Finite Group Databasebeta

← #530959

Subgroups — #530959

Order 512.

3098 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic, complemented
3#14484C214yesmaximal, complemented
4#14453C212yesmaximal, complemented
5#14417C212yesmaximal
6#14286C212yesmaximal, complemented
7#13984C214yesmaximal, complemented
8#13905C212yesmaximal, complemented
9#13885C212yesmaximal, complemented
10#13875C212yesmaximal, complemented
11#13872C212yesmaximal, complemented
12#13872C212yesmaximal, complemented
13C22⋊C4×C4⋊C4C212yesmaximal, complemented
14#13791C212yesmaximal, complemented
15C4×#380C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C4⋊C4C2212yes—
20C23×C42C2211yescharacteristic
21C4×#121C4116yescomplemented
22C2×#390C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388C2212yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#382C2212yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66#1434C2214yes—
67#1434C2212yes—
68#1434C2212yes—
69#1434C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2311yescharacteristic
83C2×#121—28no—
84C2×#121C2×C418yes—
85C2×C4×D4—44no—
86C2×C4×D4—44no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×C4×D4C2×C418yes—
90C2×C4×D4C2×C4116yescomplemented
91C22×C4⋊C4—28no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42—42no—
115C22×C42—42no—
116C22×C42C2×C418yescomplemented
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#390C2×C418yes—
126#389C2×C418yescomplemented
127#388C2×C4116yescomplemented
128#386—216no—
129#386—48no—
130#386C2×C4116yescomplemented
131#382—44no—
132#381—44no—
133#380—28no—
134#380—28no—
135#380—42no—
136#380—44no—
137#380—44no—
138#380—42no—
139#380C2×C418yescomplemented
140C4×C4⋊C4—44no—
141C4×C4⋊C4—42no—
142C4×C4⋊C4—42no—
143C4×C4⋊C4C2×C418yescomplemented
144C4×C22⋊C4—48no—
145C4×C22⋊C4—44no—
146C4×C22⋊C4C2×C4116yescomplemented
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4C22×C418yes—
183C22×D4C22×C414yes—
184C22×D4#4118yes—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—42no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—41no—
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×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#121—232no—
230C4×D4—416no—
231C4×D4—432no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
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—432no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4#4118yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C22×C418yes—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C418yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—416no—
300C2×C42—41no—
301C2×C42—41no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42C22×C414yes—
312C2×C42C22×C418yes—
313C2×C42C22×C414yes—
314C2×C42C22×C414yes—
315C2×C42C22×C414yes—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C22×C418yes—
338#95C22×C418yes—
339#95C22×C418yes—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C23×C414yes—
361C24C23×C412yes—
362C24C2×#4114yes—
363C24C2×#4114yes—
364C24H(5,𝔽2)12yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C2×D4—216no—
368C2×D4—216no—
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#117116yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—216no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—28no—
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—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4C23×C412yes—
477C22×C4C23×C412yes—
478C22×C4C23×C414yes—
479C22×C4C23×C414yes—
480C22×C4C23×C414yes—
481C22×C4C23×C412yes—
482C22×C4C23×C412yes—
483C22×C4C23×C412yes—
484C22×C4C23×C412yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+412yes—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—28no—
514C42—28no—
515C42—216no—
516C42—44no—
517C42—44no—
518C42—84no—
519C42—84no—
520C42—88no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
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—
575C23—42no—
576C23—42no—
577C23C2×#11718yes—
578C23#52014yes—
579C23C24×C411yescharacteristic
580C23C22×#4112yes—
581C23C22×#4112yes—
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)12yes—
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×2−1+411yescharacteristic
588C23C2×2−1+411yescharacteristic
589C23#58512yes—
590D4—416no—
591D4—432no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—416no—
624C2×C4—416no—
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—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—416no—
650C2×C4—432no—
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—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—832no—
673C2×C4C2×#11714yes—
674C2×C4C4×#4114yes—
675C2×C4#51814yes—
676C2×C4#51814yes—
677C2×C4#52014yes—
678C2×C4#53818yes—
679C2×C4#54018yes—
680C22—24no—
681C22—28no—
682C22—28no—
683C22—24no—
684C22—28no—
685C22—24no—
686C22—42no—
687C22—44no—
688C22—44no—
689C22—42no—
690C22—44no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22C22×#11711yescharacteristic
707C22C2×C4×#4111yescharacteristic
708C22C2×#51811yescharacteristic
709C22C2×#51811yescharacteristic
710C22C2×#52011yescharacteristic
711C22C2×#52011yescharacteristic
712C22#342811yescharacteristic
713C22C2×#53812yes—
714C22C2×#54012yes—
715C22#348112yes—
716C22#348212yes—
717C22#350512yes—
718C22#350512yes—
719C22#351312yes—
720C22#351612yes—
721C22#351912yes—
722C22#352012yes—
723C22#353611yescharacteristic
724C22#353611yescharacteristic
725C22#353711yescharacteristic
726C22#353811yescharacteristic
727C22#354612yes—
728C22#354712yes—
729C4—24no—
730C4—24no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—44no—
735C4—48no—
736C4—42no—
737C4—44no—
738C4—84no—
739C4—84no—
740C4—84no—
741C4#233318yes—
742C2—44no—
743C2—48no—
744C2C2×#233311yescharacteristic
745C2#3412711yescharacteristic
746C2#3418511yescharacteristic
747C2#3419411yescharacteristic
748C2#3424111yescharacteristic
749C2#3436011yescharacteristic
750C2#3436711yescharacteristic
751C2#4004412yes—
752C2#4004612yes—
753C2#4009212yes—
754C2#4009412yes—
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
2. #14685C2semidirect
3. #14484C2semidirect
4. #14453C2semidirect
6. #14286C2semidirect
7. #13984C2semidirect
8. #13905C2semidirect
9. #13885C2semidirect
10. #13875C2semidirect
11. #13872C2semidirect
12. #13872C2semidirect
13. C22⋊C4×C4⋊C4C2semidirect
14. #13791C2semidirect
15. C4×#380C2semidirect
21. C4×#121C4semidirect
90. C2×C4×D4C2×C4semidirect
116. C22×C42C2×C4semidirect
126. #389C2×C4semidirect
127. #388C2×C4semidirect
130. #386C2×C4semidirect
139. #380C2×C4semidirect
143. C4×C4⋊C4C2×C4semidirect
146. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#146852
#144842
#144532
#144172
#142862
#139842
#139052
#138852
#138752
#138722
C22⋊C4×C4⋊C42
#137912
C4×#3802