Finite Group Databasebeta

← #6570333

Subgroups — #6570333

Order 512.

1515 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36270C211yesmaximal, characteristic
3#30339C212yesmaximal, complemented
4#30305C211yesmaximal, characteristic, complemented
5#29554C212yesmaximal, complemented
6#29513C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic
8#29123C211yesmaximal, characteristic, complemented
9#28877C214yesmaximal
10#28264C212yesmaximal, complemented
11#28186C212yesmaximal, complemented
12#28030C214yesmaximal, complemented
13#23799C211yesmaximal, characteristic
14#23775C212yesmaximal, complemented
15#22805C212yesmaximal, complemented
16#22770C212yesmaximal, complemented
17#20569C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C2×C8×Q8C2212yes—
20C2×C4×SD16C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2862C2212yes—
24#2860C2212yes—
25#2859C2211yescharacteristic
26#2601—22no—
27#2582C2212yes—
28#2571—21no—
29#2570—22no—
30#2568—21no—
31#2562—21no—
32#2558—22no—
33#2538C2211yescharacteristic
34#2538C2211yescharacteristic
35#2536—21no—
36#2511C2214yes—
37#2506C2212yes—
38#2446—22no—
39#2445C2212yes—
40#2365—21no—
41#2355—21no—
42#2355C2212yes—
43#2354C2211yescharacteristic
44#2352—21no—
45D4×C4⋊C4C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51#2100C2212yes—
52#2096C2212yes—
53#2094C2211yescharacteristic
54#2094C2211yescharacteristic
55#2088C2214yes—
56#2084C2214yes—
57#2066C2214yes—
58#2065C2214yes—
59#2058C2212yes—
60#2058C2212yes—
61#2055C2214yes—
62#2043C2212yes—
63#2023C2212yes—
64#2023C2212yes—
65#2014C2212yes—
66#2013C2211yescharacteristic
67#2013C2211yescharacteristic
68#1968C2214yes—
69#1968C2214yes—
70#1957C2212yes—
71#1941C2214yes—
72#1934C2214yes—
73#1933C2214yes—
74#1933C2214yes—
75#1933C2214yes—
76#1927C2214yes—
77#1926C2214yes—
78#1919C2214yes—
79#1877C2214yes—
80#1844C2212yes—
81#1834C2212yes—
82#1834C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87C4×#106C2214yes—
88C8×C4⋊C4C2212yes—
89C8×C4⋊C4C2214yes—
90C4×C4⋊C8C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2214yes—
93#1744C2212yes—
94#1744C2211yescharacteristic
95C22×SD16C2312yes—
96C22×SD16C2312yes—
97#552—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#126—22no—
104C2×#125—22no—
105C2×#125—21no—
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117—22no—
121C22×C4⋊C4—21no—
122C8×Q8—28no—
123C4×SD16—28no—
124C4×SD16—28no—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C4×C8C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#400—22no—
145#400—24no—
146#398—24no—
147#398—22no—
148#396—22no—
149#396—22no—
150#394—22no—
151#394—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2314yes—
158#391C2314yes—
159#390—21no—
160#390—21no—
161#390C2311yescharacteristic
162#390C2311yescharacteristic
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2314yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—22no—
188#386—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#382—22no—
199#382—22no—
200#382—22no—
201#382—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C43C2311yescharacteristic
220C43C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232C22×Q8—22no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2412yes—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2411yescharacteristic
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C2×SD16—24no—
243C2×SD16—24no—
244C2×SD16—24no—
245C2×SD16—42no—
246C2×SD16—42no—
247C2×SD16—42no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—44no—
252C4⋊Q8—42no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255#127—22no—
256#127—41no—
257#126—48no—
258#125—48no—
259#125—44no—
260#123—48no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—84no—
268C4×Q8SD1618yescomplemented
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—42no—
273C4×D4—44no—
274C4×D4—44no—
275#117—44no—
276#117—48no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
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—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#106—24no—
342#106—24no—
343#106—44no—
344#106—44no—
345C4⋊C8—28no—
346C4⋊C8—24no—
347#103—24no—
348#103—24no—
349#103—44no—
350#103—44no—
351#102—24no—
352#102—24no—
353#102—44no—
354#102—44no—
355C4×C8—28no—
356C4×C8—44no—
357C4×C8—42no—
358C4×C8#4118yes—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—21no—
377C24—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8C2×SD1614yes—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D42−1+412yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—82no—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431SD16—48no—
432SD16—48no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8C2×#4114yes—
441C2×C8C2×#4114yes—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C2×SD1614yes—
478C4⋊C4C2×SD1618yes—
479C22⋊C4—44no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—21no—
496C42—22no—
497C42—21no—
498C42—22no—
499C42—41no—
500C42—42no—
501C42—42no—
502C42—41no—
503C42—44no—
504C42—44no—
505C42—84no—
506C42C2×SD1618yes—
507C42C2×SD1614yes—
508C42#13514yes—
509C42C2×#4114yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C23×D411yescharacteristic
525Q8—28no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—88no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—84no—
536D4—84no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—88no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4#53212yes—
582C2×C4#53212yes—
583C2×C4C22×SD1614yes—
584C2×C4C22×SD1612yes—
585C2×C4C22×SD1614yes—
586C2×C4C22×SD1612yes—
587C2×C4C22×SD1612yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4C2×#13512yes—
591C2×C4C22×#4112yes—
592C2×C4C2×2−1+411yescharacteristic
593C2×C4#58511yescharacteristic
594C2×C4#58512yes—
595C8—24no—
596C8—24no—
597C8—44no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×#53211yescharacteristic
613C22#348711yescharacteristic
614C22#348911yescharacteristic
615C22C23×SD1611yescharacteristic
616C22C22×#13511yescharacteristic
617C22#358511yescharacteristic
618C22#358611yescharacteristic
619C4—22no—
620C4—24no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—81no—
634C4—81no—
635C4#316714yes—
636C4#349312yes—
637C4#349512yes—
638C2—82no—
639C2—161no—
640C2#3606511yescharacteristic
641C2C2×#316711yescharacteristic
642C2#6072111yescharacteristic
643C2#6188311yescharacteristic
644C2#6188511yescharacteristic
645C2#6209611yescharacteristic
646C2#6209711yescharacteristic
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. #30339C2semidirect
4. #30305C2semidirect
5. #29554C2semidirect
6. #29513C2semidirect
8. #29123C2semidirect
10. #28264C2semidirect
11. #28186C2semidirect
12. #28030C2semidirect
14. #23775C2semidirect
15. #22805C2semidirect
16. #22770C2semidirect
17. #20569C2semidirect
268. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365982
#362702
#303392
#303052
#295542
#295132
#291392
#291232
#288772
#282642
#281862
#280302
#237992
#237752
#228052
#227702
#205692