Finite Group Databasebeta

← #4387620

Subgroups — #4387620

Order 512.

3934 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C211yesmaximal, characteristic, complemented
2C2×#2405C212yesmaximal
3C2×#2309C211yesmaximal, characteristic
4#16391C214yesmaximal, complemented
5#15702C218yesmaximal, complemented
6#15271C218yesmaximal, complemented
7#14907C214yesmaximal, complemented
8#14673C211yesmaximal, characteristic, complemented
9#14341C212yesmaximal, complemented
10C24×D4C2211yescharacteristic
11C22×#123C2214yes—
12C22×#117C2212yes—
13C23×C22⋊C4C2212yes—
14#2483—28no—
15#2405—216no—
16C2×#398C2214yescomplemented
17C2×#397C2214yes—
18C2×#394C2218yes—
19#2309—28no—
20C2×#390C2218yes—
21C2×#389C2214yescomplemented
22C2×#386C2214yes—
23C2×#385C2214yes—
24C2×#385C2214yescomplemented
25C2×#381C2214yescomplemented
26C2×#379C2211yescharacteristic
27C2×#379C2212yes—
28C2×#379C2214yes—
29C2×#379C2212yes—
30C2×#379C2211yescharacteristic
31#1441C2214yescomplemented
32#1441C2218yescomplemented
33#1440C2218yes—
34#1440C2218yescomplemented
35#1440C2218yescomplemented
36#1440C2218yescomplemented
37#1439C2214yes—
38#1439C2218yescomplemented
39#1439C2218yescomplemented
40#1439C2218yescomplemented
41#1436C2218yescomplemented
42#1434C2214yescomplemented
43C22⋊#95C2214yescomplemented
44C22⋊#95C2214yescomplemented
45C26C2311yescharacteristic
46C26C2311yescharacteristic
47C23×D4—24no—
48C23×D4—24no—
49C23×D4—44no—
50C23×D4—44no—
51C23×D4D418yes—
52C23×D4C2314yes—
53C24×C4C2311yescharacteristic
54C24×C4C2312yes—
55C24×C4C2312yes—
56C2×#123—216no—
57C2×#123—48no—
58C2×#123—48no—
59C2×#117—28no—
60C2×#117—44no—
61C2×#117—44no—
62C22×C4⋊C4C2318yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C22⋊C4—28no—
67C22×C22⋊C4—28no—
68C22×C22⋊C4—28no—
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2318yes—
73C22×C22⋊C4C2318yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2318yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C42C2314yes—
82#398—48no—
83#397—216no—
84#394—232no—
85#390—232no—
86#389—48no—
87#386—216no—
88#385—216no—
89#385—216no—
90#381—216no—
91#379—24no—
92#379—216no—
93#379—28no—
94#379—28no—
95#379D418yes—
96C2×#95C2318yes—
97C2×#95C2314yes—
98C2×#95C2318yes—
99C2×#95C2318yes—
100C2×#95C2314yes—
101C2×#95C2318yes—
102C2×#95C2318yes—
103C2×#95C2318yes—
104C2×#95C2318yes—
105C25—22no—
106C25—22no—
107C25—42no—
108C25—42no—
109C25—42no—
110C25—42no—
111C25—44no—
112C25—44no—
113C25—44no—
114C25—44no—
115C25C2×D414yes—
116C25C2×D414yes—
117C25C2×D414yes—
118C25#4114yes—
119C25C2412yes—
120C25C2411yescharacteristic
121C25C2412yes—
122C22×D4—232no—
123C22×D4—28no—
124C22×D4—28no—
125C22×D4—28no—
126C22×D4—28no—
127C22×D4—432no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—432no—
131C22×D4—416no—
132C22×D4—416no—
133C22×D4C2×D4116yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4C2×D414yes—
149C23×C4C2418yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155#123—464no—
156#117—432no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—216no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—216no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—216no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—216no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—48no—
194C2×C22⋊C4—432no—
195C2×C22⋊C4—48no—
196C2×C22⋊C4—48no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4C2×D418yes—
204C2×C22⋊C4C2×D4116yes—
205C2×C42—28no—
206C2×C42—28no—
207C2×C42—44no—
208#95—28no—
209#95—216no—
210#95—216no—
211#95—48no—
212#95—48no—
213#95—48no—
214#95—48no—
215#95—44no—
216#95—48no—
217C24—22no—
218C24—22no—
219C24—24no—
220C24—24no—
221C24—24no—
222C24—24no—
223C24—24no—
224C24—24no—
225C24—24no—
226C24—24no—
227C24—22no—
228C24—22no—
229C24—24no—
230C24—24no—
