Finite Group Databasebeta

← #6499927

Subgroups — #6499927

Order 512.

3316 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#3195C212yesmaximal, complemented
3C2×#2415C211yesmaximal, characteristic
4#25561C218yesmaximal, complemented
5#24453C218yesmaximal, complemented
6#24063C212yesmaximal, complemented
7#24003C212yesmaximal, complemented
8#23934C212yesmaximal, complemented
9#23921C212yesmaximal, complemented
10#21742C211yesmaximal, characteristic
11#20911C212yesmaximal, 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
26C22×#117C2211yescharacteristic
27#3195—216no—
28C2×#483C2214yes—
29C2×#483C2214yescomplemented
30C2×#470C2218yescomplemented
31C2×#451C2214yes—
32C2×#451C2214yescomplemented
33C2×#451C2214yescomplemented
34C2×#419C2214yescomplemented
35C2×#407C2212yes—
36#2415—24no—
37#2415C2218yes—
38C2×#397—21no—
39C2×#394—21no—
40C2×#391C2211yescharacteristic
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#380C2211yescharacteristic
45#2058C2214yescomplemented
46#2043C2214yescomplemented
47#2035C2218yescomplemented
48#2024C2214yescomplemented
49#2016C2214yescomplemented
50#2001C2218yescomplemented
51#1878C2218yescomplemented
52#1873C2218yescomplemented
53#1872C2214yes—
54#1872C2214yes—
55#1872C2214yescomplemented
56#1870C2218yes—
57#1865C2218yescomplemented
58#1853C2214yescomplemented
59#1817C2212yes—
60#1787C2214yescomplemented
61#1779C2212yes—
62#1736C2214yescomplemented
63C23×Q8C2311yescharacteristic
64C23×Q8C2312yes—
65C23×Q8C2312yes—
66C23×D4D411yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C22×Q16C2318yes—
70C22×M16C2314yes—
71#554—264no—
72#554—216no—
73#554—216no—
74D4×Q8—216no—
75D4×Q8—216no—
76#544—216no—
77#543—216no—
78#543—216no—
79#533—216no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×#125—24no—
84C2×#125—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8D4116yes—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106#518—216no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2314yes—
109C2×C4×D4—24no—
110C2×C4×D4—24no—
111C2×C4×D4—28no—
112C2×C4×D4D414yes—
113C2×C4×D4C2314yes—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—44no—
117C2×#117C2314yes—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#483—216no—
140#483—216no—
141#470—232no—
142#451—216no—
143#451—216no—
144#451—48no—
145C2×#106C2318yes—
146#419—48no—
147C2×#103C2318yes—
148C2×#103C2318yes—
149C2×#103C2318yes—
150C2×#103C2318yes—
151#407—28no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156#397—28no—
157#394—28no—
158#391—22no—
159#391C2314yes—
160#389—28no—
161#389—22no—
162#389C2314yes—
163#388—28no—
164#380—22no—
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×D4—24no—
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—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D412yes—
215C23×C4C2×D418yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C2×Q16—216no—
220C2×Q16—48no—
221C2×Q16—48no—
222C2×M16—28no—
223C2×M16—44no—
224C2×M16—44no—
225C22×C8C2418yes—
226C4⋊Q8—232no—
227C4⋊Q8—28no—
228C4⋊Q8—28no—
229#125—216no—
230#125—232no—
231C22⋊Q8—216no—
232C22⋊Q8—216no—
233C22⋊Q8—216no—
234C22⋊Q8—232no—
235C22⋊Q8—232no—
236C22⋊Q8—232no—
237C22⋊Q8—232no—
238C22⋊Q8—28no—
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—44no—
248C22⋊Q8—48no—
249C22⋊Q8—48no—
250C22⋊Q8C2×D4116yes—
251C22⋊Q8C2×D4116yes—
252C4×Q8—216no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×D4—232no—
256C4×D4—232no—
257C4×D4—216no—
258C4×D4—28no—
259C4×D4—28no—
260#117—216no—
261#117—28no—
262#117—44no—
263#117—48no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319#106—216no—
320#103—216no—
321#103—216no—
322#103—216no—
323#103—48no—
324C22⋊C8—28no—
325C22⋊C8—28no—
326C22⋊C8—44no—
327C22⋊C8—44no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—41no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24#12014yes—
345C24#13714yes—
346C24C22×D412yes—
347C24C22×D412yes—
348C24C22×D411yescharacteristic
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—28no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—28no—
355C2×Q8—24no—
356C2×Q8—28no—
357C2×Q8—24no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—416no—
361C2×Q8—42no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—82no—
367C2×Q8—82no—
368C2×Q8—88no—
369C2×Q8—88no—
370C2×Q8—816no—
371C2×Q8C22×D418yes—
372C2×D4—232no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—416no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
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—88no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4#12014yes—
431C22×C4#12018yes—
432C22×C4#13714yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C2511yescharacteristic
442Q16—816no—
443M16—88no—
444C2×C8—28no—
445C2×C8—28no—
446C2×C8—28no—
447C4⋊C4—24no—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4C22×D418yes—
468C4⋊C4H(5,𝔽2)14yes—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C42—216no—
479C42—24no—
480C42—24no—
481C42—48no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—28no—
499C23—28no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23C2×#12014yes—
513C23C2×#12012yes—
514C23C2×#13714yes—
515C23C2×#13712yes—
516C23C2×#13712yes—
517C23C23×D411yescharacteristic
518Q8—216no—
519Q8—48no—
520Q8—48no—
521Q8—88no—
522Q8—84no—
523Q8—88no—
524Q8—88no—
525D4—232no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—88no—
568C2×C4—88no—
569C2×C4—88no—
570C2×C4—88no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4C2×#12012yes—
575C2×C4C2×#12014yes—
576C2×C4C2×#13714yes—
577C2×C4C23×D412yes—
578C2×C4C2×H(5,𝔽2)12yes—
579C2×C4#58512yes—
580C8—48no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—88no—
602C22—88no—
603C22#300414yes—
604C22C22×#12011yescharacteristic
605C22#348411yescharacteristic
606C22#348511yescharacteristic
607C22C22×#13711yescharacteristic
608C22C22×#13711yescharacteristic
609C22#358611yescharacteristic
610C22#358611yescharacteristic
611C4—22no—
612C4—24no—
613C4—24no—
614C4—42no—
615C4—48no—
616C4—44no—
617C4—42no—
618C4—48no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#349614yes—
623C2—24no—
624C2—44no—
625C2—44no—
626C2#3534711yescharacteristic
627C2C2×#300411yescharacteristic
628C2#6037011yescharacteristic
629C2#6157311yescharacteristic
630C2#6158611yescharacteristic
631C2#6170511yescharacteristic
632C2#6172311yescharacteristic
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×#3195C2semidirect
4. #25561C2semidirect
5. #24453C2semidirect
6. #24063C2semidirect
7. #24003C2semidirect
8. #23934C2semidirect
9. #23921C2semidirect
11. #20911C2semidirect
29. C2×#483C22semidirect
30. C2×#470C22semidirect
32. C2×#451C22semidirect
33. C2×#451C22semidirect
34. C2×#419C22semidirect
45. #2058C22semidirect
46. #2043C22semidirect
47. #2035C22semidirect
48. #2024C22semidirect
49. #2016C22semidirect
50. #2001C22semidirect
51. #1878C22semidirect
52. #1873C22semidirect
55. #1872C22semidirect
57. #1865C22semidirect
58. #1853C22semidirect
60. #1787C22semidirect
62. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#31952
C2×#24152
#255612
#244532
#240632
#240032
#239342
#239212
#217422
#209112