Finite Group Databasebeta

← #722631

Subgroups — #722631

Order 512.

2728 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2396C211yesmaximal, characteristic
3C2×#2351C211yesmaximal, characteristic, complemented
4C2×#2323C211yesmaximal, characteristic
5#16567C212yesmaximal, complemented
6#16566C212yesmaximal
7#15620C214yesmaximal
8#15618C214yesmaximal, complemented
9#15149C212yesmaximal, complemented
10#15131C212yesmaximal
11#14872C212yesmaximal, complemented
12#14870C212yesmaximal
13#14340C211yesmaximal, characteristic
14#13969C212yesmaximal
15#13926C212yesmaximal, complemented
16#13925C211yesmaximal, characteristic, complemented
17#13910C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2582—28no—
24#2396—28no—
25C2×#400C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#396C2212yes—
31C2×#396C2214yes—
32C2×#395C2211yescharacteristic
33C2×#393C2211yescharacteristic
34#2351—28no—
35#2323—28no—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2214yes—
46C2×#388C2212yes—
47C2×#388C2212yes—
48C2×#387C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51C2×#382C2211yescharacteristic
52C2×#382C2212yes—
53C2×#380C2211yescharacteristic
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2212yes—
60#1446C2212yes—
61#1445C2212yes—
62#1444C2212yes—
63#1444C2212yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8Q818yescomplemented
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—28no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—44no—
101C2×C4×Q8D418yes—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—24no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141#400—24no—
142#399—24no—
143#398—216no—
144#398—28no—
145#398—48no—
146#396—28no—
147#396—48no—
148#395—24no—
149#393—24no—
150#391—24no—
151#391—28no—
152#391—28no—
153#389—28no—
154#389—24no—
155#389—28no—
156#389—216no—
157#389—28no—
158#389—28no—
159#389—48no—
160#388—28no—
161#388—28no—
162#387—28no—
163#384—24no—
164#384—44no—
165#382—24no—
166#382—28no—
167#380—24no—
168#380—44no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—28no—
172C4×C4⋊C4D418yes—
173C4×C22⋊C4—28no—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C25C2411yescharacteristic
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8C2×D414yes—
211C22×Q8C2×Q814yes—
212C22×Q8#4114yes—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4C2×Q814yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C22⋊Q8—232no—
237C4×Q8—232no—
238C4×Q8—432no—
239C4×Q8—416no—
240#117—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D418yes—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4C2×Q814yes—
303C2×C4⋊C4C2×Q818yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4C2×Q818yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—28no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42C2×D418yes—
340C2×C42C2×D418yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42#4114yes—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—21no—
379C24—22no—
380C24C22×Q812yes—
381C24C2511yescharacteristic
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8C22⋊Q8116yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×Q814yes—
486C22×C4C22×Q812yes—
487C22×C4C22×Q814yes—
488C22×C4C22×Q812yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C42−1+414yes—
497C22×C42−1+414yes—
498C22×C42−1+412yes—
499C4⋊C4—216no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—48no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—416no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—84no—
517C4⋊C4—88no—
518C4⋊C4—84no—
519C4⋊C4—88no—
520C4⋊C4—88no—
521C22⋊C4—216no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—88no—
528C42—216no—
529C42—216no—
530C42—416no—
531C42—44no—
532C42—48no—
533C42—84no—
534C42#121116yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23C23×D411yescharacteristic
559C23C23×D411yescharacteristic
560C23C23×Q811yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4111yescharacteristic
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×H(5,𝔽2)12yes—
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+412yes—
567C23C2×2−1+412yes—
568C23#58512yes—
569Q8—232no—
570Q8—416no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—416no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4C2×#12114yes—
660C2×C4C2×#12114yes—
661C2×C4C2×#12114yes—
662C2×C4C2×C22⋊Q818yes—
663C2×C4C2×C22⋊Q814yes—
664C2×C4#53718yes—
665C2×C4#53714yes—
666C2×C4#54614yes—
667C2×C4D4×Q814yes—
668C2×C4#55214yes—
669C2×C4#55414yes—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—82no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—82no—
680C22—81no—
681C22C22×#12111yescharacteristic
682C22C22×C22⋊Q811yescharacteristic
683C22C2×#53711yescharacteristic
684C22C2×#53712yes—
685C22C2×#54611yescharacteristic
686C22C2×D4×Q811yescharacteristic
687C22C2×#55211yescharacteristic
688C22C2×#55411yescharacteristic
689C22#348611yescharacteristic
690C22#348711yescharacteristic
691C22#348912yes—
692C22#349512yes—
693C22#349811yescharacteristic
694C22#349911yescharacteristic
695C22#350212yes—
696C22#350312yes—
697C22#351112yes—
698C22#351612yes—
699C22#352212yes—
700C22#355212yes—
701C22#355412yes—
702C22#355612yes—
703C22#355812yes—
704C4—24no—
705C4—28no—
706C4—28no—
707C4—24no—
708C4—44no—
709C4—42no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—81no—
714C4—84no—
715C4—82no—
716C4—84no—
717C4—81no—
718C4—82no—
719C4#241718yes—
720C2—82no—
721C2C2×#241711yescharacteristic
722C2#3527311yescharacteristic
723C2#3566111yescharacteristic
724C2#3574811yescharacteristic
725C2#3584511yescharacteristic
726C2#3802512yes—
727C2#3959912yes—
728C2#4279111yescharacteristic
729C2#4288111yescharacteristic
730C2#4527512yes—
731C2#4534612yes—
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×#2582C2semidirect
3. C2×#2351C2semidirect
5. #16567C2semidirect
8. #15618C2semidirect
9. #15149C2semidirect
11. #14872C2semidirect
15. #13926C2semidirect
16. #13925C2semidirect
17. #13910C2semidirect
92. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#23962
C2×#23512
C2×#23232
#165672
#165662
#156202
#156182
#151492
#151312
#148722
#148702
#143402
#139692
#139262
#139252
#139102