Finite Group Databasebeta

← #513652

Subgroups — #513652

Order 512.

3718 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C212yesmaximal
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#2276C211yesmaximal, characteristic, complemented
4C2×C4×#117C211yesmaximal, characteristic
5#13822C212yesmaximal
6#13818C212yesmaximal, complemented
7#13817C212yesmaximal, complemented
8#13816C212yesmaximal
9#13815C212yesmaximal, complemented
10#13810C212yesmaximal, complemented
11#13801C212yesmaximal
12#13789C214yesmaximal
13#13786C211yesmaximal, characteristic, complemented
14#13783C211yesmaximal, characteristic, complemented
15C4×#387C214yesmaximal, complemented
16#13753C211yesmaximal, characteristic
17#13750C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2362C4132yescomplemented
24C4×C22⋊Q8C4116yescomplemented
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#380C2211yescharacteristic
35#2276—28no—
36C4×#117—28no—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2214yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53C2×#376C2211yescharacteristic
54C2×#376C2212yes—
55C2×#376C2212yes—
56C2×C43C2212yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69#1434C2212yes—
70#1434C2211yescharacteristic
71#1434C2211yescharacteristic
72#1434C2214yes—
73#1434C2212yes—
74#1434C2212yes—
75#1434C2212yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C22⋊#95C2211yescharacteristic
80C4×#95C2214yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C4×#95C2214yes—
86C4×#95C2214yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8C2×C4116yes—
92C2×C22⋊Q8C2×C418yes—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8C2×C418yes—
98C2×C4×Q8C2×C4116yescomplemented
99C2×#117—28no—
100C2×#117—28no—
101C2×#117C2×C4116yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42C2×C418yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#391C2×C418yes—
137#391C2×C4116yescomplemented
138#389C2×C418yes—
139#389C2×C4116yescomplemented
140#388C2×C4132yes—
141#388C2×C4132yes—
142#387C2×C4116yes—
143#387C2×C4132yescomplemented
144#385C2×C4116yes—
145#380C2×C418yes—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—216no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4C2×C4116yes—
153C4×C4⋊C4C2×C418yes—
154C4×C4⋊C4C2×C4132yescomplemented
155C4×C4⋊C4C2×C4116yescomplemented
156C4×C4⋊C4D418yes—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4C2×C4116yes—
160C4×C22⋊C4C2×C4116yes—
161#376—24no—
162#376—24no—
163#376—28no—
164#376—28no—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C43—28no—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8C22×C414yes—
201C22×Q8C22×C418yes—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4—41no—
212C23×C4—41no—
213C23×C4—41no—
214C23×C4—41no—
215C23×C4C22×C414yes—
216C23×C4C22×C418yes—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C22⋊Q8C42164yescomplemented
236C4×Q8—264no—
237C4×Q8—232no—
238#117—264no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C4116yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C4116yes—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C4116yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4#4118yes—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4C22×C4116yes—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C4116yes—
280C2×C42—28no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—232no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C22×C4116yes—
315C2×C42C22×C418yes—
316C2×C42C22×C418yes—
317C2×C42C22×C4116yes—
318C2×C42C22×C418yes—
319C2×C42C22×C414yes—
320C2×C42C22×C414yes—
321C2×C42C22×C414yes—
322C2×C42C22×C418yes—
323C2×C42C22×C4116yes—
324C2×C42C22×C414yes—
325C2×C42C2×D418yes—
326C2×C42C2×D414yes—
327C2×C42#4114yes—
328#95—22no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—42no—
346#95C22×C418yes—
347#95C22×C418yes—
348#95C22×C418yes—
349#95C22×C418yes—
350#95C22×C418yes—
351#95C22×C418yes—
352#95C22×C418yes—
353#95C22×C4116yes—
354#95C22×C4116yes—
355#95C22×C4116yes—
356#95C22×C4116yes—
