Finite Group Databasebeta

← #798871

Subgroups — #798871

Order 512.

3126 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2583C212yesmaximal, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#16606C212yesmaximal, complemented
5#15421C214yesmaximal, complemented
6#15305C214yesmaximal, complemented
7#15158C211yesmaximal, characteristic, complemented
8#15151C211yesmaximal, characteristic, complemented
9#15145C211yesmaximal, characteristic, complemented
10#14840C212yesmaximal, complemented
11#14821C212yesmaximal, complemented
12#14151C212yesmaximal, complemented
13#14018C212yesmaximal
14#14005C212yesmaximal, complemented
15#13977C212yesmaximal
16C4×#395C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24#2583—216no—
25#2538—28no—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#396C2214yes—
31C2×#395C2212yes—
32C2×#395C2211yescharacteristic
33D4×C4⋊C4—28no—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2214yes—
37C2×#389C2214yes—
38C2×#389C2211yescharacteristic
39C2×#389C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#385C2214yes—
48C2×#384C2211yescharacteristic
49C2×#384C2212yes—
50C2×#384C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57#1446C2211yescharacteristic
58#1445C2212yes—
59#1445C2212yes—
60#1444C2212yes—
61#1444C2211yescharacteristic
62#1444C2214yes—
63#1444C2214yes—
64#1441C2212yes—
65#1441C2211yescharacteristic
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—44no—
94C2×C4×Q8—24no—
95C2×C4×Q8—44no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—44no—
101C2×C4×D4—44no—
102C2×C4×D4D418yes—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—44no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140C22×C42C2314yes—
141#398—28no—
142#398—28no—
143#398—44no—
144#398—44no—
145#396—216no—
146#395—28no—
147#395D418yescomplemented
148#390—28no—
149#390Q8116yescomplemented
150#389—216no—
151#389—24no—
152#389—28no—
153#389—24no—
154#389—28no—
155#389—216no—
156#389—44no—
157#385—28no—
158#385—216no—
159#385—28no—
160#385—216no—
161#385—48no—
162#384—24no—
163#384—28no—
164#384—24no—
165#380—28no—
166#380—28no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—44no—
169C4×C4⋊C4—44no—
170C4×C22⋊C4—28no—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2311yescharacteristic
195C2×#95C2312yes—
196C25C2412yes—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8C2×D414yes—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4C2×D414yes—
206C22×D4C2×Q818yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D418yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C22⋊Q8—432no—
241C4×Q8—416no—
242C4×D4—232no—
243C4×D4—416no—
244C4×D4—416no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—432no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—41no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D418yes—
306C2×C4⋊C4C2×Q818yes—
307C2×C4⋊C4#4114yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×Q8116yes—
326C2×C22⋊C4C2×Q8116yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—28no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D414yes—
341C2×C42C2×Q818yes—
342C2×C42#4114yes—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—42no—
358#95—44no—
359#95—42no—
360#95—42no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—42no—
365#95—44no—
366#95C2×D414yes—
367#95C2×D418yes—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—22no—
376C24—24no—
377C24C22×D414yes—
378C24C22×Q818yes—
379C24C2511yescharacteristic
380C2×Q8—28no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—216no—
388C2×D4—216no—
389C2×D4C4⋊Q8116yes—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—216no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—84no—
480C22×C4—88no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×Q814yes—
491C22×C4C22×Q818yes—
492C22×C4C22×Q814yes—
493C22×C4C22×Q814yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)14yes—
502C22×C42−1+414yes—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—416no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—416no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—88no—
524C4⋊C4—88no—
525C22⋊C4—216no—
526C22⋊C4—232no—
527C22⋊C4—48no—
528C22⋊C4—416no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—88no—
532C42—28no—
533C42—28no—
534C42—216no—
535C42—44no—
536C42—48no—
537C42—88no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23C2×C4⋊Q818yes—
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C23×Q812yes—
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×2−1+412yes—
570C23#58511yescharacteristic
571C23#58512yes—
572Q8—416no—
573D4—232no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—432no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—816no—
664C2×C4—816no—
665C2×C4#52514yes—
666C2×C4C2×C4⋊Q814yes—
667C2×C4#53414yes—
668C2×C4D4214yes—
669C2×C4D4×Q818yes—
670C2×C4#55014yes—
671C2×C4#55418yes—
672C2×C4#55418yes—
673C22—28no—
674C22—28no—
675C22—28no—
676C22—44no—
677C22—44no—
678C22—48no—
679C22—44no—
680C22—48no—
681C22—44no—
682C22—48no—
683C22—48no—
684C22C2×#52511yescharacteristic
685C22C22×C4⋊Q811yescharacteristic
686C22C2×#53411yescharacteristic
687C22C2×D4211yescharacteristic
688C22C2×D4×Q812yes—
689C22C2×#55011yescharacteristic
690C22D4×#4111yescharacteristic
691C22C2×#55412yes—
692C22C2×#55412yes—
693C22#348511yescharacteristic
694C22#348811yescharacteristic
695C22#349612yes—
696C22#349911yescharacteristic
697C22#350112yes—
698C22#350312yes—
699C22#350412yes—
700C22#350611yescharacteristic
701C22#350712yes—
702C22#351612yes—
703C22#351911yescharacteristic
704C22#354112yes—
705C22#354212yes—
706C22#354812yes—
707C4—24no—
708C4—24no—
709C4—28no—
710C4—44no—
711C4—42no—
712C4—44no—
713C4—42no—
714C4—44no—
715C4—42no—
716C4—44no—
717C4—48no—
718C4—84no—
719C4—81no—
720C4—82no—
721C4—82no—
722C4#258318yescomplemented
723C2—48no—
724C2C2×#258311yescharacteristic
725C2#3584611yescharacteristic
726C2#3595011yescharacteristic
727C2#3610111yescharacteristic
728C2D4×C4⋊Q811yescharacteristic
729C2#3933811yescharacteristic
730C2#4291611yescharacteristic
731C2#4494312yes—
732C2#4496612yes—
733C2#5982312yes—
734C2#5983112yes—
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×#2583C2semidirect
4. #16606C2semidirect
5. #15421C2semidirect
6. #15305C2semidirect
7. #15158C2semidirect
8. #15151C2semidirect
9. #15145C2semidirect
10. #14840C2semidirect
11. #14821C2semidirect
12. #14151C2semidirect
14. #14005C2semidirect
16. C4×#395C2semidirect
147. #395D4semidirect
149. #390Q8semidirect
722. C4#2583semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25832
C2×D4×C4⋊C42
#166062
#154212
#153052
#151582
#151512
#151452
#148402
#148212
#141512
#140182
#140052
#139772
C4×#3952
#138202