Finite Group Databasebeta

← #822339

Subgroups — #822339

Order 512.

3378 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C212yesmaximal, complemented
2C2×#2417C212yesmaximal
3C2×#2360C212yesmaximal, complemented
4#17070C212yesmaximal, complemented
5#15778C212yesmaximal, complemented
6#15769C214yesmaximal, complemented
7#15429C212yesmaximal
8#15182C212yesmaximal, complemented
9#15180C211yesmaximal, characteristic, complemented
10#15131C214yesmaximal, complemented
11#14224C212yesmaximal
12#13935C212yesmaximal, complemented
13#13910C211yesmaximal, characteristic, complemented
14#13907C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2429—216no—
23#2417—216no—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#394C2211yescharacteristic
29#2360—216no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2214yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2214yes—
39C2×#387C2214yes—
40C2×#386C2214yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#384C2211yescharacteristic
44C2×#380C2214yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2214yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1442C2211yescharacteristic
55#1442C2212yes—
56#1442C2212yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2211yescharacteristic
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2212yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2211yescharacteristic
71C4⋊#95C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2211yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2314yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—42no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—44no—
89C2×C22⋊Q8D4116yes—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×D4—28no—
94C2×C4×D4—42no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#398—216no—
130#398—216no—
131#396—216no—
132#396—216no—
133#394Q818yescomplemented
134#391—28no—
135#391—28no—
136#391—28no—
137#389—28no—
138#389—28no—
139#389—216no—
140#389—28no—
141#389—216no—
142#388—216no—
143#387—216no—
144#386—216no—
145#385—216no—
146#384—42no—
147#384—44no—
148#380—28no—
149#380—216no—
150#380—48no—
151C4×C4⋊C4—42no—
152C4×C4⋊C4D418yescomplemented
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—216no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C25C2412yes—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8C2×D418yes—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4C2×Q814yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D418yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8—416no—
222C22⋊Q8—264no—
223C22⋊Q8—432no—
224C4×Q8—432no—
225C4×D4—416no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—432no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4C2×D4116yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4#4118yes—
274C2×C22⋊C4—24no—
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—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C2×D4116yes—
296C2×C22⋊C4C2×Q818yes—
297C2×C22⋊C4C2×Q818yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C2×D414yes—
313C2×C42C2×D418yes—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—41no—
333#95—41no—
334#95—41no—
335#95—44no—
336#95C2×D418yes—
337#95C2×Q814yes—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24C22×D414yes—
351C24C22×Q814yes—
352C242−1+414yes—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×D4—216no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—44no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4C22×D412yes—
449C22×C4C22×D418yes—
450C22×C4C22×D414yes—
451C22×C4C22×D418yes—
452C22×C4C22×D418yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q812yes—
460C22×C4C2×#4118yes—
461C22×C4C2×#4114yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C4⋊C4—232no—
467C4⋊C4—44no—
468C4⋊C4—416no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4#121132yes—
483C22⋊C4—216no—
484C22⋊C4—232no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—88no—
491C42—28no—
492C42—48no—
493C42—416no—
494C42—48no—
495C42—44no—
496C42—44no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23D4×Q818yes—
532C23C23×D411yescharacteristic
533C23C23×D412yes—
534C23C23×Q811yescharacteristic
535C23C22×#4112yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+411yescharacteristic
539C23C2×2−1+412yes—
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542C23#58511yescharacteristic
543Q8—432no—
544D4—416no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—432no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—832no—
626C2×C4C2×#121116yes—
627C2×C4C2×#12118yes—
628C2×C4#53714yes—
629C2×C4D4214yes—
630C2×C4#54718yes—
631C2×C4D4×Q814yes—
632C2×C4#55514yes—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22C22×#12112yes—
649C22C2×#53711yescharacteristic
650C22C2×D4211yescharacteristic
651C22C2×#54712yes—
652C22C2×D4×Q812yes—
653C22C2×D4×Q811yescharacteristic
654C22C2×#55511yescharacteristic
655C22Q8×#4112yes—
656C22#348812yes—
657C22#348912yes—
658C22#348912yes—
659C22#349611yescharacteristic
660C22#349712yes—
661C22#350111yescharacteristic
662C22#350311yescharacteristic
663C22#352012yes—
664C22#352112yes—
665C22#352312yes—
666C22#353611yescharacteristic
667C22#353811yescharacteristic
668C22#355111yescharacteristic
669C22#356311yescharacteristic
670C22#356312yes—
671C4—24no—
672C4—216no—
673C4—44no—
674C4—48no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—81no—
681C4—84no—
682C4—82no—
683C4—84no—
684C4#253718yes—
685C2—48no—
686C2C2×#253711yescharacteristic
687C2#3582112yes—
688C2Q8×#12112yes—
689C2#3961411yescharacteristic
690C2#4298011yescharacteristic
691C2#4518512yes—
692C2#4524411yescharacteristic
693C2#4538411yescharacteristic
694C2#5899811yescharacteristic
695C2#5911312yes—
696C2#5993511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2429C2semidirect
3. C2×#2360C2semidirect
4. #17070C2semidirect
5. #15778C2semidirect
6. #15769C2semidirect
8. #15182C2semidirect
9. #15180C2semidirect
10. #15131C2semidirect
12. #13935C2semidirect
13. #13910C2semidirect
14. #13907C2semidirect
133. #394Q8semidirect
152. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
C2×#24172
C2×#23602
#170702
#157782
#157692
#154292
#151822
#151802
#151312
#142242
#139352
#139102
#139072
C4×#3942