Finite Group Databasebeta

← #530413

Subgroups — #530413

Order 512.

3250 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2333C212yesmaximal
2C2×C4×#121C211yesmaximal, characteristic
3#14685C211yesmaximal, characteristic, complemented
4#14465C212yesmaximal, complemented
5#14438C212yesmaximal, complemented
6#14401C214yesmaximal, complemented
7#14297C212yesmaximal, complemented
8#14118C214yesmaximal, complemented
9#13960C212yesmaximal, complemented
10#13941C212yesmaximal, complemented
11#13915C212yesmaximal, complemented
12#13899C212yesmaximal, complemented
13#13889C212yesmaximal, complemented
14#13814C212yesmaximal, complemented
15C4×#380C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C42C2211yescharacteristic
20#2333—216no—
21C4×#121C4116yescomplemented
22C2×#390C2214yes—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388C2214yes—
26C2×#388C2212yes—
27C2×#387C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#382C2212yes—
34C2×#381C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2212yes—
46#1444C2214yes—
47#1444C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C4×#95C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×#121C2×C418yes—
85C2×#121D418yes—
86C2×#121Q818yes—
87C2×C4×D4—44no—
88C2×C4×D4—44no—
89C2×C4×D4—44no—
90C2×C4×D4C2×C418yes—
91C2×C4×D4C2×C4116yescomplemented
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42—44no—
113C22×C42C2×C418yescomplemented
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#390—216no—
123#390C2×C418yes—
124#389C2×C418yescomplemented
125#388—216no—
126#388C2×C4116yescomplemented
127#387—216no—
128#386—216no—
129#386—216no—
130#386C2×C4116yescomplemented
131#385—216no—
132#385—48no—
133#382—44no—
134#381—28no—
135#380—28no—
136#380—44no—
137#380—44no—
138#380C2×C418yescomplemented
139C4×C4⋊C4—44no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4C2×C418yescomplemented
142C4×C22⋊C4—28no—
143C4×C22⋊C4—216no—
144C4×C22⋊C4C2×C4116yescomplemented
145C2×#95—44no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4C22×C414yes—
179C22×D4C22×C418yes—
180C22×D4C2×D414yes—
181C22×D4C2×D418yes—
182C22×D4C2×Q818yes—
183C22×D4C2×Q814yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C22×C414yes—
204C23×C4C22×C418yes—
205C23×C4C22×C414yes—
206C23×C4C22×C414yes—
207C23×C4C2×D414yes—
208C23×C4C2×Q814yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#121C4⋊C4164yescomplemented
223C4×D4—416no—
224C4×D4—432no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×Q814yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C2×D418yes—
280C2×C22⋊C4C2×Q818yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—416no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C22×C414yes—
298C2×C42C22×C418yes—
299C2×C42C22×C414yes—
300C2×C42C22×C414yes—
301C2×C42C22×C414yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C418yes—
323#95C22×C418yes—
324#95C22×C418yes—
325#95#4118yes—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24C23×C414yes—
341C24C23×C412yes—
342C24C22×D412yes—
343C24C22×D414yes—
344C24C22×Q814yes—
345C24C22×Q812yes—
346C24C2511yescharacteristic
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4C2×C4⋊C4132yes—
358C2×D4C2×C4⋊C4116yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4C2×C4⋊C4116yes—
436C22×C4C23×C414yes—
437C22×C4C23×C414yes—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C414yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)14yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4C2×C4⋊C4116yes—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—88no—
471C22⋊C4—84no—
472C22⋊C4—88no—
473C22⋊C4—84no—
474C22⋊C4C2×C4⋊C4132yes—
475C42—28no—
476C42—28no—
477C42—216no—
478C42—44no—
479C42—44no—
480C42—84no—
481C42—84no—
482C42—88no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23C22×C4⋊C418yes—
522C23C22×C4⋊C414yes—
523C23C24×C411yescharacteristic
524C23C23×D411yescharacteristic
525C23C23×Q811yescharacteristic
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)12yes—
529C23#58512yes—
530C23#58512yes—
531C23#58512yes—
532D4—416no—
533D4—432no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—416no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4—832no—
604C2×C4C22×C4⋊C414yes—
605C2×C4C22×C4⋊C418yes—
606C2×C4C22×C4⋊C414yes—
607C2×C4C4×#4114yes—
608C2×C4#51814yes—
609C2×C4#51814yes—
610C22—28no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—28no—
615C22—28no—
616C22—48no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22C23×C4⋊C411yescharacteristic
631C22C2×C4×#4111yescharacteristic
632C22C2×#51811yescharacteristic
633C22C2×#51811yescharacteristic
634C22#342811yescharacteristic
635C22#342811yescharacteristic
636C22#342811yescharacteristic
637C22#346014yes—
638C22D4×#4112yes—
639C22Q8×#4112yes—
640C22#349612yes—
641C22#349612yes—
642C22#350112yes—
643C22#350112yes—
644C22#351314yes—
645C22#354214yes—
646C22#356614yes—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—28no—
651C4—24no—
652C4—24no—
653C4—48no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—84no—
658C4—84no—
659C4—84no—
660C2—44no—
661C2—48no—
662C2#3408611yescharacteristic
663C2C4⋊C4×#4111yescharacteristic
664C2#3418111yescharacteristic
665C2#3418111yescharacteristic
666C2#3422811yescharacteristic
667C2#3436311yescharacteristic
668C2#3440011yescharacteristic
669C2#3944014yes—
670C2#3969614yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14685C2semidirect
4. #14465C2semidirect
5. #14438C2semidirect
6. #14401C2semidirect
7. #14297C2semidirect
8. #14118C2semidirect
9. #13960C2semidirect
10. #13941C2semidirect
11. #13915C2semidirect
12. #13899C2semidirect
13. #13889C2semidirect
14. #13814C2semidirect
15. C4×#380C2semidirect
21. C4×#121C4semidirect
91. C2×C4×D4C2×C4semidirect
113. C22×C42C2×C4semidirect
124. #389C2×C4semidirect
126. #388C2×C4semidirect
130. #386C2×C4semidirect
138. #380C2×C4semidirect
141. C4×C4⋊C4C2×C4semidirect
144. C4×C22⋊C4C2×C4semidirect
222. #121C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23332
C2×C4×#1212
#146852
#144652
#144382
#144012
#142972
#141182
#139602
#139412
#139152
#138992
#138892
#138142
C4×#3802