Finite Group Databasebeta

← #185049

Subgroups — #185049

Order 512.

2668 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×#1521C214yesmaximal, complemented
4#14683C211yesmaximal, characteristic, complemented
5#8330C214yesmaximal, complemented
6#8329C212yesmaximal, complemented
7#8328C212yesmaximal
8C2×#529—21no—
9C2×#529—21no—
10C2×#529—21no—
11C2×#529—21no—
12C2×#529—216no—
13C2×#529C412yescomplemented
14C2×#529C412yescomplemented
15C2×#529C2212yes—
16C2×#529C2212yes—
17C22×#126—21no—
18C22×#125C411yescharacteristic, complemented
19C22×#124C411yescharacteristic, complemented
20C22×#123—21no—
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C22×C4×Q8C411yescharacteristic, complemented
24C22×C4×D4—21no—
25C22×C4×D4C411yescharacteristic, complemented
26C22×#117—21no—
27C23×C42C2211yescharacteristic
28C2×#431C2212yescomplemented
29C2×#383C2212yescomplemented
30#1817C2212yescomplemented
31#1797—21no—
32#1797—21no—
33#1779C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8—21no—
36#1521—216no—
37C2×#328C2218yescomplemented
38C2×#327C2218yes—
39#1434—21no—
40#1434—21no—
41#1434C2212yescomplemented
42C22⋊#95—21no—
43C22⋊#95—21no—
44C22⋊#95C2212yes—
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4—21no—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52#529—216no—
53#529—216no—
54#529—216no—
55#529—216no—
56#529—28no—
57#529—28no—
58#529—28no—
59#529—28no—
60#529—28no—
61#529—28no—
62#529—28no—
63#529D4116yes—
64C2×#126—216no—
65C2×#126—22no—
66C2×#126—22no—
67C2×#126—22no—
68C2×#126—22no—
69C2×#125—28no—
70C2×#125—21no—
71C2×#125—21no—
72C2×#125C2×C412yes—
73C2×#125C2×C412yes—
74C2×#124—28no—
75C2×#124—21no—
76C2×#124—21no—
77C2×#124C2×C412yes—
78C2×#124C2×C412yes—
79C2×#123—216no—
80C2×#123—22no—
81C2×#123—22no—
82C2×#123—22no—
83C2×#123—22no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8C2×C412yes—
88C2×C22⋊Q8C2312yes—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121—28no—
92C2×#121C2×C412yes—
93C2×#121C2312yes—
94C2×C4×Q8—28no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8C2×C412yes—
98C2×C4×Q8C2×C412yes—
99C2×C4×D4—216no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—28no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2×C412yes—
108C2×C4×D4C2×C412yes—
109C2×#117—216no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117—22no—
113C2×#117—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2×C411yescharacteristic
119C22×C4⋊C4C2×C411yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—22no—
126C22×C42—21no—
127C22×C42—28no—
128C22×C42C2×C411yescharacteristic
129C22×C42C2×C411yescharacteristic
130C22×C42C2×C411yescharacteristic
131C22×C42C2×C411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#431—28no—
141C2×C4⋊C8—22no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8C2314yes—
145C2×C4×C8—22no—
146#383—28no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95C2314yes—
153#336—22no—
154#336—22no—
155#328—216no—
156#327—216no—
157C25—21no—
158C25C2×D411yescharacteristic
159C22×Q8—24no—
160C22×Q8—21no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8C22×C412yes—
164C22×D4—28no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—21no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4C22×C412yes—
174C23×C4—28no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
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×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C22×C8—24no—
211#126—216no—
212#126—216no—
213#126—216no—
214#126—216no—
215#125—28no—
216#125—28no—
217#125—24no—
218#125—24no—
219#125—24no—
220#125C22⋊C418yescomplemented
221#124—28no—
222#124—28no—
223#124—24no—
224#124—24no—
225#124—24no—
226#124C22⋊C418yescomplemented
227#123—216no—
228#123—216no—
229#123—216no—
230#123—216no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8C2×D418yes—
237#121—28no—
238#121—28no—
239#121—24no—
240#121—24no—
241#121—24no—
242#121C2×D418yes—
243C4×Q8—28no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8C22⋊C418yescomplemented
249C4×D4—216no—
250C4×D4—216no—
251C4×D4—28no—
252C4×D4—28no—
253C4×D4—216no—
254C4×D4—24no—
255C4×D4—216no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4C22⋊C418yescomplemented
259#117—216no—
260#117—216no—
261#117—216no—
