Finite Group Databasebeta

← #5020818

Subgroups — #5020818

Order 512.

3644 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2812C211yesmaximal, characteristic
2C2×#2848C211yesmaximal, characteristic, complemented
3#15670C213yesmaximal, complemented
4#15635C213yesmaximal, complemented
5#15579C213yesmaximal, complemented
6#15474C216yesmaximal, complemented
7#15391C216yesmaximal, complemented
8#15264C213yesmaximal, complemented
9#14694C212yesmaximal, complemented
10#14657C212yesmaximal, complemented
11#13827C211yesmaximal, characteristic, complemented
12C25×C4C2211yescharacteristic
13C22×#124C2211yescharacteristic, complemented
14C22×C22⋊Q8C2213yes—
15C22×#120C2212yes—
16#2848—28no—
17#2812—28no—
18C2×#400C2213yescomplemented
19C2×#397C2213yes—
20C2×#394C2216yes—
21C2×#394C2213yescomplemented
22C2×#392C2213yes—
23C2×#386C2216yes—
24C2×#386C2216yes—
25C2×#385C2213yes—
26C2×#385C2213yes—
27C2×#382C2213yescomplemented
28C2×#380C2213yes—
29C2×#379C2212yes—
30C2×#379C2211yescharacteristic
31C2×#379C2211yescharacteristic
32#1446C2213yescomplemented
33#1445C2213yescomplemented
34#1443C2216yescomplemented
35#1442C2216yescomplemented
36#1442C2216yescomplemented
37#1440C2216yes—
38#1440C2216yes—
39#1440C2216yescomplemented
40#1439C2216yescomplemented
41#1439C2216yescomplemented
42#1438C2216yes—
43#1438C2216yes—
44#1437C2213yes—
45#1437C2216yes—
46#1437C2213yes—
47#1437C2216yescomplemented
48#1436C2213yescomplemented
49#1436C2213yescomplemented
50C22⋊#95C2216yes—
51C22⋊#95C2216yes—
52C26C2311yescharacteristic
53C23×Q8C2311yescharacteristic
54C23×D4C2316yes—
55C23×D4C2311yescharacteristic
56C24×C4—24no—
57C24×C4—46no—
58C24×C4—46no—
59C24×C4C2313yes—
60C24×C4C2313yes—
61C2×#124—46no—
62C2×C22⋊Q8—212no—
63C2×C22⋊Q8—46no—
64C2×C22⋊Q8—46no—
65C2×#120—28no—
66C2×#120—28no—
67C2×#120—28no—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2313yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2313yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2313yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2313yes—
83C22×C22⋊C4C2313yes—
84C22×C42C2311yescharacteristic
85C22×C42C2313yes—
86#400—46no—
87#397—212no—
88#394—224no—
89#394—46no—
90#392—212no—
91#386—412no—
92#386—412no—
93#385—212no—
94#385—212no—
95#382—46no—
96#380—212no—
97#379—24no—
98#379—28no—
99#379D418yes—
100C2×#95C2313yes—
101C2×#95C2313yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2313yes—
106C2×#95C2313yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C2316yes—
114C25—43no—
115C25—43no—
116C25—46no—
117C25C2×D414yes—
118C25#4118yes—
119C25C2412yes—
120C25C2413yes—
121C22×Q8—22no—
122C22×Q8—26no—
123C22×Q8—43no—
124C22×D4—212no—
125C22×D4—212no—
126C22×D4—212no—
127C22×D4—22no—
128C22×D4—43no—
129C22×D4—46no—
130C22×D4—46no—
131C22×D4—46no—
132C22×D4—46no—
133C22×D4—43no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4—26no—
138C23×C4—212no—
139C23×C4—212no—
140C23×C4—46no—
141C23×C4—43no—
142C23×C4—43no—
143C23×C4—46no—
144C23×C4—448no—
