Finite Group Databasebeta

← #188084

Subgroups — #188084

Order 512.

2568 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2#20727C212yesmaximal, complemented
3#17364C212yesmaximal, complemented
4#17357C212yesmaximal, complemented
5#8556C212yesmaximal, complemented
6#8553C212yesmaximal, complemented
7#8552C212yesmaximal, complemented
8#8550C212yesmaximal, complemented
9C2×#529—24no—
10C2×#529—24no—
11C2×#529—24no—
12C2×#529—24no—
13C2×#529—24no—
14C2×#529—24no—
15C22×#126C411yescharacteristic
16C22×#126C2211yescharacteristic
17C22×#125C411yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#123C411yescharacteristic
20C22×#123C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C411yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C411yescharacteristic
25C22×C4×D4C411yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×#117C411yescharacteristic
28C22×#117C2211yescharacteristic
29C23×C42C411yescharacteristic
30#2920C2214yes—
31#1821C2214yes—
32#1801—24no—
33#1784C2214yes—
34C4×#101—24no—
35#1466—24no—
36#1464—24no—
37#1463C2214yes—
38#1462C2214yes—
39#1462C2214yes—
40#1462C2214yes—
41#1460—24no—
42#1458—24no—
43C23×Q8C2311yescharacteristic
44C23×D4C2×C411yescharacteristic
45C23×D4C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52#529—232no—
53#529—232no—
54#529—232no—
55#529—232no—
56C2×#126—24no—
57C2×#126—24no—
58C2×#126—24no—
59C2×#126—24no—
60C2×#126—24no—
61C2×#126—24no—
62C2×#125—24no—
63C2×#125—24no—
64C2×#125—24no—
65C2×#124—24no—
66C2×#124—24no—
67C2×#124—24no—
68C2×#123—24no—
69C2×#123—24no—
70C2×#123—24no—
71C2×#123—24no—
72C2×#123—24no—
73C2×#123—24no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×#121—24no—
78C2×#121—24no—
79C2×#121—24no—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8D414yes—
85C2×C4×Q8D414yes—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4D414yes—
103C2×C4×D4D414yes—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117D414yes—
115C2×#117D414yes—
116C22×C4⋊C4C2×C411yescharacteristic
117C22×C4⋊C4C2×C411yescharacteristic
118C22×C4⋊C4C2×C411yescharacteristic
119C22×C4⋊C4C2×C411yescharacteristic
120C22×C4⋊C4C2×C411yescharacteristic
121C22×C4⋊C4C2×C411yescharacteristic
122C22×C4⋊C4C2×C411yescharacteristic
123C22×C4⋊C4C2×C411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4C2×C411yescharacteristic
127C22×C22⋊C4C2×C411yescharacteristic
128C22×C22⋊C4C2×C411yescharacteristic
129C22×C22⋊C4C2×C411yescharacteristic
130C22×C22⋊C4C2×C411yescharacteristic
131C22×C22⋊C4C2×C411yescharacteristic
132C22×C22⋊C4C2×C411yescharacteristic
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2311yescharacteristic
136C22×C42—22no—
137C22×C42—22no—
138C22×C42—22no—
139C22×C42—22no—
140C22×C42—22no—
141C22×C42—22no—
142C22×C42C2×C411yescharacteristic
143C22×C42C2×C411yescharacteristic
144C22×C42C2×C411yescharacteristic
145C22×C42C2×C411yescharacteristic
146C22×C42C2×C411yescharacteristic
147C22×C42C2311yescharacteristic
148#432—24no—
149#432—24no—
150#431—24no—
151#431—24no—
152#423—24no—
153C2×#101—28no—
154#407C2318yes—
155C4×M16—24no—
156#343—28no—
157#324—28no—
158#323—28no—
159#323—28no—
160#323—28no—
161C25C22×C411yescharacteristic
162C25C22×C411yescharacteristic
163C25C22×C411yescharacteristic
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—44no—
167C22×Q8C2×D414yes—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4C22⋊C414yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C22⋊C412yes—
204C23×C4C22⋊C412yes—
205C23×C4C22⋊C412yes—
206C23×C4C22⋊C412yes—
207C23×C4C22⋊C412yes—
208C23×C4C22⋊C412yes—
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C2411yescharacteristic
221C2×M16—28no—
222#126—232no—
223#126—232no—
224#125—232no—
225#124—232no—
226#123—232no—
