Finite Group Databasebeta

← #515673

Subgroups — #515673

Order 512.

5142 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#519C211yesmaximal, characteristic, complemented
2C2×#2409C211yesmaximal, characteristic
3C2×C4×C22⋊Q8C212yesmaximal, complemented
4C2×#2290C211yesmaximal, characteristic
5#14174C214yesmaximal, complemented
6#14172C214yesmaximal, complemented
7#14129C211yesmaximal, characteristic, complemented
8#14064C214yesmaximal
9#14063C214yesmaximal, complemented
10#14017C212yesmaximal, complemented
11#13969C212yesmaximal, complemented
12#13922C211yesmaximal, characteristic, complemented
13#13871C211yesmaximal, characteristic
14#13786C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C2×#519—216no—
19C2×#519—28no—
20C22×C4×Q8C2214yes—
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2212yes—
23C22×C4×D4C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C22×#117C2214yes—
26C22×#117C2211yescharacteristic
27C23×C42C2211yescharacteristic
28#2409C4116yescomplemented
29C2×#395C2212yes—
30C2×#393C2212yes—
31C4×C22⋊Q8—216no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#389C2212yes—
36#2290—28no—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#385C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2214yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55#1446C2214yes—
56#1444C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1440C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2214yes—
78#1434C2214yes—
79C22⋊#95C2214yes—
80C23×Q8—28no—
81C23×Q8—28no—
82C23×Q8—24no—
83C23×Q8—24no—
84C23×Q8C2×C418yescomplemented
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8C2×C4116yes—
95#519—2128no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—216no—
99C2×C4×Q8—216no—
100C2×C4×Q8—216no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—28no—
104C2×C4×Q8D4116yescomplemented
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4C2×C418yes—
108C2×#117—24no—
109C2×#117—28no—
110C2×#117—216no—
111C2×#117—216no—
112C2×#117—216no—
113C2×#117—28no—
114C2×#117D418yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C42—24no—
136C22×C42—28no—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2314yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2314yes—
144C22×C42C2311yescharacteristic
145C22×C42C2314yes—
146C22×C42C2311yescharacteristic
147#395C2×C4116yescomplemented
148#393C2×C4116yes—
149#391—28no—
150#391—28no—
151#391C2×C4116yescomplemented
152#389—28no—
153#388—28no—
154#387—28no—
155#387—216no—
156#386—28no—
157#386C2×C4116yes—
158#385—28no—
159#385—216no—
160#385C2×C4116yescomplemented
161#380—24no—
162#380—28no—
163#380C2×C418yescomplemented
164C4×C4⋊C4—24no—
165C4×C4⋊C4—216no—
166C4×C4⋊C4—28no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—216no—
170#376—24no—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C25C2411yescharacteristic
194C25C2411yescharacteristic
195C22×Q8—232no—
196C22×Q8—232no—
197C22×Q8—264no—
198C22×Q8—264no—
199C22×Q8—232no—
200C22×Q8—24no—
201C22×Q8—28no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—28no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8C22×C418yes—
209C22×Q8C22×C418yes—
210C22×Q8C2×D418yes—
211C22×Q8#4118yes—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—42no—
216C22×D4C22×C414yes—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C22×C414yes—
235C23×C4C22×C418yes—
236C23×C4C22×C414yes—
237C23×C4C2×D414yes—
238C23×C4#4114yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C22⋊Q8—264no—
253C4×Q8—264no—
254C4×Q8—2128no—
255C4×Q8—264no—
256C4×D4—232no—
257#117—232no—
258#117—2128no—
259#117—232no—
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—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C22×C418yes—
286C2×C4⋊C4C22×C4116yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D4116yes—
289C2×C4⋊C4C2×D418yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C22×C414yes—
313C2×C22⋊C4C22×C414yes—
314C2×C22⋊C4C22×C4116yes—
315C2×C22⋊C4C22×C418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—28no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—28no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—416no—
338C2×C42C22×C414yes—
339C2×C42C2×D4116yes—
340C2×C42C2×D414yes—
