Finite Group Databasebeta

← #804199

Subgroups — #804199

Order 512.

2914 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C212yesmaximal, complemented
2C2×#2452C211yesmaximal, characteristic
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#16366C212yesmaximal, complemented
5#15385C214yesmaximal, complemented
6#15371C214yesmaximal, complemented
7#15156C211yesmaximal, characteristic, complemented
8#15153C211yesmaximal, characteristic, complemented
9#15152C211yesmaximal, characteristic, complemented
10#14843C212yesmaximal, complemented
11#14841C212yesmaximal, complemented
12#14053C212yesmaximal, complemented
13#13918C212yesmaximal, complemented
14#13916C212yesmaximal
15#13916C212yesmaximal
16C4×#400C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2503—216no—
24#2452—28no—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#400C2211yescharacteristic
30C2×#396C2212yes—
31C2×#393C2212yes—
32D4×C4⋊C4—28no—
33C2×#391C2214yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#385C2214yes—
41C2×#385C2212yes—
42C2×#384C2211yescharacteristic
43C2×#382C2214yes—
44C2×#381C2214yes—
45C2×#381C2214yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1446C2214yes—
52#1445C2214yes—
53#1445C2211yescharacteristic
54#1445C2212yes—
55#1443C2212yes—
56#1443C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2211yescharacteristic
62#1441C2212yes—
63#1441C2214yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2314yes—
89C24×C4C2312yes—
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—28no—
98C2×C4×D4—28no—
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⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132#400—28no—
133#400—28no—
134#400—28no—
135#400—44no—
136#400D418yescomplemented
137#396—28no—
138#393—28no—
139#391—216no—
140#390—24no—
141#390—24no—
142#390—28no—
143#389—28no—
144#389—24no—
145#389—24no—
146#385—28no—
147#385—216no—
148#384—24no—
149#382—216no—
150#381—216no—
151#381—216no—
152#380—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C22⋊C4—28no—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C25C2412yes—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—28no—
193C22×Q8—42no—
194C22×Q8—44no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4C2×D414yes—
199C22×D4#4118yes—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D418yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C4⋊Q8—416no—
229C22⋊Q8—432no—
230C4×D4—232no—
231C4×D4—432no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
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—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4#4118yes—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C2×D418yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—28no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42C2×D414yes—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—22no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—22no—
326#95—24no—
327#95—42no—
328#95—44no—
329#95—42no—
330#95—42no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C2×D414yes—
341#95C2×D418yes—
342#95C2×D418yes—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24C22×D414yes—
353C24C2×#4118yes—
354C24C2511yescharacteristic
355C2×Q8—48no—
356C2×Q8—416no—
357C2×Q8—48no—
358C2×Q8—84no—
359C2×Q8—88no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4#124116yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
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—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C22×C42−1+412yes—
472C22×C42−1+412yes—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—44no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4#123116yes—
488C22⋊C4—216no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—84no—
495C22⋊C4—88no—
496C42—28no—
497C42—416no—
498C42—44no—
499C42—48no—
500C42—44no—
501C42—48no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23C2×#12418yes—
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23#58512yes—
534C23#58512yes—
535C23#58511yescharacteristic
536Q8—816no—
537D4—232no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
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—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—432no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—832no—
629C2×C4C2×#12318yes—
630C2×C4C2×#12314yes—
631C2×C4C2×#12414yes—
632C2×C4#53614yes—
633C2×C4D4214yes—
634C2×C4#55218yes—
635C2×C4#55218yes—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—48no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—44no—
647C22C22×#12311yescharacteristic
648C22C22×#12411yescharacteristic
649C22C2×#53611yescharacteristic
650C22C2×D4211yescharacteristic
651C22D4×#4112yes—
652C22D4×#4112yes—
653C22C2×#55212yes—
654C22C2×#55212yes—
655C22#348511yescharacteristic
656C22#348911yescharacteristic
657C22#349711yescharacteristic
658C22#349911yescharacteristic
659C22#350011yescharacteristic
660C22#351312yes—
661C22#351412yes—
662C22#351512yes—
663C22#351612yes—
664C22#352112yes—
665C22#352412yes—
666C22#354711yescharacteristic
667C22#355011yescharacteristic
668C22#355712yes—
669C22#356512yes—
670C4—24no—
671C4—28no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—84no—
679C4—81no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#250318yescomplemented
686C2—48no—
687C2C2×#250311yescharacteristic
688C2#3598311yescharacteristic
689C2#3604211yescharacteristic
690C2D4×#12311yescharacteristic
691C2D4×#12411yescharacteristic
692C2#3934311yescharacteristic
693C2#4264711yescharacteristic
694C2#4500812yes—
695C2#4505412yes—
696C2#5949812yes—
697C2#5984212yes—
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×#2503C2semidirect
4. #16366C2semidirect
5. #15385C2semidirect
6. #15371C2semidirect
7. #15156C2semidirect
8. #15153C2semidirect
9. #15152C2semidirect
10. #14843C2semidirect
11. #14841C2semidirect
12. #14053C2semidirect
13. #13918C2semidirect
16. C4×#400C2semidirect
136. #400D4semidirect
685. C4#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#24522
C2×D4×C4⋊C42
#163662
#153852
#153712
#151562
#151532
#151522
#148432
#148412
#140532
#139182
#139162
C4×#4002
#138202