Finite Group Databasebeta

← #27182741

Subgroups — #27182741

Order 12928.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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#20265820C212yesmaximal, complemented
3#20265420C212yesmaximal, complemented
4#20265389C212yesmaximal, complemented
5#20265338C212yesmaximal, complemented
6#20265309C212yesmaximal, complemented
7C4×D808C211yesmaximal, characteristic, complemented
8#20265256C211yesmaximal, characteristic, complemented
9#495×C101C211yesmaximal, characteristic, complemented
10C2×#13108075C2212yes—
11C2×D4×D101C2212yescomplemented
12#15692282—28no—
13C2×#13108061C2212yescomplemented
14C2×D808C2211yescharacteristic
15C2×#13108059C2212yes—
16C2×#13108057C2212yescomplemented
17#15692237—22no—
18#15692231C2212yescomplemented
19#15692228—22no—
20#15692226C2212yes—
21#15692225—22no—
22C4×D404C2211yescharacteristic
23C4×D404C2212yescomplemented
24C22⋊C4×D101—22no—
25#15692189C2211yescharacteristic, complemented
26C4⋊#11717579C2211yescharacteristic, complemented
27#15692179C2212yescomplemented
28#15692177C2212yescomplemented
29#15692176C2212yescomplemented
30#15692167C2212yescomplemented
31C2×Q16×C101C2211yescharacteristic, complemented
32C2×SD16×C101C2212yescomplemented
33C2×D8×C101C2211yescharacteristic
34#124×C101C2212yescomplemented
35C4×C8×C101C2211yescharacteristic, complemented
36C23×D101—22no—
37#13108075—24no—
38#13108075—42no—
39C2×D404—22no—
40C2×D404D411yescharacteristic
41C2×D404C2312yes—
42C2×C22⋊D101—22no—
43C2×C22⋊D101—22no—
44D4×D101—24no—
45D4×D101—42no—
46C2×C4×D101—22no—
47C2×C4×D101—22no—
48C2×C4×D101C2312yes—
49#13108061—24no—
50D808D414yescomplemented
51#13108059—24no—
52#13108057—24no—
53C2×#11717579C2312yes—
54C4⋊Dic101D411yescharacteristic
55C4⋊Dic101C2312yes—
56#13108053—22no—
57C22⋊Dic101—22no—
58#13108051—22no—
59#13108051—22no—
60#13108051D412yes—
61C2×Q8×C101C2312yes—
62C2×D4×C101C2312yes—
63Q16×C101—41no—
64SD16×C101—24no—
65D8×C101—41no—
66C2×C8×C101C2311yescharacteristic
67C2×C8×C101C2311yescharacteristic
68C22⋊C4×C101—22no—
69C22⋊C4×C101—22no—
70C42×C101C2311yescharacteristic
71C23×C101—22no—
72C22×D101—22no—
73C22×D101—44no—
74C22×D101—44no—
75C22×D101—82no—
76C22×D101C2×D412yes—
77Q8×C101—22no—
78Q8×C101—22no—
79D4×C101—22no—
80D4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2411yescharacteristic
85C22⋊D101—44no—
86C22⋊D101—82no—
87C2×Dic101—22no—
88C2×Dic101C2×D412yes—
89D404—22no—
90D404—41no—
91D404—44no—
92D404C2×D414yes—
93C4×D101—22no—
94C4×D101—22no—
95C4×D101—44no—
96C4×D101—82no—
97C8×C101—21no—
98C8×C101C2×D412yes—
99#11717579—22no—
100C22×C101—42no—
101C22×C101—42no—
102C22×C101—42no—
103C22×C101C22×D411yescharacteristic
104C2×D101—22no—
105C2×D101—24no—
106C2×D101—22no—
107C2×D101—22no—
108C2×D101—42no—
109C2×D101—42no—
110C2×D101—88no—
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—42no—
114C4×C101C22×D411yescharacteristic
115C4×C101H(5,𝔽2)11yescharacteristic
116Dic101—22no—
117Dic101—42no—
118C2×C101—82no—
119C2×C101D4211yescharacteristic
120C2×C101#57511yescharacteristic
121C2×C101#57611yescharacteristic
122D101—42no—
123D101—44no—
124#3273—1011nomaximal, Sylow
125C101#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—1012no—
127#546—1012no—
128#495D10111yescharacteristic, complemented
129#468—1012no—
130#463—1012no—
