Finite Group Databasebeta

← #6524826

Subgroups — #6524826

Order 512.

3158 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3506C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C2×#3234C211yesmaximal, characteristic, complemented
4C2×#2562C211yesmaximal, characteristic, complemented
5#25893C214yesmaximal, complemented
6#25892C214yesmaximal, complemented
7#25711C214yesmaximal, complemented
8#25710C214yesmaximal, complemented
9#24007C214yesmaximal, complemented
10#24004C214yesmaximal, complemented
11#21790C211yesmaximal, characteristic, complemented
12#21395C211yesmaximal, characteristic, complemented
13#21394C211yesmaximal, characteristic, complemented
14#3506—28no—
15#3506—28no—
16C2×#559—22no—
17C2×#556C2212yes—
18C2×#555C2212yes—
19C2×#553—22no—
20C2×#550C2212yes—
21C2×#547C2212yes—
22C2×#538—22no—
23C2×#535—21no—
24C2×#533C2211yescharacteristic
25C2×#532C2211yescharacteristic
26C22×#125C2211yescharacteristic
27C2×#525C2214yes—
28C2×#520—21no—
29C22×C4×D4C2211yescharacteristic
30#3236—28no—
31#3234—28no—
32C2×#501C2212yescomplemented
33C2×#501C2212yescomplemented
34C2×#491C2212yescomplemented
35C2×#490C2212yescomplemented
36C2×#484C2212yes—
37C2×#481C2212yes—
38C2×#483C2212yes—
39C2×#480C2212yes—
40C2×#479C2214yescomplemented
41C2×#478C2214yescomplemented
42C2×#432C2211yescharacteristic
43C2×#432C2211yescharacteristic
44#2562—22no—
45#2562—22no—
46#2562C2214yes—
47#2562C2214yes—
48C2×#400—21no—
49C2×#394—21no—
50C2×#388—22no—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2036C2218yescomplemented
55#2035C2218yescomplemented
56#2034C2218yescomplemented
57#2033C2218yescomplemented
58#1988C2212yescomplemented
59#1987C2212yescomplemented
60#1982C2212yescomplemented
61#1981C2212yescomplemented
62#1955C2214yes—
63#1955C2214yes—
64#1874C2214yes—
65#1872C2214yes—
66#1854C2218yescomplemented
67#1854C2218yescomplemented
68#1788C2218yescomplemented
69#1788C2218yescomplemented
70#1779C2214yes—
71#1763C2214yescomplemented
72#1762C2214yescomplemented
73C22×#41—21no—
74C22×#41C2314yes—
75C23×D4D411yescharacteristic
76C24×C4C2312yes—
77#559—232no—
78#556—28no—
79#556—28no—
80#555—28no—
81#555—28no—
82#553—232no—
83#550—28no—
84#550—28no—
85#547—28no—
86#547—28no—
87#538—232no—
88#535—216no—
89#533—24no—
90#533D418yes—
91#532—24no—
92#532D418yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127—22no—
97C2×#127C2311yescharacteristic
98C2×#126—24no—
99C2×#125—21no—
100C2×#125—21no—
101C2×#125—21no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125C2312yes—
105C2×#125C2312yes—
106C2×#124—22no—
107C2×#124—24no—
108C2×#124—21no—
109#525—216no—
110#525—216no—
111C2×#123—24no—
112C2×#123—24no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2314yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121—21no—
119C2×#121—24no—
120C2×#121C2314yes—
121C2×#121C2312yes—
122C2×#120—24no—
123#520—216no—
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—24no—
128C2×C4×D4—24no—
129C2×C4×D4—22no—
130C2×C4×D4—24no—
131C2×C4×D4—24no—
132C2×C4×D4—21no—
133C2×C4×D4D412yes—
134C2×C4×D4D412yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×#117—21no—
139C2×#117—22no—
140C2×#117C2314yes—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C22⋊C4—22no—
149C22×C22⋊C4—22no—
150C22×C22⋊C4D412yes—
151C22×C42—21no—
152C22×C42C2311yescharacteristic
153#501—28no—
154#501—28no—
155#491—28no—
156#490—28no—
157#484—28no—
158#483—28no—
159#481—28no—
160#480—28no—
161#479—216no—
162#478—216no—
163#432—24no—
164#432—24no—
165C2×#107C2314yes—
166C2×#107C2314yes—
167C2×#106C2314yes—
168C2×#106C2314yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2318yes—
174C2×#102C2314yes—
175C2×#102C2314yes—
176C2×#102C2318yes—
177C2×C22⋊C8C2312yes—
178C2×C22⋊C8C2312yes—
179C2×C22⋊C8C2318yes—
180C2×C22⋊C8C2318yes—
181C2×#97C2312yes—
182C2×#97C2312yes—
183#400—24no—
184#400—24no—
185#394—24no—
186#394—24no—
187#388—28no—
188#388—28no—
189#380—24no—
190#380—24no—
191#380C2318yes—
192#380C2318yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C2×#95—21no—
198C2×#95—22no—
199C2×#95—22no—
200#340C2318yes—
201#340C2318yes—
202C25C2×D412yes—
203C2×#41—28no—
204C2×#41—28no—
205C2×#41—28no—
206C2×#41—48no—
207C2×#41—48no—
208C2×#41—48no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8C2412yes—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—44no—
219C22×D4C2×D412yes—
220C22×D4C2×D412yes—
221C22×D4C2412yes—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D412yes—
233C23×C4C2×D414yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8C2×D414yes—
243C4⋊Q8C2×D414yes—
244C4⋊Q8C2×D414yes—
245#127—216no—
246#127—22no—
247#127C2×D414yes—
248#126—232no—
249#125—28no—