262#117—216no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C412yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C42—28no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42C22⋊C411yescharacteristic
314C2×C42C22⋊C411yescharacteristic
315C2×C42C22⋊C411yescharacteristic
316C2×C42C22⋊C411yescharacteristic
317C2×C42C22⋊C411yescharacteristic
318C2×C42C22⋊C411yescharacteristic
319C2×C42C22⋊C411yescharacteristic
320C2×C42C22⋊C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D418yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C4⋊C8—44no—
331C22⋊C8—28no—
332C22⋊C8—44no—
333C22⋊C8—44no—
334C4×C8—44no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—41no—
348C24—41no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24C2×C22⋊C411yescharacteristic
356C24C2×C22⋊C411yescharacteristic
357C24#13514yes—
358C24C22×D411yescharacteristic
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—416no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8C2×C22⋊C414yes—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—28no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—416no—
374C2×D4—42no—
375C2×D4—432no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—42no—
381C2×D4C2×C22⋊C414yes—
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—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
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—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—432no—
444C22×C4—416no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C411yescharacteristic
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4#12014yes—
473C22×C4#12014yes—
474C22×C4C23×C411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D414yes—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8—44no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4C2×C22⋊C414yes—
493C4⋊C4C2×C22⋊C414yes—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—88no—
505C42—28no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—28no—
519C42C2×C22⋊C414yes—
520C42C2×C22⋊C414yes—
521C42#12014yes—
522C42#12014yes—
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—21no—
536C23—28no—
537C23—21no—
538C23—28no—
539C23—24no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23#41012yes—
559C23#41612yes—
560C23#41612yes—
561C23#42112yes—
562C23C22×C22⋊C411yescharacteristic
563C23C2×#12012yes—
564C23C2×#13512yes—
565C23#57512yes—
566Q8—44no—
567Q8—44no—
568Q8—48no—
569D4—44no—
570D4—44no—
571D4—48no—
572D4—416no—
573D4—416no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
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—28no—
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—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—416no—
623C2×C4—416no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#37912yes—
642C2×C4#37912yes—
643C2×C4#37912yes—
644C2×C4#37912yes—
645C2×C4#41012yes—
646C2×C4#42112yes—
647C2×C4C22×C22⋊C412yes—
648C2×C4C2×#12012yes—
649C8—84no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—416no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22#152114yescomplemented
679C22C2×#37911yescharacteristic
680C22C2×#41011yescharacteristic
681C22C2×#41011yescharacteristic
682C22C2×#41611yescharacteristic
683C22#289611yescharacteristic
684C22C2×#42111yescharacteristic
685C22C2×#42111yescharacteristic
686C22#300214yes—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—41no—
696C4—44no—
697C4—41no—
698C4—41no—
699C4—44no—
700C4—41no—
701C4—44no—
702C4—164no—
703C2—24no—
704C2—24no—
705C2—42no—
706C2—84no—
707C2C2×#152111yescharacteristic
708C2#1756711yescharacteristic
709C2#1757911yescharacteristic
710C2#1758011yescharacteristic
711C2#2058411yescharacteristic
712C2#2059711yescharacteristic
713C2#2060511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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. #20714C2semidirect
3. C2×#1521C2semidirect
4. #14683C2semidirect
5. #8330C2semidirect
6. #8329C2semidirect
13. C2×#529C4semidirect
14. C2×#529C4semidirect
18. C22×#125C4semidirect
19. C22×#124C4semidirect
23. C22×C4×Q8C4semidirect
25. C22×C4×D4C4semidirect
28. C2×#431C22semidirect
29. C2×#383C22semidirect
30. #1817C22semidirect
37. C2×#328C22semidirect
41. #1434C22semidirect
220. #125C22⋊C4semidirect
226. #124C22⋊C4semidirect
248. C4×Q8C22⋊C4semidirect
258. C4×D4C22⋊C4semidirect
678. C22#1521semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
#207142
C2×#15212
#146832
#83302
#83292
#83282