131#452—1012no—
132#447—1012no—
133C4×D8—1011no—
134#436—1011no—
135C2×#41—1012no—
136C22×D4—1012no—
137AGL(1,ℤ/8)—2028no—
138C2×Q16C2×D10111yescharacteristic, complemented
139C2×SD16—1012no—
140C2×SD16C2×D10112yescomplemented
141C2×D8—1012no—
142C2×D8—1011no—
143C2×D8C2×D10111yescharacteristic
144C2×M16—1012no—
145#124C2×D10112yescomplemented
146#123—2022no—
147C22⋊Q8—1012no—
148#121—1012no—
149#121—2022no—
150#120—2022no—
151C4×D4—1011no—
152C4×D4—1012no—
153C2×C22⋊C4—2022no—
154#107—1011no—
155C4⋊C8—1011no—
156#103—1012no—
157#102—1012no—
158#102—1012no—
159C22⋊C8—1012no—
160C4×C8C2×D10111yescharacteristic, complemented
161C24—2022no—
162#41—4042no—
163#41—2024no—
164C2×Q8C22×D10112yes—
165C2×D4—1011no—
166C2×D4—1012no—
167C2×D4—4042no—
168C2×D4—2022no—
169C2×D4—2022no—
170C2×D4—2024no—
171C2×D4—2022no—
172C2×D4C22×D10112yes—
173C22×C4—1012no—
174C22×C4—2022no—
175C22×C4—2022no—
176Q16—41no—
177SD16—24no—
178SD16—2024no—
179D8—41no—
180D8—1014no—
181D8—2024no—
182M16—2024no—
183C2×C8—1012no—
184C2×C8C22×D10111yescharacteristic
185C2×C8C22×D10111yescharacteristic
186C4⋊C4—1011no—
187C4⋊C4—1012no—
188C4⋊C4—2022no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—1012no—
192C22⋊C4—2022no—
193C22⋊C4—2022no—
194C22⋊C4—2022no—
195C42C22×D10111yescharacteristic
196C23—22no—
197C23—8082no—
198C23—1012no—
199C23—4044no—
200C23—4044no—
201C23—2022no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—22no—
206D4—8082no—
207D4—1014no—
208D4—4041no—
209D4—4044no—
210D4—4044no—
211D4—2022no—
212C2×C4—22no—
213C2×C4—8082no—
214C2×C4—1012no—
215C2×C4—4044no—
216C2×C4—2022no—
217C2×C4—2022no—
218C2×C4—2022no—
219C2×C4D4×D10111yescharacteristic
220C2×C4D4×D10111yescharacteristic
221C2×C4C23×D10111yescharacteristic
222C8—21no—
223C8—2022no—
224C8D4×D10112yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—8088no—
229C22—4042no—
230C22—4042no—
231C22—2024no—
232C22—2022no—
233C22—2022no—
234C22—2022no—
235C22C2×D4×D10111yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—4042no—
240C4—2022no—
241C4C2×D4×D10111yescharacteristic
242C4#1569230011yescharacteristic
243C2—82no—
244C2—4042no—
245C2—4044no—
246C2#2026581211yescharacteristic
247C2#2026597411yescharacteristic
248C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#495×C101C2
socleC22×C101C22×D4
hypercenter#495D101
hypocenterC101#3273
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. #20265820C2semidirect
3. #20265420C2semidirect
4. #20265389C2semidirect
5. #20265338C2semidirect
6. #20265309C2semidirect
7. C4×D808C2semidirect
8. #20265256C2semidirect
9. #495×C101C2semidirect
11. C2×D4×D101C22semidirect
13. C2×#13108061C22semidirect
16. C2×#13108057C22semidirect
18. #15692231C22semidirect
23. C4×D404C22semidirect
25. #15692189C22semidirect
26. C4⋊#11717579C22semidirect
27. #15692179C22semidirect
28. #15692177C22semidirect
29. #15692176C22semidirect
30. #15692167C22semidirect
31. C2×Q16×C101C22semidirect
32. C2×SD16×C101C22semidirect
34. #124×C101C22semidirect
35. C4×C8×C101C22semidirect
50. D808D4semidirect
125. C101#3273semidirect
128. #495D101semidirect
138. C2×Q16C2×D101semidirect
140. C2×SD16C2×D101semidirect
145. #124C2×D101semidirect
160. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
#202658202
#202654202
#202653892
#202653382
#202653092
C4×D8082
#202652562
#495×C1012
#3273101