Finite Group Databasebeta

← #628049

Subgroups — #628049

Order 512.

3232 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C2×#2618C211yesmaximal, characteristic
3C2×#2313C211yesmaximal, characteristic, complemented
4C22×#389C211yesmaximal, characteristic, complemented
5#15531C214yesmaximal, complemented
6#15517C214yesmaximal, complemented
7#15361C214yesmaximal, complemented
8#15346C214yesmaximal, complemented
9#15069C212yesmaximal, complemented
10#14912C212yesmaximal, complemented
11#14698C211yesmaximal, characteristic
12#14697C211yesmaximal, characteristic, complemented
13#14339C211yesmaximal, characteristic, complemented
14#14019C212yesmaximal, complemented
15#13996C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2618—28no—
25#2618—28no—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#396C2212yes—
29C2×#396C2214yes—
30C2×#396C2212yes—
31C2×#394C2212yes—
32C2×#394C2212yes—
33#2313—28no—
34C2×#390C2212yes—
35C2×#389—28no—
36C2×#389C2214yes—
37C2×#389C2214yes—
38C2×#389C2212yes—
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#387C2214yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#385C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C22×#95C2211yescharacteristic
54C22×#95C2211yescharacteristic
55#1445C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—44no—
92C2×#123—44no—
93C2×#123—44no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C42—24no—
136C22×C42—44no—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2314yes—
140#398—28no—
141#398—28no—
142#396—216no—
143#396—28no—
144#396—28no—
145#394—28no—
146#394—28no—
147#390—44no—
148#389—216no—
149#389—216no—
150#389—44no—
151#388—48no—
152#387—216no—
153#387—216no—
154#386—28no—
155#386—28no—
156#386—28no—
157#386—44no—
158#385—44no—
159#381—28no—
160#381—44no—
161#380—24no—
162#380—24no—
163#380D418yes—
164C4×C22⋊C4—28no—
165C2×#95—24no—
166C2×#95—24no—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—42no—
200C22×D4—42no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4#4114yes—
239C23×C4#4114yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253#123—432no—
254#123—432no—
255#121—416no—
256C2×C4⋊C4—232no—
257C2×C4⋊C4—232no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—416no—
293C2×C4⋊C4—416no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4C2×D418yes—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4C2×D418yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4C2×D418yes—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—28no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—416no—
345C2×C42—44no—
346C2×C42—44no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—24no—
351#95—24no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D418yes—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C2×#4112yes—
385C24C2×#4112yes—
386C24C2×#4112yes—
387C24H(5,𝔽2)12yes—
388C24H(5,𝔽2)12yes—
389C24C2511yescharacteristic
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4—84no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—28no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—84no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4#121116yes—
506C22×C4C22×D418yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C2×#4114yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4114yes—
517C22×C4H(5,𝔽2)14yes—
518C22×C42−1+414yes—
519C22×C42−1+414yes—
520C4⋊C4—232no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—48no—
524C4⋊C4—416no—
525C4⋊C4—416no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—84no—
534C4⋊C4—88no—
535C22⋊C4—216no—
536C22⋊C4—416no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—416no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C42—48no—
548C42—48no—
549C42—88no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23C2×#12114yes—
590C23#52514yes—
591C23#53514yes—
592C23#54614yes—
593C23#55014yes—
594C23#55314yes—
595C23#55514yes—
596C23C23×D411yescharacteristic
597C23C23×D411yescharacteristic
598C23C22×#4111yescharacteristic
599C23C22×#4111yescharacteristic
600C23C22×#4111yescharacteristic
601C23C2×H(5,𝔽2)11yescharacteristic
602C23C2×H(5,𝔽2)11yescharacteristic
603C23C2×H(5,𝔽2)12yes—
604C23C2×2−1+412yes—
605C23C2×2−1+412yes—
606C23#58512yes—
607D4—88no—
608D4—816no—
609C2×C4—232no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—416no—
640C2×C4—416no—
641C2×C4—42no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—88no—
691C2×C4—88no—
692C2×C4C2×#12114yes—
693C2×C4C2×#12114yes—
694C2×C4#53918yes—
695C22—24no—
696C22—24no—
697C22—28no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—28no—
702C22—24no—
703C22—28no—
704C22—28no—
705C22—82no—
706C22—82no—
707C22—84no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—84no—
712C22—82no—
713C22—82no—
714C22—84no—
715C22—84no—
716C22#241418yes—
717C22C22×#12111yescharacteristic
718C22C2×#52511yescharacteristic
719C22C2×#53511yescharacteristic
720C22C2×#53912yes—
721C22#346112yes—
722C22C2×#54611yescharacteristic
723C22C2×#55011yescharacteristic
724C22C2×#55311yescharacteristic
725C22C2×#55511yescharacteristic
726C22#348411yescharacteristic
727C22#348511yescharacteristic
728C22#348611yescharacteristic
729C22#348611yescharacteristic
730C22#349212yes—
731C22#349612yes—
732C22#349912yes—
733C22#350512yes—
734C22#350712yes—
735C22#350912yes—
736C22#353612yes—
737C22#353612yes—
738C22#355912yes—
739C22#355912yes—
740C4—24no—
741C4—24no—
742C4—28no—
743C4—44no—
744C4—44no—
745C4—44no—
746C4—44no—
747C4—48no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—84no—
753C2—28no—
754C2—162no—
755C2C2×#241411yescharacteristic
756C2#3498911yescharacteristic
757C2#3509211yescharacteristic
758C2#3535811yescharacteristic
759C2#3566711yescharacteristic
760C2#3670711yescharacteristic
761C2#3695811yescharacteristic
762C2#3938412yes—
763C2#3941312yes—
764C2#3950412yes—
765C2#3954912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2313C2semidirect
4. C22×#389C2semidirect
5. #15531C2semidirect
6. #15517C2semidirect
7. #15361C2semidirect
8. #15346C2semidirect
9. #15069C2semidirect
10. #14912C2semidirect
12. #14697C2semidirect
13. #14339C2semidirect
14. #14019C2semidirect
15. #13996C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C2×#23132
C22×#3892
#155312
#155172
#153612
#153462
#150692
#149122
#146982
#146972
#143392
#140192
#139962