Finite Group Databasebeta

← #356984

Subgroups — #356984

Order 512.

1718 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
1C2×#2365C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1956C211yesmaximal, characteristic
4C2×C8×C4⋊C4C211yesmaximal, characteristic
5C2×C4×#103C212yesmaximal
6C4⋊C42C211yesmaximal, characteristic
7#10932C212yesmaximal
8#10926C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10789C212yesmaximal
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C4×C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2365—24no—
22#2365C418yescomplemented
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#387—22no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1981C418yescomplemented
34#1956C418yescomplemented
35C8×C4⋊C4—24no—
36C4×#103—28no—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1436—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⋊Q8C2×C414yes—
56C2×C4⋊Q8C2×C414yes—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8C2×C418yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—22no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8C2×C414yes—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103C2×C418yes—
93C2×#103C2×C418yes—
94C2×#103C2×C418yes—
95C2×#103C2×C418yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—24no—
99C2×C4×C8—24no—
100C2×C4×C8—42no—
101C2×C4×C8C2×C414yes—
102#391—24no—
103#391C2×C418yes—
104#389—28no—
105#387—216no—
106#384—22no—
107#384C2×C414yes—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—42no—
113C4×C4⋊C4—44no—
114C4×C4⋊C4C2×C414yes—
115C4×C4⋊C4C2×C414yes—
116C4×C4⋊C4C2×C414yes—
117C4×C4⋊C4D414yes—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D412yes—
126#340—22no—
127#340—24no—
128#340—22no—
129#340—22no—
130#340—42no—
131#340D414yes—
132#336—24no—
133C22×Q8—22no—
134C22×Q8—22no—
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×Q8C22×C414yes—
145C22×Q8C22×C414yes—
146C22×Q8C22×C414yes—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C22×C412yes—
175C22×C8C22×C414yes—
176C22×C8C22×C412yes—
177C22×C8C2×D412yes—
178C22×C8#4112yes—
179C4⋊Q8—28no—
180C4⋊Q8—28no—
181C4⋊Q8—28no—
182C4⋊Q8C42116yescomplemented
183C4×Q8—216no—
184C4×Q8—48no—
185C4×Q8—48no—
186C4×Q8—48no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4C22⋊C414yes—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4SD1614yes—
228C2×C4⋊C4Q1614yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4#4112yes—
237C2×C4⋊C4#4112yes—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—28no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—82no—
254C2×C42C22×C414yes—
255C2×C42C22×C412yes—
256C2×C42C22×C412yes—
257C2×C42C22×C412yes—
258C2×C42C2×D414yes—
259C2×C42C2×D412yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262C4⋊C8—28no—
263#103—216no—
264#103—216no—
265#103—48no—
266#103—48no—
267C4×C8—28no—
268C4×C8—48no—
269#95—28no—
270#95—24no—
271#95—28no—
272#95—24no—
273#95—44no—
274#95—44no—
275#95—82no—
276#95C22⋊C418yes—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
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—42no—
294C2×Q8—42no—
295C2×Q8—42no—
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×Q8C2×C4218yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4C2×C22⋊C414yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C414yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4C2×SD1614yes—
371C22×C4C2×SD1612yes—
372C22×C4C2×Q1614yes—
373C22×C4C2×Q1612yes—
374C22×C4#13512yes—
375C22×C4#13512yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C23×C411yescharacteristic
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C42−1+411yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8C4×D414yes—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—216no—
418C4⋊C4—216no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—416no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—84no—
435C4⋊C4#103116yescomplemented
436C4⋊C4C4×D414yes—
437C4⋊C4C4×D414yes—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—416no—
443C42—48no—
444C42—48no—
