Finite Group Databasebeta

← #27180359

Subgroups — #27180359

Order 12928.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15692281C212yesmaximal, complemented
2#20265688C211yesmaximal, characteristic, complemented
3#20265328C214yesmaximal, complemented
4#20265311C214yesmaximal, complemented
5#20265182C212yesmaximal, complemented
6#20265141C211yesmaximal, characteristic, complemented
7C23×D4×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic, complemented
9#15692281—28no—
10C2×#13108062C2214yescomplemented
11C2×#13108059C2214yescomplemented
12C2×#13108058C2212yescomplemented
13#15692238—21no—
14#15692234C2212yescomplemented
15#15692229C2212yescomplemented
16#15692224—21no—
17C4×C22⋊D101—22no—
18#15692214C2211yescharacteristic, complemented
19#15692176C2218yescomplemented
20C22⋊#11717579C2214yescomplemented
21C25×C101—21no—
22C22×D4×C101—28no—
23C22×D4×C101—22no—
24C22×D4×C101—21no—
25C22×D4×C101C2214yes—
26C22×D4×C101C2212yes—
27C23×C4×C101C2211yescharacteristic
28#13108074—24no—
29C2×Dic202C2311yescharacteristic, complemented
30C2×D404C2311yescharacteristic, complemented
31C2×C22⋊D101—21no—
32C2×C22⋊D101—22no—
33C2×C4×D101—21no—
34#13108062—28no—
35#13108059—28no—
36#13108058—24no—
37C2×#11717579C2314yescomplemented
38C4⋊Dic101C2312yescomplemented
39#13108053—22no—
40C22⋊Dic101—21no—
41C22⋊Dic101—22no—
42#13108051—22no—
43C4×Dic101—21no—
44C24×C101—28no—
45C24×C101—24no—
46C24×C101—22no—
47C24×C101—48no—
48C24×C101D411yescharacteristic
49C2×D4×C101—28no—
50C2×D4×C101—216no—
51C2×D4×C101—216no—
52C2×D4×C101—24no—
53C2×D4×C101—22no—
54C2×D4×C101—24no—
55C2×D4×C101D418yescomplemented
56C2×D4×C101C2314yes—
57C22×C4×C101—24no—
58C22×C4×C101D412yes—
59C22×C4×C101D411yescharacteristic
60C22×C4×C101C2312yes—
61C22×C4×C101C2311yescharacteristic
62C23×C101—22no—
63C23×C101—24no—
64C23×C101—216no—
65C23×C101—28no—
66C23×C101—24no—
67C23×C101—432no—
68C23×C101—48no—
69C23×C101—48no—
70C23×C101—48no—
71C23×C101C2×D411yescharacteristic
72C23×C101C2×D412yes—
73C22×D101—21no—
74D4×C101—216no—
75D4×C101—28no—
76D4×C101—24no—
77D4×C101C2×D418yes—
78C2×C4×C101—22no—
79C2×C4×C101—24no—
80C2×C4×C101—24no—
81C2×C4×C101C2×D412yes—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D414yes—
84C2×C4×C101C2411yescharacteristic
85C22⋊D101—42no—
86C22⋊D101—44no—
87C2×Dic101—22no—
88C2×Dic101—21no—
89D404—22no—
90C4×D101—42no—
91Dic202—22no—
92#11717579—24no—
93C22×C101—28no—
94C22×C101—22no—
95C22×C101—22no—
96C22×C101—22no—
97C22×C101—28no—
98C22×C101—416no—
99C22×C101—432no—
100C22×C101—48no—
101C22×C101#12012yes—
102C22×C101AGL(1,ℤ/8)14yescomplemented
103C22×C101C22×D411yescharacteristic
104C2×D101—41no—
105C2×D101—42no—
106C4×C101—22no—
107C4×C101#12012yes—
108C4×C101C22×D412yes—
109Dic101—41no—
110Dic101—42no—
111C2×C101—22no—
112C2×C101—24no—
113C2×C101—48no—
114C2×C101C2×#12011yescharacteristic
115C2×C101C2×AGL(1,ℤ/8)12yes—
116D101—81no—
117#3003—1011nomaximal, Sylow
118C101#300311yescharacteristic, complemented, Sylow
