Finite Group Databasebeta

← #877266

Subgroups — #877266

Order 512.

2538 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C212yesmaximal, complemented
2#17077C212yesmaximal
3#17071C212yesmaximal
4#15531C214yesmaximal, complemented
5#15499C214yesmaximal, complemented
6#15221C212yesmaximal, complemented
7#15135C212yesmaximal, complemented
8#15131C212yesmaximal, complemented
9#14714C211yesmaximal, characteristic, complemented
10#14712C211yesmaximal, characteristic, complemented
11#14467C212yesmaximal, complemented
12#14446C212yesmaximal, complemented
13#13879C212yesmaximal, complemented
14#13871C212yesmaximal
15C4×#390C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19#2439—216no—
20C2×#399C2214yes—
21C2×#396C2214yes—
22C2×#396C2214yes—
23C2×#393C2214yes—
24C2×#391C2212yes—
25C2×#391C2214yes—
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#387C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#382C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C43C2211yescharacteristic
46#1445C2214yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1437C2214yes—
61#1437C2212yes—
62#1437C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8D418yescomplemented
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4Q818yescomplemented
85C2×#117—28no—
86C2×#117—44no—
87C2×#117—44no—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#399—48no—
124#396—216no—
125#396—48no—
126#393—216no—
127#391—216no—
128#391—28no—
129#391—44no—
130#390—24no—
131#390—24no—
132#389—28no—
133#389—28no—
134#389—28no—
135#389—44no—
136#389—44no—
137#387—216no—
138#384—24no—
139#384—42no—
140#384—42no—
141#382—28no—
142#380—28no—
143#380—44no—
144#380—44no—
145C4×C4⋊C4—42no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—28no—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C43—42no—
170C25C2412yes—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8C2×D418yes—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4C2×Q814yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4C2×Q818yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C4⋊Q8—416no—
207C4×D4—416no—
208#117—432no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×Q814yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×Q818yes—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—41no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42—41no—
307C2×C42—41no—
308C2×C42—41no—
309C2×C42C2×D414yes—
310C2×C42C2×Q814yes—
311C2×C42#4114yes—
312C2×C42#4114yes—
313C2×C42#4114yes—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C22×Q814yes—
350C24C2511yescharacteristic
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×Q8—84no—
358C2×D4—28no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4C22×D418yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q812yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C22×C42−1+412yes—
469C22×C42−1+412yes—
470C4⋊C4—216no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C42—28no—
497C42—28no—
498C42—28no—
499C42—44no—
500C42—44no—
501C42—44no—
502C42—44no—
503C42—416no—
504C42—48no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—42no—
513C23—42no—
514C23—42no—
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—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C23×D411yescharacteristic
541C23C23×Q811yescharacteristic
542C23C22×#4111yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×2−1+411yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+411yescharacteristic
553C23#58511yescharacteristic
554C23#58511yescharacteristic
555Q8—816no—
556D4—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—432no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4#53314yes—
645C2×C4#53414yes—
646C2×C4#53914yes—
647C2×C4#54314yes—
648C2×C4#54714yes—
649C2×C4D4×Q814yes—
650C2×C4#55014yes—
651C2×C4#55014yes—
652C2×C4#55414yes—
653C2×C4#55514yes—
654C2×C4#55614yes—
655C2×C4#56214yes—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22C2×#53311yescharacteristic
672C22C2×#53411yescharacteristic
673C22C2×#53911yescharacteristic
674C22C2×#54311yescharacteristic
675C22#345811yescharacteristic
676C22C2×#54711yescharacteristic
677C22C2×D4×Q811yescharacteristic
678C22C2×#55011yescharacteristic
679C22C2×#55011yescharacteristic
680C22C2×#55411yescharacteristic
681C22C2×#55511yescharacteristic
682C22C2×#55611yescharacteristic
683C22#349611yescharacteristic
684C22#349711yescharacteristic
685C22#350311yescharacteristic
686C22#350311yescharacteristic
687C22#350511yescharacteristic
688C22#350511yescharacteristic
689C22#350611yescharacteristic
690C22#350611yescharacteristic
691C22#350611yescharacteristic
692C22#350711yescharacteristic
693C22#350811yescharacteristic
694C22#350911yescharacteristic
695C22#351111yescharacteristic
696C22#351111yescharacteristic
697C22C2×#56211yescharacteristic
698C22#353611yescharacteristic
699C22#353611yescharacteristic
700C22#354911yescharacteristic
701C22#355011yescharacteristic
702C22#355811yescharacteristic
703C22#355911yescharacteristic
704C22#356111yescharacteristic
705C22#356311yescharacteristic
706C4—24no—
707C4—24no—
708C4—24no—
709C4—24no—
710C4—24no—
711C4—48no—
712C4—42no—
713C4—44no—
714C4—84no—
715C4—82no—
716C4—82no—
717C4—84no—
718C4—84no—
719C4#273218yes—
720C2—84no—
721C2#3503311yescharacteristic
722C2#3613411yescharacteristic
723C2#3619411yescharacteristic
724C2#3730611yescharacteristic
725C2#3936511yescharacteristic
726C2#3948611yescharacteristic
727C2#3949311yescharacteristic
728C2#3950411yescharacteristic
729C2#3974311yescharacteristic
730C2#3975911yescharacteristic
731C2#4601511yescharacteristic
732C2#4602311yescharacteristic
733C2C2×#273211yescharacteristic
734C2#5982111yescharacteristic
735C2#5982211yescharacteristic
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×#2439C2semidirect
4. #15531C2semidirect
5. #15499C2semidirect
6. #15221C2semidirect
7. #15135C2semidirect
8. #15131C2semidirect
9. #14714C2semidirect
10. #14712C2semidirect
11. #14467C2semidirect
12. #14446C2semidirect
13. #13879C2semidirect
81. C2×C4⋊Q8D4semidirect
84. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
#170772
#170712
#155312
#154992
#152212
#151352
#151312
#147142
#147122
#144672
#144462
#138792
#138712
C4×#3902