Finite Group Databasebeta

← #4420458

Subgroups — #4420458

Order 512.

3414 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#2820C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic, complemented
4C2×#2634C211yesmaximal, characteristic, complemented
5#16681C212yesmaximal
6#16659C214yesmaximal, complemented
7#16603C212yesmaximal, complemented
8#15928C212yesmaximal, complemented
9#15927C212yesmaximal, complemented
10#15466C212yesmaximal, complemented
11#15465C212yesmaximal, complemented
12#14891C214yesmaximal, complemented
13#14876C212yesmaximal, complemented
14#14860C212yesmaximal
15#14346C211yesmaximal, characteristic, complemented
16#14288C211yesmaximal, characteristic
17#13925C211yesmaximal, characteristic, complemented
18C2×#556—28no—
19C2×#556—28no—
20C2×#556—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126C2212yes—
23C22×#126C2212yes—
24C22×#125C2211yescharacteristic
25C22×#124C2211yescharacteristic
26C22×#124C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32C22×#117C2212yes—
33#2820—28no—
34#2634—28no—
35#2565—28no—
36C2×#400C2212yes—
37C2×#400C2212yes—
38C2×#399C2212yes—
39C2×#398C2212yes—
40C2×#398C2212yes—
41C2×#398C2212yes—
42C2×#397C2212yes—
43C2×#396C2214yes—
44C2×#395C2212yes—
45C2×#395C2212yes—
46C2×#394C2212yes—
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#387C2212yes—
50C2×#387C2214yes—
51C2×#387C2212yes—
52C2×#384C2211yescharacteristic
53C2×#383C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#1446C2212yes—
58#1446C2212yes—
59#1446C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1444C2214yes—
63#1444C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2212yes—
78#1437C2212yes—
79#1437C2212yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2212yes—
83#1436C2212yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C22⋊#95C2214yes—
88C22⋊#95C2214yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×Q8C2311yescharacteristic
92C23×D4C2311yescharacteristic
93C24×C4C2312yes—
94#556—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—24no—
107C2×#124—24no—
108C2×#124—24no—
109C2×#124—24no—
110C2×#124—24no—
111C2×#124—24no—
112C2×#124D418yes—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8—28no—
116C2×C4×Q8—24no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8D418yes—
124C2×C4×Q8D418yes—
125C2×C4×D4—24no—
126C2×C4×D4—24no—
127C2×C4×D4—24no—
128C2×#117—28no—
129C2×#117—28no—
130C2×#117—28no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2314yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2312yes—
149C22×C22⋊C4C2312yes—
150C22×C22⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2312yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2314yes—
158C22×C42C2312yes—
159C22×C42C2312yes—
160C22×C42C2311yescharacteristic
161C22×C42C2311yescharacteristic
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165#400—28no—
166#400—28no—
167#399—28no—
168#398—28no—
169#398—28no—
170#398—28no—
171#397—28no—
172#396—216no—
173#395—28no—
174#395—28no—
175#394—28no—
176#391—24no—
177#391—24no—
178#387—28no—
179#387—28no—
180#387—216no—
181#384—24no—
182#383—24no—
183#382—24no—
184#382—24no—
185C4×C4⋊C4—24no—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C25C2412yes—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—24no—
220C22×Q8—42no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8C2×D414yes—
224C22×Q8C2×D414yes—
225C22×Q8C2×D414yes—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—44no—
229C22×D4C2×D414yes—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—28no—
234C23×C4—44no—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C4⋊Q8—232no—
252#126—264no—
253#126—264no—
254#125—416no—
255#124—232no—
256#124—232no—
257C22⋊Q8—264no—
258C4×Q8—232no—
259C4×Q8—232no—
260C4×Q8—232no—
261C4×D4—232no—
262#117—264no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D418yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4C2×D418yes—
342C2×C22⋊C4C2×D418yes—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42C2×D418yes—
357C2×C42C2×D414yes—
358C2×C42C2×D414yes—
359C2×C42C2×D414yes—
360C2×C42C2×D414yes—
361C2×C42C2×D414yes—
362#95—24no—
363#95—24no—
364#95—24no—
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—24no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—28no—
379#95—44no—
380#95—44no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95—44no—
385#95—44no—
386#95—44no—
387#95—44no—
388C24—22no—
389C24—22no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24C22×D414yes—
397C24C2511yescharacteristic
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—44no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—48no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×D4—28no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—416no—
484C22×C4—416no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C42−1+414yes—
504C22×C42−1+414yes—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—216no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—416no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—84no—
522C4⋊C4—88no—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—216no—
527C22⋊C4—216no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C42—216no—
533C42—216no—
534C42—28no—
535C42—28no—
536C42—44no—
537C42—44no—
538C42#120116yes—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—24no—
543C23—24no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C2×2−1+412yes—
564C23C2×2−1+412yes—
565C23#58514yes—
566C23#58512yes—
567C23#58512yes—
568Q8—416no—
569Q8—416no—
570Q8—416no—
571D4—416no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—432no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4C2×#12014yes—
655C2×C4C2×#12014yes—
656C2×C4C2×#12014yes—
657C2×C4#53614yes—
658C22—44no—
659C22—44no—
660C22—48no—
661C22—48no—
662C22—44no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22C22×#12011yescharacteristic
668C22C2×#53611yescharacteristic
669C22#344912yes—
670C22#344911yescharacteristic
671C22#348512yes—
672C22#348712yes—
673C22#348912yes—
674C22#348912yes—
675C22#348912yes—
676C22#349712yes—
677C22#349911yescharacteristic
678C22#350011yescharacteristic
679C22#355514yes—
680C22#356414yes—
681C22#356514yes—
682C22#356614yes—
683C4—24no—
684C4—24no—
685C4—24no—
686C4—24no—
687C4—42no—
688C4—42no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—42no—
694C4—84no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—84no—
699C2—84no—
700C2#3514911yescharacteristic
701C2#3530511yescharacteristic
702C2#3566011yescharacteristic
703C2#4217512yes—
704C2#4230114yes—
705C2#4289112yes—
706C2#5089212yes—
707C2#5089612yes—
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
3. C2×#2565C2semidirect
4. C2×#2634C2semidirect
6. #16659C2semidirect
7. #16603C2semidirect
8. #15928C2semidirect
9. #15927C2semidirect
10. #15466C2semidirect
11. #15465C2semidirect
12. #14891C2semidirect
13. #14876C2semidirect
15. #14346C2semidirect
17. #13925C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#28202
C2×#25652
C2×#26342
#166812
#166592
#166032
#159282
#159272
#154662
#154652
#148912
#148762
#148602
#143462
#142882
#139252