Finite Group Databasebeta

← #27181598

Subgroups — #27181598

Order 12928.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15692288C212yesmaximal, complemented
2C2×#15692283C212yesmaximal, complemented
3#20265727C211yesmaximal, characteristic, complemented
4C2×#15692243C211yesmaximal, characteristic, complemented
5#20265571C212yesmaximal, complemented
6#20265566C212yesmaximal, complemented
7#20265561C214yesmaximal, complemented
8C4×M16×C101C211yesmaximal, characteristic, complemented
9C2×#13108074C2212yes—
10C22×Dic202C2211yescharacteristic
11C22×D404C2211yescharacteristic
12#15692288—28no—
13#15692283—28no—
14C2×Dic404C2214yescomplemented
15C2×#13108063C2214yes—
16C2×#13108063C2214yescomplemented
17C2×D808C2214yes—
18#15692255C2211yescharacteristic, complemented
19#15692252C2211yescharacteristic, complemented
20#15692243—22no—
21#15692243C2212yes—
22#15692243C2212yes—
23#15692243C2212yescomplemented
24#15692229—22no—
25C4×D404—22no—
26C2×#13108051—22no—
27C2×M16×C101C2212yes—
28C2×C42×C101C2211yescharacteristic
29#97×C101C2212yescomplemented
30C4×C8×C101C2212yescomplemented
31C23×D101—21no—
32#13108074—24no—
33#13108074—24no—
34C2×Dic202—24no—
35C2×Dic202—21no—
36C2×Dic202C2312yes—
37C2×Dic202C2312yes—
38C2×D404—24no—
39C2×D404—21no—
40C2×D404—22no—
41C2×D404C2312yes—
42C2×D404C2312yes—
43C2×C22⋊D101—22no—
44C22×Dic101—21no—
45C2×C4×D101—22no—
46Dic404—28no—
47#13108063—28no—
48#13108063—28no—
49D808—28no—
50C4⋊Dic101—22no—
51#13108051—24no—
52#13108051—24no—
53#13108051—24no—
54C22×C4×C101D412yes—
55C22×C4×C101C2311yescharacteristic
56M16×C101D418yescomplemented
57C2×C8×C101C2314yes—
58C42×C101D412yes—
59C42×C101C2311yescharacteristic
60C42×C101C2311yescharacteristic
61C23×C101C2×D411yescharacteristic
62C22×D101—22no—
63C22×D101—22no—
64C22×D101—44no—
65C22×D101—42no—
66C2×C4×C101—42no—
67C2×C4×C101C2×D412yes—
68C2×C4×C101C2×D412yes—
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101C2×D414yes—
71C2×C4×C101C2411yescharacteristic
72C22⋊D101—44no—
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—42no—
76D404—22no—
77D404—24no—
78D404—22no—
79D404—44no—
80D404—42no—
81C4×D101—44no—
82Dic202—22no—
83Dic202—24no—
84Dic202—22no—
85Dic202—42no—
86C8×C101C2×D418yes—
87C22×C101—22no—
88C22×C101#12712yes—
89C22×C101C22×D411yescharacteristic
90C2×D101—42no—
91C2×D101—42no—
92C2×D101—44no—
93C2×D101—44no—
94C2×D101—84no—
95C4×C101—22no—
96C4×C101—22no—
97C4×C101#12712yes—
98C4×C101C22×D412yes—
99Dic101—42no—
100Dic101—42no—
101C2×C101—22no—
102C2×C101C2×#12711yescharacteristic
103C2×C101#57512yes—
104D101—82no—
105D101—82no—
106#3153—1011nomaximal, Sylow
107C101#315311yescharacteristic, complemented, Sylow
108C2×#137—1012no—
109C2×AGL(1,ℤ/8)—1012no—
110#532—1011no—
111C2×#124—1011no—
112#497—1012no—
113#496—1012no—
114#495—1014no—
