Finite Group Databasebeta

← #4380254

Subgroups — #4380254

Order 512.

3652 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic, complemented
3C2×#2615C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#16256C214yesmaximal, complemented
6#15679C214yesmaximal, complemented
7#15633C214yesmaximal, complemented
8#15617C214yesmaximal, complemented
9#15316C214yesmaximal, complemented
10#14886C214yesmaximal, complemented
11#14659C211yesmaximal, characteristic, complemented
12#14297C211yesmaximal, characteristic, complemented
13#13937C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#123C2212yes—
16C22×#121C2211yescharacteristic
17C22×#121C2212yes—
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2615—28no—
21#2591—28no—
22#2406—28no—
23C2×#400C2212yes—
24C2×#396C2214yes—
25C2×#395C2212yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31C2×#392C2212yes—
32C2×#392C2214yes—
33#2316—28no—
34C2×#391C2212yes—
35C2×#390C2212yes—
36C2×#390C2214yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#388C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2214yes—
44C2×#381C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#1445C2212yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2212yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1434C2214yes—
70C22⋊#95C2214yes—
71C22×#41—24no—
72C22×#41—24no—
73C22×#41—24no—
74C22×#41—44no—
75C22×#41—44no—
76C23×Q8C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2314yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#123—28no—
87C2×#123—44no—
88C2×#123—44no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—28no—
93C2×#121—44no—
94C2×#121—44no—
95C2×#117—28no—
96C2×#117—44no—
97C2×#117—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4D418yescomplemented
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127#400—28no—
128#396—48no—
129#395—28no—
130#394—28no—
131#394—28no—
132#394—48no—
133#394—48no—
134#393—28no—
135#392—216no—
136#392—28no—
137#391—44no—
138#390—48no—
139#390—44no—
140#389—216no—
141#389—216no—
142#388—216no—
143#386—28no—
144#386—28no—
145#386—28no—
146#385—216no—
147#381—24no—
148#380—24no—
149#380—28no—
150#380—28no—
151#380D418yes—
152C4×C22⋊C4—24no—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C25C2412yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C2×#41—432no—
174C2×#41—432no—
175C2×#41—416no—
176C2×#41—416no—
177C2×#41—416no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—42no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—28no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4#4114yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244#123—432no—
245#121—432no—
246#121—416no—
247#117—432no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4C2×D418yes—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—42no—
325C2×C42—42no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C2×D418yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C2×#4112yes—
370C24H(5,𝔽2)14yes—
371C24C2511yescharacteristic
372#41—4128no—
373C2×Q8—28no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×D4—216no—
379C2×D4—28no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—416no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—416no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—84no—
398C2×D4—82no—
399C2×D4—82no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—88no—
406C2×D4—88no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4#121116yes—
479C22×C4C22×D418yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C42−1+414yes—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C42—416no—
516C42—84no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23C2×#12114yes—
577C23C23×D411yescharacteristic
578C23C23×D411yescharacteristic
579C23C22×#4111yescharacteristic
580C23C2×H(5,𝔽2)12yes—
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)12yes—
583C23C2×2−1+412yes—
584C23#58514yes—
585Q8—416no—
586D4—416no—
587D4—432no—
588D4—88no—
589D4—832no—
590D4—816no—
591C2×C4—232no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—416no—
615C2×C4—432no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—88no—
654C2×C4C2×#12114yes—
655C2×C4C2×#12114yes—
656C2×C4#53718yes—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—48no—
669C22—48no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22—84no—
676C22—82no—
677C22—84no—
678C22—82no—
679C22—84no—
680C22—82no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22C22×#12111yescharacteristic
686C22C2×#53712yes—
687C22#348412yes—
688C22#348412yes—
689C22#348411yescharacteristic
690C22#348611yescharacteristic
691C22#348712yes—
692C22#349011yescharacteristic
693C22#349011yescharacteristic
694C22#349212yes—
695C22#350012yes—
696C22#351012yes—
697C22#354214yes—
698C22#354614yes—
699C22#354714yes—
700C22#355914yes—
701C4—24no—
702C4—24no—
703C4—28no—
704C4—44no—
705C4—44no—
706C4—48no—
707C4—84no—
708C4—82no—
709C4—84no—
710C4—84no—
711C2—44no—
712C2—84no—
713C2—162no—
714C2#3522311yescharacteristic
715C2#3534811yescharacteristic
716C2#3567711yescharacteristic
717C2#4247614yes—
718C2#4642812yes—
719C2#4642912yes—
720C2#4959212yes—
721C2#4959312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2615C2
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×#2591C2semidirect
2. C2×#2406C2semidirect
4. C2×#2316C2semidirect
5. #16256C2semidirect
6. #15679C2semidirect
7. #15633C2semidirect
8. #15617C2semidirect
9. #15316C2semidirect
10. #14886C2semidirect
11. #14659C2semidirect
12. #14297C2semidirect
13. #13937C2semidirect
100. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#24062
C2×#26152
C2×#23162
#162562
#156792
#156332
#156172
#153162
#148862
#146592
#142972
#139372