Finite Group Databasebeta

← #27186485

Subgroups — #27186485

Order 12928.

230 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 64 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#20266198C211yesmaximal, characteristic, complemented
2C2×C4×#11717581C211yesmaximal, characteristic, complemented
3Q8⋊#11717581C212yesmaximal, complemented
4#20266025C212yesmaximal, complemented
5#20265713C211yesmaximal, characteristic
6C22×#11717581—21no—
7C4⋊#11717581C2212yes—
8C22⋊#11717581—21no—
9#15692310—21no—
10C4×#11717581—21no—
11C4×#11717581C2212yes—
12#15692250—21no—
13#15692246C411yescharacteristic
14#15692244C411yescharacteristic
15#15692239—21no—
16#15692229C411yescharacteristic
17Q8×Dic101C2211yescharacteristic
18#15692217C2211yescharacteristic
19#15692215—21no—
20#15692213—21no—
21C2×C4×Dic101C2211yescharacteristic
22C22⋊Q8×C101C411yescharacteristic
23C2×#11717581—24no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C2×D404C2×C411yescharacteristic
27C2×C22⋊D101—21no—
28C22×Dic101D411yescharacteristic
29C22×Dic101D411yescharacteristic
30C2×C4×D101—21no—
31C4⋊Dic101—21no—
32C4⋊Dic101—21no—
33C4⋊Dic101C2×C411yescharacteristic
34#13108053—21no—
35#13108051—21no—
36#13108051—21no—
37#13108051—21no—
38#13108051—21no—
39C4×Dic101—21no—
40C4×Dic101C2×C411yescharacteristic
41C4×Dic101D411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101C2311yescharacteristic
44C2×Q8×C101C2×C411yescharacteristic
45C22×C4×C101C2×C411yescharacteristic
46C4⋊C4×C101—21no—
47C4⋊C4×C101C2×C411yescharacteristic
48C22⋊C4×C101—21no—
49C23×C101C22⋊C411yescharacteristic
50C22×D101—21no—
51Q8×C101—22no—
52C2×C4×C101—22no—
53C2×C4×C101—21no—
54C2×C4×C101—21no—
55C2×C4×C101C22⋊C411yescharacteristic
56C2×C4×C101C22×C411yescharacteristic
57C22⋊D101—42no—
58C2×Dic101—22no—
59C2×Dic101—22no—
60C2×Dic101—21no—
61C2×Dic101C22⋊C411yescharacteristic
62C2×Dic101C22⋊C411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64C2×Dic101C2×D411yescharacteristic
65D404—22no—
66C4×D101—42no—
67#11717581—24no—
68#11717581—42no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101#10412yes—
72C22×C101C2×C22⋊C411yescharacteristic
73C2×D101—41no—
74C2×D101—41no—
75C2×D101—41no—
76C4×C101—21no—
77C4×C101—41no—
78C4×C101—41no—
79C4×C101#13112yes—
80Dic101—21no—
81Dic101—21no—
82Dic101—21no—
83Dic101—41no—
84Dic101#13512yes—
85C2×C101—22no—
86C2×C101#40811yescharacteristic
87C2×C101#41611yescharacteristic
88C2×C101C2×#10411yescharacteristic
89D101—81no—
90#1494—1011nomaximal, Sylow
91C101#149411yescharacteristic, complemented, Sylow
92#529—1011no—
93#432—1011no—
94C2×C4×C8—1011no—
95Q8⋊C8—1012no—
96D4⋊C8—1012no—
97C22×C8—2021no—
98#126—2021no—
99#125—1011no—
100#124—1011no—
101#123—2021no—
102C22⋊Q8#8793811yescharacteristic
103#121—1011no—
104C4×Q8—1011no—
105C4×D4—1011no—
106C4×D4—2021no—
107#117—2021no—
108C2×C42—1011no—
109C4⋊C8—1012no—
110C22⋊C8—2021no—
111#97—2021no—
112C4×C8—1012no—
113C4×C8—2021no—
114C2×Q8C2×#8793811yescharacteristic
115C2×D4—1011no—
116C2×D4—2021no—
117C22×C4—1011no—
118C22×C4—1011no—
119C22×C4—2021no—
120C22×C4C2×#8793811yescharacteristic
121C2×C8—2022no—
122C2×C8—2022no—
123C2×C8—2024no—
124C4⋊C4—21no—
125C4⋊C4—1011no—
126C4⋊C4—2021no—
127C4⋊C4—2021no—
128C4⋊C4—2021no—
129C4⋊C4C2×#8793811yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—2021no—
132C22⋊C4—2021no—
133C22⋊C4—2021no—
134C22⋊C4—2021no—
135C42—1011no—
136C42—1011no—
137C42—1011no—
138C42—1011no—
139C42—2021no—
140C23—2021no—
141C23C22⋊#8793811yescharacteristic
142Q8—22no—
143D4—4042no—
144D4—2022no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—1011no—
149C2×C4—1011no—
150C2×C4—1011no—
151C2×C4—1011no—
152C2×C4—4042no—
153C2×C4—2022no—
154C2×C4—2021no—
155C2×C4—2022no—
156C2×C4C22⋊#8793811yescharacteristic
157C2×C4C22×#8793811yescharacteristic
158C8—4042no—
159C8—2024no—
160C22—21no—
161C22—21no—
162C22—4041no—
163C22—4041no—
164C22—4041no—
165C22#1569232012yes—
166C22C2×C22⋊#8793811yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—1012no—
171C4—4041no—
172C4—2021no—
173C4—2021no—
174C4—2021no—
175C4#1569234212yes—
176C2—22no—
177C2—8081no—
178C2#2026615011yescharacteristic
179C2#2026616711yescharacteristic
180C2C2×#1569232011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radical#15692229C4
cosocleC4×Dic101C23
hypercenterC22⋊Q8#87938
hypocenterC101#1494
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20266198C2semidirect
2. C2×C4×#11717581C2semidirect
3. Q8⋊#11717581C2semidirect
4. #20266025C2semidirect
91. C101#1494semidirect

Maximal subgroups

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

subgroupindex
#202661982
C2×C4×#117175812
Q8⋊#117175812
#202660252
#202657132
#1494101