Finite Group Databasebeta

← #209212

Subgroups — #209212

Order 512.

1028 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic, complemented
2#21486C211yesmaximal, characteristic
3C4×#448C211yesmaximal, characteristic
4C2×#1582C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8948C211yesmaximal, characteristic, complemented
8#8944C211yesmaximal, characteristic
9#8850C211yesmaximal, characteristic, complemented
10#8846C211yesmaximal, characteristic, complemented
11#8833C211yesmaximal, characteristic, complemented
12#8311C211yesmaximal, characteristic, complemented
13#8207C211yesmaximal, characteristic, complemented
14#8203C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#448C411yescharacteristic
18Q8×M16—24no—
19Q8×M16—24no—
20C2×#446—21no—
21C2×#446—21no—
22C2×#446C2211yescharacteristic
23C2×#446C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C4×SD16C2211yescharacteristic
27C2×C4×SD16C2211yescharacteristic
28C2×#435—21no—
29C2×C8×D4C2211yescharacteristic
30C2×#423—21no—
31C2×#423C2211yescharacteristic
32C2×C4×M16—21no—
33C2×C4×M16C2211yescharacteristic
34C4×#121C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1924—21no—
38#1880C411yescharacteristic
39#1880C411yescharacteristic
40#1874C411yescharacteristic
41#1870—21no—
42#1865C411yescharacteristic
43#1865C411yescharacteristic
44#1853C411yescharacteristic
45#1797C2211yescharacteristic
46#1787C411yescharacteristic
47#1770C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#103C2211yescharacteristic
50#1752—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1582—24no—
54C8×SD16—24no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C23×Q8C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C22×M16—21no—
75C22×M16D411yescharacteristic
76C22×M16D411yescharacteristic
77C23×C8—21no—
78C2×#121C2×C411yescharacteristic
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×Q8C2311yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4C2311yescharacteristic
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95#448—24no—
96#446—28no—
97#446—28no—
98#446—22no—
99#446—22no—
100#446—22no—
101#446D414yescomplemented
102C8×Q8—22no—
103C8×Q8—22no—
104C8×Q8—22no—
105C8×Q8D414yescomplemented
106C4×SD16—22no—
107C4×SD16—22no—
108C4×SD16—22no—
109C4×SD16—22no—
110#435—44no—
111C8×D4—22no—
112C8×D4—22no—
113C2×#106C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115#423—28no—
116#423—22no—
117#423—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8C2×C411yescharacteristic
142C2×C22⋊C8C2×C411yescharacteristic
143C4×M16—28no—
144C4×M16—22no—
145C4×M16—22no—
146C2×#97—21no—
147C2×#97—21no—
148C2×#97—21no—
149C2×#97D411yescharacteristic
150C2×#97D411yescharacteristic
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#390C2×C411yescharacteristic
160#389C2×C411yescharacteristic
161#388—21no—
162#386—21no—
163#380C2×C411yescharacteristic
164C4×C4⋊C4C2311yescharacteristic
165C4×C22⋊C4—21no—
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#340C2×C411yescharacteristic
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176#334—22no—
177#334—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181Q8⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184#322—22no—
185C82—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—41no—
192C22×Q8—41no—
193C22×Q8C22×C411yescharacteristic
194C22×Q8C22×C411yescharacteristic
195C22×D4—21no—
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4#4111yescharacteristic
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—22no—
207C2×SD16—41no—
208C2×SD16M1614yescomplemented
209C2×M16—24no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—41no—
215C2×M16—42no—
216C2×M16—42no—
217C2×M16SD1614yescomplemented
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8C22×C411yescharacteristic
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235#121—22no—
236#121—41no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C2×D412yes—
250C4×Q8C2×D412yes—
251C4×Q8#4112yes—
252C4×Q8#4112yes—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—42no—
258C4×D4—42no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42C2411yescharacteristic
294#106—22no—
