Finite Group Databasebeta

← #356103

Subgroups — #356103

Order 512.

2261 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2901C211yesmaximal, characteristic
2#60070C211yesmaximal, characteristic
3#20623C211yesmaximal, characteristic
4#20405C212yesmaximal, complemented
5#20401C211yesmaximal, characteristic, complemented
6C4×#410C211yesmaximal, characteristic, complemented
7#10951C212yesmaximal, complemented
8#10945C212yesmaximal, complemented
9#10783C212yesmaximal, complemented
10#10766C211yesmaximal, characteristic, complemented
11#10758C211yesmaximal, characteristic, complemented
12C2×#585C2211yescharacteristic
13C2×#568C2211yescharacteristic
14#3428C2212yes—
15C2×C4×#41C2211yescharacteristic
16#2901—24no—
17#2901—24no—
18C2×#421C414yescomplemented
19C2×#421C2212yes—
20C2×#421C2211yescharacteristic
21C2×#421C2211yescharacteristic
22C22×#104C414yescomplemented
23C22×#104C2211yescharacteristic
24C22×#104C2211yescharacteristic
25#2896C2212yes—
26C2×#413C2211yescharacteristic
27#2884C414yescomplemented
28#2883—24no—
29#2883C414yescomplemented
30C2×#410—24no—
31C2×#410C2211yescharacteristic
32C2×#410C2211yescharacteristic
33C2×#410C2212yes—
34C2×#410C2211yescharacteristic
35#2873C2212yescomplemented
36C4×#117C2212yescomplemented
37#1796C414yescomplemented
38#1795C414yescomplemented
39#1759—24no—
40C4×#104—24no—
41#1757—24no—
42C4×#99—24no—
43C2×#344C2212yescomplemented
44#1740C2212yescomplemented
45#1733C2212yescomplemented
46#1733C2212yescomplemented
47C2×#339C2212yescomplemented
48C2×H(3,ℤ/4)C2212yescomplemented
49#585—24no—
50#585—24no—
51#585—42no—
52#585—42no—
53C2×2−1+4—21no—
54C2×2−1+4C2×C412yes—
55C2×2−1+4C2×C412yes—
56C2×H(5,𝔽2)—22no—
57C2×H(5,𝔽2)—21no—
58C2×H(5,𝔽2)C2×C412yes—
59C2×H(5,𝔽2)C2×C412yes—
60C22×#41—22no—
61C22×#41—22no—
62C22×#41C2×C414yes—
63C22×#41C2311yescharacteristic
64C22×#41C2311yescharacteristic
65C22×#41C2311yescharacteristic
66#568—24no—
67#568—42no—
68C2×#131—22no—
69C2×#131—22no—
70C22×M16C2×C414yescomplemented
71C22×M16C2312yes—
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74#520—24no—
75#520—42no—
76#519C2×C414yescomplemented
77#518—24no—
78#518C2×C414yescomplemented
79C4×#41—22no—
80C4×#41—22no—
81C4×#41—24no—
82C4×#41—42no—
83C4×#41—42no—
84C2×C4×Q8—21no—
85C2×C4×D4—22no—
86C2×C4×D4—21no—
87C2×#117—24no—
88C2×#117C2×C412yescomplemented
89C2×#117C2312yes—
90C2×#117C2312yes—
91C2×#117C2312yes—
92C2×#117C2311yescharacteristic
93C22×C42C2×C412yescomplemented
94C22×C42C2311yescharacteristic
95#421—24no—
96#421—24no—
97#421—24no—
98#421—216no—
99#421C2×C418yescomplemented
100C2×#104—24no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—28no—
104C2×#104—28no—
105C2×#104C2×C414yes—
106C2×#104C2×C412yes—
107C2×#104C2×C412yes—
108C2×#104C2×C412yescomplemented
109C2×#104C2×C414yescomplemented
110C2×#104C2×C412yescomplemented
111C2×#104D414yes—
112C2×#104D414yes—
113#419C2×C414yescomplemented
114#418C2×C414yescomplemented
115#417—42no—
116#416—42no—
117#413—22no—
118#413C2×C414yescomplemented
119C2×#101—21no—
120C2×#100—21no—
121#410—22no—
122#410—22no—
123#410—24no—
124#410—216no—
125#410C2×C414yes—
126#410C2×C414yes—
127#410C2×C414yescomplemented
128#410C2×C418yescomplemented
129#410D414yes—
130C2×#99—22no—
131C2×#99—22no—
132C2×#99—24no—
133C2×#99—216no—
134C2×#99D414yes—
135#405—24no—
136#405—42no—
137C4×M16—24no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—24no—
140#376—22no—
141C43—22no—
142#344—24no—
143#342—24no—
144#339—24no—
145#339—22no—
146#339—22no—
147#339—42no—
148H(3,ℤ/4)—42no—
1492−1+4—28no—
1502−1+4—28no—
1512−1+4—44no—
1522−1+4—44no—
1532−1+4—44no—
1542−1+4—44no—
155H(5,𝔽2)—28no—
156H(5,𝔽2)—28no—
157H(5,𝔽2)—48no—
158H(5,𝔽2)—48no—
159H(5,𝔽2)—44no—
160H(5,𝔽2)—44no—
161H(5,𝔽2)—44no—
162H(5,𝔽2)—44no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—24no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—24no—
169C2×#41—24no—
170C2×#41—24no—
