Finite Group Databasebeta

← #392270

Subgroups — #392270

Order 512.

3910 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×#41C211yesmaximal, characteristic
2C2×#1825C212yesmaximal, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4C2×#1788C212yesmaximal
5#14685C211yesmaximal, characteristic, complemented
6#10789C212yesmaximal, complemented
7#10789C212yesmaximal, complemented
8#10788C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C2×C4×#41—24no—
12C2×C4×#41—28no—
13C2×C4×#41—28no—
14C2×C4×#41C418yescomplemented
15C22×C4×Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117—21no—
22C22×#117C2211yescharacteristic
23C23×C42—21no—
24C23×C42C2211yescharacteristic
25C22×#103C2214yes—
26C22×#102C2214yes—
27C22×C22⋊C8C2212yes—
28C2×#381—21no—
29C2×#380C2212yes—
30#1825—28no—
31#1817—24no—
32#1788—28no—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39#1434—21no—
40#1434—21no—
41C22⋊#95—21no—
42C22×#41—24no—
43C22×#41—24no—
44C22×#41—24no—
45C22×#41—24no—
46C22×#41—22no—
47C22×#41—24no—
48C22×#41C2×C414yes—
49C22×#41D418yes—
50C23×Q8C2311yescharacteristic
51C23×D4—21no—
52C23×D4C2311yescharacteristic
53C24×C4—21no—
54C24×C4—21no—
55C24×C4—21no—
56C24×C4D411yescharacteristic
57C24×C4D411yescharacteristic
58C24×C4C2311yescharacteristic
59C23×C8C2314yes—
60C4×#41—264no—
61C4×#41—232no—
62C4×#41—232no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C414yes—
70C2×C4×Q8C2×C414yes—
71C2×C4×D4—216no—
72C2×C4×D4—216no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—28no—
76C2×C4×D4—28no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4C2×C414yes—
82C2×C4×D4C2×C414yes—
83C2×#117—216no—
84C2×#117—22no—
85C2×#117—24no—
86C2×#117—28no—
87C2×#117C2×C414yes—
88C2×#117D418yes—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C42—216no—
99C22×C42—21no—
100C22×C42—21no—
101C22×C42—21no—
102C22×C42—24no—
103C22×C42—24no—
104C22×C42—28no—
105C22×C42C2×C414yes—
106C22×C42D411yescharacteristic
107C22×C42D411yescharacteristic
108C22×C42D414yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C2×#103—28no—
112C2×#103—28no—
113C2×#103—28no—
114C2×#102—28no—
115C2×#102—28no—
116C2×#102—28no—
117C2×C22⋊C8—24no—
118C2×C22⋊C8—24no—
119C2×C22⋊C8—24no—
120#381—44no—
121#380—24no—
122#380—24no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128#340—28no—
129#340—28no—
130#336—24no—
131#336—24no—
132C25—21no—
133C25C2×D411yescharacteristic
134C2×#41—232no—
135C2×#41—232no—
136C2×#41—216no—
137C2×#41—216no—
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—232no—
141C2×#41—28no—
142C2×#41—28no—
143C2×#41—232no—
144C2×#41—28no—
145C2×#41—216no—
146C2×#41C22⋊C4116yes—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C22×C412yes—
152C22×Q8C2×D414yes—
153C22×Q8#4114yes—
154C22×Q8#4114yes—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—24no—
158C22×D4—28no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C22×C412yes—
164C22×D4C2×D414yes—
165C22×D4#4114yes—
166C22×D4#4114yes—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—28no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4C22⋊C414yes—
196C23×C4C22⋊C414yes—
197C23×C4C22×C412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2411yescharacteristic
207C22×C8—24no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—24no—
214C4×Q8—216no—
215C4×Q8—216no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×D4—264no—
221C4×D4—264no—
222C4×D4—216no—
223C4×D4—216no—
224C4×D4—28no—
225C4×D4—28no—
226C4×D4—28no—
227C4×D4—28no—
228#117—264no—
229#117—28no—
230#117—216no—
231#117C22⋊C4116yes—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C42—28no—
260C2×C42—28no—
261C2×C42—264no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—28no—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—216no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42C22⋊C414yes—
276C2×C42C22⋊C414yes—
277C2×C42C22×C412yes—
278C2×C42C22×C412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281#103—416no—
282#102—416no—
283C22⋊C8—48no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24—24no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—42no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24C2×C22⋊C412yes—
