Finite Group Databasebeta

← #4457608

Subgroups — #4457608

Order 512.

3244 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3C2×#2611C211yesmaximal, characteristic
4C2×#2634C211yesmaximal, characteristic, complemented
5#16683C212yesmaximal, complemented
6#16674C212yesmaximal
7#16673C212yesmaximal
8#16634C212yesmaximal, complemented
9#16607C212yesmaximal, complemented
10#16606C212yesmaximal
11#16603C212yesmaximal, complemented
12#16601C212yesmaximal
13#14931C214yesmaximal
14#14929C214yesmaximal, complemented
15#13961C211yesmaximal, characteristic, complemented
16#13938C211yesmaximal, characteristic
17#13850C211yesmaximal, characteristic, complemented
18C2×#541—28no—
19C2×#541—28no—
20C2×#541—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126C2212yes—
23C22×#126C2212yes—
24C22×#125C2211yescharacteristic
25C22×#125C2211yescharacteristic
26C22×#125C2212yes—
27C22×#125C2212yes—
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#2634—28no—
34#2611—28no—
35#2584—28no—
36C2×#400C2212yes—
37C2×#400C2212yes—
38C2×#400C2211yescharacteristic
39C2×#399C2211yescharacteristic
40C2×#398C2212yes—
41C2×#398C2212yes—
42C2×#398C2211yescharacteristic
43C2×#397C2211yescharacteristic
44C2×#395C2212yes—
45C2×#395C2212yes—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#389C2212yes—
49C2×#389C2214yes—
50C2×#388C2212yes—
51C2×#387C2214yes—
52C2×#387C2212yes—
53C2×#385C2212yes—
54C2×#383C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×#376C2211yescharacteristic
57#1446C2212yes—
58#1446C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1444C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2212yes—
76#1440C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1437C2212yes—
82#1436C2212yes—
83#1436C2214yes—
84#1434C2212yes—
85#1434C2212yes—
86#1434C2212yes—
87C22⋊#95C2212yes—
88C4×#95C2214yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4C2311yescharacteristic
92#541—2128no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×#126—28no—
97C2×#126—28no—
98C2×#126—28no—
99C2×#126—28no—
100C2×#126—28no—
101C2×#126—28no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—28no—
107C2×#125—28no—
108C2×#125—24no—
109C2×#125—28no—
110C2×#125—28no—
111C2×#125—24no—
112C2×#125—24no—
113C2×#125—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8D418yes—
125C2×C4×Q8D418yes—
126C2×#117—24no—
127C2×#117—24no—
128C2×#117D418yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2312yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2311yescharacteristic
155C22×C22⋊C4C2311yescharacteristic
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2314yes—
158C22×C42C2312yes—
159C22×C42C2311yescharacteristic
160C22×C42C2312yes—
161C22×C42C2312yes—
162C22×C42C2312yes—
163C22×C42C2311yescharacteristic
164#400—24no—
165#400—28no—
166#400—28no—
167#399—24no—
168#398—28no—
169#398—24no—
170#398—28no—
171#397—24no—
172#395—28no—
173#395—28no—
174#389—216no—
175#389—28no—
176#389—28no—
177#389—28no—
178#388—28no—
179#387—28no—
180#387—216no—
181#385—28no—
182#383—24no—
183C4×C4⋊C4—24no—
184#376—24no—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C25C2411yescharacteristic
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—42no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8C2×D414yes—
222C22×Q8C2×D414yes—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C2×D414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C4⋊Q8—232no—
245#126—264no—
246#126—264no—
247#125—232no—
248#125—264no—
249#125—264no—
250#125—416no—
251C22⋊Q8—232no—
252C22⋊Q8—232no—
253C4×Q8—232no—
254C4×Q8—232no—
255#117—232no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
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—24no—
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—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2×D418yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—42no—
350C2×C42—44no—
351C2×C42C2×D414yes—
352C2×C42C2×D418yes—
353C2×C42C2×D418yes—
354C2×C42C2×D414yes—
355C2×C42C2×D414yes—
356C2×C42C2×D414yes—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—28no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—44no—
378#95—44no—
379#95—44no—
380#95—44no—
381#95—44no—
382#95—44no—
383#95—44no—
384C24—21no—
385C24—21no—
386C24—22no—
387C24—24no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24C22×D412yes—
392C24C2511yescharacteristic
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—48no—
402C2×Q8—48no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
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×C42−1+418yes—
495C22×C42−1+414yes—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—416no—
512C4⋊C4—416no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—88no—
518C22⋊C4—216no—
519C22⋊C4—216no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—88no—
525C42—28no—
526C42—216no—
527C42—216no—
528C42—48no—
529C42—84no—
530C42#120116yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C2×2−1+414yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553C23#58512yes—
554C23#58512yes—
555Q8—416no—
556Q8—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—416no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
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—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4C2×#12014yes—
644C2×C4C2×#12014yes—
645C2×C4C2×#12014yes—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—84no—
652C22—82no—
653C22—82no—
654C22C22×#12011yescharacteristic
655C22#344911yescharacteristic
656C22#344911yescharacteristic
657C22#344912yes—
658C22#344912yes—
659C22#348612yes—
660C22#348912yes—
661C22#349712yes—
662C22#349912yes—
663C22#349912yes—
664C22#349911yescharacteristic
665C22#350011yescharacteristic
666C22#355314yes—
667C22#355514yes—
668C22#356314yes—
669C22#356414yes—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—48no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—82no—
686C2—82no—
687C2#3463211yescharacteristic
688C2#3566011yescharacteristic
689C2#3566611yescharacteristic
690C2#3719114yes—
691C2#4230512yes—
692C2#4230612yes—
693C2#4231312yes—
694C2#4238912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2584C2semidirect
4. C2×#2634C2semidirect
5. #16683C2semidirect
8. #16634C2semidirect
9. #16607C2semidirect
11. #16603C2semidirect
14. #14929C2semidirect
15. #13961C2semidirect
17. #13850C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#25842
C2×#26112
C2×#26342
#166832
#166742
#166732
#166342
#166072
#166062
#166032
#166012
#149312
#149292
#139612
#139382
#138502