171C2×#41—22no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—22no—
176C2×#41—21no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—28no—
182C2×#41—28no—
183C2×#41—22no—
184C2×#41—22no—
185C2×#41—22no—
186C2×#41—22no—
187C2×#41—44no—
188C2×#41—44no—
189C2×#41—48no—
190C2×#41—48no—
191C2×#41—48no—
192C2×#41—48no—
193C2×#41C4214yescomplemented
194C2×#41C4214yescomplemented
195C2×#41C22×C412yes—
196C2×#41C22×C412yes—
197C2×#41C22×C412yes—
198C2×#41C22×C412yes—
199C2×#41C22×C412yes—
200C2×#41C22×C412yes—
201C2×#41C2×D414yes—
202C2×#41C2×D412yes—
203C2×#41#4114yes—
204C2×#41#4112yes—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8C4214yescomplemented
208C22×Q8C22×C411yescharacteristic
209C22×Q8C22×C411yescharacteristic
210C22×Q8#4111yescharacteristic
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—24no—
218C22×D4C4214yescomplemented
219C22×D4C22×C411yescharacteristic
220C22×D4C22×C411yescharacteristic
221C22×D4C2×D412yes—
222C22×D4#4111yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4C22×C412yes—
229C23×C4C2411yescharacteristic
230#131—48no—
231#131—48no—
232C2×M16—21no—
233C2×M16—22no—
234C2×M16—24no—
235C2×M16—21no—
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—22no—
239C2×M16—22no—
240C2×M16—22no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—28no—
244C2×M16—44no—
245C2×M16C22×C412yes—
246C2×M16C22×C414yes—
247C2×M16C22×C412yes—
248C2×M16C2×D414yes—
249C22×C8—21no—
250C22×C8—24no—
251C22×C8—21no—
252C22×C8—22no—
253C4×Q8—28no—
254C4×Q8—44no—
255C4×Q8—42no—
256C4×Q8—42no—
257C4×D4—28no—
258C4×D4—44no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—48no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—48no—
265#117—22no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—24no—
272#117—22no—
273#117—22no—
274#117—28no—
275#117—22no—
276#117—22no—
277#117—42no—
278#117—44no—
279#117—44no—
280#117C22×C414yes—
281#117C22×C414yes—
282#117C22×C414yes—
283#117C2×D414yes—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—44no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4C2×D414yes—
296C2×C42—22no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C22×C412yes—
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312#104—28no—
313#104—216no—
314#104—28no—
315#104—22no—
316#104—22no—
317#104—24no—
318#104C22⋊C414yes—
319#104C22⋊C414yes—
320#104C22⋊C418yes—
321#103—44no—
322#102—44no—
323#101—24no—
324#100—24no—
325#99—28no—
326#99—28no—
327#99—216no—
328#99C22⋊C4116yes—
329#97—44no—
330C4×C8—44no—
331#95—44no—
332C24—21no—
333C24—21no—
334C24—24no—
335C24C4×D412yes—
336C24C22×D411yescharacteristic
337#41—24no—
338#41—24no—
339#41—22no—
340#41—22no—
341#41—24no—
342#41—24no—
343#41—24no—
344#41—24no—
345#41—24no—
346#41—24no—
347#41—24no—
348#41—24no—
349#41—24no—
350#41—24no—
351#41—24no—
352#41—28no—
353#41—28no—
354#41—22no—
355#41—22no—
356#41—24no—
357#41—22no—
358#41—22no—
359#41—24no—
360#41—48no—
361#41—48no—
362#41—44no—
363#41—44no—
364#41—48no—
365#41—48no—
366#41—416no—
367#41—48no—
368#41—48no—
369#41—416no—
370#41—48no—
371#41—48no—
372#41—42no—
373#41—42no—
374#41—42no—
375#41—42no—
376#41C2×C22⋊C414yes—
377#41#11714yes—
378#41C4×D418yes—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—28no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8C2×C4212yes—
392C2×Q8C2×C4212yes—
393C2×Q8#11712yes—
394C2×Q8C4×D414yes—
395C2×Q8C4×D412yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—44no—
416C2×D4—48no—
417C2×D4—416no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—416no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×C4212yes—
425C2×D4C2×C4212yes—
426C2×D4C2×C22⋊C414yes—
427C2×D4#11712yes—
428C2×D4C4×D412yes—
429C2×D4C4×D414yes—
430C2×D4C4×D414yes—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—82no—
462C22×C4—84no—
463C22×C4C2×C4214yes—
464C22×C4C4×D414yes—
465C22×C4C4×D412yes—
466C22×C4C23×C411yescharacteristic
467C22×C4C23×C411yescharacteristic
