Finite Group Databasebeta

← #2032502

Subgroups — #2032502

Order 512.

4086 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2677C213yesmaximal, complemented
2C2×#2848C213yesmaximal
3#16264C216yesmaximal, complemented
4#15467C211yesmaximal, characteristic
5#15430C213yesmaximal, complemented
6#15339C216yesmaximal, complemented
7#15284C213yesmaximal
8#15265C213yesmaximal, complemented
9#14409C213yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C22×C4⋊Q8C2213yescomplemented
12C22×C22⋊Q8C2216yes—
13C22×C22⋊Q8C2213yes—
14#2848—224no—
15#2677—224no—
16C2×#399C2213yes—
17C2×#398C2216yescomplemented
18C2×#398C2213yescomplemented
19C2×#395C2216yescomplemented
20C2×#395C2211yescharacteristic, complemented
21C2×#393C2213yes—
22C2×#388C2216yes—
23C2×#387C2216yescomplemented
24C2×#385C2216yes—
25C2×#385C2213yes—
26C2×#385C2216yes—
27C2×#385C2213yes—
28C2×#384C2211yescharacteristic, complemented
29C2×#380C2216yes—
30C2×#380C2213yes—
31C2×#379C2213yes—
32C2×#379C2213yes—
33#1446C2213yescomplemented
34#1445C2213yescomplemented
35#1444C2216yescomplemented
36#1443C2212yescomplemented
37#1442C2216yescomplemented
38#1441C2216yescomplemented
39#1441C2216yescomplemented
40#1440C2216yes—
41#1440C2216yes—
42#1440C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1437C2213yes—
46#1437C2213yes—
47#1436C2213yescomplemented
48#1436C2213yescomplemented
49C26C2311yescharacteristic
50C23×Q8C2313yes—
51C23×Q8C2311yescharacteristic
52C24×C4—212no—
53C24×C4—46no—
54C24×C4—42no—
55C24×C4C2313yes—
56C24×C4C2313yes—
57C2×C4⋊Q8—224no—
58C2×C4⋊Q8—46no—
59C2×C22⋊Q8—224no—
60C2×C22⋊Q8—224no—
61C2×C22⋊Q8—212no—
62C2×C22⋊Q8—412no—
63C2×C22⋊Q8—46no—
64C2×C22⋊Q8—46no—
65C22×C4⋊C4C2316yes—
66C22×C4⋊C4C2316yes—
67C22×C4⋊C4C2313yes—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2316yes—
72C22×C4⋊C4C2313yes—
73C22×C22⋊C4C2313yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2313yes—
79C22×C22⋊C4C2313yes—
80C22×C22⋊C4C2313yes—
81C22×C42C2313yes—
82C22×C42C2311yescharacteristic
83#399—212no—
84#398—224no—
85#398—46no—
86#395—224no—
87#395—42no—
88#393—212no—
89#388—224no—
90#387—224no—
91#385—212no—
92#385—224no—
93#385—224no—
94#385D4124yes—
95#384—42no—
96#380—212no—
97#380—224no—
98#379—212no—
99#379—212no—
100C2×#95C2316yes—
101C2×#95C2313yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2313yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2313yes—
108C2×#95C2313yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2312yes—
113C2×#95C2316yes—
114C25—26no—
115C25—212no—
116C25—43no—
117C25—41no—
118C25—42no—
119C25C2413yes—
120C22×Q8—26no—
121C22×Q8—212no—
122C22×Q8—26no—
123C22×Q8—26no—
124C22×Q8—43no—
125C22×Q8—46no—
126C22×Q8—43no—
127C22×Q8—41no—
128C23×C4—26no—
129C23×C4—212no—
130C23×C4—26no—
131C23×C4—212no—
132C23×C4—26no—
