Finite Group Databasebeta

← #4435096

Subgroups — #4435096

Order 512.

2877 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2821C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3C2×#2603C211yesmaximal, characteristic
4C2×#2660C211yesmaximal, characteristic, complemented
5C2×#2664C211yesmaximal, characteristic, complemented
6C2×#2633C211yesmaximal, characteristic, complemented
7#17062C212yesmaximal, complemented
8#16812C212yesmaximal, complemented
9#16692C212yesmaximal, complemented
10#16510C212yesmaximal, complemented
11#15513C214yesmaximal, complemented
12#15505C214yesmaximal, complemented
13#15173C212yesmaximal, complemented
14#15136C212yesmaximal, complemented
15#15020C212yesmaximal, complemented
16#14928C212yesmaximal
17#13806C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×#124C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2821—28no—
24#2664—28no—
25#2660—28no—
26#2633—28no—
27#2603—28no—
28#2547—28no—
29C2×#400C2211yescharacteristic
30C2×#400C2212yes—
31C2×#399C2212yes—
32C2×#399C2211yescharacteristic
33C2×#399C2212yes—
34C2×#399C2214yes—
35C2×#398C2212yes—
36C2×#398C2212yes—
37C2×#398C2211yescharacteristic
38C2×#398C2211yescharacteristic
39C2×#398C2214yes—
40C2×#398C2212yes—
41C2×#397C2212yes—
42C2×#397C2211yescharacteristic
43C2×#397C2212yes—
44C2×#397C2212yes—
45C2×#397C2211yescharacteristic
46C2×#397C2212yes—
47C2×#396C2212yes—
48C2×#395C2212yes—
49C2×#395C2212yes—
50C2×#395C2212yes—
51C2×#394C2212yes—
52C2×#394C2211yescharacteristic
53C2×#393C2212yes—
54C2×#392C2211yescharacteristic
55C2×#391C2212yes—
56C2×#390C2212yes—
57C2×#390C2214yes—
58C2×#389C2212yes—
59C2×#388C2212yes—
60C2×#388C2212yes—
61C2×#386C2212yes—
62C2×#385C2214yes—
63C2×#385C2212yes—
64C2×#384C2212yes—
65C2×#383C2211yescharacteristic
66C2×#382C2212yes—
67C2×#380C2214yes—
68C2×C4×C4⋊C4C2211yescharacteristic
69C2×#376C2211yescharacteristic
70#1445C2214yes—
71#1445C2214yes—
72#1444C2214yes—
73#1444C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1438C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C4×#95C2212yes—
88C4×#95C2212yes—
89C23×Q8C2312yes—
90C23×D4C2311yescharacteristic
91C23×D4C2312yes—
92C24×C4C2312yes—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×#126—24no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#124—28no—
103C2×#124—48no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C42C2312yes—
141C22×C42C2314yes—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#400—28no—
145#400—24no—
146#399—28no—
147#399—28no—
148#399—24no—
149#399—48no—
150#398—28no—
151#398—28no—
152#398—24no—
153#398—28no—
154#398—216no—
155#398—24no—
156#397—28no—
157#397—24no—
158#397—28no—
159#397—24no—
160#397—28no—
161#397—28no—
162#396—28no—
163#395—28no—
164#395—28no—
165#395—28no—
166#394—24no—
167#394—28no—
168#393—28no—
169#392—24no—
170#391—28no—
171#390—28no—
172#390—48no—
173#389—28no—
174#388—28no—
175#388—28no—
176#386—28no—
177#385—28no—
178#385—216no—
179#384—28no—
180#383D418yescomplemented
181#382—28no—
182#380—216no—
183C4×C4⋊C4D418yescomplemented
184#376D418yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2311yescharacteristic
202C2×#95C2312yes—
203C2×#95C2311yescharacteristic
204C2×#95C2311yescharacteristic
205C2×#95C2312yes—
206C2×#95C2311yescharacteristic
207C2×#95C2312yes—
208C2×#95C2312yes—
209C25C2411yescharacteristic
210C25C2412yes—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—28no—
223C22×D4—42no—
224C22×D4—44no—
225C22×D4—42no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C4⋊Q8—416no—
247#126—416no—
248#125—416no—
249#124—432no—
250C4×D4—416no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C2×D418yes—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—42no—
