Finite Group Databasebeta

← #4453298

Subgroups — #4453298

Order 512.

2934 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#397C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic
3#16429C218yesmaximal, complemented
4#16307C218yesmaximal, complemented
5#15019C212yesmaximal, complemented
6#15014C212yesmaximal, complemented
7#14936C212yesmaximal, complemented
8#14822C212yesmaximal, complemented
9#14341C211yesmaximal, characteristic
10#14127C212yesmaximal
11#13869C212yesmaximal
12C23×C4⋊C4C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C2×#400C2214yescomplemented
18C2×#397—28no—
19C2×#397C2218yes—
20#2315—28no—
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#381C2214yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#379C2212yes—
29C22×#95C2211yescharacteristic
30C22×#95C2211yescharacteristic
31C22×#95C2211yescharacteristic
32#1445C2214yescomplemented
33#1445C2214yescomplemented
34#1445C2218yescomplemented
35#1445C2214yescomplemented
36#1444C2214yescomplemented
37#1444C2214yescomplemented
38#1443C2218yescomplemented
39#1442C2214yescomplemented
40#1442C2218yescomplemented
41#1442C2218yescomplemented
42#1441C2214yescomplemented
43#1440C2218yes—
44#1440C2218yes—
45#1439C2218yes—
46C22⋊#95C2212yes—
47C22⋊#95C2212yes—
48C22⋊#95C2214yes—
49C22⋊#95C2214yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C26C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2312yes—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—44no—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2318yes—
70C22×C22⋊C4—28no—
71C22×C22⋊C4—28no—
72C22×C22⋊C4—24no—
73C22×C22⋊C4—24no—
74C22×C22⋊C4—28no—
75C22×C22⋊C4—44no—
76C22×C22⋊C4—48no—
77C22×C22⋊C4D418yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86#400—48no—
87#397—232no—
88#385—216no—
89#385—216no—
90#385—48no—
91#385—48no—
92#381—48no—
93#380—28no—
94#380—28no—
95#379—28no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2318yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2318yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2318yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C25—24no—
116C25—24no—
117C25—42no—
118C25—42no—
119C25—42no—
120C25—42no—
121C25C2×D414yes—
122C25#4118yes—
123C25C2412yes—
124C25C2411yescharacteristic
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—42no—
141C23×C4—44no—
142C23×C4—44no—
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×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C2×D414yes—
156C23×C4C2×D414yes—
157C23×C4C2418yes—
158C23×C4C2412yes—
159C23×C4C2414yes—
160C23×C4C2412yes—
161C23×C4C2414yes—
162C23×C4C2412yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—416no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C22⋊C4—232no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
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—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—416no—
204C2×C22⋊C4—432no—
205C2×C22⋊C4—416no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4C2×D4116yes—
209#95—28no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—216no—
215#95—48no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—48no—
220#95—48no—
221#95—48no—
222#95—48no—
223#95—44no—
224#95—48no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—24no—
230C24—24no—
231C24—24no—
232C24—24no—
233C24—24no—
234C24—24no—
235C24—28no—
236C24—28no—
237C24—232no—
238C24—44no—
239C24—44no—
240C24—44no—
241C24—44no—
242C24—416no—
243C24—416no—
244C24—88no—
245C24—88no—
246C24—88no—
247C24—88no—
248C24#123116yes—
249C24C22×D414yes—
250C24C22×D412yes—
251C24C2×#4114yes—
252C24C2×#4114yes—
253C24C2×#4114yes—
254C24H(5,𝔽2)14yes—
255C242−1+414yes—
256C24C2511yescharacteristic
257C22×C4—28no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—28no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—28no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—24no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—416no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—44no—
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—416no—
318C22×C4—88no—
319C22×C4—88no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4—816no—
327C22×C4—816no—
328C22×C4C22×D414yes—
329C22×C4C22×D414yes—
330C22×C4H(5,𝔽2)14yes—
331C4⋊C4—416no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C4⋊C4—816no—
337C22⋊C4—232no—
338C22⋊C4—416no—
339C22⋊C4—416no—
340C22⋊C4—416no—
341C22⋊C4—416no—
342C22⋊C4—416no—
343C22⋊C4—416no—
344C22⋊C4—88no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—28no—
350C23—28no—
351C23—28no—
352C23—28no—
353C23—28no—
354C23—28no—
355C23—24no—
356C23—24no—
357C23—28no—
358C23—28no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—28no—
364C23—28no—
365C23—28no—
366C23—28no—
367C23—232no—
368C23—416no—
369C23—416no—
370C23—48no—
371C23—48no—
372C23—48no—
373C23—48no—
374C23—88no—
375C23—88no—
376C23—88no—
377C23—864no—
378C23—88no—
379C23C2×#12318yes—
380C23C2×#12314yes—
381C23#53514yes—
382C23#53614yes—
383C23#539116yes—
384C23C23×D411yescharacteristic
385C23C22×#4112yes—
386C23C2×H(5,𝔽2)12yes—
387C23C2×H(5,𝔽2)12yes—
388C23C2×2−1+412yes—
389C23#58512yes—
390C23#58512yes—
391C23#58512yes—
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—48no—
404C2×C4—48no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
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—48no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—88no—
436C2×C4—88no—
437C2×C4—88no—
438C2×C4—88no—
439C2×C4—816no—
440C2×C4—816no—
441C2×C4—88no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—232no—
447C22—24no—
448C22—24no—
449C22—28no—
450C22—28no—
451C22—28no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—416no—
456C22—416no—
457C22—44no—
458C22—832no—
459C22#262318yes—
460C22C22×#12311yescharacteristic
461C22C2×#53511yescharacteristic
462C22C2×#53611yescharacteristic
463C22C2×#53914yes—
464C22#348411yescharacteristic
465C22#348511yescharacteristic
466C22#348511yescharacteristic
467C22#348511yescharacteristic
468C22#349114yes—
469C22#350514yes—
470C22#353514yes—
471C22#355814yes—
472C22#356014yes—
473C22#356414yes—
474C4—44no—
475C4—44no—
476C4—44no—
477C4—82no—
478C4—82no—
479C4—84no—
480C4—82no—
481C4—84no—
482C4—88no—
483C2—28no—
484C2—48no—
485C2C2×#262311yescharacteristic
486C2#3535111yescharacteristic
487C2#3544811yescharacteristic
488C2#3672011yescharacteristic
489C2#3696711yescharacteristic
490C2#3696911yescharacteristic
491C2#3697011yescharacteristic
492C2#4055918yes—
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
3. #16429C2semidirect
4. #16307C2semidirect
5. #15019C2semidirect
6. #15014C2semidirect
7. #14936C2semidirect
8. #14822C2semidirect
17. C2×#400C22semidirect
32. #1445C22semidirect
33. #1445C22semidirect
34. #1445C22semidirect
35. #1445C22semidirect
36. #1444C22semidirect
37. #1444C22semidirect
38. #1443C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3972
C2×#23152
#164292
#163072
#150192
#150142
#149362
#148222
#143412
#141272
#138692