Finite Group Databasebeta

← #358318

Subgroups — #358318

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1736C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6C2×#1729C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11293C214yesmaximal, complemented
9#11292C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2317—24no—
18#2317—24no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#385—22no—
22C2×#384—21no—
23C2×#382—21no—
24#1736—24no—
25#1736—24no—
26#1735C418yescomplemented
27C2×#340—24no—
28C2×#340C2212yes—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34#1729—24no—
35C2×#336C2214yes—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2214yes—
39C22⋊#95—22no—
40C24×C4—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16D414yes—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8C2×C414yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117C2×C414yes—
74C2×#117C2×C414yes—
75C2×#117D414yes—
76C2×#117Q814yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4D411yescharacteristic
93C22×C4⋊C4D414yes—
94C22×C4⋊C4Q814yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4D412yes—
103C22×C42—21no—
104C22×C42—21no—
105C22×C42C2312yes—
106#389—24no—
107#389—24no—
108#389—24no—
109#389—24no—
110#385—216no—
111#384—28no—
112#382—28no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—24no—
116C2×#95—24no—
117#340—28no—
118#340—28no—
119#340—24no—
120#340—24no—
121#340C2×C418yescomplemented
122#340C2×C418yescomplemented
123#336—28no—
124C25C2×D411yescharacteristic
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×Q812yes—
148C23×C4C2×Q812yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C2×M16—28no—
159C2×M16—28no—
160C2×M16—28no—
161C2×M16—44no—
162C2×M16—44no—
163C2×M16—44no—
164C2×M16—44no—
165C22×C8—28no—
166C22×C8—28no—
167C22×C8—28no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—44no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8—44no—
184C22×C8C22×C414yes—
185C22×C8C2×D414yes—
186#117—28no—
187#117—28no—
188#117—28no—
189#117—28no—
190#117—28no—
191#117—44no—
192#117—44no—
193#117—44no—
194#117—44no—
195#117—44no—
196#117—44no—
197#117C4⋊C4116yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—84no—
242C2×C4⋊C4—84no—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×Q814yes—
250C2×C4⋊C4C2×Q812yes—
251C2×C4⋊C4C2×Q812yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—42no—
273#95—28no—
274#95—28no—
275#95—84no—
276#95—84no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24C4×D412yes—
282C24#12012yes—
283C24#12112yes—
284C24C22⋊Q812yes—
285C24C22⋊Q812yes—
286C24#12312yes—
287C24#12412yes—
288C24C22×D411yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—88no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—88no—
332C22×C4—88no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—88no—
336C22×C4C2×C4⋊C414yes—
337C22×C4#11714yes—
338C22×C4#11714yes—
339C22×C4C4×D414yes—
340C22×C4C4×D414yes—
341C22×C4C4×D412yes—
342C22×C4C4×Q814yes—
343C22×C4#12012yes—
344C22×C4#12114yes—
345C22×C4#12114yes—
346C22×C4#12112yes—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q814yes—
350C22×C4C22⋊Q812yes—
351C22×C4C22⋊Q812yes—
352C22×C4#12314yes—
353C22×C4#12314yes—
354C22×C4#12314yes—
355C22×C4#12314yes—
356C22×C4#12312yes—
357C22×C4#12414yes—
358C22×C4#12414yes—
359C22×C4#12414yes—
360C22×C4#12412yes—
361C22×C4#12514yes—
362C22×C4#12514yes—
363C22×C4#12514yes—
364C22×C4#12514yes—
365C22×C4#12514yes—
366C22×C4#12614yes—
367C22×C4#12614yes—
368C22×C4#12614yes—
369C22×C4#12614yes—
370C22×C4C4⋊Q814yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C22×Q811yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385M16—416no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—416no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4C2×C4⋊C414yes—
