Finite Group Databasebeta

← #704582

Subgroups — #704582

Order 512.

2824 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2395C211yesmaximal, characteristic
3C2×#2354C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic, complemented
5#17060C212yesmaximal, complemented
6#16701C212yesmaximal, complemented
7#15929C214yesmaximal, complemented
8#15271C214yesmaximal
9#15183C212yesmaximal, complemented
10#15151C212yesmaximal, complemented
11#15133C212yesmaximal, complemented
12#15085C212yesmaximal, complemented
13#14343C211yesmaximal, characteristic, complemented
14#14046C212yesmaximal, complemented
15#13951C212yesmaximal, complemented
16#13951C211yesmaximal, characteristic, complemented
17#13924C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2486—28no—
25#2395—28no—
26C2×#400C2211yescharacteristic
27C2×#399C2211yescharacteristic
28C2×#398C2211yescharacteristic
29C2×#398C2212yes—
30C2×#398C2214yes—
31C2×#397C2212yes—
32C2×#397C2212yes—
33C2×#397C2211yescharacteristic
34C2×#394C2212yes—
35#2354—28no—
36#2317—28no—
37C2×#390C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#385C2214yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#382C2212yes—
49C2×#381C2212yes—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58C2×C4×C22⋊C4C2211yescharacteristic
59#1445C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1445C2212yes—
63#1445C2212yes—
64#1443C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2214yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#123—28no—
92C2×#123—44no—
93C2×#123—44no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4D418yes—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—28no—
102C2×#117—28no—
103C2×#117D418yes—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143#400—24no—
144#399—24no—
145#398—24no—
146#398—28no—
147#398—48no—
148#397—28no—
149#397—24no—
150#397—28no—
151#394—28no—
152#390—28no—
153#389—28no—
154#389—28no—
155#389—28no—
156#389—28no—
157#389—48no—
158#388—28no—
159#388—28no—
160#385—216no—
161#384—24no—
162#384—28no—
163#382—28no—
164#381—28no—
165#380—24no—
166#380—28no—
167#380—24no—
168#380—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—28no—
172C4×C22⋊C4—28no—
173C4×C22⋊C4—24no—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C25C2411yescharacteristic
200C25C2411yescharacteristic
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4C2×D414yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4114yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240#123—432no—
241C4×D4—232no—
242#117—232no—
243#117—416no—
244#117—416no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—416no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4#4118yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—22no—
328C2×C42—28no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—28no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—42no—
341C2×C42C2×D418yes—
342C2×C42C2×D414yes—
343#95—24no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—28no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
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#95—44no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—22no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—21no—
386C24C22×D412yes—
387C24C22×D412yes—
388C24C22×D412yes—
389C24C2×#4112yes—
390C24C2511yescharacteristic
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4#121116yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4118yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4114yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C42−1+414yes—
503C22×C42−1+414yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—48no—
507C4⋊C4—416no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—416no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—88no—
519C4⋊C4—88no—
520C4⋊C4—84no—
521C4⋊C4#123116yescomplemented
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—216no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—28no—
535C42—216no—
536C42—48no—
537C42—48no—
538C42—48no—
539C42—48no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23C2×#12114yes—
586C23C23×D411yescharacteristic
587C23C23×D411yescharacteristic
588C23C22×#4111yescharacteristic
589C23C22×#4112yes—
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×2−1+412yes—
593C23C2×2−1+412yes—
594C23#58511yescharacteristic
595C23#58511yescharacteristic
596C23#58512yes—
597D4—416no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—232no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—416no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
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—44no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—816no—
682C2×C4C2×#12114yes—
683C2×C4C2×#12114yes—
684C2×C4C2×#12318yes—
685C2×C4C2×#12314yes—
686C2×C4#53514yes—
687C2×C4#53614yes—
688C2×C4#54718yes—
689C2×C4#55218yes—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—44no—
697C22—42no—
698C22—44no—
699C22—42no—
700C22—42no—
701C22—44no—
702C22—42no—
703C22—44no—
704C22—42no—
705C22—42no—
706C22—82no—
707C22—81no—
708C22—81no—
709C22—82no—
710C22—81no—
711C22—81no—
712C22C22×#12111yescharacteristic
713C22C22×#12311yescharacteristic
714C22C2×#53511yescharacteristic
715C22C2×#53611yescharacteristic
716C22#344911yescharacteristic
717C22C2×#54712yes—
718C22C2×#55212yes—
719C22#348811yescharacteristic
720C22#348811yescharacteristic
721C22#348912yes—
722C22#349411yescharacteristic
723C22#349411yescharacteristic
724C22#349911yescharacteristic
725C22#350011yescharacteristic
726C22#350712yes—
727C22#351512yes—
728C22#351712yes—
729C22#352012yes—
730C22#352412yes—
731C22#354712yes—
732C22#355212yes—
733C22#355512yes—
734C22#356512yes—
735C4—28no—
736C4—24no—
737C4—24no—
738C4—44no—
739C4—42no—
740C4—42no—
741C4—44no—
742C4—84no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—84no—
747C4—82no—
748C4—82no—
749C4#248618yes—
750C2—44no—
751C2—82no—
752C2C2×#248611yescharacteristic
753C2#3525211yescharacteristic
754C2#3562111yescharacteristic
755C2#3599811yescharacteristic
756C2#3607611yescharacteristic
757C2#3804812yes—
758C2#3892012yes—
759C2#4291811yescharacteristic
760C2#4296511yescharacteristic
761C2#4499012yes—
762C2#4517812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#2317C2semidirect
5. #17060C2semidirect
6. #16701C2semidirect
7. #15929C2semidirect
9. #15183C2semidirect
10. #15151C2semidirect
11. #15133C2semidirect
12. #15085C2semidirect
13. #14343C2semidirect
14. #14046C2semidirect
15. #13951C2semidirect
16. #13951C2semidirect
17. #13924C2semidirect
521. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23952
C2×#23542
C2×#23172
#170602
#167012
#159292
#152712
#151832
#151512
#151332
#150852
#143432
#140462
#139512
#139242