Finite Group Databasebeta

← #393370

Subgroups — #393370

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3080C211yesmaximal, characteristic
2C4×#520C211yesmaximal, characteristic
3#21314C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic, complemented
6#10899C212yesmaximal, complemented
7#10899C212yesmaximal, complemented
8#10852C211yesmaximal, characteristic, complemented
9#10847C211yesmaximal, characteristic, complemented
10#10781C212yesmaximal
11#10781C212yesmaximal
12C22×#125C2211yescharacteristic
13C2×#520C2212yes—
14C2×C4×#41C2211yescharacteristic
15#3080—24no—
16#3080C418yescomplemented
17C2×#479—21no—
18C2×#479—21no—
19C2×#478—21no—
20C2×#478—21no—
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×#104C2212yes—
26C2×#417C2211yescharacteristic
27C2×#417C2211yescharacteristic
28#2276—21no—
29#2275—22no—
30#2275—21no—
31#2274—22no—
32#2274C2212yes—
33C42×Q8—21no—
34C42×D4—22no—
35C42×D4—21no—
36C4×#117—21no—
37C4×#117C2211yescharacteristic
38C2×C4×C4⋊C4—21no—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#1989—21no—
42#1989—21no—
43#1989—22no—
44#1806—24no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47#1736C2212yes—
48#1736C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1441—22no—
53#1436—22no—
54C4⋊#95C2212yes—
55C4⋊#95C2212yes—
56C22×#41C2311yescharacteristic
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C2×#125—21no—
61C2×#125—21no—
62C2×#125—22no—
63C2×#125—21no—
64C2×#125—21no—
65C2×#125—22no—
66C2×#125—22no—
67C2×#125C2×C414yescomplemented
68#520—24no—
69#520—24no—
70#520—24no—
71#520Q818yes—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41C2×C414yes—
75C4×#41D414yes—
76C2×C4×Q8—22no—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×D4—22no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—24no—
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×#117—22no—
86C2×#117—21no—
87C2×#117C2311yescharacteristic
88C2×#117C2312yes—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C42—21no—
97C22×C42—22no—
98C22×C42D412yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#479—28no—
102#479—28no—
103#478—28no—
104#478—28no—
105#428—22no—
106#428—22no—
107#428C2×C414yescomplemented
108#428C2×C414yescomplemented
109C2×#107—21no—
110C2×#107—21no—
111C2×#106—21no—
112C2×#106—21no—
113#423—22no—
114#423—22no—
115#423—22no—
116#423C2×C414yescomplemented
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×#104—24no—
122C2×#104—24no—
123C2×#104—24no—
124#417—22no—
125#417—22no—
126#417C2×C414yes—
127#417C2×C414yes—
128C2×#103—21no—
129C2×#103—21no—
130C2×#102—21no—
131C2×#102—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4D411yescharacteristic
147C4×C4⋊C4D411yescharacteristic
148C4×C4⋊C4D412yes—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—24no—
156#376—22no—
157#376—21no—
158C2×#95—22no—
159C2×#95—24no—
160C2×#95—21no—
161C2×#95—24no—
162C2×#95D412yes—
163C43—22no—
164C43—21no—
165#343—24no—
166#343—24no—
167#340—22no—
168#340—22no—
169#340—22no—
170#340—22no—
171#339—24no—
172#339—24no—
173#339—24no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41C22×C412yes—
179C2×#41C2×D412yes—
180C2×#41C2×Q814yes—
181C2×#41#4114yes—
182C22×Q8C2×D411yescharacteristic
183C22×D4—21no—
184C22×D4C2×D411yescharacteristic
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4C2×D414yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2411yescharacteristic
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16C22×C412yes—
209C2×M16C22×C412yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213#125—24no—
214#125—24no—
215#125—24no—
216#125—24no—
217#125—42no—
218#125—42no—
219#125—42no—
220#125—42no—
221C4×Q8—28no—
222C4×Q8—28no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8C22⋊C414yes—
226C4×Q8C22⋊C414yes—
227C4×D4—216no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—216no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4C22⋊C414yes—
238C4×D4C22⋊C414yes—
239#117—28no—
240#117—21no—
241#117—21no—
242#117—22no—
243#117—22no—
244#117—28no—
245#117—24no—
246#117—24no—
247#117C22×C412yes—
248#117C2×D412yes—
249#117C2×Q814yes—
250#117#4114yes—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×Q814yes—
287C2×C4⋊C4#4114yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C42—24no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—84no—
314C2×C42—82no—
315C2×C42C22×C412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42#4112yes—
