Finite Group Databasebeta

← #4397203

Subgroups — #4397203

Order 512.

3246 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2436C211yesmaximal, characteristic
3C2×#2434C211yesmaximal, characteristic, complemented
4C2×#2333C211yesmaximal, characteristic
5#16803C212yesmaximal, complemented
6#16506C212yesmaximal, complemented
7#15740C214yesmaximal, complemented
8#15383C214yesmaximal, complemented
9#15311C214yesmaximal, complemented
10#15300C214yesmaximal, complemented
11#15161C212yesmaximal, complemented
12#15125C212yesmaximal, complemented
13#14336C211yesmaximal, characteristic, complemented
14#14292C211yesmaximal, characteristic, complemented
15#13953C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22#2816—28no—
23#2436—28no—
24#2434—28no—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#397C2214yes—
28C2×#395C2214yes—
29C2×#394C2214yes—
30C2×#394C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34#2333—28no—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#390C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#386C2212yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#385C2214yes—
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2214yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1446C2214yes—
56#1446C2212yes—
57#1445C2214yes—
58#1445C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77#1434C2212yes—
78C22⋊#95C2214yes—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—44no—
83C23×Q8—44no—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8—44no—
97C2×#124—28no—
98C2×#124—44no—
99C2×#124—44no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×#121—24no—
104C2×#121—24no—
105C2×#121D418yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#399—28no—
133#398—28no—
134#397—48no—
135#395—216no—
136#394—28no—
137#394—216no—
138#393—28no—
139#393—28no—
140#393—28no—
141#391—28no—
142#391—24no—
143#391—24no—
144#390—28no—
145#388—28no—
146#387—28no—
147#386—216no—
148#386—28no—
149#386—28no—
150#385—216no—
151#381—24no—
152#381—24no—
153#381—44no—
154#381—44no—
155#380—48no—
156C4×C22⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4D418yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—432no—
208C22×Q8—432no—
209C22×Q8—416no—
210C22×Q8—416no—
211C22×Q8—416no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4C2×D414yes—
221C22×D4C2×D418yes—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C4⋊Q8—416no—
253C4⋊Q8—416no—
254#124—432no—
255C22⋊Q8—416no—
256#121—232no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95C2×D418yes—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24C22×D414yes—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C2×#4112yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—4128no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×D4—216no—
394C2×D4—28no—
395C2×D4—48no—
396C2×D4—416no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—88no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—216no—
410C22×C4—22no—
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—22no—
420C22×C4—24no—
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—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
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—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—88no—
479C22×C4#121116yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+414yes—
492C22×C42−1+414yes—
493C4⋊C4—28no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C4⋊C4—84no—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—82no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C22⋊C4—82no—
514C22⋊C4—88no—
515C42—44no—
516C42—44no—
517C42—48no—
518C42—84no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23C2×#12114yes—
558C23C23×D411yescharacteristic
559C23C23×D411yescharacteristic
560C23C22×#4111yescharacteristic
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+412yes—
564C23C2×2−1+412yes—
565C23#58514yes—
566Q8—432no—
567Q8—416no—
568Q8—416no—
569Q8—88no—
570D4—88no—
571D4—816no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—416no—
596C2×C4—432no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—816no—
645C2×C4C2×#12114yes—
646C2×C4C2×#12114yes—
647C2×C4#53618yes—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—44no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—84no—
665C22—82no—
666C22—82no—
667C22C22×#12111yescharacteristic
668C22C2×#53612yes—
669C22#348411yescharacteristic
670C22#348412yes—
671C22#348411yescharacteristic
672C22#348512yes—
673C22#349012yes—
674C22#349912yes—
675C22#350012yes—
676C22#350911yescharacteristic
677C22#350911yescharacteristic
678C22#351012yes—
679C22#354114yes—
680C22#354714yes—
681C22#355414yes—
682C22#356114yes—
683C4—24no—
684C4—24no—
685C4—28no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—84no—
694C4—84no—
695C4—82no—
696C2—44no—
697C2—84no—
698C2#3531611yescharacteristic
699C2#3534811yescharacteristic
700C2#3561711yescharacteristic
701C2#4049112yes—
702C2#4243112yes—
703C2#4645512yes—
704C2#4703212yes—
705C2#4794314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2434C2semidirect
5. #16803C2semidirect
6. #16506C2semidirect
7. #15740C2semidirect
8. #15383C2semidirect
9. #15311C2semidirect
10. #15300C2semidirect
11. #15161C2semidirect
12. #15125C2semidirect
13. #14336C2semidirect
14. #14292C2semidirect
15. #13953C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#24362
C2×#24342
C2×#23332
#168032
#165062
#157402
#153832
#153112
#153002
#151612
#151252
#143362
#142922
#139532