Finite Group Databasebeta

← #535401

Subgroups — #535401

Order 512.

3518 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#2328C211yesmaximal, characteristic
4#14415C214yesmaximal, complemented
5#14391C214yesmaximal, complemented
6#14141C212yesmaximal, complemented
7#14064C214yesmaximal, complemented
8#14030C212yesmaximal, complemented
9#14025C212yesmaximal
10#14019C214yesmaximal, complemented
11#13941C211yesmaximal, characteristic, complemented
12#13941C211yesmaximal, characteristic, complemented
13#13854C211yesmaximal, characteristic, complemented
14#13820C212yesmaximal, complemented
15#13779C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C23×C4⋊C4C2212yes—
22#2354—28no—
23D4×C4⋊C4—28no—
24#2328C4116yescomplemented
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yes—
48C2×#376C2211yescharacteristic
49#1443C2214yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1440C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1438C2214yes—
62#1438C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2314yes—
81C24×C4C2314yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×#127—24no—
85C2×#127—24no—
86C2×#127C2×C418yes—
87C2×C4×D4—28no—
88C2×C4×D4—28no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—44no—
94C2×C4×D4C2×C418yes—
95C2×C4×D4C2×C418yescomplemented
96C2×#117—28no—
97C2×#117—48no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—48no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129#390—28no—
130#390—28no—
131#390—44no—
132#390C2×C418yes—
133#390C2×C418yescomplemented
134#389—28no—
135#388—216no—
136#388—48no—
137#386—48no—
138#386—48no—
139#386C2×C4132yescomplemented
140#386C2×C4132yescomplemented
141#385—216no—
142#384D418yescomplemented
143#382—24no—
144#382C2×C418yescomplemented
145#380—28no—
146#380—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C22⋊C4—48no—
152#376C2×C418yescomplemented
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2412yes—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4C22×C414yes—
196C22×D4C22×C414yes—
197C22×D4C22×C4116yes—
198C22×D4#4114yes—
199C22×D4#4114yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C22×C418yes—
216C23×C4C22×C418yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#127—232no—
231C4×D4—232no—
232C4×D4—232no—
233C4×D4—432no—
234#117—432no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—432no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C22×C4116yes—
295C2×C22⋊C4C22×C4116yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C22×C414yes—
313C2×C42C22×C414yes—
314C2×C42C22×C414yes—
315C2×C42C2×D414yes—
316C2×C42#4114yes—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C22×C418yes—
339#95C22×C418yes—
340#95C22×C4116yes—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24C23×C414yes—
360C24C23×C414yes—
361C24C2×#4114yes—
362C24C2×#4114yes—
363C24C2511yescharacteristic
364C2×D4—232no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4#117116yescomplemented
385C22×C4—22no—
386C22×C4—22no—
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—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4C23×C414yes—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C414yes—
469C22×C4C23×C412yes—
470C22×C4C23×C412yes—
471C22×C4C23×C412yes—
472C22×C4C22×D418yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C42−1+412yes—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C42—28no—
508C42—28no—
509C42—416no—
510C42—48no—
511C42—48no—
512C42—88no—
513C42C4×D4116yes—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23C2×#11718yes—
560C23C24×C411yescharacteristic
561C23C23×D411yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4111yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58512yes—
571D4—232no—
572D4—416no—
573D4—832no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—432no—
602C2×C4—432no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—832no—
651C2×C4C2×#11714yes—
652C2×C4C2×C4×D414yes—
653C2×C4C2×C4×D414yes—
654C2×C4C2×C4×D414yes—
655C2×C4#52014yes—
656C2×C4#54614yes—
657C2×C4#54814yes—
658C2×C4#55214yes—
659C2×C4#55314yes—
660C22—28no—
661C22—216no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—48no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—44no—
675C22—44no—
676C22—82no—
677C22—82no—
678C22—84no—
679C22—84no—
680C22—82no—
681C22—82no—
682C22C22×#11711yescharacteristic
683C22C22×C4×D411yescharacteristic
684C22C2×#52011yescharacteristic
685C22C4×H(5,𝔽2)12yes—
686C22C4×H(5,𝔽2)12yes—
687C22C2×#54611yescharacteristic
688C22C2×#54811yescharacteristic
689C22C2×#55211yescharacteristic
690C22C2×#55311yescharacteristic
691C22#348411yescharacteristic
692C22#348411yescharacteristic
693C22#348711yescharacteristic
694C22#348711yescharacteristic
695C22#350612yes—
696C22#350612yes—
697C22#351312yes—
698C22#351412yes—
699C22#351912yes—
700C22#352112yes—
701C22#354312yes—
702C22#354712yes—
703C22#355912yes—
704C22#356212yes—
705C4—24no—
706C4—24no—
707C4—44no—
708C4—48no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—82no—
715C4—84no—
716C4—82no—
717C4#235218yes—
718C2—48no—
719C2—84no—
720C2C2×#235211yescharacteristic
721C2#3426112yes—
722C2#3426511yescharacteristic
723C2#3426511yescharacteristic
724C2#3443112yes—
725C2#4493611yescharacteristic
726C2#4493811yescharacteristic
727C2#4496812yes—
728C2#4497012yes—
729C2#4526911yescharacteristic
730C2#4527111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2354C2semidirect
2. C2×D4×C4⋊C4C2semidirect
4. #14415C2semidirect
5. #14391C2semidirect
6. #14141C2semidirect
7. #14064C2semidirect
8. #14030C2semidirect
10. #14019C2semidirect
11. #13941C2semidirect
12. #13941C2semidirect
13. #13854C2semidirect
14. #13820C2semidirect
15. #13779C2semidirect
24. #2328C4semidirect
95. C2×C4×D4C2×C4semidirect
133. #390C2×C4semidirect
139. #386C2×C4semidirect
140. #386C2×C4semidirect
142. #384D4semidirect
144. #382C2×C4semidirect
152. #376C2×C4semidirect
384. C2×D4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×D4×C4⋊C42
C2×#23282
#144152
#143912
#141412
#140642
#140302
#140252
#140192
#139412
#138542
#138202
#137792