Finite Group Databasebeta

← #641203

Subgroups — #641203

Order 512.

2634 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C212yesmaximal
2#17076C212yesmaximal, complemented
3#17073C212yesmaximal, complemented
4#16907C214yesmaximal, complemented
5#15228C212yesmaximal, complemented
6#15221C212yesmaximal, complemented
7#15218C212yesmaximal, complemented
8#15137C212yesmaximal, complemented
9#14766C214yesmaximal
10#14686C211yesmaximal, characteristic
11#14353C212yesmaximal
12#14275C212yesmaximal
13#13907C212yesmaximal
14C4×#380C211yesmaximal, characteristic
15C2×C4⋊#95C211yesmaximal, characteristic
16C22×#125C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2442—216no—
21C2×#400C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#385C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2214yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C2×#125—28no—
82C2×#125—44no—
83C2×#125—44no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4D418yes—
89C22×C4⋊C4Q818yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C42—42no—
114C22×C42—42no—
115C22×C42Q818yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#400—216no—
123#398—216no—
124#398—48no—
125#389—28no—
126#389—28no—
127#389—44no—
128#389—44no—
129#385—216no—
130#384—42no—
131#384—42no—
132#382—28no—
133#380—216no—
134#380—28no—
135#380—24no—
136#380—24no—
137#380—28no—
138#380—44no—
139#380—48no—
140C4×C4⋊C4—42no—
141C4×C4⋊C4—42no—
142C4×C22⋊C4—28no—
143C2×#95—28no—
144C2×#95—44no—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4C2×D414yes—
198C23×C4C2×D418yes—
199C23×C4C2×Q814yes—
200C23×C4C2×Q814yes—
201C23×C4C2×Q814yes—
202C23×C4C2×Q818yes—
203C23×C4C2×Q814yes—
204C23×C4#4114yes—
205C23×C4#4114yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#125—432no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
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—416no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×Q818yes—
282C2×C4⋊C4C2×Q818yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—416no—
303C2×C42—41no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42—41no—
307C2×C42C2×Q814yes—
308C2×C42C2×Q814yes—
309C2×C42C2×Q814yes—
310C2×C42C2×Q814yes—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—44no—
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—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24C22×D412yes—
345C24C22×Q812yes—
346C24C22×Q812yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24H(5,𝔽2)12yes—
350C242−1+412yes—
351C24C2511yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4#125116yescomplemented
462C22×C4C4⋊Q8116yes—
463C22×C4C22×D418yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q818yes—
468C22×C4C22×Q812yes—
469C22×C4C22×Q812yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C4⋊C4—88no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C42—28no—
509C42—28no—
510C42—44no—
511C42—44no—
512C42—48no—
513C42—48no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23C2×#12514yes—
543C23C2×C4⋊Q814yes—
544C23#54414yes—
545C23#54714yes—
546C23#55014yes—
547C23#55414yes—
548C23#55714yes—
549C23C23×D411yescharacteristic
550C23C23×Q811yescharacteristic
551C23C23×Q811yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×2−1+411yescharacteristic
560C23C2×2−1+411yescharacteristic
561C23C2×2−1+411yescharacteristic
562C23#58511yescharacteristic
563C23#58511yescharacteristic
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—232no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—816no—
655C2×C4—816no—
656C2×C4C2×#12514yes—
657C2×C4C2×#12514yes—
658C2×C4C2×C4⋊Q814yes—
659C2×C4C2×C4⋊Q814yes—
660C2×C4#53414yes—
661C2×C4#53714yes—
662C2×C4#54314yes—
663C2×C4#54314yes—
664C2×C4#54414yes—
665C2×C4#54414yes—
666C2×C4#54414yes—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22#255218yes—
682C22C22×#12511yescharacteristic
683C22C22×C4⋊Q811yescharacteristic
684C22C2×#53411yescharacteristic
685C22C2×#53711yescharacteristic
686C22C2×#54311yescharacteristic
687C22C2×#54311yescharacteristic
688C22C2×#54411yescharacteristic
689C22C2×#54411yescharacteristic
690C22C2×#54411yescharacteristic
691C22C2×#54411yescharacteristic
692C22#345811yescharacteristic
693C22C2×#54711yescharacteristic
694C22C2×#55011yescharacteristic
695C22C2×#55411yescharacteristic
696C22C2×#55711yescharacteristic
697C22#349611yescharacteristic
698C22#349711yescharacteristic
699C22#350111yescharacteristic
700C22#350211yescharacteristic
701C22#350511yescharacteristic
702C22#350511yescharacteristic
703C22#350511yescharacteristic
704C22#350511yescharacteristic
705C22#350811yescharacteristic
706C22#350911yescharacteristic
707C22#351111yescharacteristic
708C22#351111yescharacteristic
709C22#353611yescharacteristic
710C22#353611yescharacteristic
711C22#353711yescharacteristic
712C22#353811yescharacteristic
713C22#355811yescharacteristic
714C22#355911yescharacteristic
715C22#356111yescharacteristic
716C22#356311yescharacteristic
717C4—24no—
718C4—24no—
719C4—24no—
720C4—24no—
721C4—48no—
722C4—48no—
723C4—42no—
724C4—82no—
725C4—82no—
726C4—84no—
727C4—82no—
728C4—82no—
729C4—84no—
730C4#265918yes—
731C2—28no—
732C2C2×#265911yescharacteristic
733C2C2×#255211yescharacteristic
734C2#3502011yescharacteristic
735C2#3516711yescharacteristic
736C2#3528611yescharacteristic
737C2#3747711yescharacteristic
738C2#3756111yescharacteristic
739C2#3948611yescharacteristic
740C2#3948711yescharacteristic
741C2#3949511yescharacteristic
742C2#3949611yescharacteristic
743C2#3974211yescharacteristic
744C2#3974311yescharacteristic
745C2#3975111yescharacteristic
746C2#3975211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #17076C2semidirect
3. #17073C2semidirect
4. #16907C2semidirect
5. #15228C2semidirect
6. #15221C2semidirect
7. #15218C2semidirect
8. #15137C2semidirect
461. C22×C4#125semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
#170762
#170732
#169072
#152282
#152212
#152182
#151372
#147662
#146862
#143532
#142752
#139072
C4×#3802
C2×C4⋊#952