Finite Group Databasebeta

← #789622

Subgroups — #789622

Order 512.

2912 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2422C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3#16360C212yesmaximal, complemented
4#15551C214yesmaximal, complemented
5#15372C214yesmaximal, complemented
6#15242C211yesmaximal, characteristic, complemented
7#15195C211yesmaximal, characteristic, complemented
8#15156C211yesmaximal, characteristic, complemented
9#15114C212yesmaximal, complemented
10#14826C212yesmaximal, complemented
11#14135C212yesmaximal, complemented
12#13970C212yesmaximal
13#13941C212yesmaximal, complemented
14#13884C212yesmaximal, complemented
15#13875C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic, complemented
17D4×#95C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2422—28no—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#397C2212yes—
27C2×#394C2212yes—
28#2354—28no—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#386C2214yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C22×#95C2212yes—
48#1445C2214yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1444C2211yescharacteristic
52#1443C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2212yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2211yescharacteristic
67#1439C2212yes—
68#1439C2214yes—
69#1438C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2214yes—
79C22⋊#95C2212yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#121—24no—
94C2×#121—28no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—44no—
99C2×C4×D4D418yes—
100C2×#117—28no—
101C2×#117—48no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#400—44no—
135#399—24no—
136#397—28no—
137#394—28no—
138#390—28no—
139#390—24no—
140#389—24no—
141#388—216no—
142#387—48no—
143#386—48no—
144#385—28no—
145#384—24no—
146#384—24no—
147#382—44no—
148#380—28no—
149#380—24no—
150#380—44no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—28no—
156C2×#95—44no—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2411yescharacteristic
186C25C2412yes—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—42no—
194C22×D4—44no—
195C22×D4C2×D414yes—
196C22×D4#4118yes—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—28no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D418yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#123—432no—
230#121—416no—
231C4×D4—232no—
232C4×D4—432no—
233#117—432no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
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—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4#4114yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42#4118yes—
320#95—22no—
321#95—22no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—42no—
337#95—44no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24C22×D414yes—
368C24C2×#4112yes—
369C24C2×#4118yes—
370C24C2511yescharacteristic
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—216no—
374C2×D4—216no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4#123116yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—216no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—48no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C4⋊C4—216no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C22⋊C4—216no—
502C22⋊C4—28no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—28no—
515C42—216no—
516C42—416no—
517C42—44no—
518C42—48no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23C2×#12318yes—
565C23C23×D411yescharacteristic
566C23C22×#4112yes—
567C23C22×#4112yes—
568C23C22×#4111yescharacteristic
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+411yescharacteristic
573C23C2×2−1+411yescharacteristic
574C23#58512yes—
575C23#58511yescharacteristic
576D4—232no—
577D4—816no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
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—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—432no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
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—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—832no—
663C2×C4—88no—
664C2×C4C2×#12314yes—
665C2×C4#53214yes—
666C2×C4#53714yes—
667C2×C4#54614yes—
668C2×C4#55118yes—
669C2×C4#55218yes—
670C22—28no—
671C22—28no—
672C22—28no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—48no—
677C22—48no—
678C22—48no—
679C22—48no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—81no—
687C22—82no—
688C22—82no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22C22×#12311yescharacteristic
693C22C2×#53211yescharacteristic
694C22C2×#53711yescharacteristic
695C22C2×#54611yescharacteristic
696C22C2×#55112yes—
697C22C2×#55212yes—
698C22#347512yes—
699C22#348112yes—
700C22#348411yescharacteristic
701C22#348911yescharacteristic
702C22#349212yes—
703C22#349412yes—
704C22#350011yescharacteristic
705C22#350612yes—
706C22#350711yescharacteristic
707C22#351111yescharacteristic
708C22#351512yes—
709C22#351912yes—
710C22#352112yes—
711C22#354611yescharacteristic
712C22#354612yes—
713C22#355011yescharacteristic
714C22#355412yes—
715C4—24no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—42no—
723C4—82no—
724C4—82no—
725C4—84no—
726C4—82no—
727C4—84no—
728C4—81no—
729C4#249818yes—
730C2—48no—
731C2—82no—
732C2C2×#249811yescharacteristic
733C2#3592311yescharacteristic
734C2#3606711yescharacteristic
735C2#3623611yescharacteristic
736C2#3641511yescharacteristic
737C2#3946811yescharacteristic
738C2#4242911yescharacteristic
739C2#4490112yes—
740C2#4491012yes—
741C2#5926012yes—
742C2#5927812yes—
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×#2354C2semidirect
3. #16360C2semidirect
4. #15551C2semidirect
5. #15372C2semidirect
6. #15242C2semidirect
7. #15195C2semidirect
8. #15156C2semidirect
9. #15114C2semidirect
10. #14826C2semidirect
11. #14135C2semidirect
13. #13941C2semidirect
14. #13884C2semidirect
15. #13875C2semidirect
16. C4×#399C2semidirect
17. D4×#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24222
C2×#23542
#163602
#155512
#153722
#152422
#151952
#151562
#151142
#148262
#141352
#139702
#139412
#138842
#138752
C4×#3992
D4×#952