Finite Group Databasebeta

← #559893

Subgroups — #559893

Order 512.

2998 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2376C212yesmaximal
2C2×C4×#123C211yesmaximal, characteristic
3#14686C212yesmaximal, complemented
4#14674C212yesmaximal, complemented
5#14468C214yesmaximal, complemented
6#14442C214yesmaximal, complemented
7#14402C212yesmaximal, complemented
8#14401C212yesmaximal
9#13910C212yesmaximal, complemented
10#13906C211yesmaximal, characteristic
11#13884C212yesmaximal, complemented
12#13880C212yesmaximal, complemented
13#13876C212yesmaximal, complemented
14#13876C212yesmaximal, complemented
15#13872C211yesmaximal, characteristic
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2376—216no—
21C4×#123C4116yescomplemented
22C2×#390C2212yes—
23C2×#389C2214yes—
24C2×#389C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#388C2212yes—
28C2×#387C2214yes—
29C2×#387C2212yes—
30C2×#386C2212yes—
31C2×#386C2211yescharacteristic
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#385C2211yescharacteristic
35C2×#384C2212yes—
36C2×#384C2212yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#380C2214yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C2×#123C2×C418yes—
84C2×#123D418yes—
85C2×#123Q818yes—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4C2×C418yes—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C42—42no—
118C22×C42—42no—
119C22×C42C2×C418yescomplemented
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126#390—28no—
127#389—216no—
128#389—48no—
129#388—216no—
130#388—216no—
131#388C2×C4116yes—
132#387—216no—
133#387C2×C4116yescomplemented
134#386—28no—
135#386C2×C418yes—
136#385—216no—
137#385—48no—
138#385C2×C418yescomplemented
139#384—42no—
140#384—44no—
141#384—44no—
142#384—44no—
143#381C2×C418yescomplemented
144#380—216no—
145#380—48no—
146#380—44no—
147#380—44no—
148#380—44no—
149#380—42no—
150C4×C4⋊C4—42no—
151C4×C4⋊C4—44no—
152C4×C4⋊C4C2×C4116yescomplemented
153C4×C22⋊C4C2×C4116yes—
154C4×C22⋊C4C2×C418yescomplemented
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4C22×C414yes—
183C22×D4C2×D414yes—
184C22×D4C2×Q814yes—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4—44no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4C22×C414yes—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C22×C414yes—
210C23×C4C2×D414yes—
211C23×C4C2×Q814yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225#123C4⋊C4164yescomplemented
226C4×D4—416no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
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—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×Q818yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C22×C418yes—
295C2×C22⋊C4C22×C414yes—
296C2×C22⋊C4C22×C414yes—
297C2×C22⋊C4C22×C414yes—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C414yes—
300C2×C22⋊C4C2×D418yes—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×Q818yes—
303C2×C22⋊C4C2×Q814yes—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—416no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42—42no—
319C2×C42C22×C414yes—
320C2×C42C22×C418yes—
321C2×C42C22×C418yes—
322C2×C42C22×C418yes—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C22×C418yes—
340#95C22×C418yes—
341#95C22×C418yes—
342#95C22×C418yes—
343#95C22×C418yes—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24C23×C412yes—
361C24C23×C412yes—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C22×Q812yes—
365C24C22×Q812yes—
366C24C2511yescharacteristic
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4C2×C4⋊C4116yes—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—84no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4C2×C4⋊C4116yescomplemented
463C22×C4C23×C414yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C23×C414yes—
468C22×C4C23×C414yes—
469C22×C4C23×C412yes—
470C22×C4C23×C414yes—
471C22×C4C23×C412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q814yes—
477C22×C4C22×Q812yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C42−1+414yes—
483C22×C42−1+414yes—
484C4⋊C4—216no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C4⋊C4—84no—
494C4⋊C4—88no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C4⋊C4C2×C4⋊C4132yescomplemented
500C22⋊C4—216no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—84no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4C2×C4⋊C4132yes—
509C22⋊C4C2×C4⋊C4116yescomplemented
510C42—216no—
511C42—28no—
512C42—216no—
513C42—48no—
514C42—84no—
515C42—88no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23C22×C4⋊C414yes—
561C23C22×C4⋊C414yes—
562C23C24×C411yescharacteristic
563C23C23×D411yescharacteristic
564C23C23×Q811yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×2−1+412yes—
570C23C2×2−1+412yes—
571C23#58511yescharacteristic
572C23#58511yescharacteristic
573D4—416no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—416no—
623C2×C4—416no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—816no—
653C2×C4C22×C4⋊C418yes—
654C2×C4C22×C4⋊C418yes—
655C2×C4C22×C4⋊C414yes—
656C2×C4#51814yes—
657C2×C4#52018yes—
658C2×C4#52018yes—
659C2×C4#56014yes—
660C2×C4#56218yes—
661C2×C4#56314yes—
662C22—24no—
663C22—24no—
664C22—28no—
665C22—28no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—44no—
679C22—44no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22C23×C4⋊C411yescharacteristic
685C22C2×#51811yescharacteristic
686C22C2×#52012yes—
687C22C2×#52012yes—
688C22#342811yescharacteristic
689C22#349611yescharacteristic
690C22#349611yescharacteristic
691C22#349812yes—
692C22#349912yes—
693C22#350111yescharacteristic
694C22#350111yescharacteristic
695C22#350312yes—
696C22#350312yes—
697C22C2×#56011yescharacteristic
698C22C2×#56212yes—
699C22C2×#56311yescharacteristic
700C22#353612yes—
701C22#353711yescharacteristic
702C22#353811yescharacteristic
703C22#354612yes—
704C22#354712yes—
705C22#355412yes—
706C22#355512yes—
707C4—28no—
708C4—28no—
709C4—24no—
710C4—28no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—44no—
715C4—84no—
716C4—84no—
717C4—84no—
718C4—84no—
719C4#237618yescomplemented
720C2—44no—
721C2—44no—
722C2#3418111yescharacteristic
723C2#3418312yes—
724C2C2×#237611yescharacteristic
725C2#3436011yescharacteristic
726C2#3438312yes—
727C2#4548612yes—
728C2#4548711yescharacteristic
729C2#4548811yescharacteristic
730C2#4550112yes—
731C2#4550211yescharacteristic
732C2#4550311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14686C2semidirect
4. #14674C2semidirect
5. #14468C2semidirect
6. #14442C2semidirect
7. #14402C2semidirect
9. #13910C2semidirect
11. #13884C2semidirect
12. #13880C2semidirect
13. #13876C2semidirect
14. #13876C2semidirect
21. C4×#123C4semidirect
119. C22×C42C2×C4semidirect
133. #387C2×C4semidirect
138. #385C2×C4semidirect
143. #381C2×C4semidirect
152. C4×C4⋊C4C2×C4semidirect
154. C4×C22⋊C4C2×C4semidirect
225. #123C4⋊C4semidirect
462. C22×C4C2×C4⋊C4semidirect
499. C4⋊C4C2×C4⋊C4semidirect
509. C22⋊C4C2×C4⋊C4semidirect
719. C4#2376semidirect

Maximal subgroups

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

subgroupindex
C2×#23762
C2×C4×#1232
#146862
#146742
#144682
#144422
#144022
#144012
#139102
#139062
#138842
#138802
#138762
#138722