Finite Group Databasebeta

← #618035

Subgroups — #618035

Order 512.

5314 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2583C211yesmaximal, characteristic, complemented
3C2×#2434C211yesmaximal, characteristic, complemented
4C2×#2626C211yesmaximal, characteristic
5#15466C214yesmaximal, complemented
6#15276C218yesmaximal, complemented
7#15265C214yesmaximal, complemented
8#15092C212yesmaximal, complemented
9#15092C212yesmaximal, complemented
10#14693C212yesmaximal
11#14262C212yesmaximal
12#13923C211yesmaximal, characteristic, complemented
13#13922C211yesmaximal, characteristic, complemented
14#13775C211yesmaximal, characteristic
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C4⋊Q8C2212yescomplemented
18C22×C22⋊Q8—28no—
19C22×C22⋊Q8—28no—
20C22×C22⋊Q8—28no—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2214yes—
23C22×C22⋊Q8C2214yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29#2626—28no—
30#2583—28no—
31#2434—28no—
32C2×#399C2212yes—
33C2×#398C2212yescomplemented
34C2×#396C2214yes—
35C2×#395C2212yescomplemented
36C2×#395C2214yescomplemented
37C2×#393C2212yes—
38C2×#391C2212yescomplemented
39C2×#391C2214yescomplemented
40C2×#389C2214yescomplemented
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#385C2218yes—
47C2×#380C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1446C2214yescomplemented
53#1444C2218yescomplemented
54#1444C2218yescomplemented
55#1441C2214yescomplemented
56#1441C2218yescomplemented
57#1441C2218yescomplemented
58#1439C2218yes—
59#1439C2218yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2218yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—48no—
73C23×Q8D418yes—
74C23×Q8C2312yes—
75C23×Q8C2312yes—
76C23×Q8C2311yescharacteristic
77C24×C4—24no—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—48no—
81C24×C4C2311yescharacteristic
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—48no—
90C2×C22⋊Q8—2128no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—216no—
97C2×C22⋊Q8—216no—
98C2×C22⋊Q8—216no—
99C2×C22⋊Q8—216no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4D418yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—24no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4—24no—
125C22×C22⋊C4—24no—
126C22×C22⋊C4D418yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2318yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139#399—28no—
140#398—28no—
141#396—216no—
142#395—216no—
143#395—28no—
144#393—28no—
145#391—28no—
146#391—48no—
147#389—216no—
148#385—28no—
149#385—28no—
150#385—28no—
151#385—28no—
152#385—28no—
153#385—416no—
154#380—24no—
155#379—28no—
156#379Q8116yescomplemented
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C25—22no—
173C25—22no—
174C25—24no—
175C25—44no—
176C25—44no—
177C25C2×D414yes—
178C25C2×Q818yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C22×Q8—232no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—28no—
190C22×Q8—28no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—432no—
195C22×Q8—432no—
196C22×Q8—432no—
197C22×Q8C2×D418yes—
198C22×Q8C2×D418yes—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—28no—
202C23×C4—232no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—216no—
215C23×C4—48no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—432no—
222C23×C4—432no—
223C23×C4—432no—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C4⋊Q8—432no—
245C22⋊Q8—2128no—
246C22⋊Q8—2128no—
247C22⋊Q8—416no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—232no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—232no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—232no—
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—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4C2×D418yes—
307C2×C22⋊C4C2×Q818yes—
308C2×C22⋊C4C2×Q818yes—
309C2×C22⋊C4C2×Q8132yescomplemented
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—44no—
313C2×C42—44no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—216no—
