Finite Group Databasebeta

← #6803182

Subgroups — #6803182

Order 512.

1457 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34954C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32532C214yesmaximal, complemented
5#31908C214yesmaximal, complemented
6#29829C212yesmaximal, complemented
7#28508C211yesmaximal, characteristic
8#28372C212yesmaximal, complemented
9#28281C212yesmaximal, complemented
10#28265C212yesmaximal, complemented
11#28252C211yesmaximal, characteristic
12#26093C212yesmaximal, complemented
13#25295C212yesmaximal
14#24633C212yesmaximal
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22110C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#475C2212yes—
19C2×#474C2212yes—
20C2×#474C2212yes—
21C2×#462C2211yescharacteristic
22C2×#460C2212yes—
23C2×#461C2212yes—
24C2×C8×Q8C2212yes—
25C2×C4×Q16C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2593—22no—
29#2582C2211yescharacteristic
30#2582C2212yes—
31#2571—21no—
32#2571C2211yescharacteristic
33#2570—22no—
34#2568—21no—
35#2548C2211yescharacteristic
36#2548C2211yescharacteristic
37#2513—21no—
38#2513C2212yes—
39#2511C2214yes—
40#2448—21no—
41#2448C2212yes—
42#2446—22no—
43#2395—22no—
44#2365—21no—
45#2355—21no—
46#2355C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#124—21no—
51C4×#124C2212yes—
52#2293—21no—
53#2293C2212yes—
54C42×Q8C2211yescharacteristic
55C42×Q8C2211yescharacteristic
56#2097C2214yes—
57#2093C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2068C2214yes—
61#2065C2214yes—
62#2055C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2015C2214yes—
66#1988C2212yes—
67#1983C2214yes—
68#1981C2211yescharacteristic
69#1969C2214yes—
70#1968C2214yes—
71#1957C2212yes—
72#1949C2214yes—
73#1948C2212yes—
74#1948C2212yes—
75#1947C2211yescharacteristic
76#1942C2214yes—
77#1939C2214yes—
78#1939C2212yes—
79#1934C2214yes—
80#1928C2212yes—
81#1925C2214yes—
82#1918C2214yes—
83#1878C2212yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1842C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1803C2212yes—
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×SD16C2314yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#124—21no—
103C2×#124C2312yes—
104C2×C4×Q8—22no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116#475—44no—
117#474—28no—
118#474—44no—
119#462—24no—
120#461—44no—
121#460—44no—
122C8×Q8—44no—
123C4×Q16—44no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2314yes—
146#400—24no—
147#400—24no—
148#400—22no—
149#399—22no—
150#398—24no—
151#398—22no—
152#397—22no—
153#397—22no—
154#397—22no—
155#394—22no—
156#394—22no—
157#391—22no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2314yes—
164#391C2314yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—24no—
193#387—24no—
194#387—24no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#382—21no—
203#382—22no—
204#382—22no—
205#382C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C43—21no—
227C43C2311yescharacteristic
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C2×Q16—24no—
245C2×Q16—24no—
246C2×Q16—42no—
247C2×SD16—44no—
248C2×SD16—44no—
249C2×SD16—44no—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259#127—22no—
260#127—22no—
261#124—24no—
262#124—42no—
263#124—44no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—41no—
276C4×Q8—84no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—44no—
280C4×D4—82no—
281#117—84no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2414yes—
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—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—24no—
330C2×C42—21no—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—28no—
345#107—42no—
346#107—44no—
347#106—44no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C4⋊C8—22no—
351C4⋊C8—42no—
352C4⋊C8—44no—
353C4⋊C8—44no—
354C4⋊C8#4114yes—
355#103—24no—
356#103—42no—
357#103—42no—
358#103—42no—
359#103—44no—
360#103—44no—
361#102—28no—
362#102—44no—
363#102—44no—
364C4×C8—24no—
365C4×C8—44no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8#13714yes—
404C2×Q82−1+412yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—81no—
416C2×D4—81no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439Q16—48no—
440SD16—88no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8C2×#4114yes—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—84no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—21no—
505C42—22no—
506C42—21no—
507C42—22no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—44no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—81no—
517C42—84no—
518C42#13514yes—
519C42C2×#4112yes—
520C42H(5,𝔽2)12yes—
521C42H(5,𝔽2)12yes—
522C422−1+412yes—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C23×D411yescharacteristic
531Q8—28no—
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—88no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—84no—
546D4—84no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—88no—
592C2×C4#53712yes—
593C2×C4#53712yes—
594C2×C4#55012yes—
595C2×C4#55012yes—
596C2×C4C2×#13512yes—
597C2×C4C2×#13512yes—
598C2×C4C2×#13512yes—
599C2×C4C2×#13714yes—
600C2×C4C2×#13712yes—
601C2×C4#57612yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58511yescharacteristic
608C8—24no—
609C8—42no—
610C8—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#53711yescharacteristic
619C22C2×#55011yescharacteristic
620C22#348911yescharacteristic
621C22C22×#13511yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57611yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—24no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—81no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—84no—
639C4—81no—
640C4—82no—
641C4#324814yes—
642C4#339614yes—
643C4#351811yescharacteristic
644C4#352011yescharacteristic
645C4#354611yescharacteristic
646C4#355211yescharacteristic
647C2—162no—
648C2#3913611yescharacteristic
649C2C2×#324811yescharacteristic
650C2#6092811yescharacteristic
651C2C2×#339611yescharacteristic
652C2#6140411yescharacteristic
653C2#6210111yescharacteristic
654C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #33196C2semidirect
4. #32532C2semidirect
5. #31908C2semidirect
6. #29829C2semidirect
8. #28372C2semidirect
9. #28281C2semidirect
10. #28265C2semidirect
12. #26093C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22110C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#349542
#331962
#325322
#319082
#298292
#285082
#283722
#282812
#282652
#282522
#260932
#252952
#246332
Q8×C4⋊C82
#221102
C4×#4622