Finite Group Databasebeta

← #161072

Subgroups — #161072

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4#7646C212yesmaximal, complemented
5#7645C212yesmaximal, complemented
6#7623C212yesmaximal, complemented
7#7620C212yesmaximal, complemented
8#7589C212yesmaximal, complemented
9#7586C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C2×C4×#41C411yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#432—21no—
14C22×C22⋊C8—21no—
15C4×C22⋊Q8C411yescharacteristic
16C4×C22⋊Q8C411yescharacteristic
17C4×C22⋊Q8C2211yescharacteristic
18C4×C22⋊Q8C2211yescharacteristic
19C4×#123—21no—
20C4×#123—21no—
21C4×#121C411yescharacteristic
22C4×#121C411yescharacteristic
23C4×#121C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#120—21no—
26C4×#120—21no—
27#2297C2211yescharacteristic
28#2296C2211yescharacteristic
29#2290—21no—
30#2290—21no—
31#2280C411yescharacteristic
32C42×D4—21no—
33C42×D4—21no—
34C4×#117C411yescharacteristic
35#1817—21no—
36#1817—21no—
37#1817C2212yes—
38#1817C2212yes—
39#1817C2212yes—
40#1797—21no—
41#1797C2212yes—
42#1779—21no—
43#1749—21no—
44#1749C2212yes—
45C2×#336—21no—
46C2×#336C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C23×C8—22no—
56#525—24no—
57#525—24no—
58C2×#123—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×#121C2×C411yescharacteristic
62C2×#121C2×C411yescharacteristic
63C2×#120—21no—
64C4×#41—44no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C22×C42—21no—
82C22×C42—44no—
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#432—44no—
96C2×C4⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—44no—
103C2×#97—22no—
104#391C2×C411yescharacteristic
105#391C2311yescharacteristic
106#390C2×C411yescharacteristic
107#390C2311yescharacteristic
108#389C2×C411yescharacteristic
109#389C2×C411yescharacteristic
110#389C2311yescharacteristic
111#389C2311yescharacteristic
112#388—21no—
113#388—21no—
114#388—21no—
115#388—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#386—21no—
121#386—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#385—21no—
127#385—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#381—21no—
133#380C2×C411yescharacteristic
134#380C2×C411yescharacteristic
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137#379—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162#376—21no—
163C43—21no—
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336C2314yes—
169#336C2314yes—
170C25C22⋊C411yescharacteristic
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—42no—
174C2×#41—42no—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C22×C411yescharacteristic
180C23×C4—24no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—44no—
214C22×C8—44no—
215C22×C8—44no—
216#123—28no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8M1614yes—
221#121—22no—
222#121—22no—
223#121—22no—
224#121M1614yes—
225#120—28no—
226C4×Q8—42no—
227C4×Q8—42no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—44no—
233C4×D4—44no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—44no—
237C4×D4—44no—
238#117—22no—
239#117—22no—
240#117—44no—
241#117—42no—
242#117—42no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2411yescharacteristic
312C4⋊C8—44no—
313C22⋊C8—44no—
314C22⋊C8—44no—
315C22⋊C8—44no—
316C22⋊C8—44no—
317C22⋊C8—44no—
318#97—44no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—41no—
342C24—41no—
343C24C2×C22⋊C411yescharacteristic
344C24C2×C22⋊C411yescharacteristic
345C24C2×C22⋊C411yescharacteristic
346#41—48no—
347#41—48no—
348C2×Q8—21no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8C2×M1612yes—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4C2×M1612yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—416no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4#9914yes—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×M1612yes—
449C22×C4C2×M1612yes—
450C22×C4#13514yes—
451C22×C4C23×C411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C2×C8—44no—
455C2×C8—44no—
456C2×C8—44no—
457C2×C8—44no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—88no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4C2×M1612yes—
476C4⋊C4C2×M1612yes—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—42no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—28no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23#40712yes—
550C23C2×#9912yes—
551C23#41612yes—
552C23#42112yes—
553C23#42112yes—
554C23C22×C22⋊C411yescharacteristic
555Q8—42no—
556Q8—42no—
557D4—42no—
558D4—42no—
559D4—48no—
560D4—48no—
561D4—48no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4#40712yes—
636C2×C4C2×#9912yes—
637C2×C4C2×#9912yes—
638C2×C4#41312yes—
639C2×C4#41612yes—
640C2×C4C22×M1611yescharacteristic
641C2×C4C22×M1611yescharacteristic
642C2×C4C2×#13512yes—
643C2×C4#57512yes—
644C8—84no—
645C8—84no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—416no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22#148814yes—
675C22C2×#40711yescharacteristic
676C22C22×#9911yescharacteristic
677C22C2×#41311yescharacteristic
678C22C2×#41611yescharacteristic
679C22#289611yescharacteristic
680C22C2×#42111yescharacteristic
681C22C2×#42111yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—21no—
689C4—21no—
690C4—22no—
691C4—42no—
692C4—42no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—42no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—81no—
702C2—24no—
703C2—42no—
704C2—42no—
705C2—82no—
706C2#1736211yescharacteristic
707C2C2×#148811yescharacteristic
708C2#1745211yescharacteristic
709C2#1757211yescharacteristic
710C2#1758711yescharacteristic
711C2#1767811yescharacteristic
712C2#1771611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. C2×#1817C2semidirect
4. #7646C2semidirect
5. #7645C2semidirect
6. #7623C2semidirect
7. #7620C2semidirect
8. #7589C2semidirect
9. #7586C2semidirect
47. #1272C22semidirect
48. #1271C22semidirect
49. #1271C22semidirect
50. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#207152
C2×#18172
#76462
#76452
#76232
#76202
#75892
#75862