306C24#12014yes—
307C24C22×D411yescharacteristic
308#41—2128no—
309#41—232no—
310#41—232no—
311#41—232no—
312#41—232no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—28no—
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8C2×C22⋊C414yes—
323C2×Q8#11714yes—
324C2×Q8#11714yes—
325C2×Q8#123116yes—
326C2×D4—264no—
327C2×D4—216no—
328C2×D4—216no—
329C2×D4—216no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—28no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4C2×C22⋊C414yes—
340C2×D4#11714yes—
341C2×D4#11714yes—
342C2×D4#123116yes—
343C22×C4—264no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—432no—
392C22×C4—416no—
393C22×C4—416no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C414yes—
411C22×C4C2×C22⋊C414yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4#12018yes—
416C22×C4#12014yes—
417C22×C4#13514yes—
418C22×C4AGL(1,ℤ/8)12yes—
419C22×C4#13712yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—44no—
434C2×C8—44no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4C2×C22⋊C414yes—
446C4⋊C4C2×C22⋊C414yes—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—88no—
451C42—216no—
452C42—216no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—24no—
458C42—24no—
459C42—28no—
460C42—216no—
461C42—28no—
462C23—28no—
463C23—216no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—44no—
477C23—48no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—48no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23#37914yes—
496C23C22×C22⋊C411yescharacteristic
497C23C2×#12012yes—
498C23#52512yes—
499C23#52512yes—
500C23C2×#13512yes—
501C23C2×AGL(1,ℤ/8)11yescharacteristic
502C23C2×#13711yescharacteristic
503C23#57512yes—
504C23#57512yes—
505Q8—216no—
506Q8—24no—
507Q8—24no—
508Q8—24no—
509Q8#38118yescomplemented
510D4—264no—
511D4—216no—
512D4—24no—
513D4—24no—
514D4—24no—
515D4#38118yescomplemented
516C2×C4—264no—
517C2×C4—28no—
518C2×C4—216no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—216no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—416no—
545C2×C4—416no—
546C2×C4—432no—
547C2×C4—432no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4#37914yes—
566C2×C4#41614yes—
567C2×C4#41714yes—
568C2×C4C22×C22⋊C411yescharacteristic
569C2×C4C22×C22⋊C411yescharacteristic
570C2×C4C2×#11711yescharacteristic
571C2×C4C2×#11711yescharacteristic
572C2×C4C2×#11711yescharacteristic
573C2×C4C2×#11711yescharacteristic
574C2×C4C2×#12012yes—
575C2×C4C2×#12314yes—
576C2×C4C2×#12314yes—
577C2×C4#52512yes—
578C2×C4#52512yes—
579C2×C4#52514yes—
580C2×C4#52514yes—
581C8—84no—
582C22—24no—
583C22—28no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—416no—
598C22—48no—
599C22—416no—
600C22C2×#37911yescharacteristic
601C22#228011yescharacteristic
602C22#228011yescharacteristic
603C22#284914yes—
604C22C2×#41611yescharacteristic
605C22C2×#41711yescharacteristic
606C22#289611yescharacteristic
607C22#289611yescharacteristic
608C22#300214yes—
609C22#300612yes—
610C22#300712yes—
611C22#310214yes—
612C22#310612yes—
613C22#310712yes—
614C22#311114yes—
615C22#311512yes—
616C22#311612yes—
617C4—24no—
618C4—28no—
619C4—24no—
620C4—24no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—44no—
625C4—164no—
626C4C2×#38111yescharacteristic
627C4C2×#38111yescharacteristic
628C4#228011yescharacteristic
629C4#228011yescharacteristic
630C4#284914yes—
631C2—44no—
632C2—48no—
633C2#1468111yescharacteristic
634C2#2059711yescharacteristic
635C2#2060111yescharacteristic
636C2#2073611yescharacteristic
637C2#2073711yescharacteristic
638C2#2074211yescharacteristic
639C2#2074311yescharacteristic
640C2#3300614yes—
641C2#3301114yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1825C2semidirect
3. C2×#1817C2semidirect
5. #14685C2semidirect
6. #10789C2semidirect
7. #10789C2semidirect
8. #10788C2semidirect
9. #10788C2semidirect
14. C2×C4×#41C4semidirect
509. Q8#381semidirect
515. D4#381semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#412
C2×#18252
C2×#18172
C2×#17882
#146852
#107892
#107882