Finite Group Databasebeta

← #1011230

Subgroups — #1011230

Order 512.

3342 conjugacy classes of subgroups.

The subgroup lattice has 790 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2491C212yesmaximal, complemented
2#17216C212yesmaximal, complemented
3#17068C212yesmaximal, complemented
4#16680C214yesmaximal, complemented
5#15763C212yesmaximal, complemented
6#15655C214yesmaximal, complemented
7#15127C211yesmaximal, characteristic, complemented
8#15124C212yesmaximal, complemented
9#15123C212yesmaximal, complemented
10#14589C212yesmaximal, complemented
11#14161C211yesmaximal, characteristic, complemented
12#14141C212yesmaximal, complemented
13#13998C212yesmaximal, complemented
14#13973C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#124C2214yes—
19C22×#121C2212yes—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22#2491—216no—
23C2×#394C2212yes—
24C2×#394C2211yescharacteristic
25C2×#394C2214yes—
26C2×#394C2212yes—
27C2×#393C2214yes—
28C2×#392C2214yes—
29C2×#392C2214yes—
30C2×#392C2214yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2214yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46#1446C2212yes—
47#1446C2212yes—
48#1442C2211yescharacteristic
49#1442C2214yes—
50#1441C2211yescharacteristic
51#1440C2212yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1438C2212yes—
63#1438C2212yes—
64#1438C2211yescharacteristic
65#1438C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2314yes—
80C23×D4C2314yes—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C24×C4C2312yes—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8D418yescomplemented
87C2×#127—44no—
88C2×#127—42no—
89C2×#124—216no—
90C2×#124—48no—
91C2×#124—48no—
92C2×#121—28no—
93C2×#121—28no—
94C2×#121—44no—
95C2×#121—44no—
96C2×#121—44no—
97C2×#121—44no—
98C2×C4×D4—42no—
99C2×C4×D4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#394—216no—
132#394—24no—
133#394—28no—
134#394—44no—
135#393—216no—
136#392—48no—
137#392—48no—
138#392—48no—
139#390—28no—
140#390—28no—
141#390—28no—
142#390—48no—
143#390—44no—
144#389—44no—
145#388—216no—
146#388—48no—
147#380—28no—
148#380—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—42no—
151C4×C4⋊C4—42no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—44no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C25C2414yes—
175C25C2412yes—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8C2×D418yes—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—41no—
197C22×D4—42no—
198C22×D4—41no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—42no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—416no—
238#127—416no—
239#124—464no—
240#121—432no—
241#121—432no—
242C4×D4—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4#4118yes—
265C2×C4⋊C4#4114yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42C2×D414yes—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—41no—
342#95—42no—
343#95—42no—
344#95—41no—
345#95—41no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—42no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24H(5,𝔽2)14yes—
383C24C2511yescharacteristic
384C2×Q8—416no—
385C2×Q8—48no—
386C2×Q8—84no—
387C2×Q8—88no—
388C2×Q8—84no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—88no—
408C2×D4—84no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—84no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—84no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—84no—
423C2×D4—88no—
424C2×D4—88no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4C22×D418yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4118yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4114yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C4H(5,𝔽2)14yes—
510C22×C42−1+412yes—
511C4⋊C4—216no—
512C4⋊C4—216no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—88no—
519C22⋊C4—216no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—416no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—416no—
531C22⋊C4—88no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C22⋊C4—88no—
535C22⋊C4—88no—
536C42—28no—
537C42—416no—
538C42—48no—
539C42—44no—
540C42—44no—
541C42—84no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23C23×D411yescharacteristic
613C23C22×#4112yes—
614C23C22×#4111yescharacteristic
615C23C2×H(5,𝔽2)11yescharacteristic
616C23C2×H(5,𝔽2)11yescharacteristic
617C23C2×H(5,𝔽2)12yes—
618C23C2×H(5,𝔽2)11yescharacteristic
619C23C2×H(5,𝔽2)12yes—
620C23C2×H(5,𝔽2)12yes—
621C23C2×2−1+411yescharacteristic
622C23#58511yescharacteristic
623Q8—816no—
624D4—416no—
625D4—832no—
626D4—832no—
627D4—816no—
628D4—816no—
629D4—816no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—28no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—28no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—44no—
675C2×C4—41no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—41no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—42no—
702C2×C4—41no—
703C2×C4—816no—
704C2×C4#53414yes—
705C2×C4#53818yes—
706C2×C4#55014yes—
707C2×C4#55018yes—
708C2×C4#55514yes—
709C2×C4H(3,𝔽4)14yes—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—44no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22—84no—
736C22—84no—
737C22—84no—
738C22—84no—
739C22—84no—
740C22C2×#53411yescharacteristic
741C22C2×#53812yes—
742C22C2×#55011yescharacteristic
743C22C2×#55012yes—
744C22C2×#55511yescharacteristic
745C22#348312yes—
746C22#348412yes—
747C22#349711yescharacteristic
748C22#350511yescharacteristic
749C22#351111yescharacteristic
750C22#351312yes—
751C22#351912yes—
752C22#351912yes—
753C22#352212yes—
754C22C2×H(3,𝔽4)11yescharacteristic
755C22#353611yescharacteristic
756C22#353612yes—
757C22#353711yescharacteristic
758C22#354212yes—
759C22#354312yes—
760C22#354912yes—
761C22#354911yescharacteristic
762C22#355811yescharacteristic
763C4—24no—
764C4—42no—
765C4—48no—
766C4—42no—
767C4—42no—
768C4—44no—
769C4—48no—
770C4—42no—
771C4—81no—
772C4—82no—
773C4—84no—
774C4—82no—
775C4#273318yes—
776C2—84no—
777C2—164no—
778C2#3974311yescharacteristic
779C2#4012512yes—
780C2#4247511yescharacteristic
781C2#4253212yes—
782C2#4488212yes—
783C2#4537811yescharacteristic
784C2#4564511yescharacteristic
785C2C2×#273311yescharacteristic
786C2#5301511yescharacteristic
787C2#5905612yes—
788C2#5995211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2491C2semidirect
2. #17216C2semidirect
3. #17068C2semidirect
4. #16680C2semidirect
5. #15763C2semidirect
6. #15655C2semidirect
7. #15127C2semidirect
8. #15124C2semidirect
9. #15123C2semidirect
10. #14589C2semidirect
11. #14161C2semidirect
12. #14141C2semidirect
13. #13998C2semidirect
14. #13973C2semidirect
15. C4×#394C2semidirect
86. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24912
#172162
#170682
#166802
#157632
#156552
#151272
#151242
#151232
#145892
#141612
#141412
#139982
#139732
C4×#3942