Finite Group Databasebeta

← #536713

Subgroups — #536713

Order 512.

3498 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×#2360C212yesmaximal, complemented
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic
5#14548C214yesmaximal, complemented
6#14405C214yesmaximal, complemented
7#14185C214yesmaximal
8#14150C211yesmaximal, characteristic, complemented
9#14130C212yesmaximal, complemented
10#14032C214yesmaximal, complemented
11#13994C212yesmaximal, complemented
12#13969C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14#13786C211yesmaximal, characteristic, complemented
15#13779C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2536C4116yescomplemented
23C2×#398C2212yes—
24C2×#394C2212yes—
25#2360—216no—
26Q8×C4⋊C4—28no—
27D4×C4⋊C4—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#387C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2212yes—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2214yes—
52C2×C4×C22⋊C4C2212yes—
53C2×#376C2211yescharacteristic
54#1445C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2214yes—
65#1438C2212yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76#1434C2214yes—
77#1434C2214yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2314yes—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8C2×C4116yescomplemented
86C2×C4×Q8—28no—
87C2×C4×Q8—28no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—44no—
91C2×C4×Q8C2×C418yescomplemented
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—48no—
96C2×C4×D4C2×C418yes—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132#398C2×C4116yescomplemented
133#394C2×C4116yes—
134#391—24no—
135#391D418yescomplemented
136#389—28no—
137#389—216no—
138#389—28no—
139#389—28no—
140#389—216no—
141#388—48no—
142#388C2×C4132yescomplemented
143#387—216no—
144#385—216no—
145#385—216no—
146#385—48no—
147#384—28no—
148#384D418yes—
149#382C2×C418yescomplemented
150#380—28no—
151#380C2×C4116yescomplemented
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—28no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—216no—
158C4×C22⋊C4—44no—
159#376—24no—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C25C2412yes—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C22×C414yes—
189C22×Q8C2×D414yes—
190C22×Q8#4114yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4C22×C414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C22×C418yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C22⋊Q8—264no—
225C4×Q8—232no—
226C4×Q8—432no—
227C4×D4—232no—
228C4×D4—432no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—432no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
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⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C4116yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C4116yes—
297C2×C22⋊C4C22×C4116yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—28no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C22×C418yes—
317C2×C42C22×C414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42#4114yes—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—44no—
334#95—42no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C22×C4116yes—
341#95C22×C418yes—
342#95C22×C418yes—
343#95C2×D4116yes—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24C23×C414yes—
357C24H(5,𝔽2)14yes—
358C242−1+414yes—
359C24C2511yescharacteristic
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8C4×D4116yescomplemented
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—216no—
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—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4C23×C418yes—
452C22×C4C23×C414yes—
453C22×C4C23×C414yes—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C23×C414yes—
457C22×C4C23×C412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D418yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+414yes—
473C22×C42−1+412yes—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—232no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C22⋊C4—216no—
491C22⋊C4—232no—
492C22⋊C4—216no—
493C22⋊C4—232no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C42—216no—
499C42—28no—
500C42—416no—
501C42—48no—
502C42—48no—
503C42—44no—
504C42—88no—
505C42C4×D4116yes—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23#52018yes—
541C23C24×C411yescharacteristic
542C23C23×D411yescharacteristic
543C23C23×D411yescharacteristic
544C23C22×#4111yescharacteristic
545C23C22×#4111yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+412yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552Q8—232no—
553D4—232no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—432no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—832no—
634C2×C4C2×C4×D414yes—
635C2×C4C2×C4×D414yes—
636C2×C4C2×C4×D418yes—
637C2×C4C2×C4×D414yes—
638C2×C4C2×C4×D414yes—
639C2×C4#52014yes—
640C2×C4D4214yes—
641C2×C4#54714yes—
642C2×C4#54814yes—
643C2×C4#55314yes—
644C22—28no—
645C22—28no—
646C22—28no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22C22×C4×D411yescharacteristic
660C22C22×C4×D411yescharacteristic
661C22C2×#52011yescharacteristic
662C22C2×#52012yes—
663C22C4×2−1+412yes—
664C22C2×D4211yescharacteristic
665C22C2×#54711yescharacteristic
666C22C2×#54811yescharacteristic
667C22C2×#55311yescharacteristic
668C22#348411yescharacteristic
669C22#348511yescharacteristic
670C22#348611yescharacteristic
671C22#348612yes—
672C22#348611yescharacteristic
673C22#348812yes—
674C22#349212yes—
675C22#350712yes—
676C22#351612yes—
677C22#352212yes—
678C22#353812yes—
679C22#354212yes—
680C22#354412yes—
681C22#356312yes—
682C4—28no—
683C4—24no—
684C4—24no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—48no—
689C4—44no—
690C4—44no—
691C4—48no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#236118yes—
696C2—48no—
697C2C2×#236111yescharacteristic
698C2#3420011yescharacteristic
699C2#3426611yescharacteristic
700C2#3429412yes—
701C2#3446512yes—
702C2#4492112yes—
703C2#4492612yes—
704C2#4493211yescharacteristic
705C2#4493711yescharacteristic
706C2#4514311yescharacteristic
707C2#4541811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2360C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
5. #14548C2semidirect
6. #14405C2semidirect
8. #14150C2semidirect
9. #14130C2semidirect
10. #14032C2semidirect
11. #13994C2semidirect
12. #13969C2semidirect
14. #13786C2semidirect
15. #13779C2semidirect
22. #2536C4semidirect
85. C2×C22⋊Q8C2×C4semidirect
91. C2×C4×Q8C2×C4semidirect
132. #398C2×C4semidirect
135. #391D4semidirect
142. #388C2×C4semidirect
149. #382C2×C4semidirect
151. #380C2×C4semidirect
366. C2×Q8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×#23602
C2×Q8×C4⋊C42
C2×D4×C4⋊C42
#145482
#144052
#141852
#141502
#141302
#140322
#139942
#139692
#138472
#137862
#137792