Finite Group Databasebeta

← #946059

Subgroups — #946059

Order 512.

2178 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic
3#17200C214yesmaximal, complemented
4#16329C214yesmaximal, complemented
5#16307C214yesmaximal
6#15972C214yesmaximal
7#15242C211yesmaximal, characteristic, complemented
8#15240C211yesmaximal, characteristic, complemented
9#15237C211yesmaximal, characteristic, complemented
10#15236C211yesmaximal, characteristic
11#15182C211yesmaximal, characteristic
12#15153C211yesmaximal, characteristic, complemented
13#14354C211yesmaximal, characteristic, complemented
14#14261C211yesmaximal, characteristic, complemented
15#14000C211yesmaximal, characteristic, complemented
16#13910C211yesmaximal, characteristic, complemented
17#13881C211yesmaximal, characteristic
18#13880C211yesmaximal, characteristic
19#13842C211yesmaximal, characteristic
20C22×C22⋊Q8C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2502—28no—
23#2494—28no—
24C2×#401C2212yes—
25C2×#400C2214yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#397C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2214yes—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2214yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2211yescharacteristic
59C2×#376C2211yescharacteristic
60#1445C2214yes—
61#1445C2214yes—
62#1445C2214yes—
63#1445C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1444C2214yes—
68#1443C2212yes—
69#1443C2214yes—
70#1443C2214yes—
71#1442C2212yes—
72#1442C2214yes—
73#1442C2214yes—
74#1442C2214yes—
75#1442C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1441C2214yes—
79#1440C2212yes—
80#1440C2214yes—
81#1439C2212yes—
82#1439C2214yes—
83#1436C2214yes—
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89C4⋊#95C2211yescharacteristic
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—42no—
98C2×C22⋊Q8—42no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4—42no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#401—28no—
136#400—48no—
137#398—28no—
138#398—28no—
139#397—28no—
140#391—24no—
141#391—24no—
142#391—24no—
143#391—24no—
144#391—42no—
145#391—42no—
146#389—24no—
147#389—24no—
148#389—48no—
149#388—28no—
150#387—28no—
151#385—28no—
152#385—28no—
153#385—48no—
154#384—24no—
155#384—24no—
156#384—42no—
157#384—42no—
158#384—42no—
159#382—24no—
160#382—42no—
161#382—42no—
162#382—42no—
163#380—24no—
164#380—24no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4D418yes—
170#376—24no—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2311yescharacteristic
205C2×#95C2311yescharacteristic
206C25C2411yescharacteristic
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C22⋊Q8—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
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—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—416no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
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×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42#4114yes—
333C2×C42#4114yes—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—42no—
357#95—42no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95C2×D414yes—
368#95C2×D414yes—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—21no—
377C24—22no—
378C24C22×D412yes—
379C242−1+414yes—
380C24C2511yescharacteristic
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
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—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
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—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
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—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4H(5,𝔽2)14yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C42−1+414yes—
502C22×C42−1+412yes—
503C22×C42−1+412yes—
504C4⋊C4—216no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—88no—
521C4⋊C4—88no—
522C4⋊C4—88no—
523C22⋊C4—216no—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—84no—
530C42—28no—
531C42—28no—
532C42—28no—
533C42—28no—
534C42—44no—
535C42—44no—
536C42—84no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23#53614yes—
560C23C23×D411yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4111yescharacteristic
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+412yes—
567C23C2×2−1+412yes—
568C23C2×2−1+411yescharacteristic
569C23C2×2−1+411yescharacteristic
570C23#58512yes—
571Q8—88no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
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—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—416no—
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—42no—
630C2×C4—42no—
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—44no—
640C2×C4—44no—
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—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—816no—
670C2×C4#53514yes—
671C2×C4#53614yes—
672C2×C4#53914yes—
673C2×C4#54214yes—
674C2×C4#54714yes—
675C2×C4#54714yes—
676C2×C4#55314yes—
677C2×C4#56314yes—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—44no—
688C22—42no—
689C22C2×#53511yescharacteristic
690C22C2×#53611yescharacteristic
691C22C2×#53611yescharacteristic
692C22C2×#53911yescharacteristic
693C22C2×#54211yescharacteristic
694C22C2×#54711yescharacteristic
695C22C2×#54711yescharacteristic
696C22C2×#55311yescharacteristic
697C22#349212yes—
698C22#349212yes—
699C22#349512yes—
700C22#349811yescharacteristic
701C22#350011yescharacteristic
702C22#350712yes—
703C22C2×#56311yescharacteristic
704C22#354612yes—
705C22#355212yes—
706C22#355212yes—
707C22#355212yes—
708C22#355212yes—
709C22#355512yes—
710C22#355812yes—
711C22#356312yes—
712C4—24no—
713C4—44no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4#271818yes—
730C2—44no—
731C2#3730511yescharacteristic
732C2#3761711yescharacteristic
733C2#3786211yescharacteristic
734C2#3786211yescharacteristic
735C2#3786511yescharacteristic
736C2#3949011yescharacteristic
737C2#4245112yes—
738C2#4251312yes—
739C2#4512012yes—
740C2C2×#271811yescharacteristic
741C2#5300212yes—
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. #17200C2semidirect
4. #16329C2semidirect
7. #15242C2semidirect
8. #15240C2semidirect
9. #15237C2semidirect
12. #15153C2semidirect
13. #14354C2semidirect
14. #14261C2semidirect
15. #14000C2semidirect
16. #13910C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#25022
#172002
#163292
#163072
#159722
#152422
#152402
#152372
#152362
#151822
#151532
#143542
#142612
#140002
#139102
#138812
#138802
#138422