145C23×C4—448no—
146C23×C4—416no—
147C23×C4C2411yescharacteristic
148C23×C4C2413yes—
149C23×C4C2416yes—
150C23×C4C2412yes—
151C23×C4C2416yes—
152C23×C4C2413yes—
153C23×C4C2411yescharacteristic
154C23×C4C2413yes—
155C23×C4C2411yescharacteristic
156#124—416no—
157C22⋊Q8—448no—
158#120—432no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4—26no—
161C2×C4⋊C4—26no—
162C2×C4⋊C4—26no—
163C2×C4⋊C4—26no—
164C2×C4⋊C4—26no—
165C2×C4⋊C4—26no—
166C2×C4⋊C4—43no—
167C2×C4⋊C4—43no—
168C2×C4⋊C4—43no—
169C2×C4⋊C4—43no—
170C2×C4⋊C4—43no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4—26no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—212no—
177C2×C22⋊C4—212no—
178C2×C22⋊C4—212no—
179C2×C22⋊C4—212no—
180C2×C22⋊C4—212no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—46no—
183C2×C22⋊C4—43no—
184C2×C22⋊C4—43no—
185C2×C22⋊C4—43no—
186C2×C22⋊C4—43no—
187C2×C22⋊C4—46no—
188C2×C22⋊C4—46no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—46no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—46no—
196C2×C22⋊C4—46no—
197C2×C22⋊C4—46no—
198C2×C22⋊C4—46no—
199C2×C22⋊C4C2×D4112yes—
200C2×C22⋊C4C2×D4112yes—
201C2×C42—43no—
202C2×C42—43no—
203C2×C42—46no—
204#95—212no—
205#95—26no—
206#95—26no—
207#95—212no—
208#95—212no—
209#95—46no—
210#95—46no—
211#95—46no—
212#95—46no—
213#95—46no—
214#95—46no—
215#95—43no—
216#95—43no—
217#95—46no—
218C24—23no—
219C24—23no—
220C24—26no—
221C24—26no—
222C24—26no—
223C24—26no—
224C24—26no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—26no—
229C24—26no—
230C24—26no—
231C24—26no—
232C24—412no—
233C24—412no—
234C24—424no—
235C24—424no—
236C24—44no—
237C24—412no—
238C24—412no—
239C24—424no—
240C24—412no—
241C24#124116yescomplemented
242C24C22×D416yes—
243C24C2×#41112yes—
244C24C2511yescharacteristic
245C2×Q8—412no—
246C2×Q8—44no—
247C2×Q8—86no—
248C2×D4—424no—
249C2×D4—424no—
250C2×D4—44no—
251C2×D4—424no—
252C2×D4—86no—
253C2×D4—812no—
254C2×D4—812no—
255C2×D4—812no—
256C2×D4—812no—
257C2×D4—86no—
258C22×C4—21no—
259C22×C4—23no—
260C22×C4—224no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—26no—
265C22×C4—26no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—23no—
272C22×C4—26no—
273C22×C4—21no—
274C22×C4—26no—
275C22×C4—22no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—23no—
284C22×C4—23no—
285C22×C4—26no—
286C22×C4—23no—
287C22×C4—23no—
288C22×C4—26no—
289C22×C4—26no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—26no—
293C22×C4—26no—
294C22×C4—26no—
295C22×C4—26no—
296C22×C4—424no—
297C22×C4—412no—
298C22×C4—424no—
299C22×C4—424no—
300C22×C4—412no—
301C22×C4—424no—
302C22×C4—412no—
303C22×C4—4128no—
304C22×C4—412no—
305C22×C4—412no—
306C22×C4C22×D416yes—
307C22×C4C22×D412yes—
308C22×C4H(5,𝔽2)16yes—
309C22×C4H(5,𝔽2)16yes—
310C4⋊C4—412no—
311C4⋊C4—412no—
312C4⋊C4—412no—
313C4⋊C4—86no—
314C22⋊C4—412no—
315C22⋊C4—412no—
316C22⋊C4—412no—
317C22⋊C4—412no—
318C22⋊C4—48no—
319C22⋊C4—424no—
320C22⋊C4—48no—
321C22⋊C4—812no—
322C22⋊C4—812no—
323C22⋊C4—812no—
