Finite Group Databasebeta

← #216507

Subgroups — #216507

Order 512.

1303 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3#14685C211yesmaximal, characteristic, complemented
4#9913C212yesmaximal, complemented
5#9716C212yesmaximal, complemented
6#9113C212yesmaximal
7#9111C212yesmaximal, complemented
8#8572C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10C2×#529C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C22×C4⋊C8—21no—
13C22×C22⋊C8C2212yes—
14#2855C411yescharacteristic, complemented
15#2854C411yescharacteristic, complemented
16#2853—21no—
17#2852—21no—
18#2852—21no—
19#2851C411yescharacteristic, complemented
20#2851C2211yescharacteristic
21#2850C411yescharacteristic, complemented
22#2850C2211yescharacteristic
23#2849—21no—
24#2848C2211yescharacteristic
25#2847C2211yescharacteristic
26C4×C22⋊Q8C411yescharacteristic, complemented
27C4×C22⋊Q8C2211yescharacteristic
28C4×#123—21no—
29C4×#121C411yescharacteristic, complemented
30C4×#121C2211yescharacteristic
31C4×#120—21no—
32#2290—21no—
33#2290—21no—
34#2283C411yescharacteristic, complemented
35C2×#381—21no—
36C2×#380—21no—
37#1797—21no—
38#1797—21no—
39#1797C2212yes—
40#1797C2212yes—
41C2×#336—21no—
42C2×#328C2214yes—
43C2×#327C2214yes—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46#1434—21no—
47#1434C2212yes—
48C22⋊#95C2212yes—
49#1294C2214yes—
50#1294C2214yes—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55#529—44no—
56C2×#126—21no—
57C2×#125C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#123—21no—
60C2×#123—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×#121C2311yescharacteristic
67C2×#120—21no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—44no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—44no—
94C2×C22⋊C8C2314yes—
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#390C2×C411yescharacteristic
98#390C2×C411yescharacteristic
99#389C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#389C2×C411yescharacteristic
102#389C2×C411yescharacteristic
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#388—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#386—21no—
120#386—21no—
121#386—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#383—21no—
140#382C2×C411yescharacteristic
141#382C2×C411yescharacteristic
142#381—21no—
143#381—21no—
144#381—44no—
145#380—44no—
146#380C2×C411yescharacteristic
147#380C2×C411yescharacteristic
148#380C2×C411yescharacteristic
149#380C2×C411yescharacteristic
150#380C2311yescharacteristic
151#380C2311yescharacteristic
152#379—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4C2×C411yescharacteristic
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162#376—21no—
163C2×#95—22no—
164C2×#95—22no—
165C2×#95—22no—
166C2×#95—22no—
167C2×#95C2314yes—
168#336—22no—
169#336—22no—
170#328—44no—
171#327—44no—
172C25C2×D411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2411yescharacteristic
209C22×C8—24no—
210C22×C8—44no—
211C22×C8—44no—
212C22×C8—44no—
213#126—44no—
214#125—42no—
215#124—42no—
216#123—44no—
217#123—44no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222#121—22no—
223#121—22no—
224#121—42no—
225#121—42no—
226#120—42no—
227#120—42no—
228C4×Q8—42no—
229C4×D4—42no—
230C4×D4—44no—
231C4×D4—44no—
232C4×D4—44no—
233#117—44no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
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—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
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—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C4⋊C8—44no—
307C4⋊C8—44no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—44no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
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—24no—
332#95—24no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—41no—
343C24—41no—
344C24C2×C22⋊C411yescharacteristic
345C24C2×C22⋊C411yescharacteristic
346C24AGL(1,ℤ/8)12yes—
347C24#13712yes—
348C24C22×D411yescharacteristic
349C2×Q8—22no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
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—41no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4#9914yes—
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4#13514yes—
454C22×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—44no—
461C2×C8—88no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—88no—
480C22⋊C4—42no—
481C22⋊C4—42no—
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—41no—
491C22⋊C4—41no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—88no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—42no—
519C42—42no—
520C42—42no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23C2×#9912yes—
553C23#41712yes—
554C23#41712yes—
555C23#42112yes—
556C23C22×C22⋊C411yescharacteristic
557C23#53712yes—
558C23C2×#13512yes—
559C23C2×AGL(1,ℤ/8)11yescharacteristic
560C23C2×#13711yescharacteristic
561Q8—82no—
562D4—82no—
563D4—84no—
564D4—82no—
565D4—82no—
566D4—84no—
567C2×C4—24no—
568C2×C4—24no—
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—28no—
580C2×C4—28no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
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—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
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—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4C2×#9912yes—
648C2×C4C2×#9912yes—
649C2×C4#41012yes—
650C2×C4#41612yes—
651C2×C4#41612yes—
652C2×C4#42112yes—
653C2×C4#52012yes—
654C2×C4#53511yescharacteristic
655C2×C4#53611yescharacteristic
656C8—84no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—21no—
663C22—21no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—21no—
669C22—21no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—48no—
675C22—48no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22#151314yes—
688C22#231111yescharacteristic
689C22C22×#9911yescharacteristic
690C22C2×#41011yescharacteristic
691C22C2×#41611yescharacteristic
692C22C2×#41711yescharacteristic
693C22C2×#42111yescharacteristic
694C22C2×#42111yescharacteristic
695C22#324712yes—
696C22#324812yes—
697C4—21no—
698C4—21no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—81no—
714C4—81no—
715C4—81no—
716C4—164no—
717C4#151614yes—
718C2—22no—
719C2—22no—
720C2—24no—
721C2—42no—
722C2—161no—
723C2C2×#151311yescharacteristic
724C2C2×#151611yescharacteristic
725C2#1757311yescharacteristic
726C2#1758111yescharacteristic
727C2#2185711yescharacteristic
728C2#2187411yescharacteristic
729C2#2188711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. #14685C2semidirect
4. #9913C2semidirect
5. #9716C2semidirect
7. #9111C2semidirect
8. #8572C2semidirect
9. #8571C2semidirect
10. C2×#529C4semidirect
14. #2855C4semidirect
15. #2854C4semidirect
19. #2851C4semidirect
21. #2850C4semidirect
26. C4×C22⋊Q8C4semidirect
29. C4×#121C4semidirect
34. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#366212
C2×#17972
#146852
#99132
#97162
#91132
#91112
#85722
#85712