Finite Group Databasebeta

← #363285

Subgroups — #363285

Order 512.

2954 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#559C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic, complemented
3C2×#1982C211yesmaximal, characteristic
4C2×#1957C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×C4×#103C211yesmaximal, characteristic, complemented
7#13926C211yesmaximal, characteristic, complemented
8#11009C214yesmaximal, complemented
9#10788C214yesmaximal, complemented
10C2×#559—24no—
11C2×#559—24no—
12C2×#559—24no—
13C2×#559—24no—
14C2×#559—24no—
15C2×#559C418yescomplemented
16C22×#127C2211yescharacteristic
17C22×#124—21no—
18C22×#124C2212yes—
19C22×#123—22no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4—21no—
23C22×C4×D4C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#102C2214yes—
28C22×#102C2214yes—
29C22×C4×C8C2211yescharacteristic
30C2×#391—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1984—24no—
34#1982—24no—
35#1957—24no—
36#1803—24no—
37C4×#103—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2214yes—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95C2214yes—
44C23×Q8D411yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4—21no—
47C23×D4—21no—
48C23×D4C2312yes—
49C24×C4—22no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52#559—232no—
53#559—232no—
54#559—232no—
55#559—232no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127C2×C414yes—
60C2×#127D414yes—
61C2×#127D414yes—
62C2×#124—28no—
63C2×#124—28no—
64C2×#124—28no—
65C2×#124—24no—
66C2×#124—24no—
67C2×#124—24no—
68C2×#124—24no—
69C2×#124—24no—
70C2×#124C2×C418yes—
71C2×#123—216no—
72C2×#123—216no—
73C2×#123—216no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8C2×C414yes—
82C2×C4×Q8C2×C414yes—
83C2×C4×Q8D414yes—
84C2×C4×Q8D414yes—
85C2×C4×Q8D414yes—
86C2×C4×D4—28no—
87C2×C4×D4—28no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4C2×C418yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4D412yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—24no—
116C2×#103—24no—
117C2×#103—44no—
118C2×#102—28no—
119C2×#102—28no—
120C2×#102—28no—
121C2×#102—28no—
122C2×#102—44no—
123C2×#102—44no—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8—22no—
127#391—44no—
128#384—22no—
129#384—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C2×#95—24no—
133C2×#95—21no—
134C2×#95—24no—
135#340—24no—
136#340—24no—
137#336—28no—
138C25—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C22⋊C414yes—
153C22×Q8C22×C412yes—
154C22×Q8C2×D412yes—
155C22×Q8C2×D412yes—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—82no—
174C22×D4—82no—
175C22×D4C22×C414yes—
176C22×D4C2×D414yes—
177C22×D4C2×D414yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—84no—
186C23×C4—84no—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—24no—
202C22×C8—24no—
203C22×C8#4114yes—
204#127—28no—
205#127—28no—
206#127—28no—
207#127C22⋊C4116yes—
208#124—216no—
209#124—216no—
210#124—264no—
211#124—216no—
212#124—216no—
213#123—2128no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×Q8—28no—
221C4×Q8C22⋊C4116yes—
222C4×D4—216no—
223C4×D4—216no—
224C4×D4—264no—
225C4×D4—216no—
226C4×D4—216no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C22⋊C414yes—
252C2×C4⋊C4C22⋊C418yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4#4114yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42C22×C414yes—
277C2×C42C22×C412yes—
278C2×C42C2×D414yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C4⋊C8—44no—
285#103—216no—
286#103—48no—
287#102—416no—
288#102—416no—
289C4×C8—44no—
290#95—48no—
291#95—42no—
292#95—84no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—44no—
297C24—44no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—44no—
303C24—44no—
304C24C22×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—28no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8C2×C22⋊C414yes—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—216no—
340C2×D4—216no—
341C2×D4—44no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—88no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—88no—