324C22⋊C4—812no—
325C22⋊C4—812no—
326C42—412no—
327C42—44no—
328C23—26no—
329C23—212no—
330C23—26no—
331C23—212no—
332C23—212no—
333C23—212no—
334C23—212no—
335C23—212no—
336C23—212no—
337C23—212no—
338C23—448no—
339C23—416no—
340C23—448no—
341C23—496no—
342C23—432no—
343C23—43no—
344C23—43no—
345C23—46no—
346C23—46no—
347C23—43no—
348C23—46no—
349C23—46no—
350C23—46no—
351C23—43no—
352C23—46no—
353C23—42no—
354C23—46no—
355C23—46no—
356C23—46no—
357C23—46no—
358C23—46no—
359C23—46no—
360C23—42no—
361C23—46no—
362C23—46no—
363C23—46no—
364C23—46no—
365C23—46no—
366C23C2×#124112yes—
367C23C23×D411yescharacteristic
368C23C22×#4112yes—
369C23C2×H(5,𝔽2)13yes—
370C23C2×H(5,𝔽2)13yes—
371C23#58516yes—
372Q8—88no—
373D4—88no—
374D4—848no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—26no—
380C2×C4—212no—
381C2×C4—212no—
382C2×C4—448no—
383C2×C4—448no—
384C2×C4—46no—
385C2×C4—41no—
386C2×C4—43no—
387C2×C4—43no—
388C2×C4—43no—
389C2×C4—43no—
390C2×C4—46no—
391C2×C4—43no—
392C2×C4—46no—
393C2×C4—43no—
394C2×C4—43no—
395C2×C4—46no—
396C2×C4—46no—
397C2×C4—46no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—46no—
401C2×C4—46no—
402C2×C4—46no—
403C2×C4—46no—
404C2×C4—46no—
405C2×C4—46no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—46no—
409C2×C4—43no—
410C2×C4—46no—
411C2×C4—43no—
412C2×C4—46no—
413C2×C4—46no—
414C2×C4—46no—
415C2×C4—43no—
416C2×C4—46no—
417C2×C4H(3,𝔽4)14yes—
418C22—224no—
419C22—212no—
420C22—464no—
421C22—46no—
422C22—46no—
423C22—412no—
424C22—412no—
425C22—412no—
426C22—412no—
427C22—412no—
428C22—86no—
429C22—82no—
430C22—86no—
431C22—86no—
432C22—86no—
433C22—82no—
434C22—82no—
435C22C22×#12411yescharacteristic
436C22#348413yes—
437C22#348513yes—
438C22#350516yes—
439C22#350516yes—
440C22C2×H(3,𝔽4)11yescharacteristic
441C22#352913yes—
442C22#353213yes—
443C22#353916yes—
444C22#354013yes—
445C4—24no—
446C4—46no—
447C4—42no—
448C4—46no—
449C4—86no—
450C4—81no—
451C4—83no—
452C4—86no—
453C4—82no—
454C2—412no—
455C2—162no—
456C2#3537713yes—
457C2#4543911yescharacteristic
458C2#4544113yes—
459C2#4550616yes—
460C2#4550912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2812C2
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×#2848C2semidirect
3. #15670C2semidirect
4. #15635C2semidirect
5. #15579C2semidirect
6. #15474C2semidirect
7. #15391C2semidirect
8. #15264C2semidirect
9. #14694C2semidirect
10. #14657C2semidirect
11. #13827C2semidirect
13. C22×#124C22semidirect
18. C2×#400C22semidirect
21. C2×#394C22semidirect
27. C2×#382C22semidirect
32. #1446C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1442C22semidirect
36. #1442C22semidirect
39. #1440C22semidirect
40. #1439C22semidirect
41. #1439C22semidirect
47. #1437C22semidirect
48. #1436C22semidirect
49. #1436C22semidirect
241. C24#124semidirect

Maximal subgroups

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

subgroupindex
C2×#28122
C2×#28482
#156702
#156352
#155792
#154742
#153912
#152642
#146942
#146572
#138272