133C23×C4—212no—
134C23×C4—43no—
135C23×C4—43no—
136C23×C4—46no—
137C23×C4—448no—
138C23×C4—448no—
139C23×C4—416no—
140C23×C4C2×D4112yes—
141C23×C4C2413yes—
142C23×C4C2416yes—
143C23×C4C2413yes—
144C23×C4C2413yes—
145C23×C4C2416yes—
146C23×C4C2413yes—
147C23×C4C2411yescharacteristic
148C23×C4C2413yes—
149C4⋊Q8—448no—
150C22⋊Q8—496no—
151C22⋊Q8—448no—
152C2×C4⋊C4—212no—
153C2×C4⋊C4—212no—
154C2×C4⋊C4—26no—
155C2×C4⋊C4—212no—
156C2×C4⋊C4—26no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—212no—
159C2×C4⋊C4—212no—
160C2×C4⋊C4—212no—
161C2×C4⋊C4—212no—
162C2×C4⋊C4—212no—
163C2×C4⋊C4—26no—
164C2×C4⋊C4—26no—
165C2×C4⋊C4—26no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—26no—
168C2×C4⋊C4—43no—
169C2×C4⋊C4—46no—
170C2×C4⋊C4—43no—
171C2×C4⋊C4—43no—
172C2×C4⋊C4—46no—
173C2×C4⋊C4—43no—
174C2×C4⋊C4—43no—
175C2×C4⋊C4C2×D4124yes—
176C2×C22⋊C4—212no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—212no—
179C2×C22⋊C4—212no—
180C2×C22⋊C4—26no—
181C2×C22⋊C4—212no—
182C2×C22⋊C4—212no—
183C2×C22⋊C4—212no—
184C2×C22⋊C4—212no—
185C2×C22⋊C4—212no—
186C2×C22⋊C4—212no—
187C2×C22⋊C4—212no—
188C2×C22⋊C4—46no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—43no—
191C2×C22⋊C4—43no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—43no—
195C2×C22⋊C4—43no—
196C2×C22⋊C4C2×D4112yes—
197C2×C22⋊C4C2×D4112yes—
198C2×C42—212no—
199C2×C42—43no—
200C2×C42—41no—
201C2×C42—42no—
202#95—212no—
203#95—212no—
204#95—26no—
205#95—212no—
206#95—212no—
207#95—212no—
208#95—26no—
209#95—46no—
210#95—46no—
211#95—42no—
212#95—43no—
213#95—43no—
214#95—46no—
215#95C2×D4124yes—
216C24—26no—
217C24—26no—
218C24—23no—
219C24—23no—
220C24—26no—
221C24—26no—
222C24—26no—
223C24—224no—
224C24—412no—
225C24—424no—
226C24—424no—
227C24—412no—
228C24—412no—
229C24—424no—
230C24—44no—
231C24—44no—
232C24—412no—
233C24C22×D416yes—
234C242−1+4112yes—
235C24C2511yescharacteristic
236C2×Q8—424no—
237C2×Q8—412no—
238C2×Q8—412no—
239C2×Q8—412no—
240C2×Q8—812no—
241C2×Q8—86no—
242C2×Q8—86no—
243C2×Q8—82no—
244C22×C4—23no—
245C22×C4—26no—
246C22×C4—23no—
247C22×C4—224no—
248C22×C4—26no—
249C22×C4—23no—
250C22×C4—26no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—26no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—26no—
257C22×C4—23no—
258C22×C4—26no—
259C22×C4—26no—
260C22×C4—26no—
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—248no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—23no—
268C22×C4—26no—
269C22×C4—23no—
270C22×C4—26no—
271C22×C4—23no—
272C22×C4—23no—
273C22×C4—26no—
274C22×C4—23no—
275C22×C4—23no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—26no—
282C22×C4—412no—
283C22×C4—412no—
284C22×C4—424no—
285C22×C4—424no—
286C22×C4—412no—
287C22×C4—424no—
288C22×C4—412no—
289C22×C4—4128no—
290C22×C4—412no—
291C22×C4C22×D416yes—
