Finite Group Databasebeta

← #625871

Subgroups — #625871

Order 512.

3578 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic, complemented
3C2×#2504C211yesmaximal, characteristic, complemented
4C2×#2501C211yesmaximal, characteristic, complemented
5C2×#2619C211yesmaximal, characteristic
6C2×#2317C211yesmaximal, characteristic, complemented
7#15696C218yesmaximal, complemented
8#15370C218yesmaximal, complemented
9#15244C211yesmaximal, characteristic, complemented
10#15244C211yesmaximal, characteristic, complemented
11#15244C211yesmaximal, characteristic, complemented
12#15155C211yesmaximal, characteristic, complemented
13#14713C211yesmaximal, characteristic, complemented
14#14687C211yesmaximal, characteristic, complemented
15#14354C211yesmaximal, characteristic, complemented
16#14272C211yesmaximal, characteristic, complemented
17#13761C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C22×#123C2214yes—
20C22×#121C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2619—28no—
26#2504—28no—
27#2501—28no—
28#2500—28no—
29#2500—28no—
30C2×#399C2214yes—
31C2×#399C2218yes—
32C2×#397C2214yes—
33C2×#394C2214yes—
34C2×#394C2214yes—
35C2×#394C2214yes—
36C2×#392C2214yes—
37#2317—28no—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2218yes—
49C2×#389C2214yes—
50C2×#388C2214yes—
51C2×#388C2214yes—
52C2×#386C2214yes—
53C2×#385C2214yes—
54C2×#384C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C2×#382C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#382C2211yescharacteristic
64C2×#382C2211yescharacteristic
65C2×#382C2211yescharacteristic
66C2×#382C2211yescharacteristic
67C2×C4×C22⋊C4C2211yescharacteristic
68C2×C4×C22⋊C4C2211yescharacteristic
69C2×#376C2211yescharacteristic
70C2×C43C2211yescharacteristic
71#1444C2218yes—
72#1439C2218yes—
73#1439C2218yes—
74#1439C2218yes—
75#1438C2218yes—
76#1437C2214yes—
77#1436C2214yes—
78C4⋊#95C2218yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2312yes—
85C23×D4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#127—24no—
89C2×#127—24no—
90C2×#127—24no—
91C2×#123—216no—
92C2×#123—416no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117D418yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2312yes—
138C22×C42C2312yes—
139C22×C42C2314yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146#399—216no—
147#399—416no—
148#397—216no—
149#394—216no—
150#394—216no—
151#394—216no—
152#392—216no—
153#390—24no—
154#390—24no—
155#390—24no—
156#390—24no—
157#390—24no—
158#390—24no—
159#390—24no—
160#390—24no—
161#390—24no—
162#390—24no—
163#390—416no—
164#389—216no—
165#388—216no—
166#388—216no—
167#386—216no—
168#385—216no—
169#384—24no—
170#384—24no—
171#384—24no—
172#384—24no—
173#384—24no—
174#384—24no—
175#382—24no—
176#382—24no—
177#382—24no—
178#382—24no—
179#382—24no—
180#382—24no—
181#382—24no—
182C4×C22⋊C4D418yescomplemented
183C4×C22⋊C4D418yescomplemented
184#376—24no—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C43—24no—
198C25C2412yes—
199C25C2412yes—
200C25C2411yescharacteristic
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—28no—
214C22×D4—28no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—28no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—48no—
225C22×D4—48no—
226C22×D4—48no—
227C22×D4—48no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—48no—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253#127—416no—
254#123—464no—
255#121—416no—
256#121—416no—
257#117—232no—
258#117—416no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—216no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—416no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4C2×D418yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—216no—
313C2×C22⋊C4—216no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4C2×D418yes—
330C2×C22⋊C4C2×D418yes—
331C2×C22⋊C4C2×D418yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—28no—
346C2×C42—28no—
347C2×C42—44no—
348C2×C42C2×D414yes—
349C2×C42C2×D414yes—
