Finite Group Databasebeta

← #368317

Subgroups — #368317

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1956C211yesmaximal, characteristic
4C2×#1865C212yesmaximal
5C2×C4×#106C211yesmaximal, characteristic
6#14151C211yesmaximal, characteristic
7#11122C212yesmaximal
8#11067C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10834C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2365—24no—
23#2365—24no—
24C2×#391C2212yes—
25C2×#389—21no—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#387—22no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1981—24no—
35#1956C418yescomplemented
36#1865—28no—
37C4×#106—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×Q8C2312yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8Q814yes—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—22no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106D414yes—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103C2×C418yes—
101C2×#103C2×C418yes—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8C2×C414yes—
105#391—24no—
106#391—24no—
107#389—28no—
108#389—24no—
109#389—42no—
110#389—44no—
111#387—216no—
112#384—22no—
113#384C2×C414yes—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D414yes—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95D412yes—
129#340—24no—
130#340—22no—
131#340—22no—
132#340—42no—
133#336—24no—
134#336—42no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C22×C414yes—
147C22×Q8C2×D414yes—
148C22×Q8C2×Q814yes—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C22×C412yes—
178C22×C8C22×C412yes—
179C22×C8C2×D412yes—
180C22×C8#4112yes—
181C4⋊Q8—28no—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8C4⋊C4116yescomplemented
185C4×Q8—48no—
186C4×Q8—48no—
187C4×Q8—48no—
188C4×Q8—48no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4—81no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4SD1614yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4#4112yes—
242C2×C4⋊C4#4112yes—
243C2×C42—28no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264#106—28no—
265#106—28no—
266C4⋊C8—44no—
267#103—216no—
268#103—48no—
269#103—48no—
270#103—48no—
271C4×C8—28no—
272#95—28no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—44no—
277#95—42no—
278#95—44no—
279#95—82no—
280C24C22×D411yescharacteristic
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8C2×C4⋊C418yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4C22⋊Q812yes—
373C22×C4C22⋊Q812yes—
374C22×C4C2×SD1614yes—
375C22×C4C2×SD1612yes—
376C22×C4#13512yes—
377C22×C4#13712yes—
378C22×C4#13712yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C42−1+411yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8C4×D414yes—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—216no—
418C4⋊C4—216no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—416no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—84no—
440C4⋊C4—88no—
441C4⋊C4#12114yes—
442C42—416no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—48no—
448C42—84no—
449C42C2×C4⋊C414yes—
450C42C4×D414yes—
451C42C22⋊Q814yes—
452C42#12314yes—
453C23C2×C4×D411yescharacteristic
454C23C2×#12011yescharacteristic
455C23C2×C22⋊Q811yescharacteristic
456C23D4211yescharacteristic
457C23#54611yescharacteristic
458C23#54811yescharacteristic
459C23#55011yescharacteristic
460C23C22×SD1611yescharacteristic
461C23C2×#13511yescharacteristic
462C23C2×#13711yescharacteristic
463C23C2×#13711yescharacteristic
464C23#57711yescharacteristic
465C23#57711yescharacteristic
466C23#57811yescharacteristic
467C23#57811yescharacteristic
468Q8—28no—
469Q8—28no—
470Q8—44no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4#38514yes—
539C2×C4#38514yes—
540C2×C4C4×SD1614yes—
541C2×C4#44114yes—
542C2×C4#44814yes—
543C2×C4#45214yes—
544C2×C4#47514yes—
545C2×C4#47914yes—
546C2×C4C22×C4⋊C411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C4×#4111yescharacteristic
550C2×C4#51811yescharacteristic
551C2×C4#52011yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×C22⋊Q811yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C2×C4#52511yescharacteristic
556C2×C4#52911yescharacteristic
557C2×C4#53211yescharacteristic
558C2×C4#53511yescharacteristic
559C2×C4#53711yescharacteristic
560C2×C4#54011yescharacteristic
561C2×C4#54211yescharacteristic
562C2×C4D4211yescharacteristic
563C2×C4#54611yescharacteristic
564C2×C4#54611yescharacteristic
565C2×C4#54711yescharacteristic
566C2×C4D4×Q811yescharacteristic
567C2×C4#55011yescharacteristic
568C2×C4#55411yescharacteristic
569C2×C4#55411yescharacteristic
570C2×C4#55411yescharacteristic
571C2×C4#55611yescharacteristic
572C2×C4#55911yescharacteristic
573C2×C4#55911yescharacteristic
574C8—24no—
575C8—42no—
576C8—82no—
577C22C2×#38511yescharacteristic
578C22#236611yescharacteristic
579C22#236711yescharacteristic
580C22#240711yescharacteristic
581C22#240811yescharacteristic
582C22#245211yescharacteristic
583C22#245411yescharacteristic
584C22C2×C4×SD1611yescharacteristic
585C22C2×#44111yescharacteristic
586C22#294911yescharacteristic
587C22#295011yescharacteristic
588C22C2×#44811yescharacteristic
589C22C2×#45211yescharacteristic
590C22#301311yescharacteristic
591C22#301311yescharacteristic
592C22C2×#47511yescharacteristic
593C22C2×#47911yescharacteristic
594C22#308211yescharacteristic
595C22#308211yescharacteristic
596C22#327611yescharacteristic
597C22#327611yescharacteristic
598C22D4×SD1611yescharacteristic
599C22#328411yescharacteristic
600C22#329811yescharacteristic
601C22#330211yescharacteristic
602C22#332911yescharacteristic
603C22#333111yescharacteristic
604C22#335911yescharacteristic
605C22#336411yescharacteristic
606C22#336911yescharacteristic
607C22#337211yescharacteristic
608C22#339211yescharacteristic
609C22#339411yescharacteristic
610C22#339811yescharacteristic
611C22#339811yescharacteristic
612C4—24no—
613C4—44no—
614C4—44no—
615C4—48no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4—82no—
620C4—82no—
621C4#186518yescomplemented
622C4C4×#12111yescharacteristic
623C4#232411yescharacteristic
624C4D4×C4⋊C411yescharacteristic
625C4#234911yescharacteristic
626C4#244511yescharacteristic
627C4#244711yescharacteristic
628C4#253811yescharacteristic
629C4#258011yescharacteristic
630C2#1401811yescharacteristic
631C2C2×#186511yescharacteristic
632C2#2088011yescharacteristic
633C2#2318211yescharacteristic
634C2#2318711yescharacteristic
635C2#2321411yescharacteristic
636C2#2322111yescharacteristic
637C2#2418711yescharacteristic
638C2#2418911yescharacteristic
639C2#2423911yescharacteristic
640C2#2424111yescharacteristic
641C2#2646211yescharacteristic
642C2#2646511yescharacteristic
643C2#2652811yescharacteristic
644C2#2653111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
35. #1956C4semidirect
184. C4⋊Q8C4⋊C4semidirect
621. C4#1865semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#19812
C2×#19562
C2×#18652
C2×C4×#1062
#141512
#111222
#110672
#108492
#108342
#107892