445C42—88no—
446C42C2×C4214yes—
447C42#11714yes—
448C42C4×D414yes—
449C42C4×D414yes—
450C23C22×C22⋊C411yescharacteristic
451C23C2×C4×D411yescharacteristic
452C23C2×C4×D411yescharacteristic
453C23D4211yescharacteristic
454C23#54611yescharacteristic
455C23#54811yescharacteristic
456C23#55011yescharacteristic
457C23C22×SD1611yescharacteristic
458C23C22×Q1611yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×#13511yescharacteristic
461C23#57711yescharacteristic
462C23#57711yescharacteristic
463C23#57811yescharacteristic
464C23#57811yescharacteristic
465Q8—28no—
466Q8—28no—
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—44no—
472Q8—44no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4C4×C22⋊C414yes—
535C2×C4C4×C22⋊C414yes—
536C2×C4C2×#10318yes—
537C2×C4C2×#10314yes—
538C2×C4#41614yes—
539C2×C4C4×SD1614yes—
540C2×C4C4×SD1614yes—
541C2×C4C4×Q1614yes—
542C2×C4C4×Q1614yes—
543C2×C4C22×C4211yescharacteristic
544C2×C4C2×#11711yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C4×#4111yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4C4×#4111yescharacteristic
553C2×C4#51811yescharacteristic
554C2×C4#51811yescharacteristic
555C2×C4#51811yescharacteristic
556C2×C4#52011yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4#52011yescharacteristic
559C2×C4D4211yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54711yescharacteristic
562C2×C4#54811yescharacteristic
563C2×C4#55011yescharacteristic
564C2×C4#55011yescharacteristic
565C2×C4#55311yescharacteristic
566C2×C4#55311yescharacteristic
567C2×C4#55511yescharacteristic
568C2×C4#55611yescharacteristic
569C2×C4#55911yescharacteristic
570C2×C4#55911yescharacteristic
571C8—24no—
572C8—42no—
573C8—44no—
574C22C2×C4×C22⋊C411yescharacteristic
575C22D4×C22⋊C411yescharacteristic
576C22#234111yescharacteristic
577C22#236111yescharacteristic
578C22#236311yescharacteristic
579C22#236611yescharacteristic
580C22#236711yescharacteristic
581C22C22×#10311yescharacteristic
582C22C2×#41611yescharacteristic
583C22#289811yescharacteristic
584C22#289811yescharacteristic
585C22C2×C4×SD1611yescharacteristic
586C22C2×C4×SD1611yescharacteristic
587C22C2×C4×Q1611yescharacteristic
588C22C2×C4×Q1611yescharacteristic
589C22#294711yescharacteristic
590C22#294711yescharacteristic
591C22#294911yescharacteristic
592C22#295011yescharacteristic
593C22D4×SD1611yescharacteristic
594C22#328311yescharacteristic
595C22#328411yescharacteristic
596C22D4×Q1611yescharacteristic
597C22#329811yescharacteristic
598C22#329911yescharacteristic
599C22#330211yescharacteristic
600C22#330311yescharacteristic
601C22#335911yescharacteristic
602C22#336011yescharacteristic
603C22#336311yescharacteristic
604C22#336411yescharacteristic
605C22#339211yescharacteristic
606C22#339311yescharacteristic
607C22#339411yescharacteristic
608C22#339511yescharacteristic
609C4—28no—
610C4—24no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—48no—
615C4—82no—
616C4—82no—
617C4—84no—
618C4C4×#10318yescomplemented
619C4C42×D411yescharacteristic
620C4#227511yescharacteristic
621C4#235411yescharacteristic
622C4#235711yescharacteristic
623C4#236111yescharacteristic
624C4#236311yescharacteristic
625C4#236611yescharacteristic
626C4#236711yescharacteristic
627C2#1382011yescharacteristic
628C2C2×C4×#10311yescharacteristic
629C2#2041711yescharacteristic
630C2D4×#10311yescharacteristic
631C2#2241711yescharacteristic
632C2#2254111yescharacteristic
633C2#2254211yescharacteristic
634C2#2299611yescharacteristic
635C2#2299811yescharacteristic
636C2#2309411yescharacteristic
637C2#2309611yescharacteristic
638C2#2318211yescharacteristic
639C2#2318411yescharacteristic
640C2#2321411yescharacteristic
641C2#2321711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×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
22. #2365C4semidirect
33. #1981C4semidirect
34. #1956C4semidirect
182. C4⋊Q8C42semidirect
435. C4⋊C4#103semidirect
618. C4C4×#103semidirect

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×C8×C4⋊C42
C2×C4×#1032
C4⋊C422
#109322
#109262
#108492
#107892
C4×#3402