Finite Group Databasebeta

← #626002

Subgroups — #626002

Order 512.

3938 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C212yesmaximal
2C2×#2659C211yesmaximal, characteristic, complemented
3C22×#385C211yesmaximal, characteristic
4#15472C218yesmaximal, complemented
5#15426C218yesmaximal, complemented
6#15019C212yesmaximal, complemented
7#14820C212yesmaximal
8#14680C212yesmaximal
9#14339C212yesmaximal, complemented
10#14139C212yesmaximal, complemented
11#13778C211yesmaximal, characteristic, complemented
12C25×C4C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2659—28no—
19#2561—216no—
20C2×#398C2214yescomplemented
21C2×#397C2214yes—
22C2×#397C2214yes—
23C2×#395C2214yescomplemented
24C2×#393C2214yes—
25C2×#391C2214yescomplemented
26C2×#388C2218yes—
27C2×#385—28no—
28C2×#385C2218yes—
29C2×#381C2214yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2214yes—
34C2×#380C2214yes—
35C2×#379C2214yes—
36C2×C4×C22⋊C4C2212yes—
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39#1445C2214yescomplemented
40#1443C2218yescomplemented
41#1442C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2218yescomplemented
44#1441C2214yescomplemented
45#1441C2218yescomplemented
46#1440C2218yes—
47#1439C2214yes—
48#1436C2218yescomplemented
49C4⋊#95C2214yescomplemented
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2218yes—
53C22⋊#95C2212yes—
54C26C2311yescharacteristic
55C23×Q8C2312yes—
56C24×C4—24no—
57C24×C4—28no—
58C24×C4—44no—
59C24×C4—44no—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2314yes—
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—44no—
87C22×C22⋊C4D418yes—
88C22×C22⋊C4C2318yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2318yes—
94C22×C42C2314yes—
95#398—216no—
96#397—216no—
97#397—48no—
98#395—216no—
99#393—216no—
100#391—48no—
101#388—232no—
102#385—232no—
103#381—216no—
104#380—28no—
105#380—28no—
106#380—28no—
107#380—48no—
108#380—48no—
109#379—216no—
110C4×C22⋊C4—28no—
111C2×#95—24no—
112C2×#95D418yes—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2314yes—
127C25—22no—
128C25—22no—
129C25—24no—
130C25—24no—
131C25—24no—
132C25—24no—
133C25—42no—
134C25—42no—
135C25C2×D414yes—
136C25C2412yes—
137C25C2411yescharacteristic
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—44no—
142C22×Q8—44no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—28no—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—28no—
157C23×C4—28no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—416no—
170C23×C4—464no—
171C23×C4—416no—
172C23×C4—416no—
173C23×C4C2×D414yes—
174C23×C4C2×D414yes—
175C23×C4C2×D414yes—
176C23×C4C2×D414yes—
177C23×C4C2×D414yes—
178C23×C4#4114yes—
179C23×C4#4118yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2418yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C22⋊Q8—432no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—416no—
208C2×C4⋊C4—416no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C22⋊C4—232no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—216no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—216no—
226C2×C22⋊C4—216no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—216no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—416no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4C2×D4116yes—
235C2×C42—28no—
236C2×C42—44no—
237C2×C42—44no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—216no—
243#95—216no—
244#95—28no—
245#95—216no—
246#95—44no—
247#95—48no—
248#95—48no—
249#95—44no—
250#95—48no—
251#95C2×D4116yes—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—28no—
257C24—28no—
258C24—216no—
259C24—28no—
260C24—216no—
261C24—22no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—22no—
266C24—22no—
267C24—24no—
268C24—24no—
269C24—216no—
270C24—48no—
271C24—416no—
272C24—416no—
273C24—416no—
274C24—416no—
275C24—44no—
276C24—48no—
277C24—44no—
278C24—44no—
279C24—44no—
280C24—48no—
281C24C22×D414yes—
282C24C22×D412yes—
283C24C22×D412yes—
