Finite Group Databasebeta

← #827042

Subgroups — #827042

Order 512.

2602 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C212yesmaximal
2C2×#2365C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic
4#16567C212yesmaximal
5#15969C214yesmaximal
6#15963C214yesmaximal
7#15158C211yesmaximal, characteristic
8#15153C211yesmaximal, characteristic
9#15152C211yesmaximal, characteristic
10#14884C212yesmaximal
11#14876C212yesmaximal
12#14053C212yesmaximal
13#14046C212yesmaximal
14#13981C212yesmaximal
15#13918C211yesmaximal, characteristic
16#13884C212yesmaximal
17C4×#395C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22#2581—216no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#395C2212yes—
27C2×#395C2211yescharacteristic
28#2365—28no—
29Q8×C4⋊C4—28no—
30C2×#391C2214yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2214yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#387C2214yes—
42C2×#387C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#382C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#1446C2211yescharacteristic
53#1446C2214yes—
54#1445C2212yes—
55#1445C2214yes—
56#1444C2212yes—
57#1444C2211yescharacteristic
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8Q818yescomplemented
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—24no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—44no—
98C2×C4×Q8—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#400—28no—
130#400—28no—
131#400—44no—
132#395—28no—
133#395D418yes—
134#391—216no—
135#391—28no—
136#389—216no—
137#389—24no—
138#389—28no—
139#389—28no—
140#389—24no—
141#389—28no—
142#389—216no—
143#387—216no—
144#387—28no—
145#387—216no—
146#387—28no—
147#384—28no—
148#384—24no—
149#382—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4D418yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8C2×D414yes—
200C22×Q8C2×Q818yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C4⋊Q8—232no—
219C4×Q8—232no—
220C4×Q8—432no—
221C4×Q8—432no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×Q818yes—
278C2×C4⋊C4C2×Q814yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4#4114yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42C2×D418yes—
303C2×C42C2×D414yes—
304C2×C42C2×Q814yes—
305C2×C42#4118yes—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95C2×D418yes—
337#95C2×D414yes—
338#95C2×D418yes—
339C24C2511yescharacteristic
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—28no—
345C2×Q8—216no—
346C2×Q8—216no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4114yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+414yes—
453C22×C42−1+412yes—
454C22×C42−1+414yes—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—416no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—88no—
474C4⋊C4—84no—
475C4⋊C4C22⋊Q8116yes—
476C42—28no—
477C42—28no—
478C42—216no—
479C42—216no—
480C42—416no—
481C42—48no—
482C42—48no—
483C42—44no—
484C42—48no—
485C23C23×D411yescharacteristic
486C23C23×D411yescharacteristic
487C23C23×Q811yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4112yes—
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×2−1+412yes—
492C23C2×2−1+412yes—
493C23C2×2−1+411yescharacteristic
494C23#58511yescharacteristic
495C23#58512yes—
496Q8—232no—
497Q8—432no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4C2×C22⋊Q818yes—
593C2×C4C2×C22⋊Q814yes—
594C2×C4#52514yes—
595C2×C4#53614yes—
596C2×C4D4214yes—
597C2×C4#54718yes—
598C2×C4D4×Q814yes—
599C2×C4#55618yes—
600C2×C4#55718yes—
601C22C22×C22⋊Q811yescharacteristic
602C22C2×#52511yescharacteristic
603C22C2×#53611yescharacteristic
604C22C2×D4211yescharacteristic
605C22C2×#54712yes—
606C22C2×D4×Q811yescharacteristic
607C22D4×#4111yescharacteristic
608C22C2×#55612yes—
609C22C2×#55712yes—
610C22#348611yescharacteristic
611C22#348911yescharacteristic
612C22#349212yes—
613C22#349312yes—
614C22#349712yes—
615C22#350011yescharacteristic
616C22#350111yescharacteristic
617C22#350212yes—
618C22#350712yes—
619C22#351812yes—
620C22#352011yescharacteristic
621C22#355412yes—
622C22#355512yes—
623C22#355612yes—
624C4—24no—
625C4—28no—
626C4—28no—
627C4—24no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4—81no—
639C4—84no—
640C4#258118yescomplemented
641C2C2×#258111yescharacteristic
642C2#3571111yescharacteristic
643C2#3575911yescharacteristic
644C2D4×C22⋊Q811yescharacteristic
645C2#3610311yescharacteristic
646C2#3936311yescharacteristic
647C2#4281111yescharacteristic
648C2#4532912yes—
649C2#4536312yes—
650C2#5924412yes—
651C2#5982812yes—
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
90. C2×C4⋊Q8Q8semidirect
640. C4#2581semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×#23652
C2×Q8×C4⋊C42
#165672
#159692
#159632
#151582
#151532
#151522
#148842
#148762
#140532
#140462
#139812
#139182
#138842
C4×#3952