Finite Group Databasebeta

← #4380495

Subgroups — #4380495

Order 512.

4326 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2432C211yesmaximal, characteristic
4C2×#2616C211yesmaximal, characteristic
5#16251C214yesmaximal, complemented
6#15996C214yesmaximal, complemented
7#15882C212yesmaximal, complemented
8#15849C212yesmaximal, complemented
9#15762C212yesmaximal, complemented
10#15761C212yesmaximal, complemented
11#15265C214yesmaximal, complemented
12#14888C214yesmaximal, complemented
13#14660C211yesmaximal, characteristic, complemented
14#14292C211yesmaximal, characteristic, complemented
15#13923C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C2×#544—28no—
18C2×#544—28no—
19C2×#544—28no—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2212yes—
24C22×#125C2212yes—
25C22×#124C2211yescharacteristic
26C22×#123C2212yes—
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×C22⋊Q8C2212yes—
31C22×#117C2212yes—
32C23×C4⋊C4C2211yescharacteristic
33#2616—28no—
34#2432—28no—
35#2406—28no—
36C2×#395C2212yes—
37C2×#395C2212yes—
38C2×#395C2214yes—
39C2×#394C2212yes—
40C2×#394C2212yes—
41C2×#394C2214yes—
42C2×#394C2214yes—
43C2×#393C2214yes—
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#386C2212yes—
47C2×#385C2214yes—
48C2×#385C2214yes—
49C2×#381C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2212yes—
74#1436C2212yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2212yes—
78#1436C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22×#41—24no—
82C22×#41—24no—
83C22×#41—24no—
84C22×#41—44no—
85C22×#41—44no—
86C23×Q8C2311yescharacteristic
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×D4C2312yes—
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96#544—2128no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—44no—
105C2×#125—28no—
106C2×#125—28no—
107C2×#125—28no—
108C2×#125—28no—
109C2×#125—28no—
110C2×#125—28no—
111C2×#124—24no—
112C2×#124—44no—
113C2×#123—28no—
114C2×#123—44no—
115C2×#123—44no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—28no—
122C2×C22⋊Q8—28no—
123C2×C22⋊Q8—28no—
124C2×C22⋊Q8—44no—
125C2×C22⋊Q8—44no—
126C2×C22⋊Q8D418yes—
127C2×C22⋊Q8D418yes—
128C2×#117—28no—
129C2×#117—28no—
130C2×#117—28no—
131C22×C4⋊C4—24no—
132C22×C4⋊C4—24no—
133C22×C4⋊C4D418yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2314yes—
140C22×C4⋊C4C2314yes—
141C22×C4⋊C4C2314yes—
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2314yes—
151C22×C22⋊C4C2314yes—
152C22×C22⋊C4C2314yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2312yes—
158C22×C22⋊C4C2314yes—
159C22×C22⋊C4C2312yes—
160C22×C42C2312yes—
161C22×C42C2312yes—
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164#395—28no—
165#395—28no—
166#395—48no—
167#394—28no—
168#394—28no—
169#394—216no—
170#394—216no—
171#393—48no—
172#391—24no—
173#390—24no—
174#386—28no—
175#385—216no—
176#385—216no—
177#381—24no—
178#380—24no—
179#380—28no—
180C4×C22⋊C4—24no—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C25C2412yes—
201C25C2411yescharacteristic
202C2×#41—432no—
203C2×#41—432no—
204C2×#41—416no—
205C2×#41—416no—
206C2×#41—416no—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—24no—
214C22×Q8—24no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—24no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2×D414yes—
226C22×Q8C2×D414yes—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—42no—
237C22×D4—42no—
238C22×D4—42no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—28no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4C2×D414yes—
256C23×C4C2×D414yes—
257C23×C4C2×D414yes—
258C23×C4C2×D414yes—
259C23×C4C2×D414yes—
260C23×C4C2414yes—
261C23×C4C2414yes—
262C23×C4C2412yes—
263C23×C4C2412yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C23×C4C2412yes—
273C4⋊Q8—264no—
274C4⋊Q8—264no—
275C4⋊Q8—416no—
276#125—264no—
277#125—264no—
278#124—416no—
279#123—432no—
280C22⋊Q8—264no—
281C22⋊Q8—232no—
282C22⋊Q8—232no—
283C22⋊Q8—432no—
284#117—264no—
285C2×C4⋊C4—232no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D418yes—
326C2×C4⋊C4C2×D418yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D418yes—
330C2×C4⋊C4C2×D418yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—28no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—28no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—44no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4—42no—
