Finite Group Databasebeta

← #27186546

Subgroups — #27186546

Order 12928.

222 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 56 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#20266199C211yesmaximal, characteristic, complemented
2#20266121C211yesmaximal, characteristic, complemented
3#20266030C212yesmaximal, complemented
4#20266029C212yesmaximal
5#20265712C211yesmaximal, characteristic
6C2×C4⋊#87938—21no—
7C4⋊#11717581—21no—
8#15692312C2212yes—
9#15692310—21no—
10(C2×#87938)⋊C4—21no—
11(C2×#87938)⋊C4C2212yes—
12#15692250—21no—
13#15692243C411yescharacteristic, complemented
14#15692240—21no—
15#15692235C2211yescharacteristic
16#15692230C2211yescharacteristic
17#15692220C411yescharacteristic, complemented
18#15692217—21no—
19#15692217C411yescharacteristic, complemented
20#15692213—21no—
21C42×D101C2211yescharacteristic
22#125×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24C2×#11717581—22no—
25C4⋊#87938—42no—
26C2×Dic202C2×C411yescharacteristic
27C2×D404—21no—
28C2×D404C2×C411yescharacteristic
29C2×C4×D101—21no—
30C2×C4×D101D411yescharacteristic
31C2×C4×D101D411yescharacteristic
32C2×C4×D101C2311yescharacteristic
33C4⋊Dic101—21no—
34#13108053—21no—
35#13108051—21no—
36#13108051—21no—
37#13108051—21no—
38#13108051—21no—
39#13108051—21no—
40C4×Dic101—21no—
41C4×Dic101C2×C411yescharacteristic
42C4×Dic101D411yescharacteristic
43C4×Dic101D411yescharacteristic
44C4⋊C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101C2×C411yescharacteristic
47C4⋊C4×C101C2×C411yescharacteristic
48C42×C101C2×C411yescharacteristic
49C22×D101—21no—
50C22×D101C2×D411yescharacteristic
51C2×#87938—42no—
52C2×#87938—42no—
53C2×#87938—42no—
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101C22⋊C411yescharacteristic
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101C22×C411yescharacteristic
59C2×Dic101—21no—
60C2×Dic101C22⋊C411yescharacteristic
61C2×Dic101C22⋊C411yescharacteristic
62C2×Dic101C2×D411yescharacteristic
63D404—41no—
64D404—41no—
65D404—41no—
66C4×D101—21no—
67C4×D101—21no—
68C4×D101—41no—
69C4×D101—41no—
70C4×D101—42no—
71Dic202—41no—
72#11717581—42no—
73C22×C101C2×C22⋊C411yescharacteristic
74C2×D101—21no—
75C2×D101—21no—
76C2×D101—41no—
77C2×D101—41no—
78C2×D101—41no—
79C2×D101AGL(1,ℤ/8)11yescharacteristic
80C2×D101#13711yescharacteristic
81#87938—82no—
82C4×C101—21no—
83C4×C101—21no—
84C4×C101—21no—
85C4×C101—41no—
86C4×C101—41no—
87Dic101—21no—
88Dic101—21no—
89Dic101—21no—
90Dic101—41no—
91C2×C101#41011yescharacteristic
92C2×C101#41711yescharacteristic
93C2×C101#42111yescharacteristic
94D101—21no—
95D101—21no—
96D101—81no—
97#1512—1011nomaximal, Sylow
98C101#151211yescharacteristic, complemented, Sylow
99#529—1011no—
100#432—1011no—
101#382—1011no—