Finite Group Databasebeta

← #1394109

Subgroups — #1394109

Order 512.

2770 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2628C212yesmaximal, complemented
3C2×#2354C211yesmaximal, characteristic
4#16602C212yesmaximal, complemented
5#16566C212yesmaximal, complemented
6#16331C214yesmaximal, complemented
7#16304C214yesmaximal, complemented
8#15157C211yesmaximal, characteristic
9#15148C212yesmaximal
10#15146C212yesmaximal, complemented
11#15144C211yesmaximal, characteristic
12#15128C211yesmaximal, characteristic, complemented
13#14887C212yesmaximal, complemented
14#14046C212yesmaximal, complemented
15#13920C212yesmaximal
16#13918C211yesmaximal, characteristic, complemented
17C4×#395C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2628—216no—
24#2546—28no—
25C2×#400C2214yes—
26C2×#399C2214yes—
27C2×#398C2214yes—
28C2×#395C2212yes—
29C2×#395C2211yescharacteristic
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33#2354—28no—
34C2×#391C2212yes—
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382C2212yes—
48C2×#380C2214yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1444C2212yes—
57#1444C2212yes—
58#1444C2211yescharacteristic
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2212yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4Q818yescomplemented
100C2×#117—28no—
101C2×#117—28no—
102C2×#117—44no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#400—216no—
138#399—216no—
139#398—216no—
140#395—28no—
141#395D418yescomplemented
142#393—28no—
143#393—28no—
144#393—44no—
145#391—28no—
146#390—24no—
147#390—24no—
148#390—44no—
149#390—44no—
150#389—216no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—44no—
155#388—28no—
156#388—28no—
157#384—24no—
158#382—28no—
159#380—28no—
160#380—216no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—44no—
164C4×C4⋊C4D418yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2312yes—
192C25C2412yes—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8C2×D414yes—
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×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C4⋊Q8—432no—
232C4⋊Q8—416no—
233C4×Q8—416no—
234C4×D4—232no—
235#117—432no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×Q814yes—
284C2×C4⋊C4#4114yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
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—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
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⋊C4C2×Q818yes—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—42no—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328C2×C42C2×Q814yes—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—22no—
341#95—24no—
342#95—42no—
343#95—44no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—42no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95C2×D418yes—
354#95C2×D418yes—
355#95C2×D414yes—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24C22×Q814yes—
367C24C2511yescharacteristic
368C2×Q8—28no—
369C2×Q8—48no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—88no—
377C2×Q8—88no—
378C2×D4—28no—
379C2×D4—216no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—48no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
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—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
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—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q812yes—
479C22×C4C22×Q812yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C42−1+412yes—
485C22×C42−1+414yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—48no—
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—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—88no—
502C4⋊C4—84no—
503C4⋊C4C22⋊Q8116yes—
504C22⋊C4—216no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—28no—
514C42—28no—
515C42—216no—
516C42—48no—
517C42—44no—
518C42—48no—
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—44no—
533C23—44no—
534C23—42no—
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
549C23C2×2−1+412yes—
550C23C2×2−1+411yescharacteristic
551C23#58511yescharacteristic
552C23#58512yes—
553C23#58512yes—
554Q8—416no—
555Q8—816no—
556D4—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—432no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4C2×C22⋊Q818yes—
647C2×C4C2×C22⋊Q814yes—
648C2×C4#53514yes—
649C2×C4#53614yes—
650C2×C4#54218yes—
651C2×C4#54418yes—
652C2×C4D4214yes—
653C2×C4D4×Q814yes—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—48no—
658C22—44no—
659C22—82no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—82no—
665C22C22×C22⋊Q811yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#53611yescharacteristic
668C22C2×#54212yes—
669C22C2×#54412yes—
670C22C2×D4211yescharacteristic
671C22C2×D4×Q811yescharacteristic
672C22D4×#4111yescharacteristic
673C22#348711yescharacteristic
674C22#348911yescharacteristic
675C22#349612yes—
676C22#349712yes—
677C22#350011yescharacteristic
678C22#350111yescharacteristic
679C22#350412yes—
680C22#352011yescharacteristic
681C22#352412yes—
682C22#354012yes—
683C22#354212yes—
684C22#354712yes—
685C22#355412yes—
686C22#355512yes—
687C22#355712yes—
688C4—24no—
689C4—24no—
690C4—28no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—82no—
698C4—84no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4—81no—
703C4#262818yescomplemented
704C2—84no—
705C2C2×#262811yescharacteristic
706C2#3604011yescharacteristic
707C2D4×C22⋊Q811yescharacteristic
708C2#3933911yescharacteristic
709C2#3936311yescharacteristic
710C2#3937012yes—
711C2#3941112yes—
712C2#4249312yes—
713C2#4263212yes—
714C2#4288411yescharacteristic
715C2#4299511yescharacteristic
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×#2546C2semidirect
2. C2×#2628C2semidirect
4. #16602C2semidirect
5. #16566C2semidirect
6. #16331C2semidirect
7. #16304C2semidirect
10. #15146C2semidirect
12. #15128C2semidirect
13. #14887C2semidirect
14. #14046C2semidirect
16. #13918C2semidirect
17. C4×#395C2semidirect
99. C2×C4×D4Q8semidirect
141. #395D4semidirect
703. C4#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#26282
C2×#23542
#166022
#165662
#163312
#163042
#151572
#151482
#151462
#151442
#151282
#148872
#140462
#139202
#139182
C4×#3952