Finite Group Databasebeta

← #4453276

Subgroups — #4453276

Order 512.

3930 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2579C212yesmaximal, complemented
3C22×#398C211yesmaximal, characteristic, complemented
4#16659C214yesmaximal, complemented
5#16605C214yesmaximal, complemented
6#16264C214yesmaximal, complemented
7#16257C214yesmaximal, complemented
8#14914C211yesmaximal, characteristic, complemented
9#14913C211yesmaximal, characteristic
10#14912C212yesmaximal, complemented
11#14912C212yesmaximal, complemented
12#14720C211yesmaximal, characteristic, complemented
13#14719C211yesmaximal, characteristic
14#14339C211yesmaximal, characteristic, complemented
15#13872C212yesmaximal, complemented
16C2×#539—216no—
17C2×#539—28no—
18C22×#126C2212yes—
19C22×#125C2212yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×#121C2211yescharacteristic
25C22×C4×D4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30#2579—216no—
31C2×#398—28no—
32C2×#398C2214yes—
33C2×#398C2214yes—
34C2×#398C2214yes—
35C2×#396C2214yes—
36C2×#395C2212yes—
37C2×#394C2212yes—
38C2×#387C2214yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#385C2214yes—
46C2×#381C2212yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1446C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2212yes—
74C4⋊#95C2214yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90#539—2128no—
91C2×#126—28no—
92C2×#126—28no—
93C2×#126—28no—
94C2×#125—28no—
95C2×#125—28no—
96C2×#125—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—28no—
108C2×#121—28no—
109C2×#121—24no—
110C2×C4×D4—28no—
111C2×C4×D4—28no—
112C2×C4×D4D4116yes—
113C22×C4⋊C4—28no—
114C22×C4⋊C4—28no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—44no—
119C22×C4⋊C4—44no—
120C22×C4⋊C4D418yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C42C2312yes—
148#398—216no—
149#398—216no—
150#398—216no—
151#396—216no—
152#395—44no—
153#394—44no—
154#387—216no—
155#386—28no—
156#386—28no—
157#386—44no—
158#386—44no—
159#385—216no—
160#385—28no—
161#385—28no—
162#381—44no—
163#380—28no—
164#380—48no—
165C2×#95—24no—
166C2×#95—24no—
167C2×#95—24no—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C25C2412yes—
190C25C2411yescharacteristic
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4C2×D418yes—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—44no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—44no—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D418yes—
240C23×C4C2×D414yes—
241C23×C4C2×D418yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255#126—264no—
256#125—264no—
257#123—264no—
258#123—264no—
259C22⋊Q8—264no—
260C22⋊Q8—232no—
261#121—232no—
262C4×D4—264no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—416no—
301C2×C4⋊C4—416no—
302C2×C4⋊C4—416no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D418yes—
308C2×C4⋊C4C2×D418yes—
309C2×C4⋊C4C2×D418yes—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4C2×D418yes—
342C2×C22⋊C4C2×D418yes—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42C2×D418yes—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—44no—
357#95—44no—
358#95—42no—
359#95—42no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—42no—
366#95—42no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24C22×D414yes—
382C24C22×D414yes—
383C24C22×D412yes—
384C24H(5,𝔽2)12yes—
385C242−1+414yes—
386C242−1+412yes—
387C24C2511yescharacteristic
388C2×Q8—28no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8—88no—
392C2×D4—216no—
393C2×D4—216no—
394C2×D4—216no—
395C2×D4—28no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—82no—
401C2×D4—82no—
402C2×D4—88no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
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—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—416no—
475C22×C4—44no—
476C22×C4—416no—
477C22×C4—44no—
478C22×C4—416no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4—88no—
496C22×C4—88no—
497C22×C4—88no—
498C22×C4—84no—
499C22×C4—84no—
500C22×C4#120116yes—
501C22×C4C22×D414yes—
502C22×C4C22×D414yes—
503C22×C4C22×D414yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D414yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C42−1+414yes—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—28no—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—416no—
521C4⋊C4—416no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—416no—
525C4⋊C4—88no—
526C4⋊C4—88no—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—88no—
536C22⋊C4—88no—
537C22⋊C4—88no—
538C42—216no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—28no—
544C23—28no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—28no—
555C23—28no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23C2×#12014yes—
575C23#53614yes—
576C23#53618yes—
577C23#53718yes—
578C23#53714yes—
579C23C23×D411yescharacteristic
580C23C23×D412yes—
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)12yes—
583C23C2×2−1+411yescharacteristic
584C23C2×2−1+412yes—
585C23C2×2−1+412yes—
586C23#58512yes—
587C23#58512yes—
588Q8—88no—
589D4—432no—
590D4—88no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—42no—
621C2×C4—42no—
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—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
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—816no—
668C2×C4—88no—
669C2×C4—88no—
670C2×C4—88no—
671C2×C4C2×#12018yes—
672C22—24no—
673C22—24no—
674C22—216no—
675C22—216no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—44no—
681C22—44no—
682C22—48no—
683C22—48no—
684C22—44no—
685C22—82no—
686C22—84no—
687C22—84no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22#261618yes—
692C22C22×#12011yescharacteristic
693C22C2×#53611yescharacteristic
694C22C2×#53612yes—
695C22C2×#53712yes—
696C22C2×#53711yescharacteristic
697C22#348411yescharacteristic
698C22#348511yescharacteristic
699C22#348611yescharacteristic
700C22#348711yescharacteristic
701C22#348914yes—
702C22#349614yes—
703C22#353412yes—
704C22#353512yes—
705C22#355214yes—
706C22#355314yes—
707C22#355912yes—
708C22#356312yes—
709C4—28no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—44no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—82no—
719C4—84no—
720C4—84no—
721C4—82no—
722C2—28no—
723C2—84no—
724C2C2×#261611yescharacteristic
725C2#3534711yescharacteristic
726C2#3565911yescharacteristic
727C2#3672211yescharacteristic
728C2#3696811yescharacteristic
729C2#3712011yescharacteristic
730C2#3723311yescharacteristic
731C2#4051614yes—
732C2#4051814yes—
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
2. C2×#2579C2semidirect
3. C22×#398C2semidirect
4. #16659C2semidirect
5. #16605C2semidirect
6. #16264C2semidirect
7. #16257C2semidirect
8. #14914C2semidirect
10. #14912C2semidirect
11. #14912C2semidirect
12. #14720C2semidirect
14. #14339C2semidirect
15. #13872C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#25792
C22×#3982
#166592
#166052
#162642
#162572
#149142
#149132
#149122
#147202
#147192
#143392
#138722