Finite Group Databasebeta

← #1007244

Subgroups — #1007244

Order 512.

3402 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2415C212yesmaximal, complemented
2C2×#2415C212yesmaximal, complemented
3C2×#2417C212yesmaximal, complemented
4#17109C212yesmaximal, complemented
5#16314C214yesmaximal, complemented
6#16282C212yesmaximal
7#15363C212yesmaximal, complemented
8#15295C214yesmaximal, complemented
9#15189C212yesmaximal, complemented
10#15178C211yesmaximal, characteristic, complemented
11#15178C212yesmaximal, complemented
12#14402C212yesmaximal, complemented
13#14130C211yesmaximal, characteristic, complemented
14#14129C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24#2417—216no—
25#2415—216no—
26#2415—216no—
27C2×#398C2214yes—
28C2×#397C2214yes—
29C2×#397C2214yes—
30C2×#396C2214yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#392C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#386C2214yes—
45C2×#385C2212yes—
46C2×#380C2214yes—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51#1444C2211yescharacteristic
52#1444C2212yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1443C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1439C2211yescharacteristic
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1438C2212yes—
71#1438C2211yescharacteristic
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2211yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—44no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C22⋊Q8—44no—
94C2×C22⋊Q8—44no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—44no—
99C2×C4×D4—42no—
100C2×#117—28no—
101C2×#117—28no—
102C2×#117—44no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—42no—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2314yes—
136#398—216no—
137#397—216no—
138#397—216no—
139#396—216no—
140#394—216no—
141#394—216no—
142#392—24no—
143#391—28no—
144#391—28no—
145#391—28no—
146#389—28no—
147#389—28no—
148#389—28no—
149#389—216no—
150#389—216no—
151#388—216no—
152#388—216no—
153#386—216no—
154#385—28no—
155#380—216no—
156#380D418yescomplemented
157#380D4116yescomplemented
158C4×C22⋊C4—42no—
159C4×C22⋊C4—44no—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2314yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—42no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4—42no—
202C22×D4—42no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4C2×D418yes—
219C23×C4C2×D414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C4⋊Q8—416no—
234C22⋊Q8—432no—
235C22⋊Q8—432no—
236C22⋊Q8—432no—
237C4×D4—416no—
238C4×D4—432no—
239#117—432no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D4116yes—
317C2×C22⋊C4#4114yes—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—44no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—42no—
342#95—44no—
343#95—41no—
344#95—42no—
345#95—42no—
346#95—41no—
347#95—41no—
348#95C2×D4116yes—
349#95C2×D418yes—
350C24—22no—
351C24—22no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24C22×D414yes—
373C24C22×D412yes—
374C24C2×#4112yes—
375C242−1+412yes—
376C24C2511yescharacteristic
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×D4—216no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—44no—
395C2×D4—48no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—216no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—216no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4C22×D418yes—
479C22×C4C22×D418yes—
480C22×C4C22×D418yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+412yes—
490C22×C42−1+414yes—
491C22×C42−1+412yes—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—416no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C22⋊C4—216no—
507C22⋊C4—416no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C42—416no—
520C42—44no—
521C42—48no—
522C42—44no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—41no—
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—42no—
578C23—42no—
579C23—42no—
580C23#54714yes—
581C23C23×D411yescharacteristic
582C23C23×D412yes—
583C23C22×#4111yescharacteristic
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×H(5,𝔽2)12yes—
586C23C2×2−1+411yescharacteristic
587C23C2×2−1+411yescharacteristic
588C23C2×2−1+412yes—
589C23C2×2−1+411yescharacteristic
590C23#58512yes—
591C23#58511yescharacteristic
592Q8—816no—
593D4—432no—
594D4—416no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—416no—
651C2×C4—432no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—816no—
672C2×C4#53718yes—
673C2×C4#53714yes—
674C2×C4D4214yes—
675C2×C4D4218yes—
676C2×C4#56014yes—
677C22—24no—
678C22—28no—
679C22—42no—
680C22—42no—
681C22—44no—
682C22—44no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22C2×#53712yes—
704C22C2×#53711yescharacteristic
705C22C2×D4211yescharacteristic
706C22C2×D4212yes—
707C22C2×#54711yescharacteristic
708C22D4×#4112yes—
709C22#348512yes—
710C22#348612yes—
711C22#348912yes—
712C22#349711yescharacteristic
713C22#349912yes—
714C22#350511yescharacteristic
715C22#350911yescharacteristic
716C22#352012yes—
717C22C2×#56011yescharacteristic
718C22#353811yescharacteristic
719C22#354011yescharacteristic
720C22#354112yes—
721C22#354112yes—
722C22#354712yes—
723C22#355011yescharacteristic
724C22#355512yes—
725C22#356311yescharacteristic
726C4—42no—
727C4—48no—
728C4—48no—
729C4—42no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—42no—
734C4—42no—
735C4—81no—
736C4—82no—
737C4—84no—
738C4#267518yes—
739C2—44no—
740C2—84no—
741C2#3948711yescharacteristic
742C2#3995912yes—
743C2#4267512yes—
744C2#4292911yescharacteristic
745C2#4509112yes—
746C2#4532111yescharacteristic
747C2#4634211yescharacteristic
748C2C2×#267511yescharacteristic
749C2#5313711yescharacteristic
750C2#5900011yescharacteristic
751C2#5900212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2415C2semidirect
2. C2×#2415C2semidirect
3. C2×#2417C2semidirect
4. #17109C2semidirect
5. #16314C2semidirect
7. #15363C2semidirect
8. #15295C2semidirect
9. #15189C2semidirect
10. #15178C2semidirect
11. #15178C2semidirect
12. #14402C2semidirect
13. #14130C2semidirect
14. #14129C2semidirect
156. #380D4semidirect
157. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24152
C2×#24172
#171092
#163142
#162822
#153632
#152952
#151892
#151782
#144022
#141302
#141292
C4×#3922