341C2×C42C2×D418yes—
342C2×C42C2×D414yes—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C22×C418yes—
363#95C22×C4116yes—
364#95C22×C4116yes—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24C23×C412yes—
379C24C23×C412yes—
380C24C22×D412yes—
381C24C2×#4112yes—
382C242−1+414yes—
383C24C2511yescharacteristic
384C2×Q8—216no—
385C2×Q8—2256no—
386C2×Q8—216no—
387C2×Q8—232no—
388C2×Q8—232no—
389C2×Q8—216no—
390C2×Q8—216no—
391C2×Q8—216no—
392C2×Q8—216no—
393C2×Q8—216no—
394C2×Q8—216no—
395C2×Q8—216no—
396C2×Q8C4×D4132yes—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—44no—
400C2×D4—48no—
401C2×D4—48no—
402C22×C4—216no—
403C22×C4—232no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—232no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—28no—
459C22×C4—216no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4C4×D4116yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C23×C418yes—
487C22×C4C23×C412yes—
488C22×C4C23×C418yes—
489C22×C4C23×C412yes—
490C22×C4C23×C412yes—
491C22×C4C22×D418yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D418yes—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C2×#4118yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4114yes—
503C22×C4C2×#4112yes—
504C22×C42−1+414yes—
505C4⋊C4—232no—
506C4⋊C4—28no—
507C4⋊C4—216no—
508C4⋊C4—232no—
509C4⋊C4—232no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—48no—
513C4⋊C4—416no—
514C4⋊C4—48no—
515C22⋊C4—216no—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—232no—
519C22⋊C4—216no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C42—216no—
526C42—232no—
527C42—28no—
528C42—232no—
529C42—28no—
530C42—416no—
531C42—48no—
532C42—44no—
533C42—44no—
534C42#120116yes—
535C23—24no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C2×C4×D414yes—
576C23#51914yescomplemented
577C23C24×C411yescharacteristic
578C23C23×D412yes—
579C23C23×D411yescharacteristic
580C23C22×#4111yescharacteristic
581C23C22×#4112yes—
582C23C2×2−1+412yes—
583C23C2×2−1+412yes—
584C23#58512yes—
585C23#58512yes—
586Q8—264no—
587Q8—264no—
588D4—416no—
589C2×C4—264no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—232no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—432no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—416no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4C2×C4×D4116yes—
667C2×C4C2×C4×D414yes—
668C2×C4C2×C4×D414yes—
669C2×C4C2×C4×D418yes—
670C2×C4C2×#12018yes—
671C2×C4C2×#12014yes—
672C2×C4#52514yes—
673C2×C4#52518yes—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—28no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22C22×C4×D411yescharacteristic
693C22C22×C4×D412yes—
694C22C2×#51911yescharacteristic
695C22#342811yescharacteristic
696C22C4×2−1+412yes—
697C22C22×#12011yescharacteristic
698C22C2×#52511yescharacteristic
699C22C2×#52512yes—
700C22#348612yes—
701C22#348712yes—
702C22#348912yes—
703C22#348912yes—
704C22#348912yes—
705C22#348912yes—
706C22#349512yes—
707C22#350911yescharacteristic
708C22#351011yescharacteristic
709C22#351612yes—
710C22#351812yes—
711C22#352312yes—
712C22#352412yes—
713C4—28no—
714C4—216no—
715C4—24no—
716C4—48no—
717C4—44no—
718C4—44no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—48no—
723C4—82no—
724C4—84no—
725C4C4×#12018yes—
726C2—44no—
727C2—44no—
728C2C2×C4×#12011yescharacteristic
729C2#3420311yescharacteristic
730C2#3426811yescharacteristic
731C2#3429112yes—
732C2#3429312yes—
733C2#3565912yes—
734C2#3566312yes—
735C2#3566412yes—
736C2#3566512yes—
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
1. C22×#519C2semidirect
3. C2×C4×C22⋊Q8C2semidirect
5. #14174C2semidirect
6. #14172C2semidirect
7. #14129C2semidirect
9. #14063C2semidirect
10. #14017C2semidirect
11. #13969C2semidirect
12. #13922C2semidirect
14. #13786C2semidirect
15. #13777C2semidirect
28. #2409C4semidirect
84. C23×Q8C2×C4semidirect
104. C2×C4×Q8D4semidirect
147. #395C2×C4semidirect
151. #391C2×C4semidirect
160. #385C2×C4semidirect
163. #380C2×C4semidirect
576. C23#519semidirect

Maximal subgroups

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

subgroupindex
C22×#5192
C2×#24092
C2×C4×C22⋊Q82
C2×#22902
#141742
#141722
#141292
#140642
#140632
#140172
#139692
#139222
#138712
#137862
#137772