284C24C2×#4112yes—
285C24C2×#4114yes—
286C24H(5,𝔽2)14yes—
287C242−1+414yes—
288C242−1+414yes—
289C24C2511yescharacteristic
290C2×Q8—416no—
291C2×Q8—48no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—88no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—232no—
301C22×C4—28no—
302C22×C4—216no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—28no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—416no—
340C22×C4—44no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—416no—
344C22×C4—416no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—416no—
348C22×C4—416no—
349C22×C4—416no—
350C22×C4—416no—
351C22×C4—44no—
352C22×C4—4128no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—416no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4#121116yes—
364C22×C4#124116yes—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D414yes—
368C22×C4C22×D414yes—
369C22×C4C22×D414yes—
370C22×C4C2×#4114yes—
371C22×C4C2×#4114yes—
372C22×C4C2×#4114yes—
373C22×C4H(5,𝔽2)14yes—
374C4⋊C4—416no—
375C4⋊C4—48no—
376C4⋊C4—416no—
377C4⋊C4—416no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—88no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C22⋊C4—232no—
384C22⋊C4—416no—
385C22⋊C4—416no—
386C22⋊C4—432no—
387C22⋊C4—416no—
388C22⋊C4—816no—
389C42—88no—
390C23—24no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—28no—
397C23—24no—
398C23—28no—
399C23—24no—
400C23—28no—
401C23—28no—
402C23—232no—
403C23—24no—
404C23—24no—
405C23—28no—
406C23—24no—
407C23—24no—
408C23—28no—
409C23—24no—
410C23—24no—
411C23—28no—
412C23—28no—
413C23—28no—
414C23—28no—
415C23—464no—
416C23—416no—
417C23—416no—
418C23—464no—
419C23—416no—
420C23—416no—
421C23—416no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—416no—
433C23C2×#12114yes—
434C23C2×#12414yes—
435C23#53614yes—
436C23#53718yes—
437C23#54718yes—
438C23#55618yes—
439C23C23×D411yescharacteristic
440C23C23×D411yescharacteristic
441C23C22×#4111yescharacteristic
442C23C22×#4112yes—
443C23C2×H(5,𝔽2)12yes—
444C23C2×H(5,𝔽2)12yes—
445C23C2×2−1+412yes—
446C23C2×2−1+412yes—
447C23#58512yes—
448Q8—816no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—416no—
465C2×C4—416no—
466C2×C4—464no—
467C2×C4—416no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—416no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4C2×#12118yes—
500C2×C4C2×#12418yes—
501C22—24no—
502C22—24no—
503C22—216no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—216no—
509C22—28no—
510C22—216no—
511C22—464no—
512C22—44no—
513C22—44no—
514C22—48no—
515C22—48no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—48no—
521C22#254718yes—
522C22C22×#12111yescharacteristic
523C22C22×#12411yescharacteristic
524C22C2×#53611yescharacteristic
525C22C2×#53712yes—
526C22#345912yes—
527C22C2×#54712yes—
528C22C2×#55612yes—
529C22#348411yescharacteristic
530C22#348411yescharacteristic
531C22#348511yescharacteristic
532C22#348611yescharacteristic
533C22#349014yes—
534C22#349212yes—
535C22#349514yes—
536C22#349612yes—
537C22#353514yes—
538C22#353814yes—
539C4—28no—
540C4—28no—
541C4—44no—
542C4—44no—
543C4—44no—
544C4—82no—
545C4—84no—
546C4—84no—
547C4—88no—
548C2—28no—
549C2—48no—
550C2C2×#254711yescharacteristic
551C2#3498411yescharacteristic
552C2#3522311yescharacteristic
553C2#3536711yescharacteristic
554C2#3564411yescharacteristic
555C2#3670811yescharacteristic
556C2#3672011yescharacteristic
557C2#3786114yes—
558C2#3795614yes—
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×#2659C2semidirect
4. #15472C2semidirect
5. #15426C2semidirect
6. #15019C2semidirect
9. #14339C2semidirect
10. #14139C2semidirect
11. #13778C2semidirect
20. C2×#398C22semidirect
23. C2×#395C22semidirect
25. C2×#391C22semidirect
39. #1445C22semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
48. #1436C22semidirect
49. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#26592
C22×#3852
#154722
#154262
#150192
#148202
#146802
#143392
#141392
#137782