Finite Group Databasebeta

← #4453261

Subgroups — #4453261

Order 512.

5686 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2405C212yesmaximal, complemented
2C22×#392C211yesmaximal, characteristic
3C2×#2315C211yesmaximal, characteristic, complemented
4#16288C214yesmaximal, complemented
5#16282C214yesmaximal, complemented
6#16256C218yesmaximal, complemented
7#14955C212yesmaximal, complemented
8#14726C214yesmaximal, complemented
9#14724C212yesmaximal, complemented
10#14338C211yesmaximal, characteristic, complemented
11#13867C212yesmaximal, complemented
12C24×D4C2212yes—
13C24×D4C2211yescharacteristic
14C22×#123C2214yescomplemented
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19#2405—216no—
20C2×#396C2218yescomplemented
21C2×#394C2218yescomplemented
22C2×#394C2218yescomplemented
23C2×#392—28no—
24C2×#392C2214yes—
25C2×#392C2214yes—
26#2315—28no—
27C2×#386C2214yescomplemented
28C2×#386C2214yescomplemented
29C2×#386C2214yescomplemented
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2214yes—
33C2×#385C2212yescomplemented
34C2×#385C2212yescomplemented
35C2×#380C2212yes—
36C2×#380C2214yescomplemented
37C2×#379C2212yes—
38C2×#379C2214yes—
39C2×#379C2214yes—
40C22×#95C2211yescharacteristic
41#1442C2218yescomplemented
42#1441C2218yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2218yescomplemented
46#1439C2218yescomplemented
47#1439C2218yescomplemented
48#1438C2218yescomplemented
49#1438C2218yescomplemented
50C22⋊#95C2212yes—
51C22⋊#95C2214yes—
52C26C2312yes—
53C26C2311yescharacteristic
54C23×D4—28no—
55C23×D4—28no—
56C23×D4—28no—
57C23×D4—24no—
58C23×D4—48no—
59C23×D4—44no—
60C23×D4—48no—
61C23×D4—48no—
62C23×D4C2314yes—
63C23×D4C2314yes—
64C23×D4C2314yes—
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C2×#123—216no—
72C2×#123—48no—
73C2×#123—48no—
74C22×C4⋊C4C2318yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—48no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4—44no—
85C22×C22⋊C4D418yes—
86C22×C22⋊C4D418yescomplemented
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2318yes—
90C22×C22⋊C4C2318yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2318yes—
94C22×C22⋊C4C2318yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2318yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101#396—416no—
102#394—232no—
103#394—416no—
104#392—216no—
105#392—216no—
106#386—216no—
107#386—48no—
108#386—48no—
109#385—28no—
110#385—28no—
111#385—28no—
112#385—28no—
113#385—216no—
114#380—216no—
115#380—44no—
116#379—28no—
117#379—216no—
118#379—48no—
119C2×#95—24no—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C2×#95C2314yes—
128C25—24no—
129C25—24no—
130C25—24no—
131C25—28no—
132C25—24no—
133C25—24no—
134C25—28no—
135C25—28no—
136C25—44no—
137C25—44no—
138C25—44no—
139C25—44no—
140C25—44no—
141C25C2×D414yes—
142C25C2×D414yes—
143C25#4114yes—
144C25C2414yes—
145C25C2411yescharacteristic
146C25C2411yescharacteristic
147C25C2411yescharacteristic
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—216no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—44no—
161C22×D4—48no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—432no—
168C22×D4—416no—
169C22×D4—464no—
170C22×D4—464no—
171C22×D4—432no—
172C22×D4—432no—
173C22×D4—464no—
174C22×D4—432no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4C2×D418yescomplemented
197C23×C4C2418yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204#123—464no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—216no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—48no—
214C2×C22⋊C4—232no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—216no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—216no—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—216no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—216no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—432no—
244C2×C22⋊C4—416no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4C2×D418yes—
255C2×C22⋊C4C2×D418yes—
256C2×C22⋊C4C2×D418yes—
257C2×C22⋊C4C2×D418yescomplemented
258#95—28no—
259#95—28no—
260#95—28no—
261#95—216no—
262#95—216no—
263#95—44no—
264#95—48no—
265#95—48no—
266C24—24no—
267C24—24no—
268C24—22no—
269C24—22no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—22no—
275C24—22no—
276C24—28no—
277C24—28no—
278C24—216no—
279C24—216no—
280C24—216no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—28no—
285C24—24no—
286C24—24no—
287C24—28no—
288C24—28no—
289C24—24no—
290C24—28no—
291C24—48no—
292C24—48no—
293C24—416no—
294C24—416no—
295C24—416no—
296C24—416no—
297C24—432no—
298C24—432no—
