24C2×Q1614yes—
346C24C2×Q1614yes—
347C24#13714yes—
348C24C22×D412yes—
349C24C22×D412yes—
350C24C22×D411yescharacteristic
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—28no—
357C2×Q8—24no—
358C2×Q8—28no—
359C2×Q8—24no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—416no—
363C2×Q8—42no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—82no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×Q8—88no—
371C2×Q8—88no—
372C2×Q8—816no—
373C2×Q8C22×D418yes—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—232no—
377C2×D4C2×Q16116yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—416no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—88no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—84no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—816no—
436C22×C4#12014yes—
437C22×C4#12018yes—
438C22×C4C2×Q1614yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C2511yescharacteristic
448Q16—816no—
449C2×C8—28no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—88no—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—84no—
474C4⋊C4C22×D418yes—
475C4⋊C4H(5,𝔽2)14yes—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C42—216no—
485C42—24no—
486C42—24no—
487C23—21no—
488C23—22no—
489C23—21no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23#451116yescomplemented
515C23C2×#12012yes—
516C23C2×#12014yes—
517C23C22×Q1614yes—
518C23C22×Q1612yes—
519C23C22×Q1614yes—
520C23C2×#13712yes—
521C23C2×#13714yes—
522C23C23×D411yescharacteristic
523Q8—216no—
524Q8—48no—
525Q8—48no—
526Q8—88no—
527Q8—84no—
528Q8—88no—
529Q8—88no—
530D4—232no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—84no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—816no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4C2×#12012yes—
581C2×C4C2×#12014yes—
582C2×C4C22×Q1614yes—
583C2×C4C23×D412yes—
584C2×C4C2×H(5,𝔽2)12yes—
585C2×C4#58512yes—
586C8—48no—
587C22—22no—
588C22—24no—
589C22—22no—
590C22—24no—
591C22—22no—
592C22—24no—
593C22—22no—
594C22—24no—
595C22—22no—
596C22—24no—
597C22—22no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—416no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—88no—
608C22C2×#45114yes—
609C22C2×#45118yes—
610C22C22×#12011yescharacteristic
611C22#348411yescharacteristic
612C22#348511yescharacteristic
613C22C23×Q1611yescharacteristic
614C22C22×#13711yescharacteristic
615C22#358511yescharacteristic
616C22#358611yescharacteristic
617C4—22no—
618C4—24no—
619C4—24no—
620C4—42no—
621C4—48no—
622C4—44no—
623C4—42no—
624C4—48no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4#349614yes—
629C2—24no—
630C2—28no—
631C2—44no—
632C2#3534711yescharacteristic
633C2C22×#45111yescharacteristic
634C2#6037111yescharacteristic
635C2#6156311yescharacteristic
636C2#6157011yescharacteristic
637C2#6169811yescharacteristic
638C2#6170511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3189C2semidirect
4. #25545C2semidirect
5. #24437C2semidirect
6. #24064C2semidirect
7. #24006C2semidirect
8. #23934C2semidirect
9. #23921C2semidirect
11. C2×#1872C2semidirect
29. C2×#484C22semidirect
30. C2×#467C22semidirect
32. C2×#451C22semidirect
33. C2×#451C22semidirect
34. C22×#103C22semidirect
45. #2057C22semidirect
46. #2043C22semidirect
47. #2036C22semidirect
48. #2024C22semidirect
49. #2016C22semidirect
50. #2001C22semidirect
51. #1878C22semidirect
54. #1872C22semidirect
55. #1872C22semidirect
57. #1863C22semidirect
58. #1852C22semidirect
60. #1787C22semidirect
62. C2×#340C22semidirect
194. C22×D4Q16semidirect
514. C23#451semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#31892
C2×#24172
#255452
#244372
#240642
#240062
#239342
#239212
#217382
C2×#18722