292C22×C4C22×D4112yes—
293C22×C4C22×D416yes—
294C22×C4C22×D4112yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C4H(5,𝔽2)16yes—
297C4⋊C4—248no—
298C4⋊C4—424no—
299C4⋊C4—412no—
300C4⋊C4—412no—
301C4⋊C4—412no—
302C4⋊C4—424no—
303C4⋊C4—424no—
304C4⋊C4—86no—
305C22⋊C4—224no—
306C22⋊C4—412no—
307C22⋊C4—424no—
308C22⋊C4—424no—
309C22⋊C4—424no—
310C22⋊C4—424no—
311C22⋊C4—412no—
312C22⋊C4—412no—
313C42—412no—
314C42—44no—
315C23—212no—
316C23—26no—
317C23—26no—
318C23—212no—
319C23—212no—
320C23—212no—
321C23—212no—
322C23—212no—
323C23—212no—
324C23—212no—
325C23—496no—
326C23—448no—
327C23—416no—
328C23—448no—
329C23—432no—
330C23—46no—
331C23—43no—
332C23—46no—
333C23—46no—
334C23—43no—
335C23—43no—
336C23—46no—
337C23—43no—
338C23—46no—
339C23—46no—
340C23#536112yes—
341C23C23×D413yes—
342C23C2×H(5,𝔽2)13yes—
343C23C2×H(5,𝔽2)11yescharacteristic
344C23C2×2−1+416yes—
345C23#58512yes—
346Q8—824no—
347Q8—88no—
348C2×C4—212no—
349C2×C4—26no—
350C2×C4—212no—
351C2×C4—26no—
352C2×C4—212no—
353C2×C4—212no—
354C2×C4—212no—
355C2×C4—26no—
356C2×C4—26no—
357C2×C4—26no—
358C2×C4—212no—
359C2×C4—212no—
360C2×C4—448no—
361C2×C4—448no—
362C2×C4—43no—
363C2×C4—43no—
364C2×C4—46no—
365C2×C4—46no—
366C2×C4—46no—
367C2×C4—43no—
368C2×C4—41no—
369C2×C4—43no—
370C2×C4—46no—
371C2×C4—43no—
372C2×C4—46no—
373C2×C4—46no—
374C2×C4—46no—
375C2×C4—46no—
376C2×C4—46no—
377C2×C4—43no—
378C2×C4—43no—
379C2×C4—43no—
380C2×C4—46no—
381C2×C4—46no—
382C2×C4—46no—
383C2×C4—46no—
384C2×C4—46no—
385C2×C4—43no—
386C2×C4—43no—
387C2×C4—46no—
388C2×C4—43no—
389C2×C4—46no—
390C2×C4—46no—
391C2×C4D42112yes—
392C22—224no—
393C22—212no—
394C22—464no—
395C22—412no—
396C22—412no—
397C22—412no—
398C22—46no—
399C22—46no—
400C22—412no—
401C22—412no—
402C22C2×#53613yes—
403C22C2×D4213yes—
404C22#348413yes—
405C22#348613yes—
406C22#348916yes—
407C22#352911yescharacteristic
408C22#353211yescharacteristic
409C22#353816yes—
410C22#353813yes—
411C22#354116yes—
412C4—212no—
413C4—412no—
414C4—42no—
415C4—46no—
416C4—46no—
417C4—86no—
418C4—83no—
419C4—83no—
420C2—412no—
421C2#3670813yes—
422C2#3765116yes—
423C2#4545313yes—
424C2#5316812yes—
425C2#5899011yescharacteristic
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×#2677C2semidirect
3. #16264C2semidirect
5. #15430C2semidirect
6. #15339C2semidirect
8. #15265C2semidirect
9. #14409C2semidirect
11. C22×C4⋊Q8C22semidirect
17. C2×#398C22semidirect
18. C2×#398C22semidirect
19. C2×#395C22semidirect
20. C2×#395C22semidirect
23. C2×#387C22semidirect
28. C2×#384C22semidirect
33. #1446C22semidirect
34. #1445C22semidirect
35. #1444C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
47. #1436C22semidirect
48. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26772
C2×#28482
#162642
#154672
#154302
#153392
#152842
#152652
#144092