115C4×M16D10111yescharacteristic, complemented
116C2×#41—1012no—
117C22×Q8—1011no—
118C22×D4—1011no—
119#137—2028no—
120AGL(1,ℤ/8)—2028no—
121C2×Q16—1014no—
122C2×SD16—1014no—
123C2×SD16—1014no—
124C2×D8—1014no—
125C2×M16C2×D10112yes—
126C4⋊Q8—1011no—
127#127—1011no—
128#124—1012no—
129#124—1012no—
130#124—1012no—
131#124—2022no—
132#121—2022no—
133C4×D4—2022no—
134C2×C22⋊C4—2022no—
135C2×C42C2×D10111yescharacteristic
136#97C2×D10112yescomplemented
137C4×C8C2×D10112yescomplemented
138C24—2021no—
139#41—2024no—
140#41—2024no—
141C2×Q8—1012no—
142C2×Q8—1012no—
143C2×Q8—2024no—
144C2×Q8—2021no—
145C2×D4—1012no—
146C2×D4—1012no—
147C2×D4—2022no—
148C2×D4—2022no—
149C2×D4—2021no—
150C2×D4—2024no—
151C22×C4—2021no—
152C22×C4—2022no—
153C22×C4D40412yes—
154C22×C4C22×D10111yescharacteristic
155Q16—2028no—
156SD16—2028no—
157SD16—2028no—
158D8—2028no—
159M16D40418yescomplemented
160C2×C8C22×D10114yes—
161C4⋊C4—2022no—
162C22⋊C4—2024no—
163C22⋊C4—2024no—
164C22⋊C4—2024no—
165C42D40412yes—
166C42C22×D10111yescharacteristic
167C42C22×D10111yescharacteristic
168C23—4044no—
169C23—4042no—
170C23—2022no—
171C23—2022no—
172C23C2×D40411yescharacteristic
173Q8—4042no—
174Q8—2022no—
175Q8—2024no—
176Q8—2022no—
177D4—4044no—
178D4—4044no—
179D4—4042no—
180D4—2022no—
181D4—2024no—
182D4—2022no—
183C2×C4—42no—
184C2×C4—4042no—
185C2×C4—4044no—
186C2×C4—2022no—
187C2×C4—2022no—
188C2×C4C2×D40412yes—
189C2×C4C2×D40412yes—
190C2×C4C2×D40414yes—
191C2×C4C2×D40411yescharacteristic
192C2×C4C23×D10111yescharacteristic
193C8C2×D40418yes—
194C22—22no—
195C22—8084no—
196C22—4042no—
197C22—4042no—
198C22—4044no—
199C22—4044no—
200C22#1569225212yes—
201C22C22×D40411yescharacteristic
202C4—22no—
203C4—22no—
204C4—4042no—
205C4—4042no—
206C4#1569225212yes—
207C4C22×D40412yes—
208C2—22no—
209C2—8082no—
210C2—8082no—
211C2C2×#1569225211yescharacteristic
212C2#2026597712yes—
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×M16×C101C2
socleC22×C101C22×D4
hypercenterC4×M16D101
hypocenterC101#3153
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#15692288C2semidirect
2. C2×#15692283C2semidirect
3. #20265727C2semidirect
4. C2×#15692243C2semidirect
5. #20265571C2semidirect
6. #20265566C2semidirect
7. #20265561C2semidirect
8. C4×M16×C101C2semidirect
14. C2×Dic404C22semidirect
16. C2×#13108063C22semidirect
18. #15692255C22semidirect
19. #15692252C22semidirect
23. #15692243C22semidirect
29. #97×C101C22semidirect
30. C4×C8×C101C22semidirect
56. M16×C101D4semidirect
107. C101#3153semidirect
115. C4×M16D101semidirect
136. #97C2×D101semidirect
137. C4×C8C2×D101semidirect
159. M16D404semidirect

Maximal subgroups

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

subgroupindex
C2×#156922882
C2×#156922832
#202657272
C2×#156922432
#202655712
#202655662
#202655612
C4×M16×C1012
#3153101