119C23×D4D10111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1012no—
121#525—1011no—
122#450—1014no—
123#447—1014no—
124#418—1012no—
125#407—1011no—
126C25—21no—
127C2×#41—1011no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D10114yes—
132C22×D4C2×D10112yes—
133C23×C4C2×D10111yescharacteristic
134AGL(1,ℤ/8)—2028no—
135C2×SD16—1014no—
136C2×D8—1014no—
137C2×M16—1012no—
138#123—2021no—
139C22⋊Q8—1012no—
140#121—1012no—
141#120—2021no—
142C4×D4—2022no—
143#117—1011no—
144#102—1018no—
145C22⋊C8—1014no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D10111yescharacteristic
151#41—2024no—
152C2×Q8—1011no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—1011no—
160C2×D4—2022no—
161C2×D4—2021no—
162C2×D4C22⋊D10118yescomplemented
163C2×D4C22×D10114yes—
164C22×C4—24no—
165C22×C4—2021no—
166C22×C4C22⋊D10112yes—
167C22×C4C22⋊D10111yescharacteristic
168C22×C4C22×D10112yes—
169C22×C4C22×D10111yescharacteristic
170SD16—2028no—
171D8—2028no—
172M16—2024no—
173C2×C8—1014no—
174C4⋊C4—1012no—
175C4⋊C4—2022no—
176C22⋊C4—2021no—
177C22⋊C4—2022no—
178C22⋊C4—2022no—
179C42—2021no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—24no—
184C23—28no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—2021no—
190C23C2×C22⋊D10111yescharacteristic
191C23C2×C22⋊D10112yes—
192Q8—2022no—
193D4—216no—
194D4—24no—
195D4—28no—
196D4—4042no—
197D4—4044no—
198D4—2022no—
199D4C2×C22⋊D10118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—4042no—
204C2×C4—2022no—
205C2×C4—2021no—
206C2×C4C2×C22⋊D10112yes—
207C2×C4C2×C22⋊D10114yes—
208C2×C4C2×C22⋊D10111yescharacteristic
209C2×C4C23×D10111yescharacteristic
210C8—2024no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—4041no—
220C22—4042no—
221C22#1569222412yes—
222C22#1569228114yescomplemented
223C22C22×C22⋊D10111yescharacteristic
224C4—22no—
225C4—4041no—
226C4—4042no—
227C4#1569222412yes—
228C4C22×C22⋊D10112yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—8081no—
233C2C2×#1569222411yescharacteristic
234C2C2×#1569228112yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC23×D4×C101C2
socleC22×C101C22×D4
hypercenterC23×D4D101
hypocenterC101#3003
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. #20265688C2semidirect
3. #20265328C2semidirect
4. #20265311C2semidirect
5. #20265182C2semidirect
6. #20265141C2semidirect
7. C23×D4×C101C2semidirect
8. C2×#13108074C22semidirect
10. C2×#13108062C22semidirect
11. C2×#13108059C22semidirect
12. C2×#13108058C22semidirect
14. #15692234C22semidirect
15. #15692229C22semidirect
18. #15692214C22semidirect
19. #15692176C22semidirect
20. C22⋊#11717579C22semidirect
29. C2×Dic202C23semidirect
30. C2×D404C23semidirect
37. C2×#11717579C23semidirect
38. C4⋊Dic101C23semidirect
55. C2×D4×C101D4semidirect
102. C22×C101AGL(1,ℤ/8)semidirect
118. C101#3003semidirect
119. C23×D4D101semidirect
162. C2×D4C22⋊D101semidirect
222. C22#15692281semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
#202656882
#202653282
#202653112
#202651822
#202651412
C23×D4×C1012
#3003101