Finite Group Databasebeta

← #393302

Subgroups — #393302

Order 512.

1900 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic
2#36625C211yesmaximal, characteristic
3C2×#1986C212yesmaximal, complemented
4C4×#380C211yesmaximal, characteristic
5#10886C214yesmaximal, complemented
6#10851C212yesmaximal, complemented
7#10732C214yesmaximal
8C23×M16C2212yes—
9C2×#533C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×C4×M16—21no—
12C2×C4×M16—21no—
13C2×C4×M16—28no—
14C2×C4×M16C2212yes—
15C2×C4×M16C2212yes—
16C22×#97—21no—
17C22×C4×C8—21no—
18#2857C411yescharacteristic, complemented
19#2857C411yescharacteristic, complemented
20#2857C411yescharacteristic, complemented
21#2857C411yescharacteristic, complemented
22#2855—21no—
23#2855—21no—
24#2851C412yescomplemented
25#2851C412yescomplemented
26#2848—21no—
27#2848—21no—
28C4×C22⋊Q8—21no—
29C4×C22⋊Q8—21no—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32#2296C2212yes—
33#2285C2211yescharacteristic
34#2280C2212yes—
35C2×#380—21no—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38#1986—28no—
39C2×#339C2218yes—
40C2×#339C2218yes—
41C4⋊#95—21no—
42#1434—21no—
43C22⋊#95—21no—
44C4×#95C2212yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—28no—
51C22×M16—28no—
52C22×M16Q818yescomplemented
53C22×M16C2314yes—
54C23×C8—22no—
55#533—24no—
56#533—24no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117C2312yes—
69C2×#117C2312yes—
70C22×C4⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42—24no—
73C22×C42—24no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2314yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C4×M16—28no—
88C4×M16—28no—
89C4×M16—28no—
90C4×M16—24no—
91C4×M16D418yes—
92C2×#97—28no—
93C2×#97—22no—
94C2×#97—22no—
95C2×C4×C8—28no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#382C2×C412yes—
123#381—22no—
124#380—28no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380C2×C412yes—
132#380C2×C412yes—
133#380C2×C412yes—
134#380C2×C412yes—
135#379—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—24no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—28no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149#339—216no—
150#339—216no—
151C25C2×D411yescharacteristic
152C22×Q8—21no—
153C22×Q8—21no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×Q814yes—
182C23×C4#4114yes—
183C23×C4C2411yescharacteristic
184C2×M16—232no—
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—28no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—232no—
192C2×M16C2×D418yes—
193C2×M16C2×Q818yes—
194C2×M16#4118yes—
195C22×C8—28no—
196C22×C8—28no—
197C22×C8—28no—
198C22×C8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201#125—28no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204C22⋊Q8—44no—
205C22⋊Q8—44no—
206C22⋊Q8—44no—
207C22⋊Q8—44no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8—42no—
212#117—24no—
213#117—24no—
214#117—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—28no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D414yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4114yes—
283#97—28no—
284#97—28no—
285C4×C8—28no—
286C4×C8—28no—
287#95—28no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—44no—
299#95—44no—
300#95—44no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—41no—
306C24—41no—
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C24#12412yes—
310C24C4⋊Q812yes—
311C24C22×D411yescharacteristic
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—28no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4#10414yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C4×Q814yes—
392C22×C4C4×Q814yes—
393C22×C4#12114yes—
394C22×C4#12114yes—
395C22×C4C22⋊Q814yes—
396C22×C4C22⋊Q814yes—
397C22×C4#12412yes—
398C22×C4C4⋊Q812yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×Q812yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4114yes—
406M16—232no—
407M16—28no—
408M16—28no—
409M16—28no—
410M16C22⋊Q8116yescomplemented
411C2×C8—232no—
412C2×C8—28no—
413C2×C8—28no—
414C2×C8—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—416no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C22⋊C4—416no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C42—22no—
440C42—22no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42—48no—
457C42—48no—
458C42C4×D414yes—
459C42C4×D414yes—
460C42#12012yes—
461C42#12012yes—
462C42#12312yes—
463C42#12312yes—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—24no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—24no—
479C23—24no—
480C23—21no—
481C23—21no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23#39112yes—
489C23#39112yes—
490C23C2×#10412yes—
491C23#42112yes—
492C23C22×C22⋊C411yescharacteristic
493C23C2×#12112yes—
494C23C2×#12411yescharacteristic
495C23C2×C4⋊Q811yescharacteristic
496C23#53212yes—
497Q8—84no—
498Q8—84no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4#38612yes—
559C2×C4#38612yes—
560C2×C4#38612yes—
561C2×C4#38612yes—
562C2×C4#39112yes—
563C2×C4#39112yes—
564C2×C4C2×#10412yes—
565C2×C4C2×#10412yes—
566C2×C4#42112yes—
567C2×C4#42112yes—
568C2×C4C2×C4×D412yes—
569C2×C4C2×C4×Q812yes—
570C2×C4C4×#4112yes—
571C2×C4C2×#12011yescharacteristic
572C2×C4C2×C22⋊Q812yes—
573C2×C4C2×C22⋊Q814yes—
574C2×C4C2×#12311yescharacteristic
575C2×C4#52512yes—
576C2×C4#52514yes—
577C2×C4#52914yes—
578C2×C4#53314yes—
579C8—216no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—48no—
598C22—48no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22#198614yescomplemented
604C22C2×#38611yescharacteristic
605C22C2×#39111yescharacteristic
606C22#229711yescharacteristic
607C22#285012yes—
608C22#285712yes—
609C22C22×#10411yescharacteristic
610C22C2×#42111yescharacteristic
611C22C2×#42111yescharacteristic
612C22C2×#42111yescharacteristic
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—88no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#187114yes—
628C4C4×C22⋊Q814yes—
629C4#284812yes—
630C4#285512yes—
631C2—22no—
632C2—22no—
633C2—24no—
634C2—42no—
635C2#1470211yescharacteristic
636C2C2×#187111yescharacteristic
637C2#2090211yescharacteristic
638C2C2×#198611yescharacteristic
639C2#2130611yescharacteristic
640C2#2138211yescharacteristic
641C2#2138311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1986C2semidirect
5. #10886C2semidirect
6. #10851C2semidirect
18. #2857C4semidirect
19. #2857C4semidirect
20. #2857C4semidirect
21. #2857C4semidirect
24. #2851C4semidirect
25. #2851C4semidirect
52. C22×M16Q8semidirect
410. M16C22⋊Q8semidirect
603. C22#1986semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
#366252
C2×#19862
C4×#3802
#108862
#108512
#107322