Finite Group Databasebeta

← #4389776

Subgroups — #4389776

Order 512.

2912 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2583C211yesmaximal, characteristic, complemented
3C2×#2484C211yesmaximal, characteristic, complemented
4C2×#2500C211yesmaximal, characteristic
5C2×#2625C211yesmaximal, characteristic, complemented
6C2×#2618C211yesmaximal, characteristic
7#16964C212yesmaximal, complemented
8#16811C212yesmaximal, complemented
9#15825C214yesmaximal, complemented
10#15765C214yesmaximal, complemented
11#15549C214yesmaximal, complemented
12#15384C214yesmaximal, complemented
13#15170C212yesmaximal, complemented
14#15145C212yesmaximal, complemented
15#13780C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2816—28no—
22#2625—28no—
23#2618—28no—
24#2583—28no—
25#2500—28no—
26#2484—28no—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#399C2212yes—
30C2×#399C2214yes—
31C2×#398C2212yes—
32C2×#398C2212yes—
33C2×#398C2212yes—
34C2×#398C2212yes—
35C2×#398C2212yes—
36C2×#397C2214yes—
37C2×#396C2214yes—
38C2×#395C2212yes—
39C2×#394C2214yes—
40C2×#394C2214yes—
41C2×#394C2212yes—
42C2×#394C2212yes—
43C2×#393C2212yes—
44C2×#391C2212yes—
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2212yes—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#388C2214yes—
51C2×#387C2214yes—
52C2×#386C2212yes—
53C2×#385C2212yes—
54C2×#385C2212yes—
55C2×#385C2212yes—
56C2×#384C2211yescharacteristic
57C2×#384C2214yes—
58C2×#382C2211yescharacteristic
59C2×#381C2212yes—
60C2×#381C2211yescharacteristic
61C2×#380C2211yescharacteristic
62C2×C4×C22⋊C4C2211yescharacteristic
63#1445C2214yes—
64#1445C2214yes—
65#1445C2214yes—
66#1444C2214yes—
67#1444C2214yes—
68#1444C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80C4×#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#123—28no—
88C2×#123—48no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#400—28no—
135#400—28no—
136#399—28no—
137#399—48no—
138#398—28no—
139#398—28no—
140#398—28no—
141#398—28no—
142#398—28no—
143#397—216no—
144#396—216no—
145#395—28no—
146#394—216no—
147#394—216no—
148#394—28no—
149#394—28no—
150#393—28no—
151#391—28no—
152#390—24no—
153#390—24no—
154#389—28no—
155#389—28no—
156#389—28no—
157#388—216no—
158#387—216no—
159#386—28no—
160#385—28no—
161#385—28no—
162#385—28no—
163#384—24no—
164#384—48no—
165#382—24no—
166#381—28no—
167#381D418yes—
168#380D418yes—
169C4×C22⋊C4D418yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—44no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235#123—432no—
236C22⋊Q8—416no—
237#121—416no—
238#117—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—416no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D418yes—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
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×C22⋊C4C2×D418yes—
321C2×C22⋊C4C2×D418yes—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—44no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—44no—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—44no—
347#95—44no—
348#95C2×D418yes—
349#95C2×D418yes—
350#95C2×D418yes—
351#95C2×D418yes—
352C24—22no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—28no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C2511yescharacteristic
369C2×Q8—48no—
370C2×Q8—44no—
371C2×Q8—84no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—44no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—88no—
383C2×D4—88no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
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—22no—
411C22×C4—24no—
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—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—84no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4#127116yes—
459C22×C4C22×D418yes—
460C22×C4C22×D414yes—
461C22×C4C22×D418yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D414yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C42−1+418yes—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C22⋊C4—216no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—44no—
501C42—44no—
502C42—416no—
503C42—48no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23C2×#12714yes—
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C23×D411yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+414yes—
541C23#58512yes—
542C23#58512yes—
543C23#58512yes—
544Q8—88no—
545D4—88no—
546D4—816no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—416no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
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—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4C2×#12714yes—
620C2×C4C2×#12714yes—
621C2×C4#53618yes—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—44no—
630C22—84no—
631C22—82no—
632C22—84no—
633C22—84no—
634C22—88no—
635C22—84no—
636C22—82no—
637C22—82no—
638C22C22×#12711yescharacteristic
639C22C2×#53612yes—
640C22#348411yescharacteristic
641C22#348512yes—
642C22#348511yescharacteristic
643C22#348511yescharacteristic
644C22#348512yes—
645C22#348711yescharacteristic
646C22#348812yes—
647C22#348812yes—
648C22#349912yes—
649C22#349912yes—
650C22#354114yes—
651C22#354714yes—
652C22#355314yes—
653C22#356514yes—
654C4—24no—
655C4—24no—
656C4—28no—
657C4—48no—
658C4—44no—
659C4—48no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C2—44no—
668C2—162no—
669C2#3531711yescharacteristic
670C2#3546311yescharacteristic
671C2#3556511yescharacteristic
672C2#4140612yes—
673C2#4268112yes—
674C2#4681912yes—
675C2#4721412yes—
676C2#4883214yes—
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
2. C2×#2583C2semidirect
3. C2×#2484C2semidirect
5. C2×#2625C2semidirect
7. #16964C2semidirect
8. #16811C2semidirect
9. #15825C2semidirect
10. #15765C2semidirect
11. #15549C2semidirect
12. #15384C2semidirect
13. #15170C2semidirect
14. #15145C2semidirect
15. #13780C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#25832
C2×#24842
C2×#25002
C2×#26252
C2×#26182
#169642
#168112
#158252
#157652
#155492
#153842
#151702
#151452
#137802