362C2×C22⋊C4C2×D418yes—
363C2×C22⋊C4C2×D418yes—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—42no—
373C2×C42—42no—
374#95—24no—
375#95—28no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—28no—
380#95—28no—
381#95—28no—
382#95—28no—
383#95—28no—
384#95—44no—
385#95—44no—
386#95—44no—
387#95—44no—
388#95—44no—
389#95—44no—
390#95—44no—
391#95—44no—
392#95—44no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—24no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—24no—
408C24C22×D412yes—
409C24C22×D412yes—
410C24C22×D412yes—
411C24H(5,𝔽2)14yes—
412C24C2511yescharacteristic
413#41—4128no—
414C2×Q8—28no—
415C2×Q8—216no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—84no—
428C2×Q8—84no—
429C2×Q8—84no—
430C2×Q8—84no—
431C2×Q8—88no—
432C2×Q8—88no—
433C2×Q8—84no—
434C2×D4—216no—
435C2×D4—28no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—48no—
440C2×D4—44no—
441C2×D4—48no—
442C2×D4—48no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—44no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—216no—
457C22×C4—216no—
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—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—28no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—48no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—48no—
519C22×C4—84no—
520C22×C4—82no—
521C22×C4—82no—
522C22×C4#120116yes—
523C22×C4C22×D414yes—
524C22×C4C22×D414yes—
525C22×C4C22×D414yes—
526C22×C4C22×D414yes—
527C22×C4C22×D414yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C22×D414yes—
535C22×C4H(5,𝔽2)14yes—
536C22×C42−1+414yes—
537C22×C42−1+418yes—
538C4⋊C4—216no—
539C4⋊C4—216no—
540C4⋊C4—216no—
541C4⋊C4—216no—
542C4⋊C4—216no—
543C4⋊C4—216no—
544C4⋊C4—216no—
545C4⋊C4—28no—
546C4⋊C4—28no—
547C4⋊C4—216no—
548C4⋊C4—216no—
549C4⋊C4—416no—
550C4⋊C4—416no—
551C4⋊C4—416no—
552C4⋊C4—48no—
553C22⋊C4—216no—
554C22⋊C4—216no—
555C22⋊C4—416no—
556C22⋊C4—48no—
557C22⋊C4—416no—
558C22⋊C4—416no—
559C22⋊C4—416no—
560C22⋊C4—48no—
561C22⋊C4—416no—
562C22⋊C4—88no—
563C22⋊C4—84no—
564C42—216no—
565C42—216no—
566C42—44no—
567C42—44no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—22no—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—28no—
583C23—28no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—42no—
589C23—44no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23C2×#12014yes—
607C23C23×D411yescharacteristic
608C23C23×D411yescharacteristic
609C23C23×D411yescharacteristic
610C23C2×H(5,𝔽2)12yes—
611C23C2×H(5,𝔽2)12yes—
612C23C2×2−1+414yes—
613C23C2×2−1+412yes—
614C23#58512yes—
615Q8—416no—
616Q8—88no—
617Q8—88no—
618Q8—816no—
619D4—416no—
620D4—432no—
621C2×C4—232no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—28no—
651C2×C4—416no—
652C2×C4—432no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—88no—
693C2×C4C2×#12014yes—
694C2×C4C2×#12014yes—
695C2×C4#53618yes—
696C22—24no—
697C22—24no—
698C22—24no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—48no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—48no—
708C22—48no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—84no—
713C22—82no—
714C22—84no—
715C22C22×#12011yescharacteristic
716C22C2×#53612yes—
717C22#348412yes—
718C22#348411yescharacteristic
719C22#348411yescharacteristic
720C22#348412yes—
721C22#348612yes—
722C22#348612yes—
723C22#348912yes—
724C22#349712yes—
725C22#349911yescharacteristic
726C22#350011yescharacteristic
727C22#354414yes—
728C22#354714yes—
729C22#355214yes—
730C22#355914yes—
731C4—24no—
732C4—24no—
733C4—28no—
734C4—44no—
735C4—44no—
736C4—44no—
737C4—44no—
738C4—44no—
739C4—44no—
740C4—44no—
741C4—84no—
742C4—82no—
743C4—84no—
744C2—44no—
745C2—84no—
746C2#3524211yescharacteristic
747C2#3534611yescharacteristic
748C2#3566011yescharacteristic
749C2#4242314yes—
750C2#4642612yes—
751C2#4642912yes—
752C2#4954812yes—
753C2#4954912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#544C2semidirect
5. #16251C2semidirect
6. #15996C2semidirect
7. #15882C2semidirect
8. #15849C2semidirect
9. #15762C2semidirect
10. #15761C2semidirect
11. #15265C2semidirect
12. #14888C2semidirect
13. #14660C2semidirect
14. #14292C2semidirect
15. #13923C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#24062
C2×#24322
C2×#26162
#162512
#159962
#158822
#158492
#157622
#157612
#152652
#148882
#146602
#142922
#139232