Finite Group Databasebeta

← #1409262

Subgroups — #1409262

Order 512.

2540 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2512C211yesmaximal, characteristic
3C2×#2664C212yesmaximal, complemented
4#16682C212yesmaximal, complemented
5#16674C212yesmaximal, complemented
6#16550C214yesmaximal, complemented
7#16469C214yesmaximal, complemented
8#15155C211yesmaximal, characteristic
9#15152C211yesmaximal, characteristic, complemented
10#15147C211yesmaximal, characteristic
11#15145C212yesmaximal, complemented
12#15134C212yesmaximal
13#14870C212yesmaximal, complemented
14#14151C212yesmaximal, complemented
15#14141C212yesmaximal
16C4×#398C211yesmaximal, characteristic, complemented
17#13821C211yesmaximal, characteristic, complemented
18C22×#126C2212yes—
19C22×#125C2212yes—
20#2664—216no—
21#2552—28no—
22#2512—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#399C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#398C2211yescharacteristic
31C2×#397C2212yes—
32C2×#397C2212yes—
33C2×#396C2214yes—
34C2×#396C2212yes—
35C2×#396C2214yes—
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#389C2212yes—
45C2×#389C2211yescharacteristic
46C2×#388C2212yes—
47C2×#388C2214yes—
48C2×#384C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58#1445C2212yes—
59#1445C2211yescharacteristic
60#1444C2211yescharacteristic
61#1444C2214yes—
62#1444C2214yes—
63#1442C2212yes—
64#1442C2212yes—
65#1441C2211yescharacteristic
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2211yescharacteristic
83#1434C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×D4C2311yescharacteristic
87C2×#126—28no—
88C2×#126—28no—
89C2×#126—44no—
90C2×#125—28no—
91C2×#125—28no—
92C2×#125—44no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2314yes—
132#400—28no—
133#399—28no—
134#399—44no—
135#399—48no—
136#398—216no—
137#398—216no—
138#398—48no—
139#398D418yescomplemented
140#397—28no—
141#397—28no—
142#396—216no—
143#396—28no—
144#396—48no—
145#390—24no—
146#390—24no—
147#390—24no—
148#390—28no—
149#390—44no—
150#389—216no—
151#389—24no—
152#389—28no—
153#389—28no—
154#389—28no—
155#388—28no—
156#388—216no—
157#384—24no—
158#384—28no—
159#384—24no—
160#384—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—44no—
165C4×C4⋊C4—44no—
166C4×C4⋊C4D418yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C25C2412yes—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—22no—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214#126—432no—
215#125—432no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D418yes—
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—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
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×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—28no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D418yes—
322C2×C42C2×D414yes—
323C2×C42#4118yes—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—42no—
340#95—42no—
341#95—44no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C2×D414yes—
347#95C2×D418yes—
348#95C2×D418yes—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24C2511yescharacteristic
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—48no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
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—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C42−1+414yes—
460C22×C42−1+414yes—
461C22×C42−1+412yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—28no—
495C42—216no—
496C42—216no—
497C42—44no—
498C42—48no—
499C42—44no—
500C42—88no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C23×D411yescharacteristic
525C23C23×D411yescharacteristic
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+412yes—
533C23C2×2−1+411yescharacteristic
534C23#58511yescharacteristic
535D4—88no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4#53214yes—
626C2×C4#53614yes—
627C2×C4#53714yes—
628C2×C4#53918yes—
629C2×C4#54118yes—
630C2×C4D4214yes—
631C2×C4#55018yes—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22C2×#53211yescharacteristic
644C22C2×#53611yescharacteristic
645C22C2×#53711yescharacteristic
646C22C2×#53912yes—
647C22C2×#54112yes—
648C22C2×D4211yescharacteristic
649C22C2×#55012yes—
650C22#348611yescharacteristic
651C22#348911yescharacteristic
652C22#349611yescharacteristic
653C22#349811yescharacteristic
654C22#349912yes—
655C22#349911yescharacteristic
656C22#350712yes—
657C22#350812yes—
658C22#350912yes—
659C22#353612yes—
660C22#354412yes—
661C22#354711yescharacteristic
662C22#354811yescharacteristic
663C22#355212yes—
664C22#355212yes—
665C22#355912yes—
666C4—24no—
667C4—28no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—82no—
678C4—84no—
679C4—81no—
680C4—84no—
681C4#266418yescomplemented
682C2—162no—
683C2C2×#266411yescharacteristic
684C2#3593811yescharacteristic
685C2#3612811yescharacteristic
686C2#3933711yescharacteristic
687C2#3934011yescharacteristic
688C2#3938412yes—
689C2#3938512yes—
690C2#4270412yes—
691C2#4279411yescharacteristic
692C2#4291411yescharacteristic
693C2#4293912yes—
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×#2552C2semidirect
3. C2×#2664C2semidirect
4. #16682C2semidirect
5. #16674C2semidirect
6. #16550C2semidirect
7. #16469C2semidirect
9. #15152C2semidirect
11. #15145C2semidirect
13. #14870C2semidirect
14. #14151C2semidirect
16. C4×#398C2semidirect
17. #13821C2semidirect
139. #398D4semidirect
681. C4#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#25122
C2×#26642
#166822
#166742
#165502
#164692
#151552
#151522
#151472
#151452
#151342
#148702
#141512
#141412
C4×#3982
#138212