299C24—48no—
300C24—48no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24—48no—
305C24—416no—
306C24—416no—
307C24—88no—
308C24—88no—
309C24—88no—
310C24—816no—
311C24—816no—
312C24—88no—
313C24—816no—
314C24#121116yes—
315C24C22×D412yes—
316C24C22×D412yes—
317C24C22×D412yes—
318C24C22×D412yes—
319C24C22×D412yes—
320C24C22×D412yes—
321C24C2×#4112yes—
322C24C2×#4112yes—
323C24C2×#4112yes—
324C24H(5,𝔽2)18yes—
325C24H(5,𝔽2)14yes—
326C242−1+414yes—
327C24C2511yescharacteristic
328C2×D4—232no—
329C2×D4—232no—
330C2×D4—416no—
331C2×D4—416no—
332C2×D4—416no—
333C2×D4—432no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—432no—
344C2×D4—4256no—
345C2×D4—4128no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—416no—
349C22×C4—216no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—416no—
385C22×C4—416no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—416no—
389C22×C4—48no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4C22×D414yes—
401C22×C4C22×D414yes—
402C22×C4C22×D418yes—
403C22×C4H(5,𝔽2)18yes—
404C4⋊C4—432no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C4⋊C4—88no—
408C22⋊C4—216no—
409C22⋊C4—216no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—432no—
413C22⋊C4—432no—
414C22⋊C4—48no—
415C22⋊C4—416no—
416C22⋊C4—48no—
417C22⋊C4—416no—
418C22⋊C4—816no—
419C22⋊C4—88no—
420C22⋊C4—816no—
421C22⋊C4—816no—
422C23—24no—
423C23—24no—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—216no—
428C23—28no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—28no—
438C23—28no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—28no—
444C23—28no—
445C23—232no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—216no—
450C23—216no—
451C23—432no—
452C23—432no—
453C23—432no—
454C23—432no—
455C23—44no—
456C23—44no—
457C23—48no—
458C23—48no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—48no—
465C23—48no—
466C23—44no—
467C23—48no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—48no—
473C23—48no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—48no—
478C23—48no—
479C23—88no—
480C23—816no—
481C23—88no—
482C23—832no—
483C23—832no—
484C23—864no—
485C23—864no—
486C23—816no—
487C23—816no—
488C23—816no—
489C23—816no—
490C23C2×#12114yes—
491C23C2×#12114yes—
492C23C2×#12114yes—
493C23#53414yes—
494C23#53614yes—
495C23#53718yes—
496C23#53918yes—
497C23C23×D411yescharacteristic
498C23C23×D411yescharacteristic
499C23C22×#4111yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×H(5,𝔽2)14yes—
502C23C2×H(5,𝔽2)14yes—
503C23C2×2−1+412yes—
504D4—464no—
505D4—464no—
506D4—464no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—216no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—432no—
517C2×C4—416no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—816no—
543C2×C4—88no—
544C2×C4—88no—
545C2×C4—88no—
546C22—28no—
547C22—24no—
548C22—28no—
549C22—24no—
550C22—216no—
551C22—216no—
552C22—28no—
553C22—28no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—48no—
563C22—48no—
564C22—48no—
565C22—48no—
566C22—48no—
567C22—48no—
568C22—48no—
569C22—416no—
570C22—416no—
571C22—832no—
572C22—864no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—88no—
577C22—88no—
578C22—84no—
579C22#261818yes—
580C22C22×#12111yescharacteristic
581C22C2×#53411yescharacteristic
582C22C2×#53611yescharacteristic
583C22C2×#53712yes—
584C22C2×#53912yes—
585C22#348412yes—
586C22#348412yes—
587C22#348414yes—
588C22#349014yes—
589C22#353214yes—
590C22#353618yes—
591C22#354314yes—
592C4—48no—
593C4—48no—
594C4—44no—
595C4—44no—
596C4—82no—
597C4—82no—
598C4—88no—
599C2—28no—
600C2—44no—
601C2—44no—
602C2—88no—
603C2C2×#261811yescharacteristic
604C2#3534812yes—
605C2#3671411yescharacteristic
606C2#3671812yes—
607C2#3723211yescharacteristic
608C2#4058614yes—
609C2#4063814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2405C2semidirect
3. C2×#2315C2semidirect
4. #16288C2semidirect
5. #16282C2semidirect
6. #16256C2semidirect
7. #14955C2semidirect
8. #14726C2semidirect
9. #14724C2semidirect
10. #14338C2semidirect
11. #13867C2semidirect
14. C22×#123C22semidirect
20. C2×#396C22semidirect
21. C2×#394C22semidirect
22. C2×#394C22semidirect
27. C2×#386C22semidirect
28. C2×#386C22semidirect
29. C2×#386C22semidirect
33. C2×#385C22semidirect
34. C2×#385C22semidirect
36. C2×#380C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1438C22semidirect
49. #1438C22semidirect
86. C22×C22⋊C4D4semidirect
196. C23×C4C2×D4semidirect
257. C2×C22⋊C4C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24052
C22×#3922
C2×#23152
#162882
#162822
#162562
#149552
#147262
#147242
#143382
#138672