Finite Group Databasebeta

← #27169668

Subgroups — #27169668

Order 12928.

222 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 69 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#20265711C211yesmaximal, characteristic
2#20265234C211yesmaximal, characteristic, complemented
3#20264913C212yesmaximal, complemented
4#20264910C212yesmaximal
5#382×C101C211yesmaximal, characteristic, complemented
6#15692249—21no—
7#15692247C411yescharacteristic, complemented
8#15692243C411yescharacteristic, complemented
9#15692238—21no—
10#15692234C2211yescharacteristic
11#15692229C2211yescharacteristic
12C4×Dic202C411yescharacteristic, complemented
13C4×D404C411yescharacteristic, complemented
14C4×C22⋊D101—21no—
15#15692212—21no—
16C4⋊#11717579—21no—
17C22⋊#11717579C2212yes—
18#15692166—21no—
19C2×C4⋊C4×C101—21no—
20C2×C42×C101C2211yescharacteristic
21#95×C101—21no—
22#95×C101C2212yes—
23C2×Dic202C2×C411yescharacteristic
24C2×D404C2×C411yescharacteristic
25C2×C22⋊D101—21no—
26C2×C4×D101—21no—
27C2×#11717579—22no—
28C4⋊Dic101C2×C411yescharacteristic
29C4⋊Dic101C2×C411yescharacteristic
30#13108053—21no—
31#13108053—21no—
32#13108053—21no—
33#13108053—21no—
34C22⋊Dic101—21no—
35#13108051—21no—
36#13108051—21no—
37#13108051—21no—
38#13108051—21no—
39C4×Dic101—21no—
40C22×C4×C101—22no—
41C22×C4×C101D411yescharacteristic
42C22×C4×C101D411yescharacteristic
43C22×C4×C101C2311yescharacteristic
44C4⋊C4×C101—42no—
45C42×C101C2×C411yescharacteristic
46C42×C101C2×C411yescharacteristic
47C42×C101D411yescharacteristic
48C42×C101D411yescharacteristic
49C23×C101C2×D411yescharacteristic
50C22×D101—21no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101—42no—
56C2×C4×C101—42no—
57C2×C4×C101—42no—
58C2×C4×C101C22⋊C411yescharacteristic
59C2×C4×C101C22⋊C411yescharacteristic
60C2×C4×C101C22⋊C411yescharacteristic
61C2×C4×C101C22⋊C411yescharacteristic
62C2×C4×C101C22×C411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C22⋊D101—41no—
65C22⋊D101—41no—
66C2×Dic101—21no—
67C2×Dic101—21no—
68C2×Dic101—21no—
69D404—41no—
70C4×D101—42no—
71Dic202—41no—
72#11717579—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101C2×C22⋊C411yescharacteristic
76C22×C101AGL(1,ℤ/8)11yescharacteristic
77C22×C101#13711yescharacteristic
78C2×D101—41no—
79C2×D101—41no—
80C2×D101—41no—
81C4×C101—21no—
82C4×C101—21no—
83C4×C101—21no—
84C4×C101—21no—
85C4×C101—21no—
86C4×C101—21no—
87C4×C101—82no—
88Dic101—41no—
89Dic101—41no—
90Dic101—41no—
91C2×C101—21no—
92C2×C101—21no—
93C2×C101#41011yescharacteristic
94C2×C101#41711yescharacteristic
95C2×C101#42111yescharacteristic
96D101—81no—
97#1512—1011nomaximal, Sylow
98C101#151211yescharacteristic, complemented, Sylow
99#529—1011no—
100#432—1011no—
101#382D10111yescharacteristic, complemented
102#328—1012no—
103#327—1012no—
104#126—2021no—
105#125—1011no—
106#124—1011no—
107#123—2021no—
108C22⋊Q8—1011no—
109#121—1011no—
110C4×Q8—1011no—
111C4×D4—1011no—
112C4×D4—2021no—
113#117—2021no—
114C2×C4⋊C4—21no—
115C2×C42C2×D10111yescharacteristic
116C4⋊C8—2021no—
117C22⋊C8—1012no—
118#97—2021no—
119#95—21no—
120#95C2×D10112yes—
121C2×Q8—1011no—
122C2×D4—1011no—
123C2×D4—2021no—
124C22×C4—22no—
125C22×C4—2021no—
126C22×C4C22⋊D10111yescharacteristic
127C22×C4C22⋊D10111yescharacteristic
128C22×C4C22×D10111yescharacteristic
129C2×C8—2022no—
130C4⋊C4—42no—
131C4⋊C4—1011no—
132C4⋊C4—1011no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4—2021no—
136C4⋊C4—2021no—
137C22⋊C4—2021no—
138C22⋊C4—2021no—
139C22⋊C4—2021no—
140C22⋊C4—2021no—
141C22⋊C4—2021no—
142C42—2021no—
143C42C4×D10111yescharacteristic
144C42C4×D10111yescharacteristic
145C42D40411yescharacteristic
146C42D40411yescharacteristic
147C23—2021no—
148C23C2×C22⋊D10111yescharacteristic
149Q8—4041no—
150D4—4041no—
151D4—4041no—
152D4—4041no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—4042no—
161C2×C4—2021no—
162C2×C4—2021no—
163C2×C4—2021no—
164C2×C4#1310805111yescharacteristic
165C2×C4#1310805111yescharacteristic
166C2×C4#1310805111yescharacteristic
167C2×C4#1310805111yescharacteristic
168C2×C4C2×C4×D10111yescharacteristic
169C2×C4C2×D40411yescharacteristic
170C8—4042no—
171C22—21no—
172C22—21no—
173C22—4041no—
174C22—4041no—
175C22—4041no—
176C22C2×#1310805111yescharacteristic
177C22#1569228111yescharacteristic
178C22#1569228611yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—82no—
186C4—4041no—
187C4—4041no—
188C4—4041no—
189C2—21no—
190C2—21no—
191C2—8081no—
192C2#2026515011yescharacteristic
193C2#2026517611yescharacteristic
194C2#2026519511yescharacteristic
subgroupG/H
centerC22C2×#13108051
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#382×C101C2
socleC22×C101C2×C22⋊C4
prime radical#15692229C22
cosocleC22×C4×C101C23
hypercenter#382D101
hypocenterC101#1512
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #20265234C2semidirect
3. #20264913C2semidirect
5. #382×C101C2semidirect
7. #15692247C4semidirect
8. #15692243C4semidirect
12. C4×Dic202C4semidirect
13. C4×D404C4semidirect
98. C101#1512semidirect
101. #382D101semidirect

Maximal subgroups

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

subgroupindex
#202657112
#202652342
#202649132
#202649102
#382×C1012
#1512101