Finite Group Databasebeta

← #805107

Subgroups — #805107

Order 512.

3126 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C212yesmaximal, complemented
2C2×#2452C211yesmaximal, characteristic
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#16365C212yesmaximal, complemented
5#15392C214yesmaximal, complemented
6#15378C214yesmaximal, complemented
7#15151C211yesmaximal, characteristic, complemented
8#15149C211yesmaximal, characteristic, complemented
9#15145C211yesmaximal, characteristic, complemented
10#14840C212yesmaximal, complemented
11#14786C212yesmaximal, complemented
12#14053C212yesmaximal, complemented
13#14018C212yesmaximal
14#13977C212yesmaximal
15#13910C212yesmaximal, complemented
16C4×#398C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23#2502—216no—
24#2452—28no—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2211yescharacteristic
31C2×#396C2212yes—
32C2×#393C2212yes—
33D4×C4⋊C4—28no—
34C2×#391C2214yes—
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#385C2212yes—
47C2×#385C2214yes—
48C2×#384C2214yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57#1446C2214yes—
58#1445C2212yes—
59#1445C2211yescharacteristic
60#1445C2212yes—
61#1444C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1444C2211yescharacteristic
65#1442C2212yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2211yescharacteristic
70#1441C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—44no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—44no—
98C2×C4×D4—44no—
99C2×C4×D4D418yes—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#398—28no—
139#398—28no—
140#398—28no—
141#398—44no—
142#398—44no—
143#398D418yescomplemented
144#396—28no—
145#393—28no—
146#391—216no—
147#390—24no—
148#390—24no—
149#390Q8116yescomplemented
150#389—28no—
151#389—24no—
152#389—24no—
153#389—44no—
154#389—44no—
155#385—28no—
156#385—216no—
157#385—216no—
158#385—216no—
159#385—48no—
160#384—216no—
161#384—24no—
162#384—44no—
163#380—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—44no—
168C4×C22⋊C4—28no—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C25C2412yes—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—28no—
199C22×Q8—42no—
200C22×Q8—44no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2×D414yes—
205C22×D4C2×Q818yes—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C2×D418yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C4⋊Q8—416no—
235C22⋊Q8—432no—
236C4×D4—232no—
237C4×D4—432no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×Q818yes—
297C2×C4⋊C4#4118yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×D418yes—
318C2×C22⋊C4C2×Q8116yes—
319C2×C22⋊C4C2×Q8116yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—28no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—44no—
334C2×C42C2×D414yes—
335C2×C42C2×Q818yes—
336#95—24no—
337#95—22no—
338#95—24no—
339#95—22no—
340#95—28no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—24no—
348#95—28no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—42no—
360#95C2×D414yes—
361#95C2×D418yes—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24C22×D414yes—
372C24C22×Q818yes—
373C24C2511yescharacteristic
374C2×Q8—48no—
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—84no—
378C2×Q8—88no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4C4⋊Q8116yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q818yes—
483C22×C4C22×Q814yes—
484C22×C4C22×Q814yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4114yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C42−1+414yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—416no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—88no—
508C4⋊C4—84no—
509C4⋊C4—88no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C4⋊C4#123116yes—
513C22⋊C4—216no—
514C22⋊C4—232no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—84no—
521C42—28no—
522C42—216no—
523C42—416no—
524C42—44no—
525C42—44no—
526C42—48no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23C2×C4⋊Q818yes—
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C23×Q812yes—
553C23C22×#4112yes—
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×2−1+412yes—
559C23#58512yes—
560C23#58511yescharacteristic
561Q8—816no—
562D4—232no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—432no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—832no—
653C2×C4C2×#12318yes—
654C2×C4C2×#12314yes—
655C2×C4C2×C4⋊Q814yes—
656C2×C4#53514yes—
657C2×C4D4214yes—
658C2×C4D4×Q818yes—
659C2×C4#55418yes—
660C2×C4#55418yes—
661C22—28no—
662C22—28no—
663C22—28no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—44no—
672C22C22×#12311yescharacteristic
673C22C22×C4⋊Q811yescharacteristic
674C22C2×#53511yescharacteristic
675C22C2×D4211yescharacteristic
676C22C2×D4×Q812yes—
677C22D4×#4112yes—
678C22C2×#55412yes—
679C22C2×#55412yes—
680C22#348511yescharacteristic
681C22#348811yescharacteristic
682C22#349611yescharacteristic
683C22#349811yescharacteristic
684C22#349911yescharacteristic
685C22#350112yes—
686C22#350312yes—
687C22#350412yes—
688C22#351312yes—
689C22#351612yes—
690C22#352212yes—
691C22#354112yes—
692C22#354611yescharacteristic
693C22#354911yescharacteristic
694C22#355412yes—
695C4—28no—
696C4—24no—
697C4—28no—
698C4—48no—
699C4—44no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—82no—
706C4—81no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4#250218yescomplemented
711C2—48no—
712C2C2×#250211yescharacteristic
713C2#3595011yescharacteristic
714C2#3599611yescharacteristic
715C2D4×#12311yescharacteristic
716C2D4×C4⋊Q811yescharacteristic
717C2#3934211yescharacteristic
718C2#4264811yescharacteristic
719C2#4494612yes—
720C2#4505912yes—
721C2#5945812yes—
722C2#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
1. C2×#2502C2semidirect
4. #16365C2semidirect
5. #15392C2semidirect
6. #15378C2semidirect
7. #15151C2semidirect
8. #15149C2semidirect
9. #15145C2semidirect
10. #14840C2semidirect
11. #14786C2semidirect
12. #14053C2semidirect
15. #13910C2semidirect
16. C4×#398C2semidirect
143. #398D4semidirect
149. #390Q8semidirect
710. C4#2502semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#24522
C2×D4×C4⋊C42
#163652
#153922
#153782
#151512
#151492
#151452
#148402
#147862
#140532
#140182
#139772
#139102
C4×#3982
#138202