426C4⋊C4C2×C4⋊C414yes—
427C22⋊C4—216no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C42—216no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—48no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23#38514yes—
453C23#38714yes—
454C23#38814yes—
455C23#39314yes—
456C23#39414yes—
457C23#39814yes—
458C23#40014yes—
459C23C2×C4×D411yescharacteristic
460C23C2×#12011yescharacteristic
461C23C2×#12111yescharacteristic
462C23C2×C22⋊Q811yescharacteristic
463C23C2×C22⋊Q811yescharacteristic
464C23C2×#12311yescharacteristic
465C23C2×#12411yescharacteristic
466C23#57612yes—
467C23#57712yes—
468C23#57712yes—
469C23#57812yes—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—816no—
499C2×C4—816no—
500C2×C4—84no—
501C2×C4—82no—
502C2×C4—84no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—84no—
508C2×C4—82no—
509C2×C4#38218yes—
510C2×C4#38514yes—
511C2×C4#38714yes—
512C2×C4#38718yes—
513C2×C4#38814yes—
514C2×C4#38818yes—
515C2×C4#38918yes—
516C2×C4#39314yes—
517C2×C4#39414yes—
518C2×C4#39718yes—
519C2×C4#39818yes—
520C2×C4#39814yes—
521C2×C4#40018yes—
522C2×C4#40018yes—
523C2×C4#40014yes—
524C2×C4C22×C4⋊C411yescharacteristic
525C2×C4C2×#11711yescharacteristic
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×Q811yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4C2×#12311yescharacteristic
536C2×C4C2×#12311yescharacteristic
537C2×C4C2×#12311yescharacteristic
538C2×C4C2×#12311yescharacteristic
539C2×C4C2×#12411yescharacteristic
540C2×C4C2×#12411yescharacteristic
541C2×C4C2×#12411yescharacteristic
542C2×C4C2×#12511yescharacteristic
543C2×C4C2×#12511yescharacteristic
544C2×C4C2×#12511yescharacteristic
545C2×C4C2×#12511yescharacteristic
546C2×C4C2×#12511yescharacteristic
547C2×C4C2×#12611yescharacteristic
548C2×C4C2×#12611yescharacteristic
549C2×C4C2×#12611yescharacteristic
550C2×C4C2×#12611yescharacteristic
551C2×C4C2×C4⋊Q811yescharacteristic
552C8—44no—
553C8—44no—
554C22—28no—
555C22—28no—
556C22—28no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—28no—
564C22#144218yes—
565C22C2×#38511yescharacteristic
566C22C2×#38711yescharacteristic
567C22C2×#38811yescharacteristic
568C22C2×#39311yescharacteristic
569C22C2×#39411yescharacteristic
570C22C2×#39811yescharacteristic
571C22C2×#40011yescharacteristic
572C22#295112yes—
573C22#295212yes—
574C22#300811yescharacteristic
575C22#300911yescharacteristic
576C22#301011yescharacteristic
577C22#301111yescharacteristic
578C22#306212yes—
579C22#306312yes—
580C22#308112yes—
581C22#308212yes—
582C22#308312yes—
583C22#308412yes—
584C22#309612yes—
585C22#309712yes—
586C22#313612yes—
587C22#313712yes—
588C4—44no—
589C4—44no—
590C4—84no—
591C4—84no—
592C4—82no—
593C4—84no—
594C4—82no—
595C4#144218yes—
596C4C2×#38211yescharacteristic
597C4C2×#38711yescharacteristic
598C4C2×#38811yescharacteristic
599C4C2×#38911yescharacteristic
600C4C2×#39711yescharacteristic
601C4C2×#39811yescharacteristic
602C4C2×#40011yescharacteristic
603C4C2×#40011yescharacteristic
604C2—28no—
605C2C2×#144211yescharacteristic
606C2#2088111yescharacteristic
607C2#2088211yescharacteristic
608C2#2103912yes—
609C2#2108712yes—
610C2#2345611yescharacteristic
611C2#2345711yescharacteristic
612C2#2348311yescharacteristic
613C2#2348411yescharacteristic
614C2#2367112yes—
615C2#2374812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11293C2semidirect
9. #11292C2semidirect
26. #1735C4semidirect
121. #340C2×C4semidirect
122. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#17362
C2×#17352
C22×#3402
C2×#17292
#137952
#112932
#112922