Finite Group Databasebeta

← #1410027

Subgroups — #1410027

Order 512.

2668 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic, complemented
2C2×#2423C211yesmaximal, characteristic, complemented
3#17042C212yesmaximal, complemented
4#16876C214yesmaximal, complemented
5#16799C212yesmaximal
6#16548C214yesmaximal, complemented
7#15242C211yesmaximal, characteristic, complemented
8#15236C212yesmaximal, complemented
9#15181C211yesmaximal, characteristic, complemented
10#15177C212yesmaximal, complemented
11#15150C211yesmaximal, characteristic, complemented
12#14982C212yesmaximal, complemented
13#14324C212yesmaximal, complemented
14#13905C212yesmaximal, complemented
15#13876C212yesmaximal, complemented
16C4×#392C211yesmaximal, characteristic
17#13817C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2538—28no—
24#2423—28no—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#396C2214yes—
30C2×#392C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#390C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2211yescharacteristic
56#1444C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2211yescharacteristic
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2211yescharacteristic
71#1438C2212yes—
72#1438C2214yes—
73#1438C2212yes—
74#1438C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C22⋊Q8—44no—
94C2×C22⋊Q8—44no—
95C2×C4×Q8—44no—
96C2×C4×Q8D418yescomplemented
97C2×C4×D4—24no—
98C2×C4×D4—28no—
99C2×C4×D4—44no—
100C2×C4×D4—44no—
101C2×C4×D4—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136#399—28no—
137#398—28no—
138#397—28no—
139#397—44no—
140#396—216no—
141#392—24no—
142#391—24no—
143#391—28no—
144#390—28no—
145#389—24no—
146#389—28no—
147#388—28no—
148#388—48no—
149#387—28no—
150#384—24no—
151#384—24no—
152#382—44no—
153#380—24no—
154#380—28no—
155#380—44no—
156#380—44no—
157#380—44no—
158C4×C4⋊C4—44no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—28no—
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2411yescharacteristic
188C25C2412yes—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8C2×D414yes—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C22⋊Q8—416no—
234C22⋊Q8—432no—
235C4×Q8—416no—
236C4×D4—416no—
237C4×D4—432no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4#4114yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D414yes—
326C2×C42C2×D418yes—
327C2×C42#4118yes—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—22no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—22no—
344#95—44no—
345#95—42no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—42no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—21no—
378C24C2×#4112yes—
379C242−1+414yes—
380C24C2511yescharacteristic
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×D4—28no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+412yes—
495C4⋊C4—28no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C22⋊C4—216no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—88no—
520C22⋊C4—84no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C42—28no—
525C42—216no—
526C42—48no—
527C42—44no—
528C42—88no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—42no—
583C23—41no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23C23×D411yescharacteristic
588C23C22×#4112yes—
589C23C22×#4111yescharacteristic
590C23C2×H(5,𝔽2)12yes—
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)11yescharacteristic
593C23C2×H(5,𝔽2)11yescharacteristic
594C23C2×2−1+411yescharacteristic
595C23C2×2−1+412yes—
596C23#58512yes—
597C23#58511yescharacteristic
598Q8—416no—
599D4—416no—
600D4—432no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—816no—
681C2×C4—88no—
682C2×C4#53214yes—
683C2×C4#53414yes—
684C2×C4#53514yes—
685C2×C4#53818yes—
686C2×C4#54018yes—
687C2×C4#54814yes—
688C22—44no—
689C22—42no—
690C22—44no—
691C22—44no—
692C22—42no—
693C22—44no—
694C22—42no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—81no—
701C22—82no—
702C22—81no—
703C22—82no—
704C22—81no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—81no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22C2×#53211yescharacteristic
715C22C2×#53411yescharacteristic
716C22C2×#53511yescharacteristic
717C22C2×#53812yes—
718C22C2×#54012yes—
719C22C2×#54811yescharacteristic
720C22#347512yes—
721C22#348611yescharacteristic
722C22#348811yescharacteristic
723C22#349212yes—
724C22#349512yes—
725C22#350011yescharacteristic
726C22#350611yescharacteristic
727C22#350712yes—
728C22#350811yescharacteristic
729C22#352012yes—
730C22#353812yes—
731C22#353912yes—
732C22#354112yes—
733C22#354612yes—
734C22#354811yescharacteristic
735C22#355411yescharacteristic
736C22#356112yes—
737C4—24no—
738C4—44no—
739C4—42no—
740C4—42no—
741C4—44no—
742C4—42no—
743C4—44no—
744C4—44no—
745C4—82no—
746C4—81no—
747C4—82no—
748C4—84no—
749C4—82no—
750C4—84no—
751C4#266118yes—
752C2—84no—
753C2—82no—
754C2C2×#266111yescharacteristic
755C2#3592611yescharacteristic
756C2#3653411yescharacteristic
757C2#3912111yescharacteristic
758C2#3972511yescharacteristic
759C2#3992612yes—
760C2#3993212yes—
761C2#4244912yes—
762C2#4266912yes—
763C2#4278811yescharacteristic
764C2#4298511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2538C2semidirect
2. C2×#2423C2semidirect
3. #17042C2semidirect
4. #16876C2semidirect
6. #16548C2semidirect
7. #15242C2semidirect
8. #15236C2semidirect
9. #15181C2semidirect
10. #15177C2semidirect
11. #15150C2semidirect
12. #14982C2semidirect
13. #14324C2semidirect
14. #13905C2semidirect
15. #13876C2semidirect
17. #13817C2semidirect
96. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24232
#170422
#168762
#167992
#165482
#152422
#152362
#151812
#151772
#151502
#149822
#143242
#139052
#138762
C4×#3922
#138172