Finite Group Databasebeta

← #533691

Subgroups — #533691

Order 512.

3038 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2374C212yesmaximal
3#14577C212yesmaximal, complemented
4#14505C212yesmaximal
5#14248C212yesmaximal, complemented
6#14212C214yesmaximal, complemented
7#14103C214yesmaximal, complemented
8#13994C212yesmaximal, complemented
9#13994C212yesmaximal, complemented
10#13981C212yesmaximal, complemented
11#13906C211yesmaximal, characteristic
12#13821C212yesmaximal, complemented
13C22⋊C4×C4⋊C4C212yesmaximal, complemented
14C4×#380C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×#123C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2486C4116yescomplemented
21C2×#398C2212yes—
22C2×#397C2212yes—
23#2374—216no—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#386C2211yescharacteristic
31C2×#386C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#381C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2214yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2214yes—
44C2×C4×C22⋊C4C2214yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1437C2212yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2214yes—
74C22⋊#95C2211yescharacteristic
75C4×#95C2214yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×#123—28no—
84C2×#123—28no—
85C2×#123C2×C4116yes—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4C2×C418yes—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2×C418yescomplemented
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C42—28no—
113C22×C42—42no—
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122#398C2×C4116yescomplemented
123#397C2×C4116yes—
124#390—28no—
125#389—28no—
126#389C2×C4116yescomplemented
127#388—216no—
128#388—216no—
129#387—216no—
130#386—24no—
131#386D418yes—
132#384—42no—
133#381C2×C4116yescomplemented
134#380—28no—
135#380—28no—
136#380C2×C4116yes—
137#380C2×C418yescomplemented
138C4×C4⋊C4—28no—
139C4×C4⋊C4—216no—
140C4×C4⋊C4—42no—
141C4×C4⋊C4—44no—
142C4×C4⋊C4—44no—
143C4×C22⋊C4—216no—
144C4×C22⋊C4—216no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—42no—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C22×C414yes—
179C22×D4C2×D414yes—
180C22×D4#4114yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—42no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C22×C414yes—
205C23×C4C22×C414yes—
206C23×C4C22×C418yes—
207C23×C4C2×D414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221#123—264no—
222C4×D4—416no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C4116yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4#4118yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4C22×C4116yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C414yes—
280C2×C22⋊C4C22×C414yes—
281C2×C22⋊C4C2×D418yes—
282C2×C22⋊C4C2×D414yes—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—416no—
299C2×C42—42no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42—44no—
306C2×C42C22×C414yes—
307C2×C42#4118yes—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95C22×C4116yes—
330#95C22×C418yes—
331#95C22×C4116yes—
332#95C2×D418yes—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24C23×C412yes—
350C24C23×C412yes—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C2×#4112yes—
354C24C2×#4112yes—
355C24C2511yescharacteristic
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4C4×D4116yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C23×C418yes—
457C22×C4C23×C418yes—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+414yes—
473C4⋊C4—232no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C22⋊C4—216no—
484C22⋊C4—232no—
485C22⋊C4—216no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C42—28no—
494C42—216no—
495C42—48no—
496C42—416no—
497C42—48no—
498C42—88no—
499C42—84no—
500C42—88no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23C2×C4×D414yes—
546C23C2×C4×D414yes—
547C23C24×C411yescharacteristic
548C23C23×D411yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4112yes—
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×2−1+412yes—
555C23#58511yescharacteristic
556C23#58511yescharacteristic
557C23#58512yes—
558D4—416no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—816no—
640C2×C4C2×C4×D414yes—
641C2×C4C4×#4118yes—
642C2×C4#51814yes—
643C2×C4#54718yes—
644C2×C4#55114yes—
645C2×C4#55318yes—
646C2×C4#55914yes—
647C22—28no—
648C22—28no—
649C22—24no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22C22×C4×D411yescharacteristic
670C22C2×C4×#4112yes—
671C22C2×#51811yescharacteristic
672C22#342811yescharacteristic
673C22C4×2−1+412yes—
674C22C2×#54712yes—
675C22D4×#4112yes—
676C22C2×#55111yescharacteristic
677C22C2×#55312yes—
678C22#347512yes—
679C22C2×#55911yescharacteristic
680C22#348212yes—
681C22#349312yes—
682C22#349512yes—
683C22#349611yescharacteristic
684C22#349611yescharacteristic
685C22#350511yescharacteristic
686C22#350511yescharacteristic
687C22#351512yes—
688C22#352412yes—
689C22#353811yescharacteristic
690C22#354011yescharacteristic
691C22#354712yes—
692C4—28no—
693C4—24no—
694C4—44no—
695C4—44no—
696C4—48no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—48no—
701C4—48no—
702C4—82no—
703C4—84no—
704C4#237418yescomplemented
705C2—44no—
706C2—44no—
707C2C2×#237411yescharacteristic
708C2C4×#54712yes—
709C2#3420511yescharacteristic
710C2#3430012yes—
711C2#3441211yescharacteristic
712C2#5919811yescharacteristic
713C2#5920111yescharacteristic
714C2#5921312yes—
715C2#5921612yes—
716C2#5962811yescharacteristic
717C2#5963111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14577C2semidirect
5. #14248C2semidirect
6. #14212C2semidirect
7. #14103C2semidirect
8. #13994C2semidirect
9. #13994C2semidirect
10. #13981C2semidirect
12. #13821C2semidirect
13. C22⋊C4×C4⋊C4C2semidirect
14. C4×#380C2semidirect
20. #2486C4semidirect
91. C22×C4⋊C4C2×C4semidirect
122. #398C2×C4semidirect
126. #389C2×C4semidirect
133. #381C2×C4semidirect
137. #380C2×C4semidirect
704. C4#2374semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23742
#145772
#145052
#142482
#142122
#141032
#139942
#139812
#139062
#138212
C22⋊C4×C4⋊C42
C4×#3802
C4×#3862