Finite Group Databasebeta

← #4416993

Subgroups — #4416993

Order 512.

3482 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C212yesmaximal
2#16894C218yesmaximal, complemented
3#16391C214yesmaximal
4#15236C212yesmaximal, complemented
5#15236C212yesmaximal, complemented
6#15233C212yesmaximal, complemented
7#15182C212yesmaximal, complemented
8#15178C212yesmaximal, complemented
9#15178C211yesmaximal, characteristic, complemented
10#15150C211yesmaximal, characteristic, complemented
11#14627C214yesmaximal, complemented
12C4×#379C211yesmaximal, characteristic
13C25×C4C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic, complemented
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18#2848—216no—
19C2×#397C2214yes—
20C2×#391C2212yescomplemented
21C2×#391C2212yescomplemented
22C2×#391C2212yescomplemented
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#384C2211yescharacteristic, complemented
27C2×#382C2212yescomplemented
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#379C2214yes—
32C2×#379C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2214yescomplemented
38#1444C2218yescomplemented
39#1444C2214yescomplemented
40#1443C2218yescomplemented
41#1442C2218yescomplemented
42#1442C2218yescomplemented
43#1441C2218yescomplemented
44#1440C2218yes—
45#1440C2218yes—
46#1440C2218yes—
47#1439C2214yes—
48#1439C2218yes—
49#1439C2218yes—
50C4⋊#95C2212yescomplemented
51C4⋊#95C2211yescharacteristic, complemented
52C4⋊#95C2211yescharacteristic, complemented
53C4⋊#95C2212yescomplemented
54C22⋊#95C2212yes—
55C22⋊#95C2214yes—
56C26C2311yescharacteristic
57C23×Q8C2312yes—
58C24×C4—42no—
59C24×C4—44no—
60C24×C4—42no—
61C24×C4—44no—
62C24×C4D418yes—
63C24×C4C2311yescharacteristic
64C24×C4C2312yes—
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C2×C4⋊Q8—42no—
68C2×C4⋊Q8—44no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8—44no—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8—44no—
76C2×C22⋊Q8—44no—
77C2×C22⋊Q8—44no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99#397—216no—
100#391—44no—
101#391—44no—
102#391—44no—
103#385—216no—
104#385—216no—
105#385—216no—
106#384—42no—
107#382—44no—
108#380—28no—
109#380—28no—
110#380—28no—
111#379—216no—
112#379—28no—
113C4×C22⋊C4—44no—
114C4×C22⋊C4—42no—
115C4×C22⋊C4—44no—
116C4×C22⋊C4—42no—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2318yes—
127C2×#95C2314yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C25—24no—
133C25—28no—
134C25—42no—
135C25—42no—
136C25—41no—
137C25—41no—
138C25—42no—
139C25C2×D414yes—
140C25C2411yescharacteristic
141C25C2412yes—
142C22×Q8—24no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—432no—
160C23×C4—416no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—41no—
168C23×C4—42no—
169C23×C4—41no—
170C23×C4—42no—
171C23×C4—416no—
172C23×C4—432no—
173C23×C4—416no—
174C23×C4C2×D414yes—
175C23×C4C2×D418yes—
176C23×C4C2×D414yes—
177C23×C4C2×D418yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C4⋊Q8—416no—
189C22⋊Q8—432no—
190C22⋊Q8—432no—
191C22⋊Q8—432no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—216no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—216no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4#4118yes—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—216no—
254#95—44no—
255#95—48no—
256#95—44no—
257#95—42no—
258#95—44no—
259#95—42no—
260#95—48no—
261#95—48no—
262#95—48no—
263#95—48no—
264C24—24no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—28no—
280C24—216no—
281C24—48no—
282C24—44no—
283C24—44no—
284C24—416no—
285C24—416no—
286C24—48no—
287C24—48no—
288C24—48no—
289C24—48no—
290C24—416no—
291C24—416no—
292C24—44no—
293C24—48no—
294C24—44no—
295C24C22×D412yes—
296C24C22×D414yes—
297C24C2×#4114yes—
298C242−1+418yes—
299C24C2511yescharacteristic
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4—4128no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4C22×D412yes—
380C22×C4C22×D414yes—
381C22×C4C22×D412yes—
382C22×C4C2×#4114yes—
383C22×C4C2×#4114yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)14yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C4H(5,𝔽2)14yes—
388C4⋊C4—416no—
389C4⋊C4—416no—
390C4⋊C4—416no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—88no—
396C22⋊C4—48no—
397C22⋊C4—416no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—48no—
401C22⋊C4—416no—
402C22⋊C4—416no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—416no—
406C42—44no—
407C42—48no—
408C42—44no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—28no—
428C23—28no—
429C23—432no—
430C23—416no—
431C23—432no—
432C23—416no—
433C23—416no—
434C23—432no—
435C23—464no—
436C23—432no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23#53214yes—
455C23#53618yes—
456C23C23×D411yescharacteristic
457C23C22×#4112yes—
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×H(5,𝔽2)12yes—
461C23C2×H(5,𝔽2)12yes—
462C23C2×2−1+414yes—
463C23#58512yes—
464Q8—816no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—432no—
475C2×C4—416no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—416no—
490C2×C4—432no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4#53418yes—
521C2×C4#53414yes—
522C2×C4#56014yes—
523C2×C4#56018yes—
524C2×C4H(3,𝔽4)14yes—
525C22—28no—
526C22—28no—
527C22—28no—
528C22—28no—
529C22—24no—
530C22—464no—
531C22—44no—
532C22—42no—
533C22—44no—
534C22—44no—
535C22—48no—
536C22—48no—
537C22—48no—
538C22—48no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—44no—
543C22—48no—
544C22—44no—
545C22C2×#53211yescharacteristic
546C22C2×#53412yes—
547C22C2×#53411yescharacteristic
548C22C2×#53612yes—
549C22#344911yescharacteristic
550C22#350512yes—
551C22#350612yes—
552C22#352014yes—
553C22#352014yes—
554C22C2×#56011yescharacteristic
555C22C2×#56012yes—
556C22C2×H(3,𝔽4)11yescharacteristic
557C22#353812yes—
558C22#353812yes—
559C22#354114yes—
560C22#354114yes—
561C4—42no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—84no—
566C4—84no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4#281218yes—
571C2—44no—
572C2—48no—
573C2#3503211yescharacteristic
574C2#3508512yes—
575C2C2×#281211yescharacteristic
576C2#3744912yes—
577C2#3745311yescharacteristic
578C2#4562212yes—
579C2#4562414yes—
580C2#4566712yes—
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. #16894C2semidirect
4. #15236C2semidirect
5. #15236C2semidirect
6. #15233C2semidirect
7. #15182C2semidirect
8. #15178C2semidirect
9. #15178C2semidirect
10. #15150C2semidirect
11. #14627C2semidirect
14. C22×C4⋊Q8C22semidirect
20. C2×#391C22semidirect
21. C2×#391C22semidirect
22. C2×#391C22semidirect
26. C2×#384C22semidirect
27. C2×#382C22semidirect
37. #1445C22semidirect
38. #1444C22semidirect
39. #1444C22semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
50. C4⋊#95C22semidirect
51. C4⋊#95C22semidirect
52. C4⋊#95C22semidirect
53. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
#168942
#163912
#152362
#152332
#151822
#151782
#151502
#146272
C4×#3792