Finite Group Databasebeta

← #625849

Subgroups — #625849

Order 512.

3962 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C212yesmaximal
2C2×#2316C211yesmaximal, characteristic, complemented
3C22×#385C211yesmaximal, characteristic
4#15294C218yesmaximal, complemented
5#15290C218yesmaximal, complemented
6#14821C212yesmaximal, complemented
7#14820C212yesmaximal
8#14680C212yesmaximal
9#14339C211yesmaximal, characteristic, complemented
10#14275C212yesmaximal, complemented
11#13958C212yesmaximal, complemented
12C25×C4C2211yescharacteristic
13C22×#117C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2482—216no—
19C2×#399C2214yes—
20C2×#398C2214yescomplemented
21C2×#397C2214yes—
22C2×#397C2214yes—
23#2316—28no—
24C2×#389C2214yescomplemented
25C2×#389C2214yescomplemented
26C2×#388C2218yes—
27C2×#385—28no—
28C2×#385C2218yes—
29C2×#385C2214yes—
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#380C2214yes—
36C2×#379C2214yes—
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39#1445C2214yescomplemented
40#1445C2214yescomplemented
41#1444C2214yescomplemented
42#1442C2214yescomplemented
43#1441C2218yescomplemented
44#1441C2214yescomplemented
45#1441C2218yescomplemented
46#1441C2218yescomplemented
47#1440C2218yes—
48#1439C2214yes—
49#1434C2218yescomplemented
50C22⋊#95C2212yes—
51C22⋊#95C2214yes—
52C22⋊#95C2218yes—
53C22⋊#95C2214yes—
54C26C2311yescharacteristic
55C24×C4—24no—
56C24×C4—28no—
57C24×C4—44no—
58C24×C4—44no—
59C24×C4C2311yescharacteristic
60C24×C4C2314yes—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C2×#117—28no—
67C2×#117—28no—
68C2×#117—28no—
69C22×C4⋊C4—28no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4D418yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2318yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
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#399—216no—
96#398—48no—
97#397—216no—
98#397—48no—
99#389—216no—
100#389—216no—
101#388—232no—
102#385—216no—
103#385—232no—
104#381—28no—
105#380—28no—
106#380—28no—
107#380—28no—
108#380—48no—
109#380—48no—
110#379—216no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2318yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C25—22no—
128C25—22no—
129C25—24no—
130C25—24no—
131C25—24no—
132C25—24no—
133C25—42no—
134C25—42no—
135C25C2×D414yes—
136C25C2412yes—
137C25C2411yescharacteristic
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—28no—
144C23×C4—24no—
145C23×C4—28no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—28no—
152C23×C4—28no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—416no—
165C23×C4—464no—
166C23×C4—416no—
167C23×C4—416no—
168C23×C4C2×D414yes—
169C23×C4C2×D414yes—
170C23×C4C2×D414yes—
171C23×C4C2×D414yes—
172C23×C4C2×D414yes—
173C23×C4#4114yes—
174C23×C4#4118yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2418yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184#117—432no—
185C2×C4⋊C4—232no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—216no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—48no—
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—48no—
212C2×C4⋊C4C2×D4116yes—
213C2×C22⋊C4—232no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—216no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—216no—
224C2×C22⋊C4—216no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—216no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—416no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4C2×D4116yes—
233C2×C42—28no—
234C2×C42—28no—
235C2×C42—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—216no—
243#95—28no—
244#95—216no—
245#95—48no—
246#95—48no—
247#95—44no—
248#95—48no—
249#95—44no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—28no—
255C24—28no—
256C24—216no—
257C24—28no—
258C24—216no—
259C24—22no—
260C24—22no—
261C24—24no—
262C24—24no—
263C24—22no—
264C24—22no—
265C24—24no—
266C24—24no—
267C24—216no—
268C24—48no—
269C24—416no—
270C24—416no—
271C24—416no—
272C24—416no—
273C24—44no—
274C24—48no—
