Finite Group Databasebeta

← #772944

Subgroups — #772944

Order 512.

3494 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C212yesmaximal, complemented
2C2×D4×C4⋊C4C212yesmaximal, complemented
3#16339C212yesmaximal, complemented
4#15961C212yesmaximal, complemented
5#15521C212yesmaximal, complemented
6#15406C214yesmaximal, complemented
7#15181C212yesmaximal, complemented
8#15135C211yesmaximal, characteristic, complemented
9#14807C214yesmaximal, complemented
10#14171C212yesmaximal
11#14130C211yesmaximal, characteristic, complemented
12#14005C212yesmaximal, complemented
13#13973C212yesmaximal, complemented
14#13880C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2214yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2487—216no—
22C2×#400C2214yes—
23C2×#399C2214yes—
24C2×#399C2212yes—
25C2×#392C2211yescharacteristic
26D4×C4⋊C4—216no—
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#386C2214yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#384C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49#1445C2212yes—
50#1444C2212yes—
51#1444C2211yescharacteristic
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2211yescharacteristic
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2211yescharacteristic
67#1438C2212yes—
68#1438C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2314yes—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—216no—
87C2×C4×D4—28no—
88C2×C4×D4—48no—
89C2×C4×D4—42no—
90C2×C4×D4—48no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2312yes—
127#400—216no—
128#399—216no—
129#399—44no—
130#392Q818yes—
131#389—28no—
132#389—28no—
133#389—28no—
134#389—216no—
135#389—48no—
136#386—216no—
137#385—216no—
138#385—216no—
139#385—216no—
140#385—44no—
141#385—48no—
142#385—48no—
143#384—28no—
144#384—42no—
145#380—28no—
146#380—24no—
147#380—28no—
148#380—28no—
149#380D4116yescomplemented
150C4×C4⋊C4—28no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—42no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C25C2412yes—
176C25C2411yescharacteristic
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4C2×Q818yes—
188C22×D4C2×Q814yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D418yes—
212C23×C4#4114yes—
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C4×D4—264no—
227C4×D4—416no—
228C4×D4—464no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C2×D4116yes—
307C2×C22⋊C4C2×Q818yes—
308C2×C22⋊C4C2×Q814yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—22no—
317C2×C42—44no—
318C2×C42#4118yes—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—42no—
333#95—41no—
334#95—41no—
335#95—41no—
336#95—44no—
337#95—44no—
338#95—42no—
339#95C2×D4116yes—
340#95C2×Q814yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24C22×D414yes—
362C24C22×Q814yes—
363C24C22×Q812yes—
364C24C2×#4112yes—
365C24H(5,𝔽2)12yes—
366C242−1+414yes—
367C24C2511yescharacteristic
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—216no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—44no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—416no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—84no—
464C22×C4—88no—
465C22×C4—84no—
466C22×C4—88no—
467C22×C4C22×D418yes—
468C22×C4C22×D414yes—
469C22×C4C22×D418yes—
470C22×C4C22×D414yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q814yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—44no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C4⋊C4—88no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—416no—
503C22⋊C4—44no—
504C22⋊C4—416no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—88no—
510C42—216no—
511C42—416no—
512C42—44no—
513C42—44no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23D4×Q818yes—
571C23#55414yes—
572C23C23×D412yes—
573C23C23×Q811yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×2−1+411yescharacteristic
581C23C2×2−1+412yes—
582C23#58511yescharacteristic
583D4—264no—
584D4—416no—
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—22no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—432no—
640C2×C4—416no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—816no—
661C2×C4—832no—
662C2×C4#52518yes—
663C2×C4#53914yes—
664C2×C4D4214yes—
665C2×C4#54718yes—
666C2×C4#55418yes—
667C2×C4#55914yes—
668C22—28no—
669C22—28no—
670C22—28no—
671C22—28no—
672C22—24no—
673C22—42no—
674C22—44no—
675C22—44no—
676C22—42no—
677C22—42no—
678C22—44no—
679C22—44no—
680C22—42no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—42no—
694C22C2×#52512yes—
695C22C2×#53911yescharacteristic
696C22C2×D4211yescharacteristic
697C22C2×#54712yes—
698C22C2×D4×Q812yes—
699C22C2×#55412yes—
700C22C2×#55411yescharacteristic
701C22C2×#55911yescharacteristic
702C22#348412yes—
703C22#348912yes—
704C22#349212yes—
705C22#349212yes—
706C22#349212yes—
707C22#349812yes—
708C22#350111yescharacteristic
709C22#350311yescharacteristic
710C22#350611yescharacteristic
711C22#350612yes—
712C22#353711yescharacteristic
713C22#353811yescharacteristic
714C22#354911yescharacteristic
715C22#355212yes—
716C22#355911yescharacteristic
717C4—28no—
718C4—42no—
719C4—48no—
720C4—44no—
721C4—42no—
722C4—44no—
723C4—42no—
724C4—48no—
725C4—42no—
726C4—81no—
727C4—84no—
728C4—84no—
729C4#245518yes—
730C2—48no—
731C2—44no—
732C2C2×#245511yescharacteristic
733C2#3568912yes—
734C2#3654612yes—
735C2#3963111yescharacteristic
736C2#4250211yescharacteristic
737C2#4487011yescharacteristic
738C2#4525512yes—
739C2#4537511yescharacteristic
740C2#5899811yescharacteristic
741C2#5931412yes—
742C2#5986411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2487C2semidirect
2. C2×D4×C4⋊C4C2semidirect
3. #16339C2semidirect
4. #15961C2semidirect
5. #15521C2semidirect
6. #15406C2semidirect
7. #15181C2semidirect
8. #15135C2semidirect
9. #14807C2semidirect
11. #14130C2semidirect
12. #14005C2semidirect
13. #13973C2semidirect
14. #13880C2semidirect
149. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×D4×C4⋊C42
#163392
#159612
#155212
#154062
#151812
#151352
#148072
#141712
#141302
#140052
#139732
#138802
C4×#3922