359C2×D4C2×C22⋊C418yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—232no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—816no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—816no—
399C22×C4C2×C22⋊C414yes—
400C22×C4C2×C22⋊C414yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4#12012yes—
404C22×C4#12012yes—
405C22×C4#12012yes—
406C22×C4#12012yes—
407C22×C4#13512yes—
408C22×C4#13512yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4#13712yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+412yes—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—44no—
432C2×C8—44no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—216no—
437C4⋊C4—28no—
438C4⋊C4—216no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4C2×C22⋊C414yes—
449C22⋊C4—232no—
450C22⋊C4—232no—
451C22⋊C4—232no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—216no—
457C42—84no—
458C42C2×C22⋊C414yes—
459C42C2×C22⋊C414yes—
460C42#12014yes—
461C42#12014yes—
462C23—28no—
463C23—48no—
464C23—44no—
465C23—44no—
466C23—48no—
467C23—48no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—48no—
477C23—84no—
478C23—84no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23C22×C22⋊C411yescharacteristic
486C23C2×#12011yescharacteristic
487C23C2×#12011yescharacteristic
488C23#54812yes—
489C23#55012yes—
490C23C2×#13511yescharacteristic
491C23C2×#13511yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494C23#57612yes—
495C23#57712yes—
496Q8—24no—
497Q8—24no—
498Q8—44no—
499Q8—44no—
500Q8—42no—
501Q8—42no—
502Q8—416no—
503D4—216no—
504D4—44no—
505D4—48no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—432no—
510D4—816no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—832no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#37914yes—
551C2×C4#37914yes—
552C2×C4#41614yes—
553C2×C4#41714yes—
554C2×C4#44914yes—
555C2×C4#44914yes—
556C2×C4#45214yes—
557C2×C4#45214yes—
558C2×C4C22×C22⋊C411yescharacteristic
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4C4×#4112yes—
561C2×C4#52012yes—
562C2×C4C2×#12011yescharacteristic
563C2×C4C2×#12011yescharacteristic
564C2×C4#52512yes—
565C2×C4#53211yescharacteristic
566C2×C4#53211yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#53611yescharacteristic
569C2×C4#53712yes—
570C2×C4#54612yes—
571C2×C4#54612yes—
572C2×C4#54712yes—
573C2×C4#54712yes—
574C2×C4#54812yes—
575C2×C4#55012yes—
576C8—44no—
577C8—82no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—88no—
582C22—88no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—88no—
588C22—88no—
589C22C2×#37911yescharacteristic
590C22#234112yes—
591C22#240712yes—
592C22#243111yescharacteristic
593C22#243211yescharacteristic
594C22C2×#41611yescharacteristic
595C22C2×#41711yescharacteristic
596C22#289712yes—
597C22C2×#44911yescharacteristic
598C22C2×#44911yescharacteristic
599C22C2×#45211yescharacteristic
600C22C2×#45211yescharacteristic
601C22#300811yescharacteristic
602C22#300911yescharacteristic
603C22#301212yes—
604C22#336112yes—
605C22#336512yes—
606C22#337312yes—
607C22#337512yes—
608C22#339412yes—
609C22#339612yes—
610C22#339912yes—
611C22#340012yes—
612C4—24no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—84no—
618C4—162no—
619C4#178818yes—
620C4#229711yescharacteristic
621C4#231111yescharacteristic
622C4#233912yes—
623C4#240712yes—
624C4#243111yescharacteristic
625C4#243211yescharacteristic
626C2—88no—
627C2#1393711yescharacteristic
628C2C2×#178811yescharacteristic
629C2#2060211yescharacteristic
630C2#2254312yes—
631C2#2254612yes—
632C2#2418612yes—
633C2#2419012yes—
634C2#2562911yescharacteristic
635C2#2563311yescharacteristic
636C2#2564811yescharacteristic
637C2#2565211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1984C2semidirect
5. C2×#1803C2semidirect
6. C2×C4×#103C2semidirect
7. #13926C2semidirect
8. #11009C2semidirect
9. #10788C2semidirect
15. C2×#559C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5592
C2×#19842
C2×#19822
C2×#19572
C2×#18032
C2×C4×#1032
#139262
#110092
#107882