322C2×C42#4112yes—
323#107—24no—
324#107—24no—
325#106—24no—
326#106—24no—
327C4⋊C8—24no—
328C4⋊C8—24no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331#104—48no—
332#103—24no—
333#103—24no—
334#102—24no—
335#102—24no—
336#95—24no—
337#95—24no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—22no—
343#95—48no—
344#95—42no—
345#95—84no—
346C24—21no—
347C24C22×D411yescharacteristic
348#41—24no—
349#41—24no—
350#41—24no—
351#41—22no—
352#41—22no—
353#41—24no—
354#41—24no—
355#41C4×Q818yes—
356#41#12414yes—
357#41C4⋊Q814yes—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8C2×C22⋊C412yes—
361C2×Q8C22⋊Q814yes—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4C2×C22⋊C412yes—
369C2×D4C22⋊Q814yes—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—216no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4C4×D412yes—
411C22×C4C4×D412yes—
412C22×C4#12011yescharacteristic
413C22×C4#12011yescharacteristic
414C22×C4#12011yescharacteristic
415C22×C4#12011yescharacteristic
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4C22⋊Q818yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q812yes—
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4111yescharacteristic
429M16—24no—
430M16—24no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—42no—
436M16—42no—
437M16—42no—
438M16—42no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C4⋊C4—216no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—48no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4C2×C22⋊C412yes—
471C4⋊C4C2×C22⋊C412yes—
472C4⋊C4C22⋊Q814yes—
473C4⋊C4C22⋊Q814yes—
474C22⋊C4—216no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C42—216no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—24no—
487C42—24no—
488C42—42no—
489C42—44no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—82no—
496C42—82no—
497C42—84no—
498C42—84no—
499C42C2×C22⋊C412yes—
500C42C2×C22⋊C412yes—
501C42C4×D412yes—
502C42C4×D412yes—
503C42#12112yes—
504C42#12112yes—
505C42#12312yes—
506C42#12312yes—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—24no—
511C23—22no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23C2×C4×D411yescharacteristic
516C23C2×#12011yescharacteristic
517C23C2×#12111yescharacteristic
518C23C2×C22⋊Q812yes—
519C23#52512yes—
520Q8—24no—
521Q8—22no—
522Q8#39114yes—
523D4—24no—
524D4—24no—
525D4—28no—
526D4—24no—
527D4—22no—
528D4#39114yes—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—816no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4#38612yes—
580C2×C4#38612yes—
581C2×C4#38612yes—
582C2×C4#38612yes—
583C2×C4C22×C22⋊C411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×C4×Q812yes—
586C2×C4C4×#4112yes—
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×C22⋊Q812yes—
589C2×C4C2×C22⋊Q812yes—
590C2×C4C2×#12311yescharacteristic
591C2×C4#52512yes—
592C2×C4#52512yes—
593C2×C4C2×#12411yescharacteristic
594C2×C4#52912yes—
595C2×C4#52912yes—
596C2×C4C2×C4⋊Q811yescharacteristic
597C2×C4#53211yescharacteristic
598C2×C4#53211yescharacteristic
599C2×C4#53312yes—
600C2×C4#53312yes—
601C8—42no—
602C8—42no—
603C8—82no—
604C22—22no—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—22no—
610C22—21no—
611C22—21no—
612C22—24no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22C2×#38611yescharacteristic
618C22C4×C22⋊Q812yes—
619C22#284812yes—
620C22#285011yescharacteristic
621C22#285711yescharacteristic
622C22#301812yes—
623C22#301911yescharacteristic
624C22#302011yescharacteristic
625C4—24no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—48no—
631C4—44no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4C2×#39111yescharacteristic
639C4#229711yescharacteristic
640C4C4×C22⋊Q812yes—
641C4#285011yescharacteristic
642C4#285512yes—
643C4#285711yescharacteristic
644C2—21no—
645C2—21no—
646C2—22no—
647C2—44no—
648C2#1470211yescharacteristic
649C2#2094211yescharacteristic
650C2#2094311yescharacteristic
651C2#3297312yes—
652C2#3297412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21314C2semidirect
5. #14160C2semidirect
6. #10899C2semidirect
7. #10899C2semidirect
8. #10852C2semidirect
9. #10847C2semidirect
16. #3080C4semidirect
67. C2×#125C2×C4semidirect
107. #428C2×C4semidirect
108. #428C2×C4semidirect
116. #423C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30802
C4×#5202
#213142
C2×#18062
#141602
#108992
#108522
#108472
#107812