Finite Group Databasebeta

← #559891

Subgroups — #559891

Order 512.

3302 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2377C212yesmaximal
2C2×C4×#121C211yesmaximal, characteristic
3#14686C211yesmaximal, characteristic, complemented
4#14685C211yesmaximal, characteristic, complemented
5#14675C212yesmaximal, complemented
6#14488C214yesmaximal, complemented
7#14465C212yesmaximal, complemented
8#14438C212yesmaximal, complemented
9#14407C214yesmaximal, complemented
10#13906C212yesmaximal, complemented
11#13884C212yesmaximal, complemented
12#13880C212yesmaximal, complemented
13#13876C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15#13872C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C4⋊C4C2212yes—
20C23×C42C2211yescharacteristic
21#2377—216no—
22C4×#121C4116yescomplemented
23C2×#390C2214yes—
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C2214yes—
27C2×#388C2212yes—
28C2×#387C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#384C2212yes—
35C2×#384C2212yes—
36C2×#381C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C2×#121C2×C418yes—
85C2×#121D418yes—
86C2×#121Q818yes—
87C2×C4×D4—44no—
88C2×C4×D4—44no—
89C2×C4×D4—44no—
90C2×C4×D4C2×C418yes—
91C2×C4×D4C2×C4116yescomplemented
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42—44no—
120C22×C42C2×C418yescomplemented
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#390—216no—
128#390C2×C418yes—
129#389C2×C418yescomplemented
130#388—216no—
131#388C2×C4116yescomplemented
132#387—216no—
133#386—216no—
134#386—216no—
135#386C2×C4116yescomplemented
136#385—216no—
137#385—48no—
138#384—44no—
139#384—44no—
140#381—28no—
141#380—28no—
142#380—42no—
143#380—44no—
144#380—44no—
145#380—44no—
146#380—42no—
147#380—44no—
148#380—44no—
149#380C2×C418yescomplemented
150C4×C4⋊C4—44no—
151C4×C4⋊C4C2×C418yescomplemented
152C4×C22⋊C4C2×C4116yescomplemented
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4C22×C414yes—
184C22×D4C22×C418yes—
185C22×D4C2×D414yes—
186C22×D4C2×D418yes—
187C22×D4C2×Q814yes—
188C22×D4C2×Q818yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C22×C418yes—
213C23×C4C22×C414yes—
214C23×C4C22×C414yes—
215C23×C4C22×C414yes—
216C23×C4C2×D414yes—
217C23×C4C2×Q814yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#121C4⋊C4164yescomplemented
232C4×D4—416no—
233C4×D4—432no—
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—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—432no—
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—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×Q814yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4C22×C418yes—
301C2×C22⋊C4C22×C418yes—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C2×D418yes—
304C2×C22⋊C4C2×Q818yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—416no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C22×C414yes—
316C2×C42C22×C418yes—
317C2×C42C22×C414yes—
318C2×C42C22×C414yes—
319C2×C42C22×C414yes—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C22×C418yes—
340#95C22×C418yes—
341#95C22×C418yes—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24C23×C414yes—
363C24C23×C412yes—
364C24C22×D412yes—
365C24C22×D414yes—
366C24C22×Q814yes—
367C24C22×Q812yes—
368C24C2511yescharacteristic
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4C2×C4⋊C4116yes—
380C2×D4C2×C4⋊C4132yescomplemented
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
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—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—28no—
416C22×C4—216no—
417C22×C4—28no—
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—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4C2×C4⋊C4116yescomplemented
