Finite Group Databasebeta

← #27182753

Subgroups — #27182753

Order 12928.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C212yesmaximal, complemented
2#20265849C211yesmaximal, characteristic, complemented
3D4×D404C211yesmaximal, characteristic, complemented
4#20265554C211yesmaximal, characteristic, complemented
5#20265431C212yesmaximal, complemented
6#20265377C211yesmaximal, characteristic, complemented
7#20265372C211yesmaximal, characteristic, complemented
8#20265325C212yesmaximal, complemented
9#20265312C212yesmaximal, complemented
10#20265256C211yesmaximal, characteristic, complemented
11C4×SD16×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yescomplemented
13C22×D404—21no—
14C2×D4×D101C2212yes—
15#15692282—28no—
16C2×#13108061C2212yescomplemented
17C2×D808C2212yescomplemented
18C2×D808C2212yescomplemented
19C2×#13108059C2212yes—
20C2×#13108057C2212yescomplemented
21#15692252—21no—
22#15692252C2211yescharacteristic
23#15692252C2211yescharacteristic, complemented
24#15692230—22no—
25#15692230C2212yes—
26#15692230C2212yescomplemented
27#15692229—21no—
28#15692225—22no—
29C4×D404—21no—
30C4×D404C2211yescharacteristic
31C4⋊#11717579C2211yescharacteristic, complemented
32#15692175C2212yescomplemented
33#15692175C2212yescomplemented
34#15692167C2212yescomplemented
35C2×SD16×C101C2211yescharacteristic
36C4×Q8×C101C2211yescharacteristic, complemented
37C4×D4×C101C2211yescharacteristic
38#106×C101C2211yescharacteristic, complemented
39#103×C101C2211yescharacteristic, complemented
40#102×C101C2211yescharacteristic
41C4×C8×C101C2211yescharacteristic, complemented
42C23×D101—22no—
43#13108075—24no—
44#13108075—42no—
45C2×D404—22no—
46C2×D404—22no—
47C2×D404—21no—
48C2×D404—21no—
49C2×D404—22no—
50C2×D404—21no—
51C2×D404—21no—
52C2×D404—44no—
53C2×D404D411yescharacteristic
54C2×D404C2312yes—
55C2×D404C2312yes—
56C2×C22⋊D101—22no—
57D4×D101—24no—
58D4×D101—42no—
59C2×C4×D101—22no—
60C2×C4×D101C2312yes—
61#13108061—24no—
62D808—24no—
63D808—42no—
64#13108059—24no—
65#13108057—24no—
66C2×#11717579C2312yes—
67C4⋊Dic101D411yescharacteristic
68#13108051—22no—
69#13108051—22no—
70#13108051D412yes—
71C2×Q8×C101C2311yescharacteristic
72C2×D4×C101C2311yescharacteristic
73C22×C4×C101—21no—
74SD16×C101D414yescomplemented
75C2×C8×C101C2311yescharacteristic
76C2×C8×C101C2311yescharacteristic
77C4⋊C4×C101—21no—
78C4⋊C4×C101C2311yescharacteristic
79C4⋊C4×C101C2311yescharacteristic
80C22⋊C4×C101—21no—
81C42×C101—21no—
82C42×C101C2311yescharacteristic
83C23×C101—21no—
84C22×D101—22no—
85C22×D101—22no—
86C22×D101—44no—
87C22×D101—44no—
88C22×D101—82no—
89C22×D101C2×D412yes—
90Q8×C101—21no—
91Q8×C101C2×D412yescomplemented
92D4×C101—21no—
93D4×C101C2×D412yes—
94C2×C4×C101—21no—
95C2×C4×C101—21no—
96C2×C4×C101—21no—
97C2×C4×C101—42no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2411yescharacteristic
101C22⋊D101—44no—
102C2×Dic101C2×D412yes—
103D404—22no—
104D404—22no—
105D404—22no—
106D404—22no—
107D404—22no—
108D404—44no—
109D404—44no—
110D404—42no—
111D404—42no—
112D404—81no—
113D404—81no—
114D404—82no—
115C4×D101—22no—
116C4×D101—22no—
117C4×D101—44no—
118C8×C101—21no—
119C8×C101C2×D412yescomplemented
120#11717579—22no—
121C22×C101—22no—
122C22×C101—41no—
123C22×C101—41no—
124C22×C101C22×D411yescharacteristic
125C2×D101—22no—
126C2×D101—22no—
127C2×D101—22no—
128C2×D101—42no—
129C2×D101—42no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—42no—
133C2×D101—42no—
134C2×D101—88no—
135C4×C101—22no—
136C4×C101—21no—
137C4×C101—41no—
138C4×C101—41no—
139C4×C101AGL(1,ℤ/8)12yescomplemented
140C4×C101C22×D411yescharacteristic
141C4×C101H(5,𝔽2)11yescharacteristic
142Dic101—22no—
143C2×C101—42no—
144C2×C101D4211yescharacteristic
145C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C101#57611yescharacteristic
147D101—42no—
148D101—82no—
149D101—82no—
150#3274—1011nomaximal, Sylow
151C101#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1012no—
153#550—1011no—
154D42—1011no—
155#493—1011no—
156#469—1012no—
157#460—1011no—
158#459—1011no—
159#449—1012no—
160#447—1012no—
161C4×SD16D10111yescharacteristic, complemented
