Finite Group Databasebeta

← #1394110

Subgroups — #1394110

Order 512.

2866 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2628C212yesmaximal, complemented
3C2×#2354C211yesmaximal, characteristic
4#16564C212yesmaximal, complemented
5#16511C212yesmaximal, complemented
6#16327C214yesmaximal, complemented
7#16303C214yesmaximal, complemented
8#15155C211yesmaximal, characteristic
9#15152C211yesmaximal, characteristic, complemented
10#15148C211yesmaximal, characteristic
11#15144C212yesmaximal
12#15143C212yesmaximal, complemented
13#14893C212yesmaximal, complemented
14#14005C212yesmaximal, complemented
15#13920C212yesmaximal
16#13918C211yesmaximal, characteristic, complemented
17C4×#400C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2628—216no—
23#2551—28no—
24C2×#400C2212yes—
25C2×#400C2211yescharacteristic
26C2×#399C2212yes—
27C2×#399C2212yes—
28C2×#399C2212yes—
29C2×#399C2214yes—
30C2×#395C2214yes—
31C2×#395C2214yes—
32C2×#395C2214yes—
33C2×#393C2214yes—
34C2×#393C2214yes—
35#2354—28no—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#384C2212yes—
50C2×#382C2212yes—
51C2×#380C2214yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#1445C2211yescharacteristic
58#1445C2212yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1443C2212yes—
63#1442C2212yes—
64#1442C2211yescharacteristic
65#1442C2212yes—
66#1442C2214yes—
67#1441C2211yescharacteristic
68#1441C2212yes—
69#1441C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81#1434C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2314yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4Q818yescomplemented
98C2×#117—28no—
99C2×#117—28no—
100C2×#117—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#400—28no—
137#400D418yescomplemented
138#399—28no—
139#399—28no—
140#399—44no—
141#399—48no—
142#395—216no—
143#395—216no—
144#395—48no—
145#393—216no—
146#393—48no—
147#390—24no—
148#390—24no—
149#390—44no—
150#390—44no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—216no—
155#389—44no—
156#388—28no—
157#388—28no—
158#384—24no—
159#384—24no—
160#384—44no—
161#382—28no—
162#380—28no—
163#380—216no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4D418yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2311yescharacteristic
195C25C2412yes—
196C22×Q8—28no—
197C22×Q8—28no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4C2×Q814yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4C2×Q818yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C4⋊Q8—432no—
232C4⋊Q8—432no—
233C4×D4—232no—
234#117—432no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×Q814yes—
286C2×C4⋊C4#4114yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×Q818yes—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42C2×D418yes—
329C2×C42C2×D414yes—
330C2×C42C2×Q814yes—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—42no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×D418yes—
355#95C2×D418yes—
356#95C2×D414yes—
357#95C2×D418yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24C22×Q814yes—
369C24C2511yescharacteristic
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—88no—
373C2×Q8—88no—
374C2×Q8—88no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×D4—28no—
378C2×D4—216no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—48no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q814yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q812yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C42−1+414yes—
483C22×C42−1+412yes—
484C22×C42−1+414yes—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—416no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—88no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C4⋊C4C22⋊Q8116yes—
504C22⋊C4—216no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—28no—
514C42—28no—
515C42—216no—
516C42—48no—
517C42—48no—
518C42—44no—
519C42—48no—
520C23—24no—
521C23—28no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C23×D411yescharacteristic
544C23C23×D411yescharacteristic
545C23C23×Q811yescharacteristic
546C23C22×#4111yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+412yes—
553C23#58511yescharacteristic
554Q8—832no—
555D4—416no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
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—24no—
594C2×C4—432no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4C2×C22⋊Q818yes—
646C2×C4C2×C22⋊Q814yes—
647C2×C4#53514yes—
648C2×C4#53614yes—
649C2×C4#54218yes—
650C2×C4#54418yes—
651C2×C4D4214yes—
652C2×C4D4×Q814yes—
653C22—48no—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—44no—
658C22—82no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22C22×C22⋊Q811yescharacteristic
665C22C2×#53511yescharacteristic
666C22C2×#53611yescharacteristic
667C22C2×#54212yes—
668C22C2×#54412yes—
669C22C2×D4211yescharacteristic
670C22C2×D4×Q811yescharacteristic
671C22D4×#4111yescharacteristic
672C22#348611yescharacteristic
673C22#348911yescharacteristic
674C22#349811yescharacteristic
675C22#349812yes—
676C22#349912yes—
677C22#350111yescharacteristic
678C22#350312yes—
679C22#352011yescharacteristic
680C22#352212yes—
681C22#354312yes—
682C22#354412yes—
683C22#354412yes—
684C22#354712yes—
685C22#355212yes—
686C22#356112yes—
687C4—24no—
688C4—24no—
689C4—28no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—44no—
696C4—42no—
697C4—44no—
698C4—82no—
699C4—84no—
700C4—81no—
701C4—84no—
702C4#262818yescomplemented
703C2—84no—
704C2C2×#262811yescharacteristic
705C2#3574811yescharacteristic
706C2D4×C22⋊Q811yescharacteristic
707C2#3933711yescharacteristic
708C2#3936111yescharacteristic
709C2#3936912yes—
710C2#3940612yes—
711C2#4249712yes—
712C2#4263512yes—
713C2#4281211yescharacteristic
714C2#4296811yescharacteristic
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×#2551C2semidirect
2. C2×#2628C2semidirect
4. #16564C2semidirect
5. #16511C2semidirect
6. #16327C2semidirect
7. #16303C2semidirect
9. #15152C2semidirect
12. #15143C2semidirect
13. #14893C2semidirect
14. #14005C2semidirect
16. #13918C2semidirect
17. C4×#400C2semidirect
97. C2×C4×D4Q8semidirect
137. #400D4semidirect
702. C4#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#26282
C2×#23542
#165642
#165112
#163272
#163032
#151552
#151522
#151482
#151442
#151432
#148932
#140052
#139202
#139182
C4×#4002