Finite Group Databasebeta

← #27180496

Subgroups — #27180496

Order 12928.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#20265975C212yesmaximal, complemented
2#20265751C211yesmaximal, characteristic, complemented
3#20265360C214yesmaximal, complemented
4#20265350C214yesmaximal, complemented
5#20265194C212yesmaximal, complemented
6C2×#15692174C211yesmaximal, characteristic, complemented
7C2×2−1+4×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9#15692286—22no—
10#15692285C2214yescomplemented
11#15692281—22no—
12#15692280C2214yescomplemented
13#15692275—24no—
14C2×#13108058C2212yes—
15#15692253C2212yescomplemented
16#15692244C2212yescomplemented
17#15692238—22no—
18#15692231—22no—
19#15692214C2211yescharacteristic
20#15692182C2218yescomplemented
21#15692174C2214yescomplemented
222−1+4×C101—24no—
232−1+4×C101—22no—
242−1+4×C101C2214yes—
25C2×#41×C101—22no—
26C2×#41×C101—22no—
27C2×#41×C101C2212yes—
28C22×Q8×C101—22no—
29C22×Q8×C101C2211yescharacteristic
30#13108074—22no—
31#13108074—22no—
32C2×Dic202C2311yescharacteristic
33C2×D404C2311yescharacteristic
34C2×C22⋊D101—21no—
35C2×C4×D101—21no—
36#13108064—24no—
37#13108062—24no—
38#13108061—24no—
39#13108059—24no—
40#13108058—22no—
41#13108058C2314yes—
42C2×#11717579—22no—
43C4⋊Dic101—21no—
44#13108053—24no—
45C22⋊Dic101—21no—
46#13108051—24no—
47C4×Dic101C2312yes—
48#41×C101—28no—
49#41×C101—28no—
50#41×C101—28no—
51#41×C101—28no—
52#41×C101—22no—
53#41×C101—22no—
54#41×C101D414yes—
55#41×C101C2314yes—
56C2×Q8×C101—22no—
57C2×Q8×C101—28no—
58C2×Q8×C101—28no—
59C2×Q8×C101—22no—
60C2×Q8×C101—22no—
61C2×Q8×C101D412yes—
62C2×Q8×C101D412yes—
63C2×Q8×C101C2312yes—
64C2×D4×C101—22no—
65C2×D4×C101—22no—
66C2×D4×C101D412yes—
67C22×C4×C101—24no—
68C22×C4×C101—22no—
69C22×C4×C101D412yes—
70C22×C4×C101C2311yescharacteristic
71C23×C101—22no—
72C23×C101C2×D411yescharacteristic
73C22×D101—21no—
74Q8×C101—28no—
75Q8×C101—22no—
76Q8×C101—24no—
77Q8×C101—24no—
78Q8×C101C2×D414yes—
79D4×C101—28no—
80D4×C101—24no—
81D4×C101—24no—
82D4×C101—22no—
83D4×C101C2×D414yes—
84C2×C4×C101—216no—
85C2×C4×C101—22no—
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101—24no—
89C2×C4×C101—24no—
90C2×C4×C101C2×D414yes—
91C2×C4×C101C2×D411yescharacteristic
92C2×C4×C101C2×D412yes—
93C2×C4×C101C2×D412yes—
94C2×C4×C101C2411yescharacteristic
95C22⋊D101—41no—
96C22⋊D101—41no—
97C2×Dic101—22no—
98C2×Dic101—21no—
99D404—22no—
100C4×D101—42no—
101Dic202—22no—
102#11717579—24no—
103C22×C101—24no—
104C22×C101—41no—
105C22×C101—44no—
106C22×C101#12011yescharacteristic
107C22×C101#12011yescharacteristic
108C22×C101C22×D411yescharacteristic
109C2×D101—42no—
110C2×D101—41no—
111C4×C101—24no—
112C4×C101—24no—
113C4×C101#12012yes—
114C4×C101C22×D412yes—
115Dic101—42no—
116Dic101—41no—
117C2×C101—21no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101—44no—
121C2×C101C2×#12011yescharacteristic
122D101—81no—
123#3020—1011nomaximal, Sylow
124C101#302011yescharacteristic, complemented, Sylow
125C2×2−1+4D10111yescharacteristic, complemented
126#575—1012no—
127#536—1011no—
128#456—1014no—
129#454—1014no—
130#421—1012no—
131C2×#101—1011no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D10114yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1011no—
138C2×#41C2×D10112yes—
139C22×Q8—22no—
140C22×Q8C2×D10111yescharacteristic
141#137—1014no—
142#137—2022no—
143AGL(1,ℤ/8)—1014no—
144AGL(1,ℤ/8)—2022no—
145#135—2024no—
146C2×M16—1012no—
147C4⋊Q8—1012no—
148#124—1012no—
149#123—2022no—
150C22⋊Q8—2022no—
151#117—1011no—
152#104—1018no—
153#101—1014no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—2022no—
161#41—2022no—
162#41C22⋊D10114yes—
163#41C22×D10114yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1011no—
170C2×Q8C22⋊D10112yes—
171C2×Q8C22⋊D10112yes—
172C2×Q8C22×D10112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—1011no—
176C2×D4—2021no—
177C2×D4C22⋊D10112yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—2021no—
181C22×C4C22⋊D10112yes—
182C22×C4C22×D10111yescharacteristic
183Q16—2024no—
184SD16—2024no—
185SD16—2024no—
186D8—2024no—
187M16—1014no—
188M16—2022no—
189C2×C8—2022no—
190C4⋊C4—2024no—
191C4⋊C4—2021no—
192C22⋊C4—2021no—
193C22⋊C4—2024no—
194C42—1012no—
195C23—22no—
196C23—2021no—
197C23C2×C22⋊D10111yescharacteristic
198Q8—28no—
199Q8—24no—
200Q8—22no—
201Q8—24no—
202Q8—2022no—
203Q8C2×C22⋊D10114yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—4041no—
209D4—4041no—
210D4—2022no—
211D4C2×C22⋊D10114yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—4042no—
219C2×C4—2022no—
220C2×C4—2021no—
221C2×C4C2×C22⋊D10114yes—
222C2×C4C2×C22⋊D10112yes—
223C2×C4C2×C22⋊D10112yes—
224C2×C4C2×C22⋊D10111yescharacteristic
225C2×C4C23×D10111yescharacteristic
226C8—2024no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—4042no—
231C22—4041no—
232C22#1569222411yescharacteristic
233C22#1569222411yescharacteristic
234C22C22×C22⋊D10111yescharacteristic
235C4—24no—
236C4—24no—
237C4—4042no—
238C4—4041no—
239C4#1569222412yes—
240C4C22×C22⋊D10112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—8081no—
246C2C2×#1569222411yescharacteristic
subgroupG/H
centerC2C2×#15692224
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×2−1+4×C101C2
socleC2×C101C2×#120
hypercenterC2×2−1+4D101
hypocenterC101#3020
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20265975C2semidirect
2. #20265751C2semidirect
3. #20265360C2semidirect
4. #20265350C2semidirect
5. #20265194C2semidirect
6. C2×#15692174C2semidirect
7. C2×2−1+4×C101C2semidirect
10. #15692285C22semidirect
12. #15692280C22semidirect
15. #15692253C22semidirect
16. #15692244C22semidirect
20. #15692182C22semidirect
21. #15692174C22semidirect
124. C101#3020semidirect
125. C2×2−1+4D101semidirect

Maximal subgroups

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

subgroupindex
#202659752
#202657512
#202653602
#202653502
#202651942
C2×#156921742
C2×2−1+4×C1012
#3020101