Finite Group Databasebeta

← #4380717

Subgroups — #4380717

Order 512.

3298 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#2817C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4C2×#2659C211yesmaximal, characteristic
5#16683C212yesmaximal, complemented
6#16603C212yesmaximal, complemented
7#15911C214yesmaximal
8#15911C214yesmaximal
9#15881C212yesmaximal, complemented
10#15850C212yesmaximal, complemented
11#15849C212yesmaximal, complemented
12#15849C212yesmaximal, complemented
13#14879C212yesmaximal, complemented
14#14876C212yesmaximal, complemented
15#14292C211yesmaximal, characteristic
16#14159C211yesmaximal, characteristic
17#13958C211yesmaximal, characteristic
18C2×#541—28no—
19C2×#541—28no—
20C2×#541—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126C2212yes—
23C22×#126C2212yes—
24C22×#125C2212yes—
25C22×#125C2211yescharacteristic
26C22×#125C2212yes—
27C22×C22⋊Q8C2211yescharacteristic
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#2817—28no—
34#2659—28no—
35#2499—28no—
36C2×#400C2212yes—
37C2×#400C2212yes—
38C2×#398C2212yes—
39C2×#398C2212yes—
40C2×#397C2214yes—
41C2×#397C2212yes—
42C2×#397C2212yes—
43C2×#395C2212yes—
44C2×#395C2212yes—
45C2×#395C2212yes—
46C2×#395C2212yes—
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#387C2212yes—
50C2×#387C2212yes—
51C2×#384C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#381C2211yescharacteristic
54C2×#380C2214yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C22×#95C2211yescharacteristic
57#1446C2212yes—
58#1446C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1440C2214yes—
76#1436C2214yes—
77#1436C2212yes—
78#1436C2212yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2212yes—
83#1436C2212yes—
84#1434C2212yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C22⋊#95C2212yes—
88C22⋊#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×Q8C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94#541—2128no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×#126—28no—
99C2×#126—28no—
100C2×#126—28no—
101C2×#126—28no—
102C2×#126—28no—
103C2×#126—28no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125—28no—
107C2×#125—28no—
108C2×#125—28no—
109C2×#125—28no—
110C2×#125—28no—
111C2×#125—28no—
112C2×#125—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8D418yes—
121C2×C22⋊Q8D418yes—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—24no—
127C2×C4×Q8—24no—
128C2×#117—24no—
129C2×#117—24no—
130C2×#117D418yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C22⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2311yescharacteristic
153C22×C22⋊C4C2311yescharacteristic
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2312yes—
157C22×C42C2312yes—
158C22×C42C2312yes—
159C22×C42C2311yescharacteristic
160C22×C42C2311yescharacteristic
161C22×C42C2311yescharacteristic
162C22×C42C2311yescharacteristic
163#400—28no—
164#400—28no—
165#398—28no—
166#398—28no—
167#397—216no—
168#397—28no—
169#397—28no—
170#395—28no—
171#395—28no—
172#395—28no—
173#395—28no—
174#391—24no—
175#391—24no—
176#387—28no—
177#387—28no—
178#384—24no—
179#382—24no—
180#381—24no—
181#380—216no—
182C4×C22⋊C4—24no—
183C2×#95—24no—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C25C2411yescharacteristic
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×D414yes—
227C22×Q8C2×D414yes—
228C23×C4—24no—
229C23×C4—28no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2414yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C4⋊Q8—232no—
255#126—264no—
256#126—264no—
257#125—264no—
258#125—264no—
259#125—232no—
260C22⋊Q8—232no—
261C22⋊Q8—232no—
262C22⋊Q8—416no—
263C4×Q8—232no—
264C4×Q8—232no—
265#117—232no—
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—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
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—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D418yes—
322C2×C4⋊C4C2×D418yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D418yes—
343C2×C22⋊C4C2×D418yes—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42C2×D414yes—
359C2×C42C2×D414yes—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—28no—
364#95—28no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—28no—
379#95—28no—
380#95—28no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95—44no—
385#95—44no—
386#95—44no—
387#95—44no—
388C24—22no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24C22×D412yes—
394C24C22×D412yes—
395C24C22×D412yes—
396C24C2511yescharacteristic
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—84no—
411C2×Q8—84no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4#120116yes—
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D414yes—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C42−1+418yes—
505C22×C42−1+414yes—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—88no—
523C4⋊C4—88no—
524C4⋊C4—88no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—84no—
532C42—216no—
533C42—216no—
534C42—28no—
535C42—28no—
536C42—44no—
537C42—44no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C2×#12014yes—
554C23C23×D411yescharacteristic
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C2×2−1+414yes—
558C23C2×2−1+412yes—
559C23#58512yes—
560C23#58512yes—
561C23#58512yes—
562Q8—416no—
563Q8—416no—
564Q8—88no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—416no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
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—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—816no—
649C2×C4C2×#12014yes—
650C2×C4C2×#12014yes—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—44no—
658C22—44no—
659C22C22×#12011yescharacteristic
660C22#344911yescharacteristic
661C22#344911yescharacteristic
662C22#348512yes—
663C22#348512yes—
664C22#348612yes—
665C22#348912yes—
666C22#349712yes—
667C22#349912yes—
668C22#349911yescharacteristic
669C22#349912yes—
670C22#350011yescharacteristic
671C22#355514yes—
672C22#356314yes—
673C22#356414yes—
674C22#356414yes—
675C4—24no—
676C4—24no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—44no—
685C4—42no—
686C4—48no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—84no—
691C2—44no—
692C2#3508811yescharacteristic
693C2#3566011yescharacteristic
694C2#3566611yescharacteristic
695C2#4230612yes—
696C2#4231512yes—
697C2#4681512yes—
698C2#4721112yes—
699C2#5154014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #16683C2semidirect
6. #16603C2semidirect
9. #15881C2semidirect
10. #15850C2semidirect
11. #15849C2semidirect
12. #15849C2semidirect
13. #14879C2semidirect
14. #14876C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#28172
C2×#24992
C2×#26592
#166832
#166032
#159112
#158812
#158502
#158492
#148792
#148762
#142922
#141592
#139582