357#95C2×D418yes—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24C23×C414yes—
368C24C23×C412yes—
369C24C2511yescharacteristic
370C2×Q8—28no—
371C2×Q8—216no—
372C2×Q8—216no—
373C2×Q8—28no—
374C2×Q8C2×C42116yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
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—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4C2×C42116yes—
460C22×C4C23×C414yes—
461C22×C4C23×C414yes—
462C22×C4C23×C414yes—
463C22×C4C23×C418yes—
464C22×C4C23×C414yes—
465C22×C4C23×C412yes—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C412yes—
469C22×C4C23×C418yes—
470C22×C4C23×C412yes—
471C22×C4C23×C412yes—
472C22×C4C23×C412yes—
473C22×C4C23×C414yes—
474C22×C4C23×C414yes—
475C22×C4C23×C418yes—
476C22×C4C23×C414yes—
477C22×C4C23×C418yes—
478C22×C4C23×C412yes—
479C22×C4C23×C414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D418yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4118yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C42−1+414yes—
490C4⋊C4—216no—
491C4⋊C4—232no—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4C2×C42116yes—
498C4⋊C4C2×C42132yes—
499C4⋊C4C4×D4132yes—
500C22⋊C4—232no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4C2×C42132yes—
505C42—28no—
506C42—28no—
507C42—28no—
508C42—232no—
509C42—232no—
510C42—216no—
511C42—216no—
512C42—216no—
513C42—48no—
514C42—416no—
515C42—48no—
516C42—416no—
517C42—48no—
518C42—88no—
519C42—88no—
520C42C4×D4116yes—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C22×C4214yes—
544C23C24×C411yescharacteristic
545C23C24×C412yes—
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4112yes—
549C23C2×2−1+412yes—
550C23#58511yescharacteristic
551C23#58511yescharacteristic
552C23#58512yes—
553C23#58511yescharacteristic
554C23#58511yescharacteristic
555Q8—232no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—232no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4C22×C4218yes—
646C2×C4C22×C4218yes—
647C2×C4C22×C4214yes—
648C2×C4C22×C4214yes—
649C2×C4C2×C4×D414yes—
650C2×C4C2×C4×D4116yes—
651C2×C4C2×C4×D414yes—
652C2×C4C2×C4×D414yes—
653C2×C4C2×C4×D418yes—
654C2×C4C4×#4118yes—
655C2×C4C4×#4114yes—
656C2×C4#51914yes—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22C23×C4211yescharacteristic
669C22C22×C4×D411yescharacteristic
670C22C22×C4×D412yes—
671C22C2×C4×#4112yes—
672C22C2×C4×#4111yescharacteristic
673C22C2×#51911yescharacteristic
674C22#342812yes—
675C22#342811yescharacteristic
676C22#342811yescharacteristic
677C22#342811yescharacteristic
678C22#342812yes—
679C22C4×2−1+412yes—
680C22C4×2−1+412yes—
681C22#348511yescharacteristic
682C22#348711yescharacteristic
683C22#348912yes—
684C22#349311yescharacteristic
685C22#349311yescharacteristic
686C22#349512yes—
687C22#351512yes—
688C22#351812yes—
689C22#352112yes—
690C22#352412yes—
691C4—24no—
692C4—24no—
693C4—24no—
694C4—24no—
695C4—28no—
696C4—28no—
697C4—216no—
698C4—44no—
699C4—44no—
700C4—42no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—48no—
705C4—44no—
706C4C42×D418yes—
707C2—44no—
708C2C2×C42×D411yescharacteristic
709C2#3399911yescharacteristic
710C2C4×#51911yescharacteristic
711C2#3420111yescharacteristic
712C2#3420311yescharacteristic
713C2#3423111yescharacteristic
714C2#3423711yescharacteristic
715C2#3428912yes—
716C2#3429312yes—
717C2#3430012yes—
718C2#3430712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC22C23×C42
prime radicalC25C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2276C2semidirect
6. #13818C2semidirect
7. #13817C2semidirect
9. #13815C2semidirect
10. #13810C2semidirect
13. #13786C2semidirect
14. #13783C2semidirect
15. C4×#387C2semidirect
17. #13750C2semidirect
23. #2362C4semidirect
24. C4×C22⋊Q8C4semidirect
98. C2×C4×Q8C2×C4semidirect
137. #391C2×C4semidirect
139. #389C2×C4semidirect
143. #387C2×C4semidirect
154. C4×C4⋊C4C2×C4semidirect
155. C4×C4⋊C4C2×C4semidirect
235. C22⋊Q8C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×C4×C22⋊Q82
C2×#22762
C2×C4×#1172
#138222
#138182
#138172
#138162
#138152
#138102
#138012
#137892
#137862
#137832
C4×#3872
#137532
#137502