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476M16—22no—
477M16—24no—
478M16—22no—
479M16—24no—
480M16—24no—
481M16—22no—
482M16—48no—
483M16—42no—
484M16—42no—
485M16—42no—
486M16C2×C22⋊C414yes—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—48no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4C2×C22⋊C418yes—
513C42—24no—
514C42—21no—
515C42—21no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—42no—
524C42—42no—
525C42—48no—
526C42—42no—
527C42—42no—
528C42—88no—
529C42C2×C22⋊C411yescharacteristic
530C42C2×C22⋊C411yescharacteristic
531C42C2×C22⋊C412yes—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—48no—
538C23—42no—
539C23—81no—
540C23—82no—
541C23—161no—
542C23—161no—
543C23—164no—
544C23C4×C22⋊C414yes—
545C23C22×C22⋊C411yescharacteristic
546C23C2×C4×D411yescharacteristic
547C23C2×C4×D411yescharacteristic
548C23D4212yes—
549C23#54612yes—
550Q8—24no—
551Q8—24no—
552Q8—24no—
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—48no—
557Q8—42no—
558Q8C4×C22⋊C414yes—
559D4—24no—
560D4—28no—
561D4—24no—
562D4—28no—
563D4—28no—
564D4—24no—
565D4—24no—
566D4—24no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—48no—
572D4—48no—
573D4—42no—
574D4—44no—
575D4—42no—
576D4—42no—
577D4C4×C22⋊C414yes—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—82no—
613C2×C4—168no—
614C2×C4—162no—
615C2×C4C4×C22⋊C414yes—
616C2×C4C22×C4211yescharacteristic
617C2×C4C22×C22⋊C411yescharacteristic
618C2×C4C2×#11711yescharacteristic
619C2×C4C2×#11711yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C4×D412yes—
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C2×C4×D412yes—
624C2×C4#51811yescharacteristic
625C2×C4#51811yescharacteristic
626C2×C4#51812yes—
627C2×C4#51812yes—
628C2×C4D4212yes—
629C2×C4#54612yes—
630C2×C4#54612yes—
631C2×C4#54612yes—
632C2×C4#55112yes—
633C2×C4#55112yes—
634C8—24no—
635C8—42no—
636C8—44no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—41no—
641C22—42no—
642C22—41no—
643C22—41no—
644C22—44no—
645C22—42no—
646C22—82no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—162no—
652C22—168no—
653C22C2×C4×C22⋊C411yescharacteristic
654C22D4×C22⋊C411yescharacteristic
655C22D4×C22⋊C412yes—
656C22#234211yescharacteristic
657C22#235812yes—
658C4—22no—
659C4—24no—
660C4—24no—
661C4—42no—
662C4—41no—
663C4—48no—
664C4—42no—
665C4—84no—
666C4—82no—
667C4—84no—
668C4C2×C4×C22⋊C411yescharacteristic
669C4#227311yescharacteristic
670C4D4×C22⋊C411yescharacteristic
671C4#234211yescharacteristic
672C4#234212yes—
673C4#235812yes—
674C2—21no—
675C2—22no—
676C2—22no—
677C2—21no—
678C2—21no—
679C2—44no—
680C2—44no—
681C2—82no—
682C2C22⋊C4211yescharacteristic
subgroupG/H
centerC4D4×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC2C22⋊C42
prime radicalC2×#585C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20405C2semidirect
5. #20401C2semidirect
6. C4×#410C2semidirect
7. #10951C2semidirect
8. #10945C2semidirect
9. #10783C2semidirect
10. #10766C2semidirect
11. #10758C2semidirect
18. C2×#421C4semidirect
22. C22×#104C4semidirect
27. #2884C4semidirect
29. #2883C4semidirect
35. #2873C22semidirect
36. C4×#117C22semidirect
37. #1796C4semidirect
38. #1795C4semidirect
43. C2×#344C22semidirect
44. #1740C22semidirect
45. #1733C22semidirect
46. #1733C22semidirect
47. C2×#339C22semidirect
48. C2×H(3,ℤ/4)C22semidirect
70. C22×M16C2×C4semidirect
76. #519C2×C4semidirect
78. #518C2×C4semidirect
88. C2×#117C2×C4semidirect
93. C22×C42C2×C4semidirect
99. #421C2×C4semidirect
108. C2×#104C2×C4semidirect
109. C2×#104C2×C4semidirect
110. C2×#104C2×C4semidirect
113. #419C2×C4semidirect
114. #418C2×C4semidirect
118. #413C2×C4semidirect
127. #410C2×C4semidirect
128. #410C2×C4semidirect
193. C2×#41C42semidirect
194. C2×#41C42semidirect
207. C22×Q8C42semidirect
218. C22×D4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#29012
#600702
#206232
#204052
#204012
C4×#4102
#109512
#109452
#107832
#107662
#107582