162#436—1011no—
163C2×#41—1012no—
164C22×D4—1012no—
165C22×D4—2021no—
166AGL(1,ℤ/8)—2028no—
167C2×SD16—1012no—
168C2×SD16C2×D10111yescharacteristic
169C2×D8—1012no—
170C2×D8—1012no—
171C2×D8—1012no—
172C2×M16—1012no—
173#127—1011no—
174#127—1011no—
175#127—2021no—
176#121—1012no—
177#121—1012no—
178#121—2022no—
179#121—2021no—
180#120—2022no—
181C4×Q8C2×D10111yescharacteristic, complemented
182C4×D4—1011no—
183C4×D4—2021no—
184C4×D4C2×D10111yescharacteristic
185#106C2×D10111yescharacteristic, complemented
186C4⋊C8—1011no—
187#103C2×D10111yescharacteristic, complemented
188#102—1012no—
189#102—1012no—
190#102C2×D10111yescharacteristic
191C22⋊C8—1012no—
192C4×C8C2×D10111yescharacteristic, complemented
193C24—2022no—
194#41—4042no—
195#41—2024no—
196C2×Q8C22×D10111yescharacteristic
197C2×D4—1011no—
198C2×D4—1012no—
199C2×D4—1012no—
200C2×D4—4044no—
201C2×D4—4042no—
202C2×D4—2022no—
203C2×D4—2022no—
204C2×D4—2022no—
205C2×D4—2022no—
206C2×D4—2024no—
207C2×D4—2021no—
208C2×D4—2021no—
209C2×D4—2021no—
210C2×D4—2021no—
211C2×D4C22×D10111yescharacteristic
212C22×C4—21no—
213C22×C4—1012no—
214C22×C4—2022no—
215SD16—2024no—
216SD16D40414yescomplemented
217D8—4042no—
218D8—2024no—
219D8—2024no—
220M16—2024no—
221C2×C8—1012no—
222C2×C8C22×D10111yescharacteristic
223C2×C8C22×D10111yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—1011no—
226C4⋊C4C22×D10111yescharacteristic
227C4⋊C4C22×D10111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—1012no—
230C22⋊C4—2022no—
231C22⋊C4—2022no—
232C42—21no—
233C42C22×D10111yescharacteristic
234C23—21no—
235C23—8082no—
236C23—1012no—
237C23—4044no—
238C23—4044no—
239C23—2022no—
240C23—2022no—
241Q8—21no—
242Q8C2×D40412yescomplemented
243D4—21no—
244D4—8081no—
245D4—8081no—
246D4—8082no—
247D4—4044no—
248D4—4044no—
249D4—4044no—
250D4—4042no—
251D4—4042no—
252D4—2022no—
253D4—2022no—
254D4—2022no—
255D4—2022no—
256D4—2022no—
257D4C2×D40412yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1012no—
263C2×C4—4044no—
264C2×C4—2022no—
265C2×C4—2022no—
266C2×C4D4×D10111yescharacteristic
267C2×C4D4×D10111yescharacteristic
268C2×C4C23×D10111yescharacteristic
269C8—21no—
270C8—2022no—
271C8C2×D40412yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—8088no—
276C22—4042no—
277C22—4042no—
278C22—4042no—
279C22—4042no—
280C22—4042no—
281C22—4042no—
282C22—2022no—
283C22—2022no—
284C22—2022no—
285C22C2×D4×D10111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—2022no—
291C4#1569228212yescomplemented
292C4C22×D40411yescharacteristic
293C4#1569230111yescharacteristic
294C2—42no—
295C2—8082no—
296C2—8082no—
297C2—4042no—
298C2D4×D40411yescharacteristic
299C2C2×#1569228211yescharacteristic
300C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×SD16×C101C2
socleC22×C101C22×D4
hypercenterC4×SD16D101
hypocenterC101#3274
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. #20265849C2semidirect
3. D4×D404C2semidirect
4. #20265554C2semidirect
5. #20265431C2semidirect
6. #20265377C2semidirect
7. #20265372C2semidirect
8. #20265325C2semidirect
9. #20265312C2semidirect
10. #20265256C2semidirect
11. C4×SD16×C101C2semidirect
12. C2×#13108075C22semidirect
16. C2×#13108061C22semidirect
17. C2×D808C22semidirect
18. C2×D808C22semidirect
20. C2×#13108057C22semidirect
23. #15692252C22semidirect
26. #15692230C22semidirect
31. C4⋊#11717579C22semidirect
32. #15692175C22semidirect
33. #15692175C22semidirect
34. #15692167C22semidirect
36. C4×Q8×C101C22semidirect
38. #106×C101C22semidirect
39. #103×C101C22semidirect
41. C4×C8×C101C22semidirect
74. SD16×C101D4semidirect
91. Q8×C101C2×D4semidirect
119. C8×C101C2×D4semidirect
139. C4×C101AGL(1,ℤ/8)semidirect
151. C101#3274semidirect
161. C4×SD16D101semidirect
181. C4×Q8C2×D101semidirect
185. #106C2×D101semidirect
187. #103C2×D101semidirect
192. C4×C8C2×D101semidirect
216. SD16D404semidirect
242. Q8C2×D404semidirect
271. C8C2×D404semidirect
291. C4#15692282semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
#202658492
D4×D4042
#202655542
#202654312
#202653772
#202653722
#202653252
#202653122
#202652562
C4×SD16×C1012
#3274101