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⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C42—28no—
345C2×C42—28no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42C2×D418yes—
350C2×C42C2×D418yes—
351C2×C42C2×D414yes—
352C2×C42C2×D414yes—
353C2×C42C2×D414yes—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—24no—
359#95—28no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—44no—
369#95—44no—
370#95C2×D418yes—
371#95C2×D414yes—
372#95C2×D414yes—
373#95C2×D414yes—
374#95C2×D414yes—
375#95C2×D418yes—
376#95C2×D418yes—
377#95C2×D418yes—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—24no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—21no—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24H(5,𝔽2)14yes—
397C24C2511yescharacteristic
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—88no—
402C2×Q8—88no—
403C2×D4—28no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—44no—
407C2×D4—48no—
408C2×D4—416no—
409C2×D4—44no—
410C2×D4—48no—
411C2×D4—84no—
412C2×D4—88no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—416no—
478C22×C4—48no—
479C22×C4—84no—
480C22×C4—88no—
481C22×C4—84no—
482C22×C4C22×D414yes—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C42−1+418yes—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—88no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—88no—
513C4⋊C4—88no—
514C4⋊C4—88no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—84no—
526C22⋊C4—88no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—88no—
530C42—216no—
531C42—48no—
532C42—88no—
533C42#127116yes—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—28no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23C23×D411yescharacteristic
574C23C23×D411yescharacteristic
575C23C23×D411yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×2−1+414yes—
578C23#58512yes—
579C23#58512yes—
580C23#58512yes—
581Q8—816no—
582D4—416no—
583D4—816no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—816no—
660C2×C4C2×#12714yes—
661C2×C4C2×#12714yes—
662C2×C4C2×#12714yes—
663C2×C4#53618yes—
664C22—48no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—82no—
670C22—84no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—81no—
676C22—82no—
677C22—82no—
678C22—84no—
679C22—82no—
680C22—82no—
681C22—81no—
682C22—81no—
683C22C22×#12711yescharacteristic
684C22C2×#53612yes—
685C22#344912yes—
686C22#348411yescharacteristic
687C22#348711yescharacteristic
688C22#348812yes—
689C22#348812yes—
690C22#349912yes—
691C22#349912yes—
692C22#350012yes—
693C22#350011yescharacteristic
694C22#350011yescharacteristic
695C22#353514yes—
696C22#355914yes—
697C22#356414yes—
698C22#356514yes—
699C4—24no—
700C4—28no—
701C4—24no—
702C4—24no—
703C4—44no—
704C4—44no—
705C4—48no—
706C4—44no—
707C4—81no—
708C4—82no—
709C4—82no—
710C4—81no—
711C4—82no—
712C4—84no—
713C4—81no—
714C2—84no—
715C2—161no—
716C2#3531711yescharacteristic
717C2#3532911yescharacteristic
718C2#3556511yescharacteristic
719C2#4050412yes—
720C2#4193812yes—
721C2#4231212yes—
722C2#4294412yes—
723C2#5087014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2821C2semidirect
2. C2×#2547C2semidirect
4. C2×#2660C2semidirect
5. C2×#2664C2semidirect
6. C2×#2633C2semidirect
7. #17062C2semidirect
8. #16812C2semidirect
9. #16692C2semidirect
10. #16510C2semidirect
11. #15513C2semidirect
12. #15505C2semidirect
13. #15173C2semidirect
14. #15136C2semidirect
15. #15020C2semidirect
17. #13806C2semidirect
180. #383D4semidirect
183. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
C2×#25472
C2×#26032
C2×#26602
C2×#26642
C2×#26332
#170622
#168122
#166922
#165102
#155132
#155052
#151732
#151362
#150202
#149282
#138062