Finite Group Databasebeta

← #358582

Subgroups — #358582

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 644 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#11373C212yesmaximal, complemented
9#11372C212yesmaximal, complemented
10#11372C212yesmaximal, complemented
11#11371C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2317—24no—
20#2317C418yescomplemented
21C2×#389—22no—
22C2×#385—22no—
23C2×#384C2212yes—
24C2×#382C2212yes—
25#1736—24no—
26#1736—24no—
27#1735—24no—
28C2×#340—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39#1729—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16D414yes—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8D414yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2312yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117C2×C414yes—
81C2×#117C2×C414yes—
82C2×#117D414yes—
83C2×#117Q814yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4C2×C414yes—
98C22×C4⋊C4D412yes—
99C22×C4⋊C4Q814yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4D412yes—
109C22×C42—21no—
110C22×C42—21no—
111C22×C42C2312yes—
112#389—216no—
113#385—216no—
114#384—24no—
115#384C2×C418yescomplemented
116#382—24no—
117#382C2×C418yescomplemented
118C2×#95—24no—
119C2×#95—24no—
120C2×#95—24no—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#336—24no—
132#336—24no—
133C25C2×D411yescharacteristic
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C22×C412yes—
151C23×C4C2×D414yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×Q812yes—
157C23×C4C2×Q812yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—44no—
171C2×M16—44no—
172C2×M16—44no—
173C2×M16—44no—
174C22×C8—28no—
175C22×C8—28no—
176C22×C8—28no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—44no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8—44no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D414yes—
202#117—28no—
203#117—28no—
204#117—28no—
205#117—28no—
206#117—28no—
207#117—28no—
208#117—28no—
209#117—28no—
210#117—44no—
211#117—44no—
212#117—44no—
213#117—44no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—84no—
262C2×C4⋊C4—84no—
263C2×C4⋊C4C22⋊C418yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C2×Q814yes—
268C2×C4⋊C4C2×Q812yes—
269C2×C4⋊C4C2×Q812yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4C22⋊C418yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C22×C414yes—
290C2×C42C2×D414yes—
291#95—216no—
292#95—84no—
293#95—84no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C2×C22⋊C412yes—
299C24C22⋊Q812yes—
300C24C22⋊Q812yes—
301C24#12312yes—
302C24#12312yes—
303C24#12412yes—
304C24#12712yes—
305C24C22×D411yescharacteristic
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4C2×C22⋊C418yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4#11714yes—
355C22×C4#11714yes—
356C22×C4C4×D414yes—
357C22×C4C4×D414yes—
358C22×C4C4×Q814yes—
359C22×C4C4×Q814yes—
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4C22⋊Q814yes—
363C22×C4C22⋊Q814yes—
364C22×C4C22⋊Q814yes—
365C22×C4C22⋊Q814yes—
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4#12314yes—
369C22×C4#12314yes—
370C22×C4#12314yes—
371C22×C4#12314yes—
372C22×C4#12314yes—
373C22×C4#12314yes—
374C22×C4#12312yes—
375C22×C4#12312yes—
376C22×C4#12414yes—
377C22×C4#12412yes—
378C22×C4#12514yes—
379C22×C4#12514yes—
380C22×C4#12514yes—
381C22×C4#12514yes—
382C22×C4#12614yes—
383C22×C4#12614yes—
384C22×C4#12614yes—
385C22×C4#12614yes—
386C22×C4#12712yes—
387C22×C4C4⋊Q814yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×Q811yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402M16—416no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—416no—
