Finite Group Databasebeta

← #515784

Subgroups — #515784

Order 512.

3736 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2327C211yesmaximal, characteristic, complemented
3C2×#2323C212yesmaximal, complemented
4C2×C4×C22⋊Q8C212yesmaximal, complemented
5C2×#2290C211yesmaximal, characteristic
6#14231C214yesmaximal, complemented
7#14230C214yesmaximal, complemented
8#14155C211yesmaximal, characteristic, complemented
9#14084C214yesmaximal
10#14049C214yesmaximal, complemented
11#13997C212yesmaximal, complemented
12#13997C212yesmaximal, complemented
13#13954C211yesmaximal, characteristic, complemented
14#13905C211yesmaximal, characteristic
15#13750C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2542C4116yescomplemented
24C2×#399C2212yes—
25C2×#399C2212yes—
26#2327—28no—
27#2323—216no—
28C4×C22⋊Q8—216no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36#2290—28no—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#388C2212yes—
40C2×#387C2214yes—
41C2×#387C2214yes—
42C2×#387C2214yes—
43C2×#387C2214yes—
44C2×#387C2212yes—
45C2×#386C2212yes—
46C2×#385C2214yes—
47C2×#385C2212yes—
48C2×#384C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2214yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C22⋊C4C2214yes—
59C2×C4×C22⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4C2211yescharacteristic
61C2×#376C2211yescharacteristic
62#1445C2214yes—
63#1445C2212yes—
64#1445C2214yes—
65#1445C2214yes—
66#1444C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2212yes—
73#1437C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2214yes—
77C4⋊#95C2214yes—
78C4×#95C2214yes—
79C4×#95C2214yes—
80C23×Q8C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8C2×C418yescomplemented
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8C2×C4116yes—
93C2×C22⋊Q8C2×C4116yescomplemented
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4C2×C418yes—
100C2×#117—28no—
101C2×#117—24no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42—24no—
125C22×C42—28no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#399C2×C4116yes—
138#399C2×C4116yescomplemented
139#391—28no—
140#391—28no—
141#391—28no—
142#390C2×C4116yes—
143#389—28no—
144#389—28no—
145#389—216no—
146#388—216no—
147#388—216no—
148#388—28no—
149#387—216no—
150#387—216no—
151#387—216no—
152#387—28no—
153#387—216no—
154#386—28no—
155#385—28no—
156#385—216no—
157#384C2×C4116yescomplemented
158#382—24no—
159#380—28no—
160#380—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—216no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4C2×C418yescomplemented
166C4×C4⋊C4D4116yescomplemented
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—216no—
170#376D418yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C25C2411yescharacteristic
193C25C2411yescharacteristic
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8C22×C418yes—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4C22×C414yes—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C4⋊Q8—232no—
237C22⋊Q8—264no—
238C22⋊Q8—264no—
239C4×Q8—432no—
240C4×D4—232no—
241#117—416no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C4116yes—
274C2×C4⋊C4C22×C4116yes—
275C2×C4⋊C4C22×C418yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4#4118yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C22×C414yes—
305C2×C22⋊C4C22×C414yes—
306C2×C22⋊C4C22×C4116yes—
307C2×C22⋊C4C22×C4116yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—416no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C22×C418yes—
329C2×C42C22×C414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D4116yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D418yes—
335C2×C42#4114yes—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95C22×C418yes—
354#95C2×D418yes—
355#95C2×D418yes—
356#95C2×D4116yes—
357C24—22no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24C23×C412yes—
373C24C23×C412yes—
374C242−1+414yes—
375C24C2511yescharacteristic
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—48no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4C23×C418yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C23×C418yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D418yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C42−1+414yes—
486C4⋊C4—28no—
487C4⋊C4—232no—
488C4⋊C4—232no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4C4×D4132yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—232no—
502C22⋊C4—232no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—88no—
508C42—28no—
509C42—48no—
510C42—416no—
511C42—48no—
512C42—44no—
513C42—44no—
514C42—416no—
515C42—84no—
516C42—88no—
517C42C4×D4116yes—
518C42#127116yes—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23#51914yes—
561C23C24×C411yescharacteristic
562C23C23×D412yes—
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4112yes—
566C23C2×2−1+412yes—
567C23C2×2−1+412yes—
568C23#58512yes—
569C23#58512yes—
570Q8—432no—
571D4—416no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—432no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—416no—
632C2×C4—416no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4C2×C4×D414yes—
648C2×C4C2×C4×D414yes—
649C2×C4C2×C4×D4116yes—
650C2×C4C2×C4×D414yes—
651C2×C4C2×C4×D418yes—
652C2×C4C2×#12718yes—
653C2×C4C2×#12714yes—
654C2×C4#53214yes—
655C2×C4#53218yes—
656C22—24no—
657C22—28no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—42no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22C22×C4×D411yescharacteristic
675C22C22×C4×D412yes—
676C22C2×#51911yescharacteristic
677C22#342811yescharacteristic
678C22C4×2−1+412yes—
679C22C22×#12711yescharacteristic
680C22C2×#53211yescharacteristic
681C22C2×#53212yes—
682C22#348512yes—
683C22#348612yes—
684C22#348912yes—
685C22#348912yes—
686C22#348912yes—
687C22#348912yes—
688C22#349512yes—
689C22#350911yescharacteristic
690C22#351011yescharacteristic
691C22#351512yes—
692C22#351612yes—
693C22#352112yes—
694C22#352312yes—
695C4—28no—
696C4—24no—
697C4—24no—
698C4—216no—
699C4—44no—
700C4—44no—
701C4—48no—
702C4—42no—
703C4—42no—
704C4—48no—
705C4—82no—
706C4—84no—
707C4C4×#12718yes—
708C2—44no—
709C2—82no—
710C2C2×C4×#12711yescharacteristic
711C2#3420311yescharacteristic
712C2#3426811yescharacteristic
713C2#3428912yes—
714C2#3429112yes—
715C2#3563412yes—
716C2#3563512yes—
717C2#3563612yes—
718C2#3563712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2327C2semidirect
3. C2×#2323C2semidirect
4. C2×C4×C22⋊Q8C2semidirect
6. #14231C2semidirect
7. #14230C2semidirect
8. #14155C2semidirect
10. #14049C2semidirect
11. #13997C2semidirect
12. #13997C2semidirect
13. #13954C2semidirect
15. #13750C2semidirect
23. #2542C4semidirect
87. C2×C4⋊Q8C2×C4semidirect
93. C2×C22⋊Q8C2×C4semidirect
138. #399C2×C4semidirect
157. #384C2×C4semidirect
165. C4×C4⋊C4C2×C4semidirect
166. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23272
C2×#23232
C2×C4×C22⋊Q82
C2×#22902
#142312
#142302
#141552
#140842
#140492
#139972
#139542
#139052
#137502