250#125—24no—
251#125—24no—
252#125—24no—
253#125—24no—
254#125—44no—
255#125—44no—
256#124—216no—
257#124—232no—
258#124—28no—
259#123—232no—
260#123—232no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—24no—
264C22⋊Q8—28no—
265C22⋊Q8C2×D418yes—
266#121—216no—
267#121—216no—
268#121—232no—
269#121—28no—
270#121—28no—
271#121—24no—
272#121—28no—
273#121C2×D418yes—
274#120—232no—
275C4×Q8—216no—
276C4×Q8—28no—
277C4×Q8—24no—
278C4×Q8C2×D418yes—
279C4×D4—232no—
280C4×D4—216no—
281C4×D4—232no—
282C4×D4—28no—
283C4×D4—28no—
284C4×D4—28no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4C2×D418yes—
291#117—216no—
292#117—28no—
293#117—28no—
294#117—28no—
295C2×C4⋊C4—216no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—48no—
335C2×C22⋊C4—84no—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—82no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—28no—
350#107—28no—
351#106—28no—
352#106—28no—
353C4⋊C8—24no—
354C4⋊C8—24no—
355#103—28no—
356#103—28no—
357#103—48no—
358#102—28no—
359#102—28no—
360#102—48no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363C22⋊C8—48no—
364C22⋊C8—48no—
365#97—24no—
366#97—24no—
367#95—22no—
368#95—22no—
369#95—28no—
370#95—28no—
371C24—24no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24C22×D414yes—
380C24C22×D411yescharacteristic
381#41—432no—
382#41—816no—
383#41—816no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—44no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8C22×D414yes—
392C2×D4—28no—
393C2×D4—216no—
394C2×D4—216no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—28no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—416no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—84no—
412C2×D4C22×D414yes—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4#12014yes—
449C22×C4#12014yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C2511yescharacteristic
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—28no—
462C2×C8—28no—
463C4⋊C4—216no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4C22×D414yes—
488C4⋊C4C22×D414yes—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—416no—
498C22⋊C4—88no—
499C42—216no—
500C42—28no—
501C42—82no—
502C42—82no—
503C42#12014yes—
504C42#12014yes—
505C42C22×D412yes—
506C42C22×D412yes—
507C42C22×D412yes—
508C42C22×D412yes—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—48no—
526C23—44no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23C2×#12012yes—
532C23#57514yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—48no—
536Q8—44no—
537Q8—84no—
538D4—28no—
539D4—48no—
540D4—48no—
541D4—48no—
542D4—416no—
543D4—44no—
544D4—84no—
545D4—816no—
546D4—816no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—44no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—88no—
580C2×C4—84no—
581C2×C4—816no—
582C2×C4—88no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4C2×#12012yes—
586C2×C4C2×#12012yes—
587C2×C4C2×#12012yes—
588C2×C4C2×#12012yes—
589C2×C4C2×#12012yes—
590C2×C4C23×D411yescharacteristic
591C2×C4C23×D411yescharacteristic
592C2×C4#58514yes—
593C8—44no—
594C8—44no—
595C22—24no—
596C22—24no—
597C22—22no—
598C22—22no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—816no—
610C22—84no—
611C22—84no—
612C22#300514yes—
613C22C22×#12011yescharacteristic
614C22#348512yes—
615C22C2×#57512yes—
616C22#358511yescharacteristic
617C22#358611yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—84no—
628C4—82no—
629C4—82no—
630C4#350012yes—
631C4#350012yes—
632C2—24no—
633C2—84no—
634C2—84no—
635C2#3561611yescharacteristic
636C2C2×#300511yescharacteristic
637C2#6037111yescharacteristic
638C2#6171112yes—
639C2#6171312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3506C2
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×#3236C2semidirect
3. C2×#3234C2semidirect
4. C2×#2562C2semidirect
5. #25893C2semidirect
6. #25892C2semidirect
7. #25711C2semidirect
8. #25710C2semidirect
9. #24007C2semidirect
10. #24004C2semidirect
11. #21790C2semidirect
12. #21395C2semidirect
13. #21394C2semidirect
32. C2×#501C22semidirect
33. C2×#501C22semidirect
34. C2×#491C22semidirect
35. C2×#490C22semidirect
40. C2×#479C22semidirect
41. C2×#478C22semidirect
54. #2036C22semidirect
55. #2035C22semidirect
56. #2034C22semidirect
57. #2033C22semidirect
58. #1988C22semidirect
59. #1987C22semidirect
60. #1982C22semidirect
61. #1981C22semidirect
66. #1854C22semidirect
67. #1854C22semidirect
68. #1788C22semidirect
69. #1788C22semidirect
71. #1763C22semidirect
72. #1762C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35062
C2×#32362
C2×#32342
C2×#25622
#258932
#258922
#257112
#257102
#240072
#240042
#217902
#213952
#213942