231C24—28no—
232C24—28no—
233C24—28no—
234C24—48no—
235C24—48no—
236C24—48no—
237C24—44no—
238C24—44no—
239C24—416no—
240C24—416no—
241C24—48no—
242C24—48no—
243C24—416no—
244C24—48no—
245C24—44no—
246C24—44no—
247C24—88no—
248C24—84no—
249C24—84no—
250C24—88no—
251C24—88no—
252C24—88no—
253C24—88no—
254C24—816no—
255C24—816no—
256C24#121116yes—
257C24C22×D414yes—
258C24C22×D412yes—
259C24C22×D412yes—
260C24C22×D414yes—
261C24C22×D414yes—
262C24C2×#4112yes—
263C24C2×#4114yes—
264C24C2511yescharacteristic
265C2×D4—232no—
266C2×D4—232no—
267C2×D4—416no—
268C2×D4—416no—
269C2×D4—416no—
270C2×D4—416no—
271C2×D4—416no—
272C2×D4—4128no—
273C22×C4—216no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—216no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—416no—
309C22×C4—416no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4C22×D418yes—
319C22×C4C22×D414yes—
320C22×C4H(5,𝔽2)14yes—
321C22×C4H(5,𝔽2)14yes—
322C4⋊C4—416no—
323C4⋊C4—416no—
324C4⋊C4—432no—
325C4⋊C4—88no—
326C22⋊C4—416no—
327C22⋊C4—48no—
328C22⋊C4—416no—
329C22⋊C4—48no—
330C22⋊C4—416no—
331C22⋊C4—432no—
332C22⋊C4—416no—
333C22⋊C4—416no—
334C22⋊C4—816no—
335C22⋊C4—88no—
336C22⋊C4—88no—
337C22⋊C4—816no—
338C42—416no—
339C23—28no—
340C23—24no—
341C23—24no—
342C23—28no—
343C23—28no—
344C23—28no—
345C23—28no—
346C23—28no—
347C23—28no—
348C23—28no—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—28no—
361C23—216no—
362C23—416no—
363C23—432no—
364C23—416no—
365C23—48no—
366C23—44no—
367C23—48no—
368C23—44no—
369C23—48no—
370C23—48no—
371C23—48no—
372C23—44no—
373C23—48no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23—48no—
379C23—48no—
380C23—416no—
381C23—816no—
382C23—816no—
383C23—88no—
384C23—88no—
385C23—832no—
386C23—832no—
387C23—832no—
388C23—816no—
389C23—88no—
390C23—88no—
391C23—832no—
392C23C2×#12114yes—
393C23C2×#12118yes—
394C23C23×D411yescharacteristic
395C23C23×D411yescharacteristic
396C23C22×#4111yescharacteristic
397C23C2×H(5,𝔽2)12yes—
398C23C2×H(5,𝔽2)12yes—
399C23#58514yes—
400C23#58514yes—
401D4—464no—
402C2×C4—232no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—216no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—816no—
437C2×C4—816no—
438C2×C4#53418yes—
439C22—28no—
440C22—28no—
441C22—28no—
442C22—24no—
443C22—28no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—42no—
449C22—44no—
450C22—42no—
451C22—416no—
452C22—48no—
453C22—416no—
454C22—48no—
455C22—48no—
456C22—44no—
457C22—416no—
458C22—832no—
459C22—832no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—88no—
464C22C22×#12111yescharacteristic
465C22C2×#53412yes—
466C22#348412yes—
467C22#348514yes—
468C22#348514yes—
469C22#349011yescharacteristic
470C22#349011yescharacteristic
471C22#349112yes—
472C22#349112yes—
473C22#354014yes—
474C22#355114yes—
475C22#356218yes—
476C4—28no—
477C4—48no—
478C4—84no—
479C4—84no—
480C4—88no—
481C4—84no—
482C2—48no—
483C2—44no—
484C2—84no—
485C2#3498111yescharacteristic
486C2#3544712yes—
487C2#4267314yes—
488C2#4639012yes—
489C2#4699811yescharacteristic
490C2#4719511yescharacteristic
491C2#5016514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2483C2semidirect
4. #16391C2semidirect
5. #15702C2semidirect
6. #15271C2semidirect
7. #14907C2semidirect
8. #14673C2semidirect
9. #14341C2semidirect
16. C2×#398C22semidirect
21. C2×#389C22semidirect
24. C2×#385C22semidirect
25. C2×#381C22semidirect
31. #1441C22semidirect
32. #1441C22semidirect
34. #1440C22semidirect
35. #1440C22semidirect
36. #1440C22semidirect
38. #1439C22semidirect
39. #1439C22semidirect
40. #1439C22semidirect
41. #1436C22semidirect
42. #1434C22semidirect
43. C22⋊#95C22semidirect
44. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C2×#24052
C2×#23092
#163912
#157022
#152712
#149072
#146732
#143412