Finite Group Databasebeta

← #4462521

Subgroups — #4462521

Order 512.

3220 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2617C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic, complemented
3#17160C214yesmaximal, complemented
4#17092C218yesmaximal, complemented
5#17088C214yesmaximal, complemented
6#15091C212yesmaximal, complemented
7#14946C212yesmaximal, complemented
8#14821C212yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14718C214yesmaximal, complemented
11#14341C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#120C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18#2617—28no—
19C2×#399—28no—
20C2×#399C2214yes—
21C2×#399C2214yes—
22C2×#398C2214yescomplemented
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#394C2218yes—
26C2×#394C2214yescomplemented
27C2×#392C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#380C2211yescharacteristic
33C2×#379C2212yes—
34C22×#95C2211yescharacteristic
35#1445C2218yescomplemented
36#1442C2218yescomplemented
37#1441C2214yescomplemented
38#1441C2218yescomplemented
39#1441C2214yescomplemented
40#1440C2214yes—
41#1440C2214yes—
42#1440C2218yes—
43#1439C2218yes—
44#1439C2218yescomplemented
45#1439C2218yescomplemented
46#1439C2218yescomplemented
47#1438C2218yes—
48C22⋊#95C2212yes—
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C26C2311yescharacteristic
52C23×D4C2312yes—
53C23×D4C2311yescharacteristic
54C23×D4C2314yes—
55C24×C4C2312yes—
56C24×C4C2312yes—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#121—24no—
60C2×#121—44no—
61C2×#120—28no—
62C2×#120—48no—
63C22×C4⋊C4—28no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—48no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2318yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2314yes—
73C22×C22⋊C4—28no—
74C22×C22⋊C4—48no—
75C22×C22⋊C4—44no—
76C22×C22⋊C4D418yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2318yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2318yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2318yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89#399—216no—
90#399—216no—
91#398—48no—
92#397—28no—
93#394—216no—
94#394—416no—
95#394—48no—
96#392—28no—
97#386—28no—
98#386—48no—
99#386—48no—
100#385—48no—
101#380—24no—
102#379—44no—
103C2×#95—24no—
104C2×#95C2312yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2314yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2318yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C25—24no—
116C25—24no—
117C25—24no—
118C25—44no—
119C25—44no—
120C25C2×D414yes—
121C25C2412yes—
122C25C2412yes—
123C25C2411yescharacteristic
124C22×D4—24no—
125C22×D4—28no—
126C22×D4—28no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—42no—
133C22×D4—48no—
134C22×D4—44no—
135C22×D4—48no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—28no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4C2×D414yes—
150C23×C4C2×D414yes—
151C23×C4C2418yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155C23×C4C2411yescharacteristic
156C23×C4C2412yes—
157C23×C4C2411yescharacteristic
158C23×C4C2414yes—
159#121—416no—
160#120—432no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—216no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—432no—
175C2×C4⋊C4—416no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C22⋊C4—216no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—216no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—216no—
190C2×C22⋊C4—416no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—48no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—432no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—48no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—48no—
208C2×C22⋊C4—48no—