425C4⋊C4—216no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—816no—
453C22⋊C4—216no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—48no—
461C42—48no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23#38114yes—
479C23#39014yes—
480C23#39114yes—
481C23#39914yes—
482C23#39914yes—
483C23#40014yes—
484C23#40014yes—
485C23C22×C22⋊C411yescharacteristic
486C23C2×C22⋊Q811yescharacteristic
487C23C2×C22⋊Q811yescharacteristic
488C23C2×#12311yescharacteristic
489C23C2×#12311yescharacteristic
490C23C2×#12411yescharacteristic
491C23C2×#12711yescharacteristic
492C23#57612yes—
493C23#57712yes—
494C23#57712yes—
495C23#57812yes—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—816no—
516C2×C4—816no—
517C2×C4—84no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#38114yes—
536C2×C4#38718yes—
537C2×C4#38718yes—
538C2×C4#38718yes—
539C2×C4#38718yes—
540C2×C4#39014yes—
541C2×C4#39114yes—
542C2×C4#39718yes—
543C2×C4#39718yes—
544C2×C4#39818yes—
545C2×C4#39818yes—
546C2×C4#39914yes—
547C2×C4#39914yes—
548C2×C4#40014yes—
549C2×C4#40014yes—
550C2×C4C2×#11711yescharacteristic
551C2×C4C2×#11711yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×Q811yescharacteristic
555C2×C4C2×C4×Q811yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12111yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×C22⋊Q811yescharacteristic
561C2×C4C2×C22⋊Q811yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4C2×#12311yescharacteristic
564C2×C4C2×#12311yescharacteristic
565C2×C4C2×#12311yescharacteristic
566C2×C4C2×#12311yescharacteristic
567C2×C4C2×#12311yescharacteristic
568C2×C4C2×#12411yescharacteristic
569C2×C4C2×#12511yescharacteristic
570C2×C4C2×#12511yescharacteristic
571C2×C4C2×#12511yescharacteristic
572C2×C4C2×#12511yescharacteristic
573C2×C4C2×#12611yescharacteristic
574C2×C4C2×#12611yescharacteristic
575C2×C4C2×#12611yescharacteristic
576C2×C4C2×#12611yescharacteristic
577C2×C4C2×C4⋊Q811yescharacteristic
578C8—42no—
579C8—42no—
580C8—44no—
581C22—28no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—28no—
587C22—28no—
588C22—24no—
589C22—24no—
590C22—28no—
591C22#144518yes—
592C22C2×#38111yescharacteristic
593C22C2×#39011yescharacteristic
594C22C2×#39111yescharacteristic
595C22C2×#39911yescharacteristic
596C22C2×#39911yescharacteristic
597C22C2×#40011yescharacteristic
598C22C2×#40011yescharacteristic
599C22#289712yes—
600C22#289812yes—
601C22#308112yes—
602C22#308212yes—
603C22#308312yes—
604C22#308412yes—
605C22#309612yes—
606C22#309712yes—
607C22#309812yes—
608C22#309912yes—
609C22#313612yes—
610C22#313712yes—
611C22#315411yescharacteristic
612C22#315411yescharacteristic
613C22#315511yescharacteristic
614C22#315611yescharacteristic
615C4—48no—
616C4—84no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—84no—
622C4#144518yes—
623C4C2×#38711yescharacteristic
624C4C2×#38711yescharacteristic
625C4C2×#38711yescharacteristic
626C4C2×#38711yescharacteristic
627C4C2×#39711yescharacteristic
628C4C2×#39711yescharacteristic
629C4C2×#39811yescharacteristic
630C4C2×#39811yescharacteristic
631C2—28no—
632C2C2×#144511yescharacteristic
633C2#2070312yes—
634C2#2120511yescharacteristic
635C2#2120611yescharacteristic
636C2#2131512yes—
637C2#2373511yescharacteristic
638C2#2373611yescharacteristic
639C2#2373711yescharacteristic
640C2#2373811yescharacteristic
641C2#2374812yes—
642C2#2374912yes—
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. #11373C2semidirect
9. #11372C2semidirect
10. #11372C2semidirect
11. #11371C2semidirect
20. #2317C4semidirect
115. #384C2×C4semidirect
117. #382C2×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
#113732
#113722
#113712