Finite Group Databasebeta

← #6482126

Subgroups — #6482126

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic
3C2×#2369C211yesmaximal, characteristic, complemented
4#33110C212yesmaximal, complemented
5#33109C211yesmaximal, characteristic, complemented
6#33107C211yesmaximal, characteristic
7#29761C212yesmaximal, complemented
8#29757C211yesmaximal, characteristic, complemented
9#29755C211yesmaximal, characteristic
10#25464C214yesmaximal, complemented
11#25462C214yesmaximal, complemented
12#25287C214yesmaximal, complemented
13#25285C214yesmaximal, complemented
14#21955C211yesmaximal, characteristic, complemented
15#21391C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529C2212yes—
18#3239—216no—
19C2×#501C2214yes—
20C2×#489C2214yes—
21C2×#484C2212yes—
22C2×#484C2212yes—
23C2×#481C2212yes—
24C2×#481C2212yes—
25C2×#475C2214yes—
26C2×#474C2214yes—
27C2×#432C2212yes—
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2850C2212yes—
31#2618—21no—
32#2618—21no—
33#2601—22no—
34#2601—22no—
35#2566C2212yes—
36#2565C2212yes—
37#2545C2212yes—
38#2544C2212yes—
39#2369—22no—
40#2369—22no—
41#2369C2214yes—
42#2369C2214yes—
43#2325—21no—
44#2325—21no—
45C2×#391—21no—
46C2×#391—21no—
47C2×#389—21no—
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2085C2214yes—
56#2084C2212yes—
57#2083C2212yes—
58#2068C2214yes—
59#2067C2214yes—
60#2066C2214yes—
61#2065C2214yes—
62#2043C2212yes—
63#2042C2214yes—
64#2041C2212yes—
65#1989C2214yes—
66#1983C2214yes—
67#1983C2214yes—
68#1982C2212yes—
69#1981C2212yes—
70#1949C2214yes—
71#1948C2212yes—
72#1947C2212yes—
73#1940C2214yes—
74#1940C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1926C2214yes—
78#1925C2214yes—
79#1874C2212yes—
80#1874C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1763C2214yes—
87#1762C2214yes—
88C23×Q8—21no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8D418yescomplemented
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2314yes—
94C2×#127C2311yescharacteristic
95#529—28no—
96#529—28no—
97C2×#126—22no—
98C2×#125C2312yes—
99C2×#124C2312yes—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×C22⋊Q8C2312yes—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#501—216no—
126#489—216no—
127#484—28no—
128#484—28no—
129#481—28no—
130#481—28no—
131#475—216no—
132#474—216no—
133#432—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×#97C2314yes—
153#398—22no—
154#398—22no—
155#396—22no—
156#396—22no—
157#396—22no—
158#396—22no—
159#394—22no—
160#394—22no—
161#391—24no—
162#391—24no—
163#391—24no—
164#391—24no—
165#391C2312yes—
166#391C2312yes—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#389—24no—
172#389—24no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382C2312yes—
196#380—21no—
197#380—21no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4—24no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95D411yescharacteristic
218C2×#95D411yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2314yes—
226C22×Q8—28no—
227C22×Q8—24no—
228C22×Q8—82no—
229C22×Q8—82no—
230C22×Q8C2412yes—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—216no—
250C4⋊Q8—28no—
251C4⋊Q8—42no—
252C4⋊Q8C2×D4116yes—
253#127—42no—
254#126—48no—
255#125—24no—
256#125—24no—
257#124—24no—
258#124—24no—
259#124—44no—
260#123—48no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263#121—24no—
264#121—24no—
265#121—44no—
266#121—44no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—48no—
272#117—48no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42—82no—
332C2×C42—82no—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D414yes—
338C2×C42#4114yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343#107—28no—
344#107—28no—
345#106—28no—
346#106—28no—
347C4⋊C8—28no—
348C4⋊C8—44no—
349C4⋊C8—42no—
350C4⋊C8—42no—
351#103—28no—
352#103—28no—
353#103—28no—
354#103—44no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—44no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361#97—28no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24C22×D411yescharacteristic
381C2×Q8—216no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8—816no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—81no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4—81no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4#13714yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2511yescharacteristic
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C4⋊C4—24no—
463C4⋊C4—216no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—84no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4C22×D414yes—
493C4⋊C4C22×D414yes—
494C4⋊C42−1+414yes—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—41no—
510C42—41no—
511C42—42no—
512C42—48no—
513C42—44no—
514C42—82no—
515C42—82no—
516C42—84no—
517C42#12118yes—
518C42C22×D414yes—
519C42C2×#4114yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×#12112yes—
532C23C2×#13712yes—
533C23#57512yes—
534C23C23×D411yescharacteristic
535Q8—44no—
536Q8—44no—
537Q8—816no—
538D4—44no—
539D4—44no—
540D4—88no—
541D4—162no—
542D4—162no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—88no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—84no—
599C2×C4C2×#12114yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12114yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C23×D411yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×2−1+412yes—
607C2×C4#58512yes—
608C8—44no—
609C8—44no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22#303714yes—
624C22C22×#12111yescharacteristic
625C22#348511yescharacteristic
626C22#348611yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57511yescharacteristic
629C22#358511yescharacteristic
630C22#358511yescharacteristic
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—24no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—44no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#346112yes—
648C4#349712yes—
649C2—24no—
650C2—162no—
651C2#3515311yescharacteristic
652C2C2×#303711yescharacteristic
653C2#6042311yescharacteristic
654C2#6170911yescharacteristic
655C2#6171011yescharacteristic
656C2#6183011yescharacteristic
657C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3239C2semidirect
3. C2×#2369C2semidirect
4. #33110C2semidirect
5. #33109C2semidirect
7. #29761C2semidirect
8. #29757C2semidirect
10. #25464C2semidirect
11. #25462C2semidirect
12. #25287C2semidirect
13. #25285C2semidirect
14. #21955C2semidirect
15. #21391C2semidirect
91. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
#351032
C2×#23692
#331102
#331092
#331072
#297612
#297572
#297552
#254642
#254622
#252872
#252852
#219552
#213912