209C2×C22⋊C4—48no—
210C2×C22⋊C4C2×D418yes—
211C2×C22⋊C4C2×D418yes—
212#95—28no—
213#95—24no—
214#95—28no—
215#95—48no—
216#95—48no—
217#95—48no—
218#95—48no—
219#95—44no—
220#95—48no—
221#95—44no—
222#95—48no—
223C24—24no—
224C24—24no—
225C24—28no—
226C24—28no—
227C24—22no—
228C24—22no—
229C24—24no—
230C24—24no—
231C24—28no—
232C24—28no—
233C24—28no—
234C24—22no—
235C24—28no—
236C24—24no—
237C24—24no—
238C24—24no—
239C24—28no—
240C24—48no—
241C24—48no—
242C24—432no—
243C24—416no—
244C24—88no—
245C24—88no—
246C24—88no—
247C24—88no—
248C24C22×D414yes—
249C24C22×D412yes—
250C24H(5,𝔽2)14yes—
251C24H(5,𝔽2)14yes—
252C242−1+414yes—
253C24C2511yescharacteristic
254C2×D4—44no—
255C2×D4—416no—
256C2×D4—48no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—432no—
260C2×D4—84no—
261C2×D4—88no—
262C2×D4—88no—
263C2×D4—88no—
264C2×D4—816no—
265C2×D4—816no—
266C22×C4—216no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—416no—
306C22×C4—416no—
307C22×C4—48no—
308C22×C4—416no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—88no—
314C22×C4—816no—
315C22×C4C22×D414yes—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4H(5,𝔽2)18yes—
319C4⋊C4—416no—
320C4⋊C4—416no—
321C4⋊C4—416no—
322C4⋊C4—44no—
323C4⋊C4—84no—
324C4⋊C4—816no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—432no—
328C22⋊C4—48no—
329C22⋊C4—416no—
330C22⋊C4—48no—
331C22⋊C4—416no—
332C22⋊C4—88no—
333C22⋊C4—816no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C22⋊C4—816no—
337C23—24no—
338C23—24no—
339C23—28no—
340C23—28no—
341C23—28no—
342C23—216no—
343C23—28no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—28no—
348C23—28no—
349C23—432no—
350C23—416no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—48no—
356C23—48no—
357C23—416no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—48no—
366C23—48no—
367C23—48no—
368C23—48no—
369C23—416no—
370C23—48no—
371C23—88no—
372C23—88no—
373C23—816no—
374C23—864no—
375C23#53414yes—
376C23#53514yes—
377C23#53614yes—
378C23#562116yes—
379C23C23×D411yescharacteristic
380C23C2×H(5,𝔽2)12yes—
381C23C2×H(5,𝔽2)12yes—
382C23C2×H(5,𝔽2)14yes—
383C23C2×2−1+412yes—
384C23#58514yes—
385D4—88no—
386D4—832no—
387D4—816no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—416no—
398C2×C4—432no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—48no—
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—88no—
429C2×C4—816no—
430C22—24no—
431C22—28no—
432C22—24no—
433C22—28no—
434C22—232no—
435C22—44no—
436C22—44no—
437C22—48no—
438C22—44no—
439C22—48no—
440C22—832no—
441C22—88no—
442C22—88no—
443C22—82no—
444C22—82no—
445C22—84no—
446C22—82no—
447C22—84no—
448C22—816no—
449C22#281618yes—
450C22C2×#53411yescharacteristic
451C22C2×#53511yescharacteristic
452C22C2×#53611yescharacteristic
453C22#348412yes—
454C22#348512yes—
455C22C2×#56214yes—
456C22#353314yes—
457C22#353414yes—
458C22#353414yes—
459C22#354118yes—
460C22#354314yes—
461C4—44no—
462C4—42no—
463C4—42no—
464C4—48no—
465C4—82no—
466C4—84no—
467C4—84no—
468C4—88no—
469C2—28no—
470C2—84no—
471C2—162no—
472C2C2×#281611yescharacteristic
473C2#3670512yes—
474C2#3672112yes—
475C2#3695511yescharacteristic
476C2#3723211yescharacteristic
477C2#5242518yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#399C2semidirect
3. #17160C2semidirect
4. #17092C2semidirect
5. #17088C2semidirect
6. #15091C2semidirect
7. #14946C2semidirect
8. #14821C2semidirect
9. #14719C2semidirect
10. #14718C2semidirect
11. #14341C2semidirect
22. C2×#398C22semidirect
26. C2×#394C22semidirect
35. #1445C22semidirect
36. #1442C22semidirect
37. #1441C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26172
C22×#3992
#171602
#170922
#170882
#150912
#149462
#148212
#147192
#147182
#143412