Finite Group Databasebeta

← #1395696

Subgroups — #1395696

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#16443C214yesmaximal, complemented
4#16399C212yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#16280C214yesmaximal, complemented
7#15197C211yesmaximal, characteristic, complemented
8#15152C211yesmaximal, characteristic, complemented
9#15149C211yesmaximal, characteristic, complemented
10#15147C212yesmaximal
11#15138C212yesmaximal
12#14892C212yesmaximal, complemented
13#14258C212yesmaximal, complemented
14#14053C212yesmaximal, complemented
15#14045C212yesmaximal, complemented
16#13920C211yesmaximal, characteristic
17C4×#398C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2452—28no—
23C2×#400C2212yes—
24C2×#398C2211yescharacteristic
25C2×#398C2212yes—
26C2×#397C2214yes—
27C2×#397C2214yes—
28C2×#396C2212yes—
29C2×#393C2212yes—
30#2354—28no—
31C2×#391C2214yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2212yes—
40C2×#388C2214yes—
41C2×#385C2212yes—
42C2×#384C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49#1446C2214yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1444C2211yescharacteristic
53#1444C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2211yescharacteristic
63#1441C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2214yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C4×D4—28no—
94C2×C4×D4D418yes—
95C2×#117—28no—
96C2×#117—48no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130#400—44no—
131#398—24no—
132#398—44no—
133#397—48no—
134#397—48no—
135#396—28no—
136#393—28no—
137#391—48no—
138#390—28no—
139#390—24no—
140#390—24no—
141#390—44no—
142#389—24no—
143#389—28no—
144#389—24no—
145#389—44no—
146#388—216no—
147#385—28no—
148#384—24no—
149#380—28no—
150#380—44no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—44no—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—28no—
188C22×Q8—42no—
189C22×Q8—44no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4C2×D414yes—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—44no—
199C23×C4C2×D418yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—416no—
217C22⋊Q8—432no—
218C4×D4—232no—
219#117—432no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4#4114yes—
268C2×C4⋊C4#4118yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×D418yes—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—44no—
309C2×C42C2×D414yes—
310#95—28no—
311#95—22no—
312#95—24no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—22no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—42no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24C22×D414yes—
350C24C2511yescharacteristic
351C2×Q8—48no—
352C2×Q8—416no—
353C2×Q8—48no—
354C2×Q8—84no—
355C2×Q8—88no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—48no—
360C2×D4—48no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+414yes—
460C22×C42−1+414yes—
461C22×C42−1+412yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C4⋊C4#123116yes—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—88no—
486C22⋊C4—84no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—28no—
490C42—416no—
491C42—44no—
492C42—48no—
493C42—44no—
494C42—48no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23C23×D411yescharacteristic
519C23C22×#4112yes—
520C23C22×#4111yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)12yes—
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+412yes—
526C23C2×2−1+411yescharacteristic
527C23C2×2−1+412yes—
528C23#58511yescharacteristic
529Q8—816no—
530D4—416no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—432no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4C2×#12318yes—
622C2×C4C2×#12314yes—
623C2×C4#53614yes—
624C2×C4#53714yes—
625C2×C4#54118yes—
626C2×C4#54218yes—
627C2×C4#54614yes—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22C22×#12311yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#53711yescharacteristic
642C22C2×#54112yes—
643C22C2×#54212yes—
644C22C2×#54611yescharacteristic
645C22#348112yes—
646C22#348411yescharacteristic
647C22#348811yescharacteristic
648C22#349911yescharacteristic
649C22#350611yescharacteristic
650C22#351111yescharacteristic
651C22#351612yes—
652C22#351912yes—
653C22#352212yes—
654C22#352312yes—
655C22#353612yes—
656C22#354412yes—
657C22#354711yescharacteristic
658C22#354812yes—
659C22#354911yescharacteristic
660C22#355012yes—
661C22#355412yes—
662C4—24no—
663C4—28no—
664C4—42no—
665C4—44no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4—81no—
676C4—84no—
677C4#262518yes—
678C2—84no—
679C2C2×#262511yescharacteristic
680C2#3599611yescharacteristic
681C2#3623611yescharacteristic
682C2#3895411yescharacteristic
683C2#3946911yescharacteristic
684C2#4004712yes—
685C2#4016712yes—
686C2#4242811yescharacteristic
687C2#4247911yescharacteristic
688C2#4258212yes—
689C2#4258512yes—
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. #16443C2semidirect
4. #16399C2semidirect
5. #16365C2semidirect
6. #16280C2semidirect
7. #15197C2semidirect
8. #15152C2semidirect
9. #15149C2semidirect
12. #14892C2semidirect
13. #14258C2semidirect
14. #14053C2semidirect
15. #14045C2semidirect
17. C4×#398C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
C2×#23542
#164432
#163992
#163652
#162802
#151972
#151522
#151492
#151472
#151382
#148922
#142582
#140532
#140452
#139202
C4×#3982