Finite Group Databasebeta

← #874297

Subgroups — #874297

Order 512.

2358 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#2421C211yesmaximal, characteristic
3#16710C214yesmaximal, complemented
4#15929C214yesmaximal
5#15723C214yesmaximal, complemented
6#15241C211yesmaximal, characteristic, complemented
7#15240C211yesmaximal, characteristic, complemented
8#15236C211yesmaximal, characteristic
9#15231C211yesmaximal, characteristic, complemented
10#15184C211yesmaximal, characteristic, complemented
11#15178C211yesmaximal, characteristic
12#14714C211yesmaximal, characteristic, complemented
13#14689C211yesmaximal, characteristic, complemented
14#14383C214yesmaximal
15#14139C211yesmaximal, characteristic
16#14024C211yesmaximal, characteristic, complemented
17#13942C211yesmaximal, characteristic, complemented
18#13880C211yesmaximal, characteristic
19C4×#388C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2503—28no—
26#2421—28no—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#397C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#387C2212yes—
44C2×#385C2214yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#381C2212yes—
52C2×#381C2212yes—
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×C43C2211yescharacteristic
59#1445C2214yes—
60#1445C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1443C2214yes—
64#1443C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1440C2214yes—
74#1439C2214yes—
75#1437C2212yes—
76#1436C2214yes—
77#1436C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86#1434C2214yes—
87#1434C2214yes—
88#1434C2214yes—
89#1434C2212yes—
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—42no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—42no—
104C2×C22⋊Q8—42no—
105C2×C22⋊Q8—42no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—42no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—42no—
111C22×C4⋊C4—42no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#400—28no—
144#400—28no—
145#400—28no—
146#397—28no—
147#391—24no—
148#391—24no—
149#391—24no—
150#391—42no—
151#391—42no—
152#391—42no—
153#391—42no—
154#391—42no—
155#391—42no—
156#389—24no—
157#388—28no—
158#388—28no—
159#387—28no—
160#385—216no—
161#384—42no—
162#384—42no—
163#382—24no—
164#382—24no—
165#382—24no—
166#382—42no—
167#381—28no—
168#381—28no—
169#380—24no—
170C4×C4⋊C4—42no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—42no—
173C4×C22⋊C4D418yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2311yescharacteristic
209C2×#95C2311yescharacteristic
210C43—24no—
211C25C2411yescharacteristic
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8#4114yes—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—28no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—44no—
237C23×C4C2×D414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C4⋊Q8—416no—
257C22⋊Q8—416no—
258C22⋊Q8—416no—
259C4×Q8—416no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—416no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
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—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4#4118yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42C2×D414yes—
348C2×C42C2×D414yes—
349C2×C42#4114yes—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—28no—
366#95—28no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—28no—
371#95—28no—
372#95—44no—
373#95—42no—
374#95—44no—
375#95—42no—
376#95—42no—
377#95—44no—
378#95—44no—
379#95—44no—
380#95—44no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95C2×D414yes—
385#95C2×D414yes—
386C24—21no—
387C24—21no—
388C24—21no—
389C24—21no—
390C24—21no—
391C24—22no—
392C24—22no—
393C24—21no—
394C24—22no—
395C24C22×D412yes—
396C24C2×#4114yes—
397C24C2511yescharacteristic
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—88no—
511C22×C4—88no—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C2×#4114yes—
519C22×C4C2×#4114yes—
520C22×C4C2×#4114yes—
521C22×C4C2×#4114yes—
522C22×C4C2×#4112yes—
523C22×C4C2×#4112yes—
524C22×C4H(5,𝔽2)12yes—
525C22×C4H(5,𝔽2)12yes—
526C22×C4H(5,𝔽2)12yes—
527C22×C42−1+414yes—
528C22×C42−1+414yes—
529C22×C42−1+412yes—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—416no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—88no—
544C4⋊C4—88no—
545C22⋊C4—28no—
546C22⋊C4—216no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C42—28no—
553C42—28no—
554C42—28no—
555C42—48no—
556C42—48no—
557C42—44no—
558C42—44no—
559C42—48no—
560C42—44no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23#53214yes—
584C23C23×D411yescharacteristic
585C23C22×#4112yes—
586C23C22×#4111yescharacteristic
587C23C22×#4111yescharacteristic
588C23C2×H(5,𝔽2)11yescharacteristic
589C23C2×H(5,𝔽2)11yescharacteristic
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×2−1+411yescharacteristic
592C23C2×2−1+412yes—
593C23C2×2−1+412yes—
594C23#58512yes—
595Q8—416no—
596Q8—88no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
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—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—416no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—816no—
697C2×C4#53514yes—
698C2×C4#53914yes—
699C2×C4#54014yes—
700C2×C4#54614yes—
701C2×C4#54714yes—
702C2×C4#55114yes—
703C2×C4#55218yes—
704C2×C4#56214yes—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—42no—
712C22—44no—
713C22—42no—
714C22—44no—
715C22—42no—
716C22C2×#53211yescharacteristic
717C22C2×#53511yescharacteristic
718C22#344911yescharacteristic
719C22C2×#53911yescharacteristic
720C22C2×#54011yescharacteristic
721C22C2×#54611yescharacteristic
722C22C2×#54711yescharacteristic
723C22C2×#55111yescharacteristic
724C22C2×#55212yes—
725C22#348212yes—
726C22#349512yes—
727C22#349911yescharacteristic
728C22#349911yescharacteristic
729C22#350912yes—
730C22#351412yes—
731C22#351412yes—
732C22#351612yes—
733C22#352012yes—
734C22C2×#56211yescharacteristic
735C22#354712yes—
736C22#355212yes—
737C22#355212yes—
738C22#355512yes—
739C4—28no—
740C4—44no—
741C4—44no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—44no—
747C4—42no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4#270618yes—
757C2—44no—
758C2#3513111yescharacteristic
759C2#3618111yescharacteristic
760C2#3637611yescharacteristic
761C2#3739011yescharacteristic
762C2#3786811yescharacteristic
763C2#3893012yes—
764C2#3894611yescharacteristic
765C2#4022412yes—
766C2#4605512yes—
767C2C2×#270611yescharacteristic
768C2#5925312yes—
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
3. #16710C2semidirect
5. #15723C2semidirect
6. #15241C2semidirect
7. #15240C2semidirect
9. #15231C2semidirect
10. #15184C2semidirect
12. #14714C2semidirect
13. #14689C2semidirect
16. #14024C2semidirect
17. #13942C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#24212
#167102
#159292
#157232
#152412
#152402
#152362
#152312
#151842
#151782
#147142
#146892
#143832
#141392
#140242
#139422
#138802
C4×#3882