295#106—22no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—24no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—42no—
308C4⋊C8C2×D412yes—
309C4⋊C8C2×D412yes—
310#103—22no—
311#103—22no—
312#103—22no—
313#103—22no—
314#103—22no—
315#103—41no—
316#102—22no—
317#102—22no—
318C22⋊C8—22no—
319C22⋊C8—41no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—42no—
323C22⋊C8—42no—
324C22⋊C8—42no—
325#97—24no—
326#97—24no—
327#97—22no—
328#97—22no—
329#97—42no—
330C4×C8—24no—
331C4×C8—24no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8C2×D412yes—
343C4×C8C2×D412yes—
344#95—21no—
345#95—21no—
346#95—21no—
347C24—21no—
348C24C4×D411yescharacteristic
349C2×Q8—24no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8C4×D412yes—
365C2×Q8C4×D412yes—
366C2×Q8C2×M1612yes—
367C2×Q8#13112yes—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4C2×M1612yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C2×SD1612yes—
414C22×C4#13512yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418SD16—28no—
419SD16—44no—
420M16—28no—
421M16—44no—
422M16—48no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—84no—
460C2×C8#12012yes—
461C2×C8#12012yes—
462C2×C8C2×M1612yes—
463C2×C8C2×SD1612yes—
464C2×C8C2×SD1612yes—
465C2×C8#13712yes—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4C4×D412yes—
483C4⋊C4C4×D412yes—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C42—24no—
491C42—24no—
492C42—21no—
493C42—21no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—22no—
498C42—22no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—41no—
503C42—41no—
504C42C22×D411yescharacteristic
505C42C22×D411yescharacteristic
506C42C2×#4111yescharacteristic
507C42C2×#4111yescharacteristic
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C4×SD1612yes—
522C23C2×C4×D411yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8—22no—
528Q8—48no—
529Q8—41no—
530Q8—41no—
531Q8#43514yescomplemented
532D4—22no—
533D4—22no—
534D4—41no—
535D4—41no—
536D4—84no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—84no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4C4×SD1612yes—
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×#12011yescharacteristic
588C2×C4#52511yescharacteristic
589C2×C4#52511yescharacteristic
590C2×C4#52511yescharacteristic
591C2×C4C22×M1611yescharacteristic
592C2×C4C2×#13111yescharacteristic
593C2×C4#56811yescharacteristic
594C2×C4#56811yescharacteristic
595C2×C4C22×SD1611yescharacteristic
596C2×C4C2×#13511yescharacteristic
597C2×C4C2×#13711yescharacteristic
598C2×C4#57511yescharacteristic
599C8—22no—
600C8—22no—
601C8—22no—
602C8—42no—
603C8—41no—
604C8—41no—
605C8—41no—
606C8—41no—
607C8—42no—
608C8—42no—
609C8—81no—
610C8—81no—
611C8—81no—
612C8#44814yescomplemented
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22C4×#12011yescharacteristic
628C22#293311yescharacteristic
629C22D4×M1611yescharacteristic
630C22C2×C4×SD1611yescharacteristic
631C22C4×#13711yescharacteristic
632C22#295611yescharacteristic
633C22#295711yescharacteristic
634C4—24no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4C2×#43511yescharacteristic
647C4#293011yescharacteristic
648C4#293311yescharacteristic
649C4D4×M1611yescharacteristic
650C4C2×#44811yescharacteristic
651C4#300211yescharacteristic
652C4#310311yescharacteristic
653C4#310811yescharacteristic
654C2—42no—
655C2—82no—
656C2#1799711yescharacteristic
657C2#1801011yescharacteristic
658C2M16×SD1611yescharacteristic
659C2#1804711yescharacteristic
660C2#2143911yescharacteristic
661C2C4×#44811yescharacteristic
662C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×M16C2semidirect
4. C2×#1582C2semidirect
5. C2×C8×SD16C2semidirect
6. #8952C2semidirect
7. #8948C2semidirect
9. #8850C2semidirect
10. #8846C2semidirect
11. #8833C2semidirect
12. #8311C2semidirect
13. #8207C2semidirect
14. #8203C2semidirect
15. #8146C2semidirect
101. #446D4semidirect
105. C8×Q8D4semidirect
208. C2×SD16M16semidirect
217. C2×M16SD16semidirect
531. Q8#435semidirect
612. C8#448semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
#214862
C4×#4482
C2×#15822
C2×C8×SD162
#89522
#89482
#89442
#88502
#88462
#88332
#83112
#82072
#82032
#81462