Finite Group Databasebeta

← #4929112

Subgroups — #4929112

Order 512.

2682 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic
3C2×#2399C211yesmaximal, characteristic
4C2×#2276C211yesmaximal, characteristic
5#14903C212yesmaximal
6#14901C212yesmaximal
7#14872C212yesmaximal
8#14871C214yesmaximal
9#14818C212yesmaximal
10#14816C214yesmaximal
11#14354C211yesmaximal, characteristic
12#14151C212yesmaximal
13#14150C212yesmaximal
14#13942C212yesmaximal
15#13926C211yesmaximal, characteristic
16#13926C212yesmaximal
17#13925C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2582—28no—
23#2581—28no—
24#2399—28no—
25C2×#400C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#398C2211yescharacteristic
29C2×#398C2211yescharacteristic
30C2×#395C2211yescharacteristic
31C2×#395C2211yescharacteristic
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2211yescharacteristic
43C2×#389C2214yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#382C2212yes—
49C2×#382C2211yescharacteristic
50#2276—28no—
51C2×C4×C4⋊C4C2214yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×#376C2212yes—
58C2×#376C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1444C2214yes—
62#1444C2212yes—
63#1444C2212yes—
64#1444C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2212yes—
73#1441C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C2×C4×Q8—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8D418yes—
101C2×C4×Q8Q8116yescomplemented
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#400—28no—
135#398—216no—
136#398—24no—
137#398—28no—
138#398—24no—
139#395—28no—
140#395—24no—
141#395—24no—
142#395—28no—
143#391—24no—
144#391—24no—
145#391—28no—
146#391—28no—
147#391—44no—
148#389—24no—
149#389—24no—
150#389—28no—
151#389—28no—
152#389—48no—
153#387—28no—
154#387—28no—
155#384—24no—
156#384—28no—
157#382—24no—
158#382—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—216no—
164C4×C4⋊C4—28no—
165#376—28no—
166#376D418yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2311yescharacteristic
195C2×#95C2312yes—
196C2×#95C2311yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—41no—
210C22×Q8—42no—
211C22×Q8—41no—
212C22×Q8C2×D414yes—
213C22×Q8C2×Q818yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C4×Q8—232no—
232C4×Q8—264no—
233C4×Q8—416no—
234C4×Q8—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×Q818yes—
300C2×C4⋊C4C2×Q8116yes—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—44no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—42no—
320C2×C42C2×D418yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D418yes—
324C2×C42C2×Q8116yes—
325C2×C42C2×Q818yes—
326C2×C42#4114yes—
327C2×C42#4118yes—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95C2×D414yes—
356#95C2×D414yes—
357#95C2×D418yes—
358C24C2511yescharacteristic
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—216no—
363C2×Q8—28no—
364C2×Q8—216no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8C4⋊Q8116yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×Q818yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q818yes—
471C22×C4C22×Q814yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4114yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+414yes—
480C22×C42−1+412yes—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—232no—
484C4⋊C4—216no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C42—216no—
505C42—232no—
506C42—48no—
507C42—44no—
508C42—44no—
509C42—88no—
510C42—84no—
511C42#121116yes—
512C42C22⋊Q8132yes—
513C23C23×D411yescharacteristic
514C23C23×D411yescharacteristic
515C23C23×Q812yes—
516C23C22×#4111yescharacteristic
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×2−1+412yes—
520C23C2×2−1+411yescharacteristic
521C23#58512yes—
522C23#58511yescharacteristic
523C23#58511yescharacteristic
524Q8—232no—
525Q8—416no—
526Q8—416no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
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×C4C2×#12114yes—
623C2×C4C2×#12114yes—
624C2×C4C2×#12114yes—
625C2×C4C2×C22⋊Q818yes—
626C2×C4C2×C22⋊Q818yes—
627C2×C4C2×C22⋊Q818yes—
628C2×C4C2×C4⋊Q818yes—
629C2×C4C2×C4⋊Q814yes—
630C2×C4#53214yes—
631C2×C4#53318yes—
632C22C22×#12111yescharacteristic
633C22C22×C22⋊Q812yes—
634C22C22×C4⋊Q811yescharacteristic
635C22C2×#53211yescharacteristic
636C22C2×#53312yes—
637C22#348611yescharacteristic
638C22#348711yescharacteristic
639C22#348811yescharacteristic
640C22#348811yescharacteristic
641C22#348912yes—
642C22#348911yescharacteristic
643C22#348911yescharacteristic
644C22#349211yescharacteristic
645C22#349311yescharacteristic
646C22#349512yes—
647C22#350212yes—
648C22#350212yes—
649C22#350312yes—
650C22#350412yes—
651C22#350612yes—
652C22#350712yes—
653C22#350912yes—
654C22#351012yes—
655C4—28no—
656C4—24no—
657C4—28no—
658C4—24no—
659C4—28no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—48no—
665C4—81no—
666C4—81no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4#285718yes—
672C2#3551811yescharacteristic
673C2#3551912yes—
674C2#3566111yescharacteristic
675C2#3566212yes—
676C2#3593612yes—
677C2#3593711yescharacteristic
678C2#3593811yescharacteristic
679C2#3595611yescharacteristic
680C2#3595711yescharacteristic
681C2#3595812yes—
682C2C2×#285711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
101. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25812
C2×#23992
C2×#22762
#149032
#149012
#148722
#148712
#148182
#148162
#143542
#141512
#141502
#139422
#139262
#139252