350C2×C42C2×D414yes—
351C2×C42C2×D414yes—
352C2×C42C2×D414yes—
353C2×C42C2×D414yes—
354C2×C42#4114yes—
355C2×C42#4114yes—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—44no—
366#95C2×D418yes—
367#95C2×D418yes—
368C24—21no—
369C24—21no—
370C24—24no—
371C24—21no—
372C24—21no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—28no—
385C24—28no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—28no—
391C24—28no—
392C24C22×D412yes—
393C24C22×D412yes—
394C24C22×D412yes—
395C24C2511yescharacteristic
396C2×D4—432no—
397C2×D4—416no—
398C2×D4—416no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—416no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—816no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—88no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—416no—
479C22×C4—48no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4#127116yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)18yes—
500C22×C42−1+412yes—
501C22×C42−1+412yes—
502C4⋊C4—216no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—416no—
512C4⋊C4—88no—
513C4⋊C4—88no—
514C4⋊C4—88no—
515C4⋊C4—88no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C42—28no—
533C42—28no—
534C42—28no—
535C42—28no—
536C42—28no—
537C42—416no—
538C42—48no—
539C42#123116yescomplemented
540C23—22no—
541C23—24no—
542C23—22no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—48no—
575C23—48no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—48no—
586C23—48no—
587C23—48no—
588C23—48no—
589C23—48no—
590C23C2×#12714yes—
591C23C23×D411yescharacteristic
592C23C23×D411yescharacteristic
593C23C23×D411yescharacteristic
594C23C22×#4111yescharacteristic
595C23C22×#4111yescharacteristic
596C23C2×H(5,𝔽2)14yes—
597C23C2×2−1+411yescharacteristic
598C23C2×2−1+411yescharacteristic
599C23#58514yes—
600D4—832no—
601D4—816no—
602D4—816no—
603D4—88no—
604D4—88no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—232no—
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—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
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—816no—
665C2×C4C2×#12318yes—
666C2×C4C2×#12314yes—
667C2×C4C2×#12714yes—
668C2×C4C2×#12714yes—
669C2×C4#53518yes—
670C2×C4#53614yes—
671C2×C4#54714yes—
672C2×C4#54714yes—
673C2×C4#54714yes—
674C2×C4#54714yes—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—82no—
685C22—82no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22—84no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22C22×#12311yescharacteristic
703C22C22×#12711yescharacteristic
704C22C2×#53512yes—
705C22C2×#53611yescharacteristic
706C22#344912yes—
707C22C2×#54711yescharacteristic
708C22C2×#54711yescharacteristic
709C22C2×#54711yescharacteristic
710C22C2×#54711yescharacteristic
711C22#349414yes—
712C22#349812yes—
713C22#349812yes—
714C22#350012yes—
715C22#350012yes—
716C22#350614yes—
717C22#354714yes—
718C22#355914yes—
719C4—24no—
720C4—28no—
721C4—28no—
722C4—44no—
723C4—44no—
724C4—44no—
725C4—44no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4#250118yes—
733C2—44no—
734C2—162no—
735C2—162no—
736C2C2×#250111yescharacteristic
737C2#3527411yescharacteristic
738C2#3532911yescharacteristic
739C2#3553112yes—
740C2#3730412yes—
741C2#3786412yes—
742C2#3786912yes—
743C2#3948312yes—
744C2#3949112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2619C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2500C2semidirect
2. C2×#2500C2semidirect
3. C2×#2504C2semidirect
4. C2×#2501C2semidirect
6. C2×#2317C2semidirect
7. #15696C2semidirect
8. #15370C2semidirect
9. #15244C2semidirect
10. #15244C2semidirect
11. #15244C2semidirect
12. #15155C2semidirect
13. #14713C2semidirect
14. #14687C2semidirect
15. #14354C2semidirect
16. #14272C2semidirect
17. #13761C2semidirect
182. C4×C22⋊C4D4semidirect
183. C4×C22⋊C4D4semidirect
539. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#25042
C2×#25012
C2×#26192
C2×#23172
#156962
#153702
#152442
#151552
#147132
#146872
#143542
#142722
#137612