Finite Group Databasebeta

← #188032

Subgroups — #188032

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2#20722C212yesmaximal, complemented
3#20722C212yesmaximal, complemented
4#20688C212yesmaximal, complemented
5#8545C212yesmaximal, complemented
6#8544C212yesmaximal, complemented
7#8543C212yesmaximal, complemented
8#8542C212yesmaximal
9C2×#529—24no—
10C2×#529—24no—
11C2×#529—24no—
12C2×#529—24no—
13C2×#529—24no—
14C2×#529—24no—
15C22×#126C411yescharacteristic, complemented
16C22×#126C2211yescharacteristic
17C22×#125C411yescharacteristic, complemented
18C22×#124C2211yescharacteristic
19C22×#123C411yescharacteristic, complemented
20C22×#123C2211yescharacteristic
21C22×C22⋊Q8C411yescharacteristic, complemented
22C22×#121C2211yescharacteristic
23C22×C4×Q8C411yescharacteristic, complemented
24C22×C4×D4C411yescharacteristic, complemented
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×#117C411yescharacteristic, complemented
28C22×#117C2211yescharacteristic
29C23×C42C411yescharacteristic, complemented
30#2280C2214yes—
31#1820—24no—
32#1819C2214yes—
33#1819C2214yes—
34#1819C2214yes—
35#1819C2214yes—
36#1799—24no—
37#1799—24no—
38#1781C2214yes—
39#1781C2214yes—
40C4×#99—24no—
41C4×#99—24no—
42#1740—24no—
43C23×Q8C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C23×D4C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2311yescharacteristic
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×Q8—22no—
85C2×C4×Q8—22no—
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×D4D414yes—
101C2×C4×D4D414yes—
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×#117—24no—
115C2×#117D414yes—
116C2×#117D414yes—
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⋊C4C2×C411yescharacteristic
125C22×C4⋊C4C2×C411yescharacteristic
126C22×C4⋊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×C22⋊C4C2311yescharacteristic
137C22×C42—22no—
138C22×C42—22no—
139C22×C42—22no—
140C22×C42—22no—
141C22×C42—22no—
142C22×C42—22no—
143C22×C42C2×C411yescharacteristic
144C22×C42C2×C411yescharacteristic
145C22×C42C2×C411yescharacteristic
146C22×C42C2×C411yescharacteristic
147C22×C42C2×C411yescharacteristic
148C22×C42C2×C411yescharacteristic
149C2×#99—28no—
150C2×#99—28no—
151#381—24no—
152#380—24no—
153#380—24no—
154#379C2318yes—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—24no—
157#342—28no—
158#342—28no—
159#342—28no—
160#342—28no—
161C25C22×C411yescharacteristic
162C25C22×C411yescharacteristic
163C25C2411yescharacteristic
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—44no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4C2×D414yes—
179C22×D4C2×D414yes—
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—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C22⋊C412yes—
200C23×C4C22⋊C412yes—
201C23×C4C22⋊C412yes—
202C23×C4C22⋊C412yes—
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221#126—232no—
222#126—232no—
223#125—232no—
224#124—232no—
225#123—232no—
226#123—232no—
227C22⋊Q8—232no—
228#121—232no—
229C4×Q8—28no—
230C4×Q8—28no—
231C4×Q8—28no—
232C4×Q8—28no—
233C4×D4—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—
245#117—28no—
246#117—28no—
247#117—28no—
248#117—28no—
249#117—28no—
250#117—28no—
251#117—28no—
252#117—28no—
253#117—416no—
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⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C22⋊C414yes—
282C2×C4⋊C4C22⋊C414yes—
283C2×C4⋊C4C22⋊C414yes—
284C2×C4⋊C4C22⋊C414yes—
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—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4C22⋊C414yes—
313C2×C22⋊C4C22⋊C414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D414yes—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—28no—
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—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C22⋊C412yes—
346C2×C42C22⋊C412yes—
347C2×C42C22⋊C412yes—
348C2×C42C22⋊C412yes—
349C2×C42C22⋊C412yes—
350C2×C42C22⋊C412yes—
351C2×C42C22⋊C412yes—
352C2×C42C22⋊C412yes—
353#99—416no—
354#99—416no—
355#95—44no—
356#95—44no—
357C24—21no—
358C24—21no—
359C24—24no—
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—
372C24C23×C411yescharacteristic
373C24C22×D412yes—
374C24C22×D412yes—
375C24C22×D412yes—
376C2×Q8—28no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—44no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
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—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—48no—
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—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—48no—
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⋊C412yes—
482C22×C4C2×C22⋊C412yes—
483C22×C4C2×C22⋊C411yescharacteristic
484C22×C4#12014yes—
485C22×C4#12014yes—
486C22×C4#12014yes—
487C22×C4#12014yes—
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—
497C4⋊C4—28no—
498C4⋊C4—88no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C42—28no—
513C42—28no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—88no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—22no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—416no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—88no—
565C23—88no—
566C23#41014yes—
567C23C22×C22⋊C411yescharacteristic
568C23C22×C22⋊C411yescharacteristic
569C23C22×C22⋊C411yescharacteristic
570C23C2×#12014yes—
571Q8—416no—
572D4—416no—
573D4—416no—
574D4—416no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
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—416no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4#37911yescharacteristic
641C2×C4#37911yescharacteristic
642C2×C4#37914yes—
643C2×C4#37911yescharacteristic
644C2×C4#37911yescharacteristic
645C2×C4#41014yes—
646C2×C4#41014yes—
647C22—28no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—44no—
659C22—48no—
660C22—48no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22C2×#37911yescharacteristic
674C22C2×#41012yes—
675C22C2×#41012yes—
676C22C2×#41012yes—
677C4—21no—
678C4—21no—
679C4—24no—
680C4—24no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—168no—
691C4#178212yes—
692C4#178212yes—
693C2—22no—
694C2—22no—
695C2—22no—
696C2—44no—
697C2—44no—
698C2—82no—
699C2C2×#178211yescharacteristic
700C2#2058411yescharacteristic
701C2#2058411yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC24C23×C4
Frattini, cosocleC25C24
socleC22C2×#379
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20722C2semidirect
3. #20722C2semidirect
4. #20688C2semidirect
5. #8545C2semidirect
6. #8544C2semidirect
7. #8543C2semidirect
15. C22×#126C4semidirect
17. C22×#125C4semidirect
19. C22×#123C4semidirect
21. C22×C22⋊Q8C4semidirect
23. C22×C4×Q8C4semidirect
24. C22×C4×D4C4semidirect
27. C22×#117C4semidirect
29. C23×C42C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
#207222
#206882
#85452
#85442
#85432
#85422