Finite Group Databasebeta

← #530972

Subgroups — #530972

Order 512.

2974 conjugacy classes of subgroups.

The subgroup lattice has 751 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#14500C214yesmaximal, complemented
4#14452C212yesmaximal, complemented
5#14418C212yesmaximal
6#14278C212yesmaximal, complemented
7#14007C214yesmaximal, complemented
8#13906C212yesmaximal, complemented
9#13884C212yesmaximal, complemented
10#13876C212yesmaximal, complemented
11#13871C212yesmaximal, complemented
12#13871C212yesmaximal, complemented
13#13801C212yesmaximal, complemented
14#13792C212yesmaximal, complemented
15C4×#380C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×#117C2212yes—
20C23×C42C2211yescharacteristic
21C4×#121C4116yescomplemented
22C2×#390C2214yes—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388C2214yes—
26C2×#388C2212yes—
27C2×#387C2214yes—
28C2×#386C2212yes—
29C2×#384C2212yes—
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68#1434C2212yes—
69#1434C2212yes—
70#1434C2214yes—
71#1434C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2214yes—
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#121—28no—
83C2×#121C2×C418yes—
84C2×C4×D4—44no—
85C2×C4×D4—44no—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4C2×C4116yes—
89C2×C4×D4C2×C418yes—
90C2×#117—28no—
91C2×#117—44no—
92C2×#117—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42—42no—
116C22×C42—42no—
117C22×C42C2×C418yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#390—48no—
128#390C2×C418yes—
129#389C2×C418yes—
130#388—48no—
131#388C2×C4116yes—
132#387—216no—
133#386C2×C4116yes—
134#384—44no—
135#381—44no—
136#380—28no—
137#380—28no—
138#380—42no—
139#380—44no—
140#380—44no—
141#380—42no—
142#380C2×C418yes—
143C4×C4⋊C4—44no—
144C4×C4⋊C4—42no—
145C4×C4⋊C4—42no—
146C4×C4⋊C4C2×C418yes—
147C4×C22⋊C4—44no—
148C4×C22⋊C4C2×C4116yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
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—24no—
178C22×D4—42no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4C22×C418yes—
184C22×D4C22×C414yes—
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—42no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4C22×C418yes—
206C23×C4C22×C414yes—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#121—232no—
223C4×D4—416no—
224C4×D4—432no—
225#117—432no—
226C2×C4⋊C4—24no—
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—28no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4#4118yes—
253C2×C4⋊C4#4118yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C22×C418yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C22×C418yes—
289C2×C22⋊C4C22×C418yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—416no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—41no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42—41no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42C22×C418yes—
314C2×C42C22×C414yes—
315C2×C42C22×C414yes—
316C2×C42C22×C414yes—
317C2×C42C22×C414yes—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—42no—
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—21no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24C23×C414yes—
363C24C23×C412yes—
364C24H(5,𝔽2)12yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C2×D4—216no—
368C2×D4—216no—
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—216no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—28no—
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—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C23×C414yes—
470C22×C4C23×C414yes—
471C22×C4C23×C412yes—
472C22×C4C23×C414yes—
473C22×C4C23×C412yes—
474C22×C4C23×C412yes—
475C22×C4C23×C412yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C4⋊C4—216no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C4⋊C4#117116yescomplemented
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—88no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—28no—
509C42—28no—
510C42—216no—
511C42—44no—
512C42—44no—
513C42—416no—
514C42—88no—
515C42—84no—
516C42—84no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
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#52014yes—
575C23C24×C411yescharacteristic
576C23C22×#4112yes—
577C23C22×#4112yes—
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)12yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×2−1+411yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23#58512yes—
586D4—432no—
587D4—416no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
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—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—416no—
623C2×C4—416no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—416no—
645C2×C4—432no—
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—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4C2×#11718yes—
668C2×C4C2×#11714yes—
669C2×C4C4×#4114yes—
670C2×C4#51814yes—
671C2×C4#51814yes—
672C2×C4#52014yes—
673C2×C4#53818yes—
674C2×C4#54018yes—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—44no—
682C22—44no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22C22×#11711yescharacteristic
702C22C2×C4×#4111yescharacteristic
703C22C2×#51811yescharacteristic
704C22C2×#51811yescharacteristic
705C22C2×#52011yescharacteristic
706C22C2×#52011yescharacteristic
707C22#342811yescharacteristic
708C22C2×#53812yes—
709C22C2×#54012yes—
710C22#348212yes—
711C22#348312yes—
712C22#350612yes—
713C22#350612yes—
714C22#351312yes—
715C22#351612yes—
716C22#351912yes—
717C22#352012yes—
718C22#353611yescharacteristic
719C22#353611yescharacteristic
720C22#353711yescharacteristic
721C22#353811yescharacteristic
722C22#354612yes—
723C22#354712yes—
724C4—24no—
725C4—24no—
726C4—28no—
727C4—42no—
728C4—44no—
729C4—42no—
730C4—44no—
731C4—48no—
732C4—42no—
733C4—84no—
734C4—84no—
735C4—84no—
736C4#233318yes—
737C2—44no—
738C2—84no—
739C2C2×#233311yescharacteristic
740C2#3412911yescharacteristic
741C2#3418611yescharacteristic
742C2#3419411yescharacteristic
743C2#3424111yescharacteristic
744C2#3436011yescharacteristic
745C2#3436711yescharacteristic
746C2#4006812yes—
747C2#4007012yes—
748C2#4011612yes—
749C2#4011812yes—
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. #14500C2semidirect
4. #14452C2semidirect
6. #14278C2semidirect
7. #14007C2semidirect
8. #13906C2semidirect
9. #13884C2semidirect
10. #13876C2semidirect
11. #13871C2semidirect
12. #13871C2semidirect
13. #13801C2semidirect
14. #13792C2semidirect
15. C4×#380C2semidirect
21. C4×#121C4semidirect
495. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#146852
#145002
#144522
#144182
#142782
#140072
#139062
#138842
#138762
#138712
#138012
#137922
C4×#3802