Finite Group Databasebeta

← #647411

Subgroups — #647411

Order 512.

2328 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#126C211yesmaximal, characteristic
3#16965C212yesmaximal, complemented
4#16889C212yesmaximal, complemented
5#16812C212yesmaximal
6#16673C212yesmaximal
7#15221C212yesmaximal, complemented
8#15220C212yesmaximal
9#15151C211yesmaximal, characteristic, complemented
10#15150C211yesmaximal, characteristic
11#15128C211yesmaximal, characteristic, complemented
12#15125C211yesmaximal, characteristic
13#14882C212yesmaximal
14#14879C212yesmaximal, complemented
15#14689C211yesmaximal, characteristic, complemented
16#14308C212yesmaximal, complemented
17#14306C212yesmaximal
18#14041C212yesmaximal, complemented
19#13993C212yesmaximal
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2211yescharacteristic
22C22×C4×Q8C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#395C2212yes—
28C2×#393C2212yes—
29C4×C4⋊Q8—28no—
30C4×#126—28no—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2211yescharacteristic
37C2×#388C2211yescharacteristic
38C2×#388C2211yescharacteristic
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#383C2211yescharacteristic
44C2×#382C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C43C2211yescharacteristic
56#1446C2212yes—
57#1446C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2214yes—
63#1445C2214yes—
64#1444C2212yes—
65#1444C2214yes—
66#1443C2214yes—
67#1443C2214yes—
68#1442C2212yes—
69#1442C2214yes—
70#1442C2214yes—
71#1442C2214yes—
72#1441C2212yes—
73#1441C2214yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2212yes—
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C4⋊#95C2212yes—
88C4⋊#95C2212yes—
89#1434C2212yes—
90#1434C2212yes—
91#1434C2212yes—
92#1434C2212yes—
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#126—24no—
100C2×C4×Q8—28no—
101C2×C4×Q8—44no—
102C2×C4×Q8—44no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138#398—28no—
139#398—44no—
140#397—28no—
141#397—44no—
142#395—44no—
143#393—44no—
144#391—28no—
145#389—28no—
146#389—28no—
147#389—28no—
148#389—44no—
149#388—24no—
150#388—24no—
151#388—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#384—24no—
156#383—24no—
157#382—44no—
158#380—28no—
159#380—44no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—24no—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C43D418yes—
203C25C2411yescharacteristic
204C22×Q8—24no—
205C22×Q8—28no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—44no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4⋊Q8—232no—
233#126—232no—
234C4×Q8—432no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4#4114yes—
279C2×C4⋊C4#4118yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4#4114yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—28no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—42no—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—24no—
348#95—28no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24C2×#4112yes—
374C24C2511yescharacteristic
375C2×Q8—216no—
376C2×Q8—416no—
377C2×Q8—416no—
378C2×Q8—48no—
379C2×Q8—48no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C42−1+414yes—
485C22×C42−1+414yes—
486C22×C42−1+412yes—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—44no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—28no—
515C42—28no—
516C42—28no—
517C42—44no—
518C42—416no—
519C42—44no—
520C42—44no—
521C42—44no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546C23C22×#4112yes—
547C23C22×#4111yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×2−1+412yes—
553C23C2×2−1+412yes—
554C23C2×2−1+411yescharacteristic
555C23#58512yes—
556Q8—432no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—88no—
652C2×C4#52514yes—
653C2×C4#52514yes—
654C2×C4#53514yes—
655C2×C4#53614yes—
656C2×C4#53614yes—
657C2×C4#53714yes—
658C2×C4#54118yes—
659C2×C4#54218yes—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—82no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22C2×#52511yescharacteristic
672C22C2×#52511yescharacteristic
673C22C2×#53511yescharacteristic
674C22C2×#53611yescharacteristic
675C22C2×#53611yescharacteristic
676C22C2×#53711yescharacteristic
677C22#344911yescharacteristic
678C22C2×#54112yes—
679C22C2×#54212yes—
680C22#348212yes—
681C22#348212yes—
682C22#350811yescharacteristic
683C22#350811yescharacteristic
684C22#351111yescharacteristic
685C22#351111yescharacteristic
686C22#351312yes—
687C22#351312yes—
688C22#351512yes—
689C22#351812yes—
690C22#355212yes—
691C22#355212yes—
692C22#355412yes—
693C22#355512yes—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—48no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—84no—
705C4—84no—
706C4—81no—
707C4—81no—
708C4—84no—
709C4—81no—
710C4—81no—
711C4#263718yes—
712C2—82no—
713C2C2×#263711yescharacteristic
714C2#3497111yescharacteristic
715C2#3520011yescharacteristic
716C2#3523211yescharacteristic
717C2#3529211yescharacteristic
718C2#3761711yescharacteristic
719C2#3762211yescharacteristic
720C2#4020212yes—
721C2#4020312yes—
722C2#4022612yes—
723C2#4022712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×C4×C4⋊Q8C2semidirect
3. #16965C2semidirect
4. #16889C2semidirect
7. #15221C2semidirect
9. #15151C2semidirect
11. #15128C2semidirect
14. #14879C2semidirect
15. #14689C2semidirect
16. #14308C2semidirect
18. #14041C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×C4×#1262
#169652
#168892
#168122
#166732
#152212
#152202
#151512
#151502
#151282
#151252
#148822
#148792
#146892
#143082
#143062
#140412
#139932