Finite Group Databasebeta

← #4381098

Subgroups — #4381098

Order 512.

3228 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#2822C211yesmaximal, characteristic, complemented
3C2×#2611C211yesmaximal, characteristic
4C2×#2660C211yesmaximal, characteristic, complemented
5#16606C212yesmaximal
6#16601C212yesmaximal
7#15928C212yesmaximal
8#15927C212yesmaximal
9#15912C214yesmaximal
10#15881C212yesmaximal, complemented
11#15850C212yesmaximal, complemented
12#15848C214yesmaximal, complemented
13#14879C212yesmaximal, complemented
14#14876C212yesmaximal, complemented
15#14282C211yesmaximal, characteristic, complemented
16#14170C211yesmaximal, characteristic, complemented
17#13957C211yesmaximal, characteristic
18C2×#541—28no—
19C2×#541—28no—
20C2×#541—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126C2212yes—
23C22×#126C2212yes—
24C22×#125C2212yes—
25C22×#125C2211yescharacteristic
26C22×#125C2212yes—
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#2822—28no—
34#2660—28no—
35#2611—28no—
36C2×#400C2211yescharacteristic
37C2×#400C2212yes—
38C2×#400C2212yes—
39C2×#399C2211yescharacteristic
40C2×#398C2212yes—
41C2×#398C2212yes—
42C2×#398C2214yes—
43C2×#398C2214yes—
44C2×#398C2211yescharacteristic
45C2×#397C2211yescharacteristic
46C2×#395C2212yes—
47C2×#395C2212yes—
48C2×#389C2212yes—
49C2×#388C2212yes—
50C2×#387C2212yes—
51C2×#385C2212yes—
52C2×#384C2212yes—
53C2×#382C2212yes—
54C2×#382C2211yescharacteristic
55C2×#376C2211yescharacteristic
56C2×#376C2211yescharacteristic
57#1446C2212yes—
58#1446C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1440C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1437C2212yes—
78#1436C2214yes—
79#1436C2212yes—
80#1436C2212yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2214yes—
85#1436C2212yes—
86C4⋊#95C2214yes—
87#1434C2212yes—
88C22⋊#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4C2311yescharacteristic
92#541—2128no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8D418yes—
96C2×#126—28no—
97C2×#126—28no—
98C2×#126—28no—
99C2×#126—28no—
100C2×#126—28no—
101C2×#126—28no—
102C2×#125—28no—
103C2×#125—28no—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—28no—
107C2×#125—28no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—24no—
113C2×#125D418yes—
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×Q8—24no—
125C2×C4×Q8—24no—
126C2×#117—24no—
127C2×#117—24no—
128C2×#117D418yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2314yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2312yes—
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×C42C2312yes—
160C22×C42C2312yes—
161C22×C42C2312yes—
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164#400—28no—
165#400—28no—
166#400—24no—
167#399—24no—
168#398—28no—
169#398—28no—
170#398—216no—
171#398—24no—
172#398—216no—
173#397—24no—
174#395—28no—
175#395—28no—
176#389—28no—
177#388—28no—
178#387—28no—
179#385—28no—
180#384—28no—
181#382—28no—
182#382—24no—
183#376—24no—
184#376—24no—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2311yescharacteristic
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
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×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
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—264no—
248#125—264no—
249#125—232no—
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—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D418yes—
317C2×C4⋊C4C2×D418yes—
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—28no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—42no—
353C2×C42C2×D414yes—
354C2×C42C2×D414yes—
355C2×C42C2×D414yes—
356C2×C42C2×D414yes—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—24no—
361#95—22no—
362#95—28no—
363#95—28no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
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—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
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—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—28no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C42−1+418yes—
491C22×C42−1+414yes—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—84no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C4⋊C4—88no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C42—216no—
522C42—216no—
523C42—216no—
524C42—28no—
525C42—48no—
526C42#120116yes—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C23×D411yescharacteristic
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C2×2−1+414yes—
547C23C2×2−1+412yes—
548C23#58512yes—
549C23#58512yes—
550C23#58512yes—
551Q8—416no—
552Q8—416no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—416no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4C2×#12014yes—
636C2×C4C2×#12014yes—
637C2×C4C2×#12014yes—
638C22—44no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—82no—
644C22—82no—
645C22—84no—
646C22C22×#12011yescharacteristic
647C22#344911yescharacteristic
648C22#344911yescharacteristic
649C22#348512yes—
650C22#348612yes—
651C22#348712yes—
652C22#348912yes—
653C22#349712yes—
654C22#349912yes—
655C22#349911yescharacteristic
656C22#349912yes—
657C22#350011yescharacteristic
658C22#356314yes—
659C22#356314yes—
660C22#356414yes—
661C22#356514yes—
662C4—24no—
663C4—24no—
664C4—24no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—44no—
670C4—44no—
671C4—42no—
672C4—44no—
673C4—48no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—84no—
678C2—82no—
679C2#3514911yescharacteristic
680C2#3566011yescharacteristic
681C2#3566611yescharacteristic
682C2#4230612yes—
683C2#4231612yes—
684C2#4782214yes—
685C2#5154712yes—
686C2#5154812yes—
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×#2822C2semidirect
4. C2×#2660C2semidirect
10. #15881C2semidirect
11. #15850C2semidirect
12. #15848C2semidirect
13. #14879C2semidirect
14. #14876C2semidirect
15. #14282C2semidirect
16. #14170C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#28222
C2×#26112
C2×#26602
#166062
#166012
#159282
#159272
#159122
#158812
#158502
#158482
#148792
#148762
#142822
#141702
#139572