227#123—232no—
228C22⋊Q8—232no—
229#121—232no—
230C4×Q8—28no—
231C4×Q8—28no—
232C4×Q8—28no—
233C4×Q8—28no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—28no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—28no—
246#117—28no—
247#117—28no—
248#117—28no—
249#117—28no—
250#117—28no—
251#117—28no—
252#117—28no—
253#117—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
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—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C22⋊C414yes—
279C2×C4⋊C4C22⋊C414yes—
280C2×C4⋊C4C22⋊C414yes—
281C2×C4⋊C4C22⋊C414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
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—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4C22⋊C414yes—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42C22⋊C412yes—
337C2×C42C22⋊C412yes—
338C2×C42C22⋊C412yes—
339C2×C42C22⋊C412yes—
340C2×C42C22⋊C412yes—
341C2×C42C22⋊C412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C4⋊C8—44no—
349C4⋊C8—44no—
350#101—416no—
351C22⋊C8—28no—
352C22⋊C8—28no—
353#97—44no—
354C4×C8—44no—
355C24—21no—
356C24—21no—
357C24—24no—
358C24—21no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24C2×C22⋊C412yes—
371C24C2×C22⋊C412yes—
372C24C2×C22⋊C412yes—
373C24C23×C411yescharacteristic
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—44no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—44no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4C2×C22⋊C412yes—
465C22×C4C2×C22⋊C412yes—
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C412yes—
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C412yes—
473C22×C4C2×C22⋊C412yes—
474C22×C4C2×C22⋊C412yes—
475C22×C4C2×C22⋊C412yes—
476C22×C4C2×C22⋊C412yes—
477C22×C4C2×C22⋊C412yes—
478C22×C4C2×C22⋊C412yes—
479C22×C4C2×C22⋊C412yes—
480C22×C4C2×C22⋊C412yes—
481C22×C4C2×C22⋊C411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485M16—416no—
486C2×C8—48no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊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—28no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C42—28no—
508C42—28no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42#12014yes—
522C42#12014yes—
523C42#12014yes—
524C42#12014yes—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—416no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—88no—
560C23—88no—
561C23#41314yes—
562C23C22×C22⋊C411yescharacteristic
563C23C22×C22⋊C411yescharacteristic
564C23C22×C22⋊C411yescharacteristic
565Q8—416no—
566D4—416no—
567D4—416no—
568D4—416no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—416no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4#37911yescharacteristic
631C2×C4#37911yescharacteristic
632C2×C4#37914yes—
633C2×C4#37911yescharacteristic
634C2×C4#37911yescharacteristic
635C2×C4#41014yes—
636C2×C4#41314yes—
637C2×C4C2×#12014yes—
638C8—88no—
639C22—28no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—24no—
647C22—44no—
648C22—48no—
649C22—48no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22C2×#37911yescharacteristic
666C22C2×#41012yes—
667C22C2×#41312yes—
668C22C2×#41312yes—
669C4—24no—
670C4—24no—
671C4—21no—
672C4—21no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4#178512yes—
683C4#178512yes—
684C2—22no—
685C2—22no—
686C2—22no—
687C2—44no—
688C2—82no—
689C2—82no—
690C2#2058411yescharacteristic
691C2C2×#178511yescharacteristic
692C2#2059111yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#379
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #17364C2semidirect
4. #17357C2semidirect
5. #8556C2semidirect
6. #8553C2semidirect
7. #8552C2semidirect
8. #8550C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
#207272
#173642
#173572
#85562
#85532
#85522
#85502