Finite Group Databasebeta

← #719550

Subgroups — #719550

Order 512.

2702 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic
2C2×#2453C212yesmaximal
3C2×#2451C212yesmaximal
4C2×#2365C211yesmaximal, characteristic
5#17075C212yesmaximal
6#16813C212yesmaximal
7#15856C214yesmaximal
8#15804C214yesmaximal
9#15193C212yesmaximal
10#15193C212yesmaximal
11#15193C212yesmaximal
12#15082C212yesmaximal
13#14274C211yesmaximal, characteristic
14#14155C211yesmaximal, characteristic
15#13954C211yesmaximal, characteristic
16#13856C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2550—28no—
23#2453—216no—
24#2451—216no—
25C2×#400C2214yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31#2365—28no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2214yes—
38C2×#391C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2214yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#387C2214yes—
48C2×#387C2214yes—
49C2×#387C2214yes—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59#1445C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1445C2214yes—
63#1445C2214yes—
64#1445C2214yes—
65#1445C2214yes—
66#1445C2212yes—
67#1445C2214yes—
68#1445C2212yes—
69#1445C2214yes—
70#1444C2211yescharacteristic
71#1444C2211yescharacteristic
72#1444C2211yescharacteristic
73#1444C2212yes—
74#1444C2211yescharacteristic
75#1444C2212yes—
76#1444C2212yes—
77#1444C2214yes—
78#1444C2214yes—
79#1444C2212yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2214yes—
86#1434C2212yes—
87C23×Q8C2312yes—
88C23×Q8C2312yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8Q818yescomplemented
94C2×#125—28no—
95C2×#125—24no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#400—216no—
132#398—216no—
133#398—216no—
134#398—28no—
135#395—216no—
136#395—28no—
137#391—28no—
138#391—216no—
139#391—28no—
140#391—28no—
141#391—28no—
142#391—28no—
143#391—28no—
144#389—28no—
145#389—216no—
146#389—28no—
147#389—24no—
148#389—24no—
149#389—28no—
150#389—28no—
151#389D4116yes—
152#387—216no—
153#387—216no—
154#387—216no—
155#382—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4D418yes—
163#376—24no—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8—42no—
206C22×Q8C2×Q818yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C4⋊Q8—232no—
225C4⋊Q8—416no—
226#125—416no—
227C4×Q8—432no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D4116yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4C2×Q818yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42C2×D418yes—
307C2×C42C2×D418yes—
308C2×C42C2×D414yes—
309C2×C42C2×Q814yes—
310C2×C42#4114yes—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—22no—
315#95—24no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—44no—
339#95—44no—
340#95C2×D418yes—
341#95C2×D4116yes—
342C24C2511yescharacteristic
343C2×Q8—216no—
344C2×Q8—216no—
345C2×Q8—216no—
346C2×Q8—48no—
347C2×Q8—416no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—84no—
354C2×Q8—88no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D418yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D418yes—
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C22×Q818yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q812yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C42−1+414yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C22×C42−1+414yes—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—216no—
460C4⋊C4—416no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—416no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—84no—
476C4⋊C4—88no—
477C4⋊C4#127116yes—
478C42—216no—
479C42—48no—
480C42—416no—
481C42—48no—
482C42—44no—
483C42—44no—
484C42—84no—
485C42C22⋊Q8116yes—
486C23C23×D411yescharacteristic
487C23C23×D412yes—
488C23C23×Q811yescharacteristic
489C23C22×#4111yescharacteristic
490C23C2×2−1+412yes—
491C23C2×2−1+412yes—
492C23C2×2−1+411yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23#58512yes—
495C23#58511yescharacteristic
496C23#58511yescharacteristic
497Q8—432no—
498Q8—816no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4C2×C22⋊Q814yes—
591C2×C4C2×C22⋊Q814yes—
592C2×C4C2×C22⋊Q814yes—
593C2×C4C2×#12718yes—
594C2×C4C2×#12714yes—
595C2×C4#53614yes—
596C2×C4#54718yes—
597C2×C4D4×Q818yes—
598C22C22×C22⋊Q811yescharacteristic
599C22C22×#12711yescharacteristic
600C22C2×#53611yescharacteristic
601C22#346112yes—
602C22C2×#54712yes—
603C22C2×D4×Q812yes—
604C22#348512yes—
605C22#348511yescharacteristic
606C22#348611yescharacteristic
607C22#348612yes—
608C22#348911yescharacteristic
609C22#348911yescharacteristic
610C22#348912yes—
611C22#348912yes—
612C22#350211yescharacteristic
613C22#350212yes—
614C22#350211yescharacteristic
615C22#350911yescharacteristic
616C22#351011yescharacteristic
617C22#355512yes—
618C22#355512yes—
619C22#355612yes—
620C22#355712yes—
621C4—24no—
622C4—24no—
623C4—28no—
624C4—44no—
625C4—42no—
626C4—48no—
627C4—44no—
628C4—44no—
629C4—48no—
630C4—44no—
631C4—42no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4#254218yes—
638C2C2×#254211yescharacteristic
639C2#3515411yescharacteristic
640C2#3566211yescharacteristic
641C2#3592711yescharacteristic
642C2#3606011yescharacteristic
643C2#3779312yes—
644C2#3779612yes—
645C2#4267611yescharacteristic
646C2#4281111yescharacteristic
647C2#4527712yes—
648C2#4538912yes—
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
93. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#24532
C2×#24512
C2×#23652
#170752
#168132
#158562
#158042
#151932
#150822
#142742
#141552
#139542
#138562
#137832