Finite Group Databasebeta

← #674294

Subgroups — #674294

Order 512.

2984 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2C2×#2448C212yesmaximal
3C2×#2446C212yesmaximal, complemented
4C2×#2351C211yesmaximal, characteristic, complemented
5#15808C214yesmaximal, complemented
6#15765C214yesmaximal, complemented
7#15193C212yesmaximal, complemented
8#15148C212yesmaximal
9#15020C214yesmaximal, complemented
10#14708C211yesmaximal, characteristic
11#14231C214yesmaximal, complemented
12#14153C211yesmaximal, characteristic
13#13954C211yesmaximal, characteristic, complemented
14C4×#384C211yesmaximal, characteristic, complemented
15C4×#391C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21#2545—28no—
22#2448—216no—
23#2446—216no—
24C2×#400C2214yes—
25C2×#399C2214yes—
26C2×#399C2214yes—
27C2×#398C2214yes—
28C2×#397C2214yes—
29C2×#396C2214yes—
30#2351—28no—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#388C2214yes—
45C2×#387C2214yes—
46C2×#387C2214yes—
47C2×#384C2212yes—
48C2×#382C2211yescharacteristic
49C2×#382C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C43C2211yescharacteristic
57#1445C2214yes—
58#1445C2214yes—
59#1445C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1437C2212yes—
67#1437C2212yes—
68#1437C2212yes—
69#1437C2214yes—
70#1437C2212yes—
71#1436C2212yes—
72#1436C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4×#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—24no—
84C2×#124—28no—
85C2×#124—24no—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124—24no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8D418yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#400—216no—
130#399—216no—
131#399—216no—
132#398—216no—
133#397—216no—
134#396—216no—
135#391—28no—
136#391—24no—
137#391—28no—
138#391D4116yes—
139#390—28no—
140#390—24no—
141#390—28no—
142#389—28no—
143#389—216no—
144#389—28no—
145#389—28no—
146#388—216no—
147#388—216no—
148#388—216no—
149#387—216no—
150#387—216no—
151#384—28no—
152#382—28no—
153#382—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C43—24no—
183C25C2412yes—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8C2×D414yes—
193C22×Q8C2×D418yes—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—24no—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214#124—416no—
215#124—416no—
216C4×Q8—232no—
217C4×Q8—432no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4#4118yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42C2×D418yes—
311C2×C42C2×D418yes—
312C2×C42C2×D414yes—
313C2×C42#4114yes—
314C2×C42#4114yes—
315#95—22no—
316#95—24no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—28no—
334#95—44no—
335#95C2×D4116yes—
336#95C2×D4116yes—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24C2511yescharacteristic
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—216no—
350C2×Q8—216no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8#127116yes—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—48no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4C22×D414yes—
428C22×C4C22×D418yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D418yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C42−1+414yes—
447C22×C42−1+414yes—
448C4⋊C4—216no—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—28no—
472C42—216no—
473C42—216no—
474C42—48no—
475C42—44no—
476C42—416no—
477C42—48no—
478C42—44no—
479C42—44no—
480C42—84no—
481C42#123116yes—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C23×D411yescharacteristic
504C23C23×D412yes—
505C23C22×#4111yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4112yes—
508C23C2×2−1+412yes—
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58512yes—
512Q8—232no—
513Q8—88no—
514D4—88no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4C2×#12314yes—
593C2×C4C2×#12314yes—
594C2×C4C2×#12314yes—
595C2×C4C2×#12718yes—
596C2×C4C2×#12714yes—
597C2×C4#53614yes—
598C2×C4#54718yes—
599C2×C4#54718yes—
600C2×C4#54818yes—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—82no—
609C22—82no—
610C22C22×#12311yescharacteristic
611C22C22×#12711yescharacteristic
612C22#344411yescharacteristic
613C22C2×#53611yescharacteristic
614C22#345511yescharacteristic
615C22C2×#54712yes—
616C22C2×#54712yes—
617C22C2×#54812yes—
618C22D4×#4112yes—
619C22D4×#4112yes—
620C22#348012yes—
621C22#349512yes—
622C22#349712yes—
623C22#349712yes—
624C22#350012yes—
625C22#350712yes—
626C22#350911yescharacteristic
627C22#351011yescharacteristic
628C22#352312yes—
629C22#355512yes—
630C22#355712yes—
631C4—28no—
632C4—24no—
633C4—24no—
634C4—28no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—48no—
639C4—44no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4#250118yes—
645C2—162no—
646C2#3474411yescharacteristic
647C2C2×#250111yescharacteristic
648C2#3567311yescharacteristic
649C2#3614012yes—
650C2#3646512yes—
651C2#3934712yes—
652C2#3995112yes—
653C2#5965312yes—
654C2#5982912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2446C2semidirect
4. C2×#2351C2semidirect
5. #15808C2semidirect
6. #15765C2semidirect
7. #15193C2semidirect
9. #15020C2semidirect
11. #14231C2semidirect
13. #13954C2semidirect
14. C4×#384C2semidirect
15. C4×#391C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
C2×#24482
C2×#24462
C2×#23512
#158082
#157652
#151932
#151482
#150202
#147082
#142312
#141532
#139542
C4×#3842
C4×#3912