Finite Group Databasebeta

← #657580

Subgroups — #657580

Order 512.

4470 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2487C211yesmaximal, characteristic
4C22×#384C211yesmaximal, characteristic, complemented
5#15850C214yesmaximal, complemented
6#15430C214yesmaximal, complemented
7#15182C212yesmaximal, complemented
8#15143C212yesmaximal, complemented
9#14720C214yesmaximal, complemented
10#14702C211yesmaximal, characteristic, complemented
11#14015C214yesmaximal
12#13976C212yesmaximal, complemented
13#13924C211yesmaximal, characteristic, complemented
14#13880C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C2×#554—28no—
17C2×#554—28no—
18C2×#554—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C23×C4⋊C4C2214yes—
32C23×C42C2211yescharacteristic
33#2502—28no—
34#2487—28no—
35C2×#400C2212yes—
36C2×#399C2212yes—
37C2×#398C2214yes—
38C2×#395C2218yes—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#389C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#385C2214yes—
46C2×#385C2218yes—
47C2×#384—28no—
48C2×#384C2212yes—
49C2×#384C2212yes—
50C2×#384C2212yes—
51C2×#384C2214yes—
52C2×#380C2212yes—
53C2×#380C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56#1445C2214yes—
57#1444C2218yes—
58#1444C2218yes—
59#1444C2218yes—
60#1441C2218yes—
61#1439C2218yes—
62#1437C2212yes—
63#1436C2214yes—
64#1436C2218yes—
65#1436C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82#554—2128no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×#125—28no—
87C2×#125—28no—
88C2×#125—28no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8D418yes—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×D4—28no—
105C2×C4×D4—28no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—28no—
109C2×C4×D4D418yes—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—24no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—216no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—416no—
120C22×C4⋊C4D418yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2318yes—
129C22×C4⋊C4C2318yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2318yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2311yescharacteristic
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2311yescharacteristic
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2314yes—
146C22×C42—24no—
147C22×C42—28no—
148C22×C42C2312yes—
149C22×C42C2312yes—
150C22×C42C2312yes—
151C22×C42C2311yescharacteristic
152C22×C42C2312yes—
153#400—28no—
154#399—28no—
155#398—216no—
156#395—416no—
157#391—28no—
158#391—28no—
159#389—28no—
160#386—28no—
161#386Q8116yescomplemented
162#385—28no—
163#385—216no—
164#385—416no—
165#384—28no—
166#384—28no—
167#384—216no—
168#384—28no—
169#380—28no—
170#380—24no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—28no—
173C2×#95C2314yes—
174C2×#95C2318yes—
175C2×#95C2312yes—
176C2×#95C2318yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2318yes—
181C2×#95C2314yes—
182C2×#95C2318yes—
183C2×#95C2318yes—
184C2×#95C2314yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—48no—
195C22×Q8C2×D414yes—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4C2×D414yes—
200C22×D4C2×Q818yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—48no—
220C23×C4—48no—
221C23×C4—48no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D418yes—
227C23×C4C2×Q818yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C4⋊Q8—232no—
242#125—264no—
243C22⋊Q8—232no—
244C22⋊Q8—264no—
245C22⋊Q8—264no—
246C22⋊Q8—232no—
247C4×Q8—232no—
248C4×D4—264no—
249C4×D4—232no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
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—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—216no—
287C2×C4⋊C4—216no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—216no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—464no—
294C2×C4⋊C4—416no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D418yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D418yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4C2×D418yes—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×Q818yes—
325C2×C22⋊C4C2×Q8116yes—
326C2×C42—232no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—28no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42C2×D414yes—
340C2×C42#4118yes—
341#95—24no—
342#95—28no—
343#95—24no—
344#95—216no—
345#95—28no—
346#95—28no—
347#95—216no—
348#95—216no—
349#95—28no—
350#95—48no—
351#95—48no—
352#95C2×Q8116yes—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×Q814yes—
367C24C22×Q814yes—
368C242−1+414yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—88no—
377C2×Q8—88no—
378C2×Q8—88no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4C4⋊Q8116yes—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—232no—
387C22×C4—216no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—232no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—28no—
431C22×C4—216no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—416no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4#120116yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×Q818yes—
470C22×C4C22×Q814yes—
471C22×C4C22×Q818yes—
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)18yes—
475C22×C42−1+414yes—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—216no—
486C4⋊C4—432no—
487C4⋊C4—432no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—816no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—28no—
496C22⋊C4—216no—
497C22⋊C4—28no—
498C22⋊C4—232no—
499C22⋊C4—48no—
500C22⋊C4—88no—
501C42—216no—
502C42—216no—
503C42—28no—
504C42—216no—
505C42—48no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—24no—
512C23—28no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—24no—
524C23—28no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23C2×#12014yes—
533C23C2×C4⋊Q814yes—
534C23C2×C4⋊Q814yes—
535C23#53614yes—
536C23D4×Q818yes—
537C23D4×Q818yes—
538C23C23×D411yescharacteristic
539C23C23×D411yescharacteristic
540C23C23×D411yescharacteristic
541C23C23×Q812yes—
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)14yes—
544C23C2×2−1+412yes—
545C23C2×2−1+412yes—
546Q8—416no—
547Q8—816no—
548D4—232no—
549C2×C4—264no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—232no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—416no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—816no—
610C2×C4—816no—
611C2×C4—816no—
612C2×C4C2×#12014yes—
613C2×C4C2×#12014yes—
614C2×C4#52514yes—
615C2×C4C2×C4⋊Q814yes—
616C2×C4#53614yes—
617C2×C4#54718yes—
618C2×C4#54718yes—
619C2×C4#554116yescomplemented
620C22—28no—
621C22—28no—
622C22—24no—
623C22—216no—
624C22—28no—
625C22—24no—
626C22—28no—
627C22—28no—
628C22—24no—
629C22—44no—
630C22—42no—
631C22—48no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22#243418yes—
636C22C22×#12011yescharacteristic
637C22C2×#52511yescharacteristic
638C22C22×C4⋊Q811yescharacteristic
639C22C2×#53611yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#54712yes—
642C22C2×#54712yes—
643C22C2×D4×Q812yes—
644C22C2×D4×Q812yes—
645C22C2×#55414yes—
646C22#348412yes—
647C22#349214yes—
648C22#349812yes—
649C22#349812yes—
650C22#350314yes—
651C22#355214yes—
652C4—24no—
653C4—24no—
654C4—216no—
655C4—42no—
656C4—44no—
657C4—48no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—84no—
663C4—84no—
664C4#240618yes—
665C2—28no—
666C2—44no—
667C2C2×#243411yescharacteristic
668C2C2×#240611yescharacteristic
669C2#3552711yescharacteristic
670C2#3618012yes—
671C2#3654612yes—
672C2#3786212yes—
673C2#3959212yes—
674C2#5931414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2502C2semidirect
4. C22×#384C2semidirect
5. #15850C2semidirect
6. #15430C2semidirect
7. #15182C2semidirect
8. #15143C2semidirect
9. #14720C2semidirect
10. #14702C2semidirect
12. #13976C2semidirect
13. #13924C2semidirect
14. #13880C2semidirect
161. #386Q8semidirect
619. C2×C4#554semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25022
C2×#24872
C22×#3842
#158502
#154302
#151822
#151432
#147202
#147022
#140152
#139762
#139242
#138802
C4×#3862