275C24—44no—
276C24—44no—
277C24—44no—
278C24—48no—
279C24C22×D414yes—
280C24C22×D412yes—
281C24C22×D412yes—
282C24C2×#4112yes—
283C24C2×#4114yes—
284C24H(5,𝔽2)14yes—
285C242−1+414yes—
286C242−1+414yes—
287C24C2511yescharacteristic
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—216no—
291C22×C4—216no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—232no—
298C22×C4—28no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—22no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—416no—
335C22×C4—48no—
336C22×C4—416no—
337C22×C4—416no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—416no—
341C22×C4—416no—
342C22×C4—416no—
343C22×C4—416no—
344C22×C4—44no—
345C22×C4—48no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—4128no—
349C22×C4—44no—
350C22×C4—416no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4#121116yes—
357C22×C4#123116yes—
358C22×C4C22×D414yes—
359C22×C4C22×D414yes—
360C22×C4C22×D414yes—
361C22×C4C22×D414yes—
362C22×C4C22×D414yes—
363C22×C4C2×#4114yes—
364C22×C4C2×#4114yes—
365C22×C4C2×#4114yes—
366C22×C4H(5,𝔽2)14yes—
367C4⋊C4—232no—
368C4⋊C4—416no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—816no—
377C22⋊C4—232no—
378C22⋊C4—416no—
379C22⋊C4—416no—
380C22⋊C4—432no—
381C22⋊C4—416no—
382C22⋊C4—816no—
383C42—416no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—28no—
391C23—24no—
392C23—28no—
393C23—24no—
394C23—28no—
395C23—28no—
396C23—232no—
397C23—24no—
398C23—24no—
399C23—28no—
400C23—24no—
401C23—24no—
402C23—28no—
403C23—24no—
404C23—24no—
405C23—28no—
406C23—28no—
407C23—28no—
408C23—28no—
409C23—464no—
410C23—416no—
411C23—464no—
412C23—416no—
413C23—416no—
414C23—416no—
415C23—416no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—416no—
425C23—44no—
426C23—44no—
427C23C2×#12114yes—
428C23C2×#12314yes—
429C23#53614yes—
430C23#53718yes—
431C23#54718yes—
432C23#55218yes—
433C23C23×D411yescharacteristic
434C23C23×D411yescharacteristic
435C23C22×#4111yescharacteristic
436C23C22×#4112yes—
437C23C2×H(5,𝔽2)12yes—
438C23C2×H(5,𝔽2)12yes—
439C23C2×2−1+412yes—
440C23C2×2−1+412yes—
441C23#58512yes—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—232no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—416no—
459C2×C4—464no—
460C2×C4—416no—
461C2×C4—416no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—88no—
491C2×C4—88no—
492C2×C4C2×#12118yes—
493C2×C4C2×#12318yes—
494C22—24no—
495C22—216no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—216no—
502C22—28no—
503C22—216no—
504C22—464no—
505C22—44no—
506C22—48no—
507C22—48no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—48no—
514C22#248618yes—
515C22C22×#12111yescharacteristic
516C22C22×#12311yescharacteristic
517C22C2×#53611yescharacteristic
518C22C2×#53712yes—
519C22#346012yes—
520C22C2×#54712yes—
521C22C2×#55212yes—
522C22#348411yescharacteristic
523C22#348411yescharacteristic
524C22#348611yescharacteristic
525C22#348711yescharacteristic
526C22#349014yes—
527C22#349212yes—
528C22#349514yes—
529C22#349612yes—
530C22#353514yes—
531C22#353814yes—
532C4—28no—
533C4—28no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4—84no—
538C4—82no—
539C4—84no—
540C4—88no—
541C2—28no—
542C2—48no—
543C2C2×#248611yescharacteristic
544C2#3516811yescharacteristic
545C2#3522311yescharacteristic
546C2#3535011yescharacteristic
547C2#3567211yescharacteristic
548C2#3670811yescharacteristic
549C2#3671111yescharacteristic
550C2#3786114yes—
551C2#3793314yes—
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×#2316C2semidirect
4. #15294C2semidirect
5. #15290C2semidirect
6. #14821C2semidirect
9. #14339C2semidirect
10. #14275C2semidirect
11. #13958C2semidirect
20. C2×#398C22semidirect
24. C2×#389C22semidirect
25. C2×#389C22semidirect
39. #1445C22semidirect
40. #1445C22semidirect
41. #1444C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
49. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C2×#23162
C22×#3852
#152942
#152902
#148212
#148202
#146802
#143392
#142752
#139582