323#95—48no—
324#95—48no—
325#95—48no—
326#95—48no—
327C24—24no—
328C24—24no—
329C24—28no—
330C24—28no—
331C24—216no—
332C24—216no—
333C24—22no—
334C24—22no—
335C24—28no—
336C24—24no—
337C24—28no—
338C24—48no—
339C24—48no—
340C24—432no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—432no—
346C24C4⋊Q8116yescomplemented
347C24C22×D414yes—
348C24C22×D412yes—
349C24C22×D412yes—
350C24C22×D412yes—
351C24C22×Q818yes—
352C24C22×Q814yes—
353C242−1+414yes—
354C24C2511yescharacteristic
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—216no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—416no—
362C2×Q8—416no—
363C2×Q8—416no—
364C2×Q8—416no—
365C2×Q8—4128no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—84no—
369C2×Q8—84no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—216no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—432no—
426C22×C4—432no—
427C22×C4—4128no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4#120116yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×Q818yes—
451C22×C4C22×Q818yes—
452C22×C4H(5,𝔽2)18yes—
453C4⋊C4—232no—
454C4⋊C4—232no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—84no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C22⋊C4—232no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—432no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—48no—
476C42—84no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—216no—
484C23—216no—
485C23—24no—
486C23—24no—
487C23—232no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—216no—
493C23—216no—
494C23—432no—
495C23—416no—
496C23—416no—
497C23—432no—
498C23—464no—
499C23—464no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—416no—
505C23C2×#12014yes—
506C23C2×C4⋊Q818yes—
507C23C2×C4⋊Q814yes—
508C23#53614yes—
509C23D4×Q818yes—
510C23D4×Q818yes—
511C23C23×D411yescharacteristic
512C23C23×D411yescharacteristic
513C23C23×D411yescharacteristic
514C23C23×Q812yes—
515C23C2×H(5,𝔽2)14yes—
516C23C2×2−1+412yes—
517C23#58514yes—
518Q8—432no—
519Q8—432no—
520Q8—88no—
521C2×C4—232no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—216no—
545C2×C4—216no—
546C2×C4—416no—
547C2×C4—464no—
548C2×C4—416no—
549C2×C4—416no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—416no—
578C2×C4C2×#12014yes—
579C2×C4C2×#12014yes—
580C2×C4#53418yes—
581C22—28no—
582C22—24no—
583C22—28no—
584C22—24no—
585C22—216no—
586C22—216no—
587C22—216no—
588C22—28no—
589C22—464no—
590C22—44no—
591C22—48no—
592C22—44no—
593C22—416no—
594C22—416no—
595C22#243418yes—
596C22C22×#12011yescharacteristic
597C22C22×C4⋊Q811yescharacteristic
598C22C2×#53412yes—
599C22C2×#53611yescharacteristic
600C22#344912yes—
601C22C2×D4×Q812yes—
602C22C2×D4×Q812yes—
603C22#348412yes—
604C22#348412yes—
605C22#348512yes—
606C22#348712yes—
607C22#350118yes—
608C22#354118yes—
609C4—28no—
610C4—24no—
611C4—24no—
612C4—44no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—48no—
617C4—48no—
618C4—82no—
619C4—84no—
620C4—82no—
621C2—28no—
622C2—48no—
623C2C2×#243411yescharacteristic
624C2#3498011yescharacteristic
625C2#3514911yescharacteristic
626C2#3539112yes—
627C2#3670512yes—
628C2#3958814yes—
629C2#3959514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2583C2semidirect
3. C2×#2434C2semidirect
5. #15466C2semidirect
6. #15276C2semidirect
7. #15265C2semidirect
8. #15092C2semidirect
9. #15092C2semidirect
12. #13923C2semidirect
13. #13922C2semidirect
17. C22×C4⋊Q8C22semidirect
33. C2×#398C22semidirect
35. C2×#395C22semidirect
36. C2×#395C22semidirect
38. C2×#391C22semidirect
39. C2×#391C22semidirect
40. C2×#389C22semidirect
52. #1446C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
62. #1436C22semidirect
156. #379Q8semidirect
309. C2×C22⋊C4C2×Q8semidirect
346. C24C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#25832
C2×#24342
C2×#26262
#154662
#152762
#152652
#150922
#146932
#142622
#139232
#139222
#137752