Finite Group Databasebeta

← #696725

Subgroups — #696725

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic
3C2×#2345C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic
5#16396C214yesmaximal
6#15819C214yesmaximal, complemented
7#15813C214yesmaximal, complemented
8#15242C211yesmaximal, characteristic, complemented
9#15237C211yesmaximal, characteristic, complemented
10#15231C211yesmaximal, characteristic, complemented
11#15184C211yesmaximal, characteristic, complemented
12#14837C214yesmaximal
13#14258C211yesmaximal, characteristic
14#14003C211yesmaximal, characteristic, complemented
15#14000C211yesmaximal, characteristic, complemented
16#13939C211yesmaximal, characteristic
17#13911C211yesmaximal, characteristic, complemented
18#13909C211yesmaximal, characteristic, complemented
19#13844C211yesmaximal, characteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2503—28no—
25#2499—28no—
26C2×#400C2214yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29#2345—28no—
30#2317—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#389C2214yes—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#385C2214yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#381C2214yes—
54C2×#380C2212yes—
55C2×#380C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4C2212yes—
62C2×C4×C22⋊C4C2211yescharacteristic
63#1445C2214yes—
64#1444C2212yes—
65#1443C2214yes—
66#1443C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2212yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1440C2214yes—
79#1440C2214yes—
80#1439C2212yes—
81#1439C2212yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2214yes—
89C4⋊#95C2211yescharacteristic
90C4⋊#95C2211yescharacteristic
91#1434C2214yes—
92#1434C2214yes—
93C4×#95C2211yescharacteristic
94C4×#95C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2312yes—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117—24no—
106C2×#117—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4D418yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143C22×C42C2312yes—
144#400—216no—
145#397—28no—
146#397—28no—
147#391—24no—
148#391—24no—
149#391—24no—
150#391—24no—
151#391—24no—
152#391—24no—
153#389—216no—
154#389—24no—
155#389—24no—
156#387—28no—
157#387—28no—
158#387—28no—
159#387—28no—
160#385—216no—
161#384—24no—
162#384—24no—
163#384—24no—
164#382—24no—
165#382—24no—
166#382—24no—
167#382—24no—
168#382—24no—
169#381—216no—
170#380—24no—
171#380—28no—
172C4×C4⋊C4—24no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—24no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4—28no—
178C4×C22⋊C4D418yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2314yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2311yescharacteristic
212C2×#95C2311yescharacteristic
213C25C2411yescharacteristic
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D418yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C22⋊Q8—416no—
252#117—232no—
253#117—416no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—22no—
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—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D418yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D414yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42#4114yes—
347C2×C42#4114yes—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—28no—
369#95—28no—
370#95—24no—
371#95—42no—
372#95—44no—
373#95—44no—
374#95—42no—
375#95—42no—
376#95—44no—
377#95—44no—
378#95—44no—
379#95—44no—
380#95C2×D414yes—
381#95C2×D414yes—
382C24—21no—
383C24—21no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24C22×D412yes—
391C24C22×D412yes—
392C24C2×#4112yes—
393C24C2511yescharacteristic
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—216no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—416no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—88no—
497C22×C4#121116yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D414yes—
501C22×C4C22×D414yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C2×#4114yes—
509C22×C4C2×#4114yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C42−1+414yes—
515C22×C42−1+412yes—
516C22×C42−1+412yes—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—88no—
532C4⋊C4—88no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C22⋊C4—216no—
536C22⋊C4—28no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C42—28no—
543C42—28no—
544C42—28no—
545C42—28no—
546C42—48no—
547C42—44no—
548C42—44no—
549C42—84no—
550C42—84no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—24no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23C2×#12114yes—
574C23C23×D411yescharacteristic
575C23C23×D411yescharacteristic
576C23C22×#4111yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4111yescharacteristic
579C23C2×2−1+411yescharacteristic
580C23C2×2−1+411yescharacteristic
581C23C2×2−1+412yes—
582C23#58512yes—
583C23#58512yes—
584C23#58512yes—
585Q8—88no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—232no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—816no—
683C2×C4C2×#12114yes—
684C2×C4C2×#12114yes—
685C2×C4#52514yes—
686C2×C4#53614yes—
687C2×C4#53614yes—
688C2×C4#54714yes—
689C2×C4#54714yes—
690C2×C4#55214yes—
691C2×C4#55214yes—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—42no—
696C22—44no—
697C22—42no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—42no—
703C22C22×#12111yescharacteristic
704C22C2×#52511yescharacteristic
705C22C2×#53611yescharacteristic
706C22C2×#53611yescharacteristic
707C22#344911yescharacteristic
708C22C2×#54711yescharacteristic
709C22C2×#54711yescharacteristic
710C22C2×#55211yescharacteristic
711C22C2×#55211yescharacteristic
712C22#348512yes—
713C22#349312yes—
714C22#349512yes—
715C22#349712yes—
716C22#350011yescharacteristic
717C22#350011yescharacteristic
718C22#350712yes—
719C22#350712yes—
720C22#351012yes—
721C22#351212yes—
722C22#355512yes—
723C22#356312yes—
724C22#356512yes—
725C22#356512yes—
726C4—24no—
727C4—24no—
728C4—44no—
729C4—44no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—44no—
734C4—42no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4#241318yes—
744C2—44no—
745C2C2×#241311yescharacteristic
746C2#3517811yescharacteristic
747C2#3561711yescharacteristic
748C2#3575812yes—
749C2#3786511yescharacteristic
750C2#3787011yescharacteristic
751C2#3948511yescharacteristic
752C2#3949011yescharacteristic
753C2#4264412yes—
754C2#4511912yes—
755C2#4514512yes—
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
6. #15819C2semidirect
7. #15813C2semidirect
8. #15242C2semidirect
9. #15237C2semidirect
10. #15231C2semidirect
11. #15184C2semidirect
14. #14003C2semidirect
15. #14000C2semidirect
17. #13911C2semidirect
18. #13909C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#25032
C2×#23452
C2×#23172
#163962
#158192
#158132
#152422
#152372
#152312
#151842
#148372
#142582
#140032
#140002
#139392
#139112
#139092
#138442