Finite Group Databasebeta

← #4420168

Subgroups — #4420168

Order 512.

2546 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2610C211yesmaximal, characteristic
2C2×#2640C211yesmaximal, characteristic, complemented
3C2×#2628C211yesmaximal, characteristic, complemented
4C2×#2346C211yesmaximal, characteristic, complemented
5#16803C212yesmaximal, complemented
6#16801C212yesmaximal
7#16518C212yesmaximal, complemented
8#16401C212yesmaximal, complemented
9#15633C214yesmaximal, complemented
10#15631C214yesmaximal, complemented
11#15164C212yesmaximal, complemented
12#15107C212yesmaximal, complemented
13#14894C212yesmaximal, complemented
14#14842C212yesmaximal, complemented
15#14662C211yesmaximal, characteristic, complemented
16#14274C211yesmaximal, characteristic, complemented
17#13963C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2640—28no—
22#2628—28no—
23#2610—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#395C2211yescharacteristic
31C2×#394C2211yescharacteristic
32C2×#394C2214yes—
33C2×#393C2212yes—
34C2×#393C2214yes—
35C2×#393C2211yescharacteristic
36C2×#392C2211yescharacteristic
37#2346—28no—
38C2×#391C2212yes—
39C2×#391C2214yes—
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#388C2212yes—
48C2×#386C2212yes—
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×#376C2211yescharacteristic
57C2×#376C2211yescharacteristic
58C2×#376C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1444C2212yes—
65#1442C2214yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1438C2212yes—
79#1438C2214yes—
80#1437C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86#1434C2214yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C23×D4C2312yes—
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×#124—24no—
96C2×#124—24no—
97C2×#124—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#400—28no—
134#399—28no—
135#397—28no—
136#397—28no—
137#396—48no—
138#395—24no—
139#395—28no—
140#394—24no—
141#394—48no—
142#393—24no—
143#393—28no—
144#393—216no—
145#392—24no—
146#391—28no—
147#391—48no—
148#390—28no—
149#390—28no—
150#389—28no—
151#389—28no—
152#388—28no—
153#388—28no—
154#388—28no—
155#388—28no—
156#386—28no—
157#382—24no—
158#382—24no—
159#382—24no—
160#380—28no—
161#380—24no—
162C4×C4⋊C4—28no—
163C4×C22⋊C4—24no—
164#376—24no—
165#376—24no—
166#376D418yescomplemented
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—42no—
214C22×D4—44no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C4⋊Q8—416no—
236#124—416no—
237#117—232no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D414yes—
306C2×C22⋊C4C2×D414yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—44no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42C2×D418yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42#4114yes—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
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#95C2×D414yes—
349#95C2×D414yes—
350#95C2×D418yes—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—22no—
368C24—24no—
369C24C22×D412yes—
370C24C2511yescharacteristic
371C2×Q8—48no—
372C2×Q8—416no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—88no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×Q8—84no—
379C2×Q8—84no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—88no—
387C2×D4—84no—
388C2×D4—88no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C42−1+414yes—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—88no—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—84no—
493C22⋊C4—88no—
494C22⋊C4—84no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C42—28no—
498C42—48no—
499C42—48no—
500C42—48no—
501C42—48no—
502C42#121116yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23C23×D411yescharacteristic
543C23C23×D411yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×2−1+412yes—
547C23#58512yes—
548C23#58514yes—
549C23#58512yes—
550Q8—816no—
551Q8—88no—
552D4—88no—
553D4—816no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4C2×#12114yes—
629C2×C4C2×#12114yes—
630C2×C4C2×#12114yes—
631C2×C4#53614yes—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—82no—
638C22—84no—
639C22—82no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22C22×#12111yescharacteristic
652C22C2×#53611yescharacteristic
653C22#344912yes—
654C22#344911yescharacteristic
655C22#348812yes—
656C22#348912yes—
657C22#349512yes—
658C22#349811yescharacteristic
659C22#350012yes—
660C22#350011yescharacteristic
661C22#350612yes—
662C22#351012yes—
663C22#354514yes—
664C22#354714yes—
665C22#356514yes—
666C22#356614yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—84no—
680C4—82no—
681C4—82no—
682C2—82no—
683C2—162no—
684C2#3530611yescharacteristic
685C2#3532711yescharacteristic
686C2#3553311yescharacteristic
687C2#4215712yes—
688C2#4230312yes—
689C2#4239012yes—
690C2#4296512yes—
691C2#5162214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2640C2semidirect
3. C2×#2628C2semidirect
4. C2×#2346C2semidirect
5. #16803C2semidirect
7. #16518C2semidirect
8. #16401C2semidirect
9. #15633C2semidirect
10. #15631C2semidirect
11. #15164C2semidirect
12. #15107C2semidirect
13. #14894C2semidirect
14. #14842C2semidirect
15. #14662C2semidirect
16. #14274C2semidirect
17. #13963C2semidirect
166. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26102
C2×#26402
C2×#26282
C2×#23462
#168032
#168012
#165182
#164012
#156332
#156312
#151642
#151072
#148942
#148422
#146622
#142742
#139632