Finite Group Databasebeta

← #874318

Subgroups — #874318

Order 512.

2230 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic
3#16789C214yesmaximal, complemented
4#15414C214yesmaximal, complemented
5#15290C214yesmaximal
6#15239C211yesmaximal, characteristic, complemented
7#15233C211yesmaximal, characteristic
8#15231C211yesmaximal, characteristic, complemented
9#15228C211yesmaximal, characteristic, complemented
10#15196C211yesmaximal, characteristic, complemented
11#15150C211yesmaximal, characteristic
12#14712C211yesmaximal, characteristic, complemented
13#14689C211yesmaximal, characteristic, complemented
14#14541C214yesmaximal
15#14160C211yesmaximal, characteristic, complemented
16#14049C211yesmaximal, characteristic, complemented
17#13924C211yesmaximal, characteristic
18#13880C211yesmaximal, characteristic
19C4×#388C211yesmaximal, characteristic
20C22×#125C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2494—28no—
23#2442—28no—
24C2×#401C2212yes—
25C2×#400C2212yes—
26C2×#398C2212yes—
27C2×#397C2214yes—
28C2×#397C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2214yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C43C2211yescharacteristic
56#1445C2214yes—
57#1445C2214yes—
58#1445C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1437C2212yes—
72#1436C2214yes—
73#1436C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2214yes—
86C4⋊#95C2211yescharacteristic
87#1434C2212yes—
88#1434C2214yes—
89#1434C2214yes—
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—42no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#401—28no—
137#400—28no—
138#398—28no—
139#397—28no—
140#397—216no—
141#389—24no—
142#389—24no—
143#389—24no—
144#389—24no—
145#389—48no—
146#388—28no—
147#388—28no—
148#387—28no—
149#387—48no—
150#385—28no—
151#385—48no—
152#384—24no—
153#384—24no—
154#384—42no—
155#382—24no—
156#382—24no—
157#382—42no—
158#380—24no—
159#380—24no—
160#380—24no—
161#380—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—42no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—42no—
166C4×C22⋊C4D418yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C43—24no—
202C25C2411yescharacteristic
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233#125—416no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4#4114yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4#4118yes—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—42no—
326C2×C42—44no—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4114yes—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
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—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—44no—
352#95—42no—
353#95—44no—
354#95—42no—
355#95—42no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24C22×D412yes—
374C24C2×#4114yes—
375C24C2511yescharacteristic
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
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—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—88no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+414yes—
487C22×C42−1+412yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C4⋊C4—28no—
491C4⋊C4—416no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—416no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—88no—
509C22⋊C4—28no—
510C22⋊C4—216no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C42—28no—
517C42—28no—
518C42—28no—
519C42—48no—
520C42—44no—
521C42—44no—
522C42—48no—
523C42—44no—
524C42—84no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23#53214yes—
548C23C23×D411yescharacteristic
549C23C22×#4112yes—
550C23C22×#4111yescharacteristic
551C23C22×#4111yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)12yes—
554C23C2×2−1+411yescharacteristic
555C23C2×2−1+411yescharacteristic
556C23C2×2−1+411yescharacteristic
557C23C2×2−1+412yes—
558C23#58512yes—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
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—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—816no—
655C2×C4#53614yes—
656C2×C4#54114yes—
657C2×C4#54214yes—
658C2×C4#54614yes—
659C2×C4#54714yes—
660C2×C4#55214yes—
661C2×C4#55318yes—
662C2×C4#56314yes—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—44no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—42no—
674C22C2×#53211yescharacteristic
675C22C2×#53611yescharacteristic
676C22#344911yescharacteristic
677C22C2×#54111yescharacteristic
678C22C2×#54211yescharacteristic
679C22C2×#54611yescharacteristic
680C22C2×#54711yescharacteristic
681C22C2×#55211yescharacteristic
682C22C2×#55312yes—
683C22#348212yes—
684C22#349512yes—
685C22#349811yescharacteristic
686C22#349911yescharacteristic
687C22#351112yes—
688C22#351312yes—
689C22#351512yes—
690C22#351512yes—
691C22#352012yes—
692C22C2×#56311yescharacteristic
693C22#354412yes—
694C22#354612yes—
695C22#355212yes—
696C22#355512yes—
697C4—28no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—44no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#269818yes—
715C2—44no—
716C2#3513911yescharacteristic
717C2#3618211yescharacteristic
718C2#3637611yescharacteristic
719C2#3730311yescharacteristic
720C2#3786811yescharacteristic
721C2#3894511yescharacteristic
722C2#3914612yes—
723C2#4020212yes—
724C2#4617412yes—
725C2C2×#269811yescharacteristic
726C2#5920912yes—
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. #16789C2semidirect
4. #15414C2semidirect
6. #15239C2semidirect
8. #15231C2semidirect
9. #15228C2semidirect
10. #15196C2semidirect
12. #14712C2semidirect
13. #14689C2semidirect
15. #14160C2semidirect
16. #14049C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#24422
#167892
#154142
#152902
#152392
#152332
#152312
#152282
#151962
#151502
#147122
#146892
#145412
#141602
#140492
#139242
#138802
C4×#3882