Finite Group Databasebeta

← #27171454

Subgroups — #27171454

Order 12928.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15692229C211yesmaximal, characteristic
2C2×#15692175C212yesmaximal, complemented
3C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
4#20264939C212yesmaximal, complemented
5C2×#117×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic
7C22×C22⋊D101—21no—
8C22×#11717579C2212yes—
9#15692229—22no—
10#15692229C414yescomplemented
11C2×C4⋊Dic101C2211yescharacteristic
12C2×#13108051—21no—
13#15692175—24no—
14C22⋊#11717579—22no—
15C23×C4×C101C2211yescharacteristic
16#117×C101—24no—
17C2×C4⋊C4×C101C2212yes—
18C2×C22⋊C4×C101—21no—
19C2×C42×C101—21no—
20C23×D101—21no—
21C2×D404—22no—
22C2×D404—22no—
23C2×D404—21no—
24C2×D404C2×C412yes—
25C2×C22⋊D101—24no—
26C2×C22⋊D101—44no—
27C22×Dic101—21no—
28C2×#11717579—22no—
29C2×#11717579—22no—
30C2×#11717579—22no—
31C4⋊Dic101—21no—
32C4⋊Dic101C2×C412yes—
33#13108051—24no—
34C24×C101D411yescharacteristic
35C22×C4×C101—22no—
36C22×C4×C101—22no—
37C22×C4×C101—22no—
38C22×C4×C101C2×C412yes—
39C22×C4×C101D412yes—
40C22×C4×C101D411yescharacteristic
41C22×C4×C101C2311yescharacteristic
42C4⋊C4×C101—22no—
43C4⋊C4×C101D414yescomplemented
44C22⋊C4×C101—24no—
45C42×C101—24no—
46C23×C101—21no—
47C23×C101—21no—
48C23×C101—22no—
49C23×C101—22no—
50C23×C101C22⋊C412yes—
51C23×C101C2×D411yescharacteristic
52C22×D101—22no—
53C22×D101—41no—
54C22×D101—41no—
55C22×D101—42no—
56C22×D101—42no—
57C2×C4×C101—21no—
58C2×C4×C101—21no—
59C2×C4×C101—24no—
60C2×C4×C101—24no—
61C2×C4×C101—24no—
62C2×C4×C101—42no—
63C2×C4×C101—42no—
64C2×C4×C101C22⋊C412yes—
65C2×C4×C101C22×C411yescharacteristic
66C2×C4×C101C2×D412yes—
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101#4112yes—
69C22⋊D101—48no—
70C2×Dic101—22no—
71C2×Dic101—42no—
72D404—24no—
73D404—42no—
74#11717579—42no—
75C22×C101—22no—
76C22×C101—22no—
77C22×C101—22no—
78C22×C101—42no—
79C22×C101—42no—
80C22×C101C2×C22⋊C411yescharacteristic
81C22×C101#12011yescharacteristic
82C22×C101#12311yescharacteristic
83C22×C101#13512yes—
84C22×C101AGL(1,ℤ/8)12yes—
85C2×D101—42no—
86C2×D101—42no—
87C2×D101—42no—
88C2×D101—82no—
89C2×D101—82no—
90C4×C101—22no—
91C4×C101—44no—
92C4×C101C4×D412yes—
93C4×C101#12112yes—
94Dic101—42no—
95C2×C101—42no—
96C2×C101#38611yescharacteristic
97C2×C101#41611yescharacteristic
98C2×C101#41811yescharacteristic
99C2×C101#44912yes—
100C2×C101#48212yes—
101D101—82no—
102#1875—1011nomaximal, Sylow
103C101#187511yescharacteristic, complemented, Sylow
104C2×#121—1011no—
105C2×#117D10111yescharacteristic, complemented
106C2×#102—1012no—
107C2×C22⋊C8—1011no—
108#340—1012no—
109C22×D4—1011no—
110C22×D4—2021no—
111C23×C4C2×D10111yescharacteristic
112C22×C8—1012no—
113#121—1014no—
114#121—2022no—
115#117—24no—
116C2×C4⋊C4—1011no—
117C2×C4⋊C4C2×D10112yes—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—2021no—
120C2×C42—21no—
121#102—2024no—
122C22⋊C8—2022no—
123C24—2021no—
124C24C22⋊D10111yescharacteristic
125C2×D4—1012no—
126C2×D4—4044no—
127C2×D4—2024no—
128C2×D4—2022no—
129C2×D4—2021no—
130C2×D4—2022no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—2021no—
135C22×C4C4×D10112yes—
136C22×C4D40412yes—
137C22×C4C22⋊D10111yescharacteristic
138C22×C4C22×D10111yescharacteristic
139C2×C8—2022no—
140C2×C8—2022no—
141C2×C8—2022no—
142C4⋊C4—22no—
143C4⋊C4—1012no—
144C4⋊C4—2021no—
145C4⋊C4C22⋊D10114yescomplemented
146C22⋊C4—24no—
147C22⋊C4—2024no—
148C42—24no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—4041no—
154C23—4041no—
155C23—4042no—
156C23—4042no—
157C23—2022no—
158C23#1310805112yes—
159C23C2×C22⋊D10111yescharacteristic
160D4—4048no—
161D4—4042no—
162D4—2024no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—4042no—
171C2×C4—2022no—
172C2×C4#1310805112yes—
173C2×C4C2×C4×D10111yescharacteristic
174C2×C4C2×C22⋊D10112yes—
175C2×C4C2×D40411yescharacteristic
176C2×C4#1310807412yes—
177C8—4042no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—8082no—
184C22—8082no—
185C22—4042no—
186C22—4042no—
187C22—4042no—
188C22C2×#1310805111yescharacteristic
189C22#1569222411yescharacteristic
190C22#1569223811yescharacteristic
191C22#1569227512yes—
192C22#1569228012yes—
193C4—22no—
194C4—44no—
195C4—4042no—
196C4C4×C22⋊D10112yes—
197C4#1569222912yes—
198C2—42no—
199C2—8082no—
200C2#2026505611yescharacteristic
201C2#2026517111yescharacteristic
202C2#2026518111yescharacteristic
203C2#2026532212yes—
204C2#2026550112yes—
subgroupG/H
centerC23C2×C22⋊D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#117×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692229C2
cosocleC22×C4×C101C23
hypercenterC2×#117D101
hypocenterC101#1875
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#15692175C2semidirect
3. C2×C22⋊#11717579C2semidirect
4. #20264939C2semidirect
5. C2×#117×C101C2semidirect
10. #15692229C4semidirect
43. C4⋊C4×C101D4semidirect
103. C101#1875semidirect
105. C2×#117D101semidirect
145. C4⋊C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922292
C2×#156921752
C2×C22⋊#117175792
#202649392
C2×#117×C1012
#1875101