Finite Group Databasebeta

← #4462518

Subgroups — #4462518

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic, complemented
2C22×#397C211yesmaximal, characteristic
3#17130C218yesmaximal, complemented
4#17097C218yesmaximal, complemented
5#15015C212yesmaximal, complemented
6#14914C212yesmaximal, complemented
7#14821C212yesmaximal, complemented
8#14820C212yesmaximal
9#14734C212yesmaximal
10#14717C212yesmaximal
11#14349C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17#2659—28no—
18C2×#399C2214yes—
19C2×#398C2214yescomplemented
20C2×#398C2214yescomplemented
21C2×#397—28no—
22C2×#397C2218yes—
23C2×#397C2214yes—
24C2×#395C2214yescomplemented
25C2×#393C2214yes—
26C2×#385C2214yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#379C2214yes—
30C22×#95C2211yescharacteristic
31C22×#95C2211yescharacteristic
32C22×#95C2211yescharacteristic
33#1446C2214yescomplemented
34#1445C2214yescomplemented
35#1444C2214yescomplemented
36#1443C2218yescomplemented
37#1443C2218yescomplemented
38#1442C2214yescomplemented
39#1441C2218yescomplemented
40#1441C2214yescomplemented
41#1441C2218yescomplemented
42#1440C2218yes—
43#1439C2218yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2218yes—
47#1439C2214yes—
48C22⋊#95C2212yes—
49C22⋊#95C2212yes—
50C22⋊#95C2212yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C26C2311yescharacteristic
54C23×Q8C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—48no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—44no—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2318yes—
73C22×C22⋊C4—24no—
74C22×C22⋊C4—24no—
75C22×C22⋊C4—24no—
76C22×C22⋊C4—44no—
77C22×C22⋊C4—44no—
78C22×C22⋊C4—44no—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2318yes—
85C22×C22⋊C4C2318yes—
86#399—48no—
87#398—216no—
88#398—48no—
89#397—232no—
90#397—48no—
91#395—216no—
92#393—48no—
93#385—48no—
94#380—28no—
95#380—28no—
96#379—48no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95D418yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2318yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2318yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2314yes—
116C25—22no—
117C25—24no—
118C25—24no—
119C25—24no—
120C25—24no—
121C25—42no—
122C25—42no—
123C25—42no—
124C25C2412yes—
125C25C2411yescharacteristic
126C22×Q8—24no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8—44no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C2×D414yes—
152C23×C4C2×D414yes—
153C23×C4C2×D414yes—
154C23×C4C2414yes—
155C23×C4C2414yes—
156C23×C4C2418yes—
157C23×C4C2412yes—
158C23×C4C2412yes—
159C23×C4C2412yes—
160C23×C4C2412yes—
161C23×C4C2412yes—
162C23×C4C2412yes—
163C22⋊Q8—432no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—416no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—216no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—416no—
201C2×C22⋊C4—416no—
202C2×C22⋊C4—416no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—48no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—48no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—28no—
212#95—44no—
213#95—44no—
214#95—48no—
215#95—48no—
216#95—44no—
217#95—48no—
218#95—44no—
219#95—48no—
220#95—44no—
221#95—48no—
222#95—44no—
223#95C2×D4116yes—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—28no—
229C24—22no—
230C24—24no—
231C24—22no—
232C24—24no—
233C24—24no—
234C24—24no—
235C24—24no—
236C24—24no—
237C24—416no—
238C24—416no—
239C24—416no—
240C24—44no—
241C24—44no—
242C24—44no—
243C24—44no—
244C24—44no—
245C24—44no—
246C24—416no—
247C24—88no—
248C24—88no—
249C24—88no—
250C24C22×D412yes—
251C24H(5,𝔽2)14yes—
252C24H(5,𝔽2)14yes—
253C242−1+414yes—
254C242−1+414yes—
255C24C2511yescharacteristic
256C2×Q8—48no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×Q8—816no—
260C22×C4—28no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—28no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—28no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—416no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—88no—
318C22×C4—88no—
319C22×C4—88no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4C22×D414yes—
323C22×C4C22×D414yes—
324C22×C4C22×D414yes—
325C22×C4H(5,𝔽2)14yes—
326C4⋊C4—416no—
327C4⋊C4—48no—
328C4⋊C4—416no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4—816no—
334C22⋊C4—416no—
335C22⋊C4—416no—
336C22⋊C4—416no—
337C22⋊C4—416no—
338C22⋊C4—416no—
339C22⋊C4—816no—
340C22⋊C4—816no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—28no—
346C23—28no—
347C23—28no—
348C23—28no—
349C23—28no—
350C23—28no—
351C23—24no—
352C23—28no—
353C23—28no—
354C23—28no—
355C23—416no—
356C23—416no—
357C23—416no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—48no—
365C23—48no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—48no—
370C23—48no—
371C23—864no—
372C23—88no—
373C23—88no—
374C23—88no—
375C23—88no—
376C23#53514yes—
377C23#53614yes—
378C23#53614yes—
379C23#563116yes—
380C23C23×D411yescharacteristic
381C23C2×H(5,𝔽2)12yes—
382C23C2×H(5,𝔽2)12yes—
383C23C2×H(5,𝔽2)12yes—
384C23C2×2−1+412yes—
385C23C2×2−1+412yes—
386C23#58512yes—
387C23#58512yes—
388Q8—816no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—416no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—48no—
432C2×C4—88no—
433C2×C4—88no—
434C2×C4—88no—
435C2×C4—88no—
436C2×C4—88no—
437C22—232no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—416no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—832no—
449C22—88no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22#282118yes—
455C22C2×#53511yescharacteristic
456C22C2×#53611yescharacteristic
457C22C2×#53611yescharacteristic
458C22#348411yescharacteristic
459C22#348411yescharacteristic
460C22#348511yescharacteristic
461C22#348711yescharacteristic
462C22C2×#56314yes—
463C22#353414yes—
464C22#353414yes—
465C22#353514yes—
466C22#353614yes—
467C22#356114yes—
468C22#356414yes—
469C4—44no—
470C4—44no—
471C4—44no—
472C4—44no—
473C4—88no—
474C4—84no—
475C4—82no—
476C4—82no—
477C4—84no—
478C2—28no—
479C2—84no—
480C2C2×#282111yescharacteristic
481C2#3671011yescharacteristic
482C2#3671111yescharacteristic
483C2#3671811yescharacteristic
484C2#3672111yescharacteristic
485C2#3696211yescharacteristic
486C2#3696611yescharacteristic
487C2#5249018yes—
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
1. C2×#2659C2semidirect
3. #17130C2semidirect
4. #17097C2semidirect
5. #15015C2semidirect
6. #14914C2semidirect
7. #14821C2semidirect
11. #14349C2semidirect
19. C2×#398C22semidirect
20. C2×#398C22semidirect
24. C2×#395C22semidirect
33. #1446C22semidirect
34. #1445C22semidirect
35. #1444C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C22×#3972
#171302
#170972
#150152
#149142
#148212
#148202
#147342
#147172
#143492