468C22×C4C23×C412yes—
469C22×C4C23×C414yes—
470C22×C4C23×C414yes—
471C22×C4C23×C412yes—
472C22×C4C23×C412yes—
473C22×C4C23×C414yes—
474C22×C4C23×C412yes—
475C22×C4C23×C412yes—
476C22×C4C23×C412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×Q812yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+412yes—
488C22×C42−1+412yes—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C4⋊C4C2×C4⋊C4116yescomplemented
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4C2×C4⋊C4132yescomplemented
513C42—28no—
514C42—28no—
515C42—216no—
516C42—44no—
517C42—44no—
518C42—88no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C22×C4⋊C418yes—
576C23C22×C4⋊C414yes—
577C23C24×C411yescharacteristic
578C23C23×D411yescharacteristic
579C23C23×Q811yescharacteristic
580C23C2×H(5,𝔽2)12yes—
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23C2×2−1+411yescharacteristic
586C23#58512yes—
587C23#58512yes—
588D4—416no—
589D4—432no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—416no—
621C2×C4—416no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—416no—
635C2×C4—432no—
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—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—832no—
665C2×C4C22×C4⋊C414yes—
666C2×C4C22×C4⋊C418yes—
667C2×C4C22×C4⋊C414yes—
668C2×C4#51814yes—
669C2×C4#51814yes—
670C2×C4#52014yes—
671C2×C4#52014yes—
672C2×C4#56018yes—
673C2×C4#56218yes—
674C22—24no—
675C22—24no—
676C22—28no—
677C22—28no—
678C22—28no—
679C22—24no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22C23×C4⋊C411yescharacteristic
701C22C2×#51811yescharacteristic
702C22C2×#51811yescharacteristic
703C22C2×#52011yescharacteristic
704C22C2×#52011yescharacteristic
705C22#342811yescharacteristic
706C22#342811yescharacteristic
707C22#349612yes—
708C22#349612yes—
709C22#349811yescharacteristic
710C22#349911yescharacteristic
711C22#350112yes—
712C22#350112yes—
713C22#350311yescharacteristic
714C22#350311yescharacteristic
715C22C2×#56012yes—
716C22C2×#56212yes—
717C22#354112yes—
718C22#354212yes—
719C22#354612yes—
720C22#354712yes—
721C22#356212yes—
722C22#356512yes—
723C4—24no—
724C4—24no—
725C4—24no—
726C4—28no—
727C4—42no—
728C4—44no—
729C4—42no—
730C4—44no—
731C4—84no—
732C4—84no—
733C4—84no—
734C4—84no—
735C4#237718yescomplemented
736C2—44no—
737C2—48no—
738C2#3418111yescharacteristic
739C2#3418111yescharacteristic
740C2#3418311yescharacteristic
741C2C2×#237711yescharacteristic
742C2#3436211yescharacteristic
743C2#3436711yescharacteristic
744C2#3438511yescharacteristic
745C2#4548212yes—
746C2#4548312yes—
747C2#4549712yes—
748C2#4549812yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
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. #14685C2semidirect
5. #14675C2semidirect
6. #14488C2semidirect
7. #14465C2semidirect
8. #14438C2semidirect
9. #14407C2semidirect
10. #13906C2semidirect
11. #13884C2semidirect
12. #13880C2semidirect
13. #13876C2semidirect
14. #13872C2semidirect
15. #13872C2semidirect
22. C4×#121C4semidirect
91. C2×C4×D4C2×C4semidirect
120. C22×C42C2×C4semidirect
129. #389C2×C4semidirect
131. #388C2×C4semidirect
135. #386C2×C4semidirect
149. #380C2×C4semidirect
151. C4×C4⋊C4C2×C4semidirect
152. C4×C22⋊C4C2×C4semidirect
231. #121C4⋊C4semidirect
380. C2×D4C2×C4⋊C4semidirect
467. C22×C4C2×C4⋊C4semidirect
501. C4⋊C4C2×C4⋊C4semidirect
512. C22⋊C4C2×C4⋊C4semidirect
735. C4#2377semidirect

Maximal subgroups

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

subgroupindex
C2×#23772
C2×C4×#1212
#146862
#146852
#146752
#144882
#144652
#144382
#144072
#139062
#138842
#138802
#138762
#138722