Finite Group Databasebeta

← #4380480

Subgroups — #4380480

Order 512.

3986 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic
2C2×#2817C211yesmaximal, characteristic, complemented
3C2×#2561C211yesmaximal, characteristic
4C2×#2621C211yesmaximal, characteristic, complemented
5#16364C214yesmaximal, complemented
6#15672C214yesmaximal, complemented
7#15671C212yesmaximal, complemented
8#15670C212yesmaximal, complemented
9#15465C212yesmaximal, complemented
10#15279C214yesmaximal, complemented
11#15265C212yesmaximal, complemented
12#14727C214yesmaximal
13#14292C211yesmaximal, characteristic, complemented
14#14138C211yesmaximal, characteristic, complemented
15#13868C211yesmaximal, characteristic
16C2×#543—28no—
17C2×#543—28no—
18C2×#543—28no—
19C22×C4⋊Q8C2211yescharacteristic, complemented
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2212yes—
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C23×C22⋊C4C2212yes—
30C23×C22⋊C4C2211yescharacteristic
31#2817—28no—
32#2621—28no—
33#2561—28no—
34C2×#400C2212yescomplemented
35C2×#400C2212yescomplemented
36C2×#398C2214yescomplemented
37C2×#397C2212yes—
38C2×#397C2214yes—
39C2×#397C2214yes—
40C2×#395C2212yescomplemented
41C2×#395C2212yescomplemented
42C2×#393C2212yes—
43C2×#391C2211yescharacteristic, complemented
44C2×#391C2211yescharacteristic, complemented
45C2×#391C2212yescomplemented
46C2×#388C2214yes—
47C2×#385C2214yes—
48C2×#382C2211yescharacteristic, complemented
49C2×#381C2212yes—
50C2×#381C2211yescharacteristic
51C2×#379C2212yes—
52C2×#379C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54#1446C2214yescomplemented
55#1446C2214yescomplemented
56#1446C2212yescomplemented
57#1445C2212yescomplemented
58#1444C2214yescomplemented
59#1442C2214yescomplemented
60#1442C2214yescomplemented
61#1442C2214yescomplemented
62#1441C2214yescomplemented
63#1441C2214yescomplemented
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1437C2212yes—
72#1437C2212yes—
73#1436C2212yescomplemented
74#1436C2212yescomplemented
75#1436C2214yescomplemented
76C4⋊#95C2212yescomplemented
77C22⋊#95C2212yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C26C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90#543—2128no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—44no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8—28no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8D418yes—
116C2×C22⋊Q8D418yes—
117C2×#117—24no—
118C2×#117—44no—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2314yes—
131C22×C22⋊C4—28no—
132C22×C22⋊C4—24no—
133C22×C22⋊C4—24no—
134C22×C22⋊C4—28no—
135C22×C22⋊C4—28no—
136C22×C22⋊C4D418yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2312yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2312yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2312yes—
151C22×C42C2311yescharacteristic
152C22×C42C2311yescharacteristic
153C22×C42C2312yes—
154#400—28no—
155#400—28no—
156#398—216no—
157#397—28no—
158#397—216no—
159#397—216no—
160#395—28no—
161#395—28no—
162#393—28no—
163#391—24no—
164#391—24no—
165#391—44no—
166#388—216no—
167#385—48no—
168#382—24no—
169#381—28no—
170#381—24no—
171#379—28no—
172#379—44no—
173C4×C22⋊C4—24no—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C25—22no—
196C25—22no—
197C25—28no—
198C25—44no—
199C25—42no—
200C25—42no—
201C25C2×D414yes—
202C25C2411yescharacteristic
203C25C2412yes—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—44no—
221C22×Q8—44no—
222C22×Q8C2×D414yes—
223C22×Q8C2×D414yes—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C4⋊Q8—416no—
257C22⋊Q8—264no—
258C22⋊Q8—264no—
259C22⋊Q8—264no—
260C22⋊Q8—264no—
261C22⋊Q8—264no—
262C22⋊Q8—232no—
263C22⋊Q8—232no—
264C22⋊Q8—416no—
265#117—416no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D418yes—
301C2×C4⋊C4C2×D414yes—
302C2×C22⋊C4—264no—
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—232no—
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—28no—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4C2×D418yes—
342C2×C22⋊C4C2×D418yes—
343C2×C22⋊C4C2×D418yes—
344C2×C22⋊C4C2×D418yes—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42—42no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373C24—216no—
