Finite Group Databasebeta

← #1393989

Subgroups — #1393989

Order 512.

3000 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2422C211yesmaximal, characteristic
2C2×#2623C212yesmaximal, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
4#16394C212yesmaximal, complemented
5#16360C212yesmaximal, complemented
6#16306C214yesmaximal, complemented
7#16267C214yesmaximal, complemented
8#15156C211yesmaximal, characteristic, complemented
9#15153C211yesmaximal, characteristic, complemented
10#15135C211yesmaximal, characteristic, complemented
11#15133C212yesmaximal, complemented
12#15132C212yesmaximal, complemented
13#14843C212yesmaximal, complemented
14#14018C212yesmaximal, complemented
15#13970C212yesmaximal
16#13916C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23#2623—216no—
24#2422—28no—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#399C2211yescharacteristic
30C2×#398C2214yes—
31C2×#397C2214yes—
32C2×#397C2212yes—
33C2×#394C2212yes—
34D4×C4⋊C4—28no—
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2212yes—
40C2×#386C2214yes—
41C2×#385C2212yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#381C2212yes—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1445C2211yescharacteristic
57#1445C2214yes—
58#1444C2211yescharacteristic
59#1444C2212yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2211yescharacteristic
74#1439C2214yes—
75#1438C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C4×#95C2211yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2314yes—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—44no—
94C2×#121—24no—
95C2×#121—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—44no—
99C2×C4×D4—44no—
100C2×C4×D4D418yes—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#400—28no—
138#400—28no—
139#400—44no—
140#399—28no—
141#399D418yescomplemented
142#398—216no—
143#397—216no—
144#397—28no—
145#394—28no—
146#390—24no—
147#390—28no—
148#389—28no—
149#389—24no—
150#389—44no—
151#386—216no—
152#385—28no—
153#385—216no—
154#385—216no—
155#384—24no—
156#384—24no—
157#381—28no—
158#380—28no—
159#380—24no—
160#380—44no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—44no—
163C4×C22⋊C4—24no—
164C4×C22⋊C4—44no—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C25C2411yescharacteristic
193C25C2412yes—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4C2×D414yes—
203C22×D4#4118yes—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D418yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#123—432no—
237#121—416no—
238C4×D4—232no—
239C4×D4—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—432no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D418yes—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42C2×D414yes—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—22no—
344#95—42no—
345#95—42no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D414yes—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24C22×D412yes—
378C24C22×D414yes—
379C24C2×#4118yes—
380C24C2511yescharacteristic
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—216no—
384C2×D4—216no—
385C2×D4—48no—
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—84no—
389C2×D4—88no—
390C2×D4#123116yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C42−1+412yes—
495C22×C42−1+412yes—
496C22×C42−1+412yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—84no—
511C4⋊C4—88no—
512C4⋊C4—84no—
513C22⋊C4—216no—
514C22⋊C4—28no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C42—28no—
527C42—44no—
528C42—48no—
529C42—48no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—28no—
534C23—24no—
535C23—28no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—42no—
575C23C2×#12318yes—
576C23C23×D411yescharacteristic
577C23C23×D411yescharacteristic
578C23C22×#4112yes—
579C23C2×H(5,𝔽2)12yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×2−1+411yescharacteristic
582C23C2×2−1+411yescharacteristic
583C23C2×2−1+411yescharacteristic
584C23#58512yes—
585C23#58511yescharacteristic
586C23#58512yes—
587D4—232no—
588D4—816no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—432no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
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—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
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—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—832no—
674C2×C4—88no—
675C2×C4C2×#12314yes—
676C2×C4#53514yes—
677C2×C4#53614yes—
678C2×C4#53918yes—
679C2×C4#53918yes—
680C2×C4D4214yes—
681C22—28no—
682C22—28no—
683C22—28no—
684C22—48no—
685C22—48no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—48no—
691C22—48no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22—44no—
696C22—42no—
697C22—81no—
698C22—82no—
699C22—82no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22C22×#12311yescharacteristic
704C22C2×#53511yescharacteristic
705C22C2×#53611yescharacteristic
706C22C2×#53912yes—
707C22C2×#53912yes—
708C22C2×D4211yescharacteristic
709C22D4×#4112yes—
710C22#348511yescharacteristic
711C22#348911yescharacteristic
712C22#349612yes—
713C22#349711yescharacteristic
714C22#349911yescharacteristic
715C22#350011yescharacteristic
716C22#351312yes—
717C22#351512yes—
718C22#352112yes—
719C22#353912yes—
720C22#354112yes—
721C22#354612yes—
722C22#354711yescharacteristic
723C22#355011yescharacteristic
724C22#355712yes—
725C22#356512yes—
726C4—24no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—42no—
731C4—44no—
732C4—44no—
733C4—81no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—84no—
738C4—84no—
739C4—82no—
740C4#262318yescomplemented
741C2—48no—
742C2—82no—
743C2C2×#262311yescharacteristic
744C2#3606711yescharacteristic
745C2D4×#12311yescharacteristic
746C2#3934311yescharacteristic
747C2#3936211yescharacteristic
748C2#3940012yes—
749C2#3940912yes—
750C2#4257212yes—
751C2#4263012yes—
752C2#4264711yescharacteristic
753C2#4267911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2623C2semidirect
3. C2×D4×C4⋊C4C2semidirect
4. #16394C2semidirect
5. #16360C2semidirect
6. #16306C2semidirect
7. #16267C2semidirect
8. #15156C2semidirect
9. #15153C2semidirect
10. #15135C2semidirect
11. #15133C2semidirect
12. #15132C2semidirect
13. #14843C2semidirect
14. #14018C2semidirect
16. #13916C2semidirect
17. C4×#399C2semidirect
141. #399D4semidirect
740. C4#2623semidirect

Maximal subgroups

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

subgroupindex
C2×#24222
C2×#26232
C2×D4×C4⋊C42
#163942
#163602
#163062
#162672
#151562
#151532
#151352
#151332
#151322
#148432
#140182
#139702
#139162
C4×#3992