374C24—216no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—28no—
387C24—216no—
388C24—48no—
389C24—48no—
390C24—82no—
391C24—82no—
392C24—84no—
393C24—84no—
394C24—84no—
395C24—88no—
396C24—88no—
397C24—88no—
398C24—88no—
399C24C22×D412yes—
400C24C22×D412yes—
401C24C22×D412yes—
402C24C22×D414yes—
403C242−1+418yes—
404C24C2511yescharacteristic
405C2×Q8—216no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—84no—
417C2×Q8—84no—
418C2×Q8—84no—
419C2×Q8—84no—
420C2×Q8—84no—
421C2×Q8—88no—
422C2×Q8—88no—
423C2×Q8—84no—
424C2×Q8—84no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
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—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
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—216no—
454C22×C4—216no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4—84no—
495C22×C4—84no—
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×D414yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4H(5,𝔽2)14yes—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—216no—
520C4⋊C4—48no—
521C4⋊C4—416no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—88no—
525C22⋊C4—216no—
526C22⋊C4—216no—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—216no—
531C22⋊C4—416no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—416no—
536C22⋊C4—88no—
537C22⋊C4—84no—
538C22⋊C4—88no—
539C42—28no—
540C42—44no—
541C42—84no—
542C23—232no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—28no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—24no—
555C23—28no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—28no—
560C23—28no—
561C23—28no—
562C23—28no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—88no—
578C23—88no—
579C23—832no—
580C23—816no—
581C23—816no—
582C23—832no—
583C23C2×#12014yes—
584C23#53618yes—
585C23C23×D411yescharacteristic
586C23C23×D411yescharacteristic
587C23C23×D411yescharacteristic
588C23C2×H(5,𝔽2)12yes—
589C23C2×2−1+414yes—
590C23#58512yes—
591C23#58514yes—
592Q8—88no—
593Q8—88no—
594Q8—88no—
595Q8—816no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—416no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
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—88no—
665C2×C4—816no—
666C2×C4—816no—
667C2×C4C2×#12014yes—
668C2×C4C2×#12014yes—
669C22—24no—
670C22—24no—
671C22—216no—
672C22—24no—
673C22—28no—
674C22—44no—
675C22—44no—
676C22—48no—
677C22—44no—
678C22—48no—
679C22—48no—
680C22—44no—
681C22—48no—
682C22—48no—
683C22—48no—
684C22—48no—
685C22—832no—
686C22C22×#12011yescharacteristic
687C22C2×#53612yes—
688C22#348411yescharacteristic
689C22#348511yescharacteristic
690C22#348512yes—
691C22#348612yes—
692C22#348612yes—
693C22#349612yes—
694C22#349612yes—
695C22#350012yes—
696C22#350012yes—
697C22#353514yes—
698C22#356114yes—
699C22#356414yes—
700C22#356414yes—
701C4—24no—
702C4—24no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—82no—
710C4—82no—
711C4—84no—
712C4—82no—
713C4—84no—
714C2—44no—
715C2—48no—
716C2#3524211yescharacteristic
717C2#3534711yescharacteristic
718C2#3561611yescharacteristic
719C2#4051114yes—
720C2#4635712yes—
721C2#4681512yes—
722C2#4785514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2817C2semidirect
4. C2×#2621C2semidirect
5. #16364C2semidirect
6. #15672C2semidirect
7. #15671C2semidirect
8. #15670C2semidirect
9. #15465C2semidirect
10. #15279C2semidirect
11. #15265C2semidirect
13. #14292C2semidirect
14. #14138C2semidirect
19. C22×C4⋊Q8C22semidirect
34. C2×#400C22semidirect
35. C2×#400C22semidirect
36. C2×#398C22semidirect
40. C2×#395C22semidirect
41. C2×#395C22semidirect
43. C2×#391C22semidirect
44. C2×#391C22semidirect
45. C2×#391C22semidirect
48. C2×#382C22semidirect
54. #1446C22semidirect
55. #1446C22semidirect
56. #1446C22semidirect
57. #1445C22semidirect
58. #1444C22semidirect
59. #1442C22semidirect
60. #1442C22semidirect
61. #1442C22semidirect
62. #1441C22semidirect
63. #1441C22semidirect
73. #1436C22semidirect
74. #1436C22semidirect
75. #1436C22semidirect
76. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#28172
C2×#25612
C2×#26212
#163642
#156722
#156712
#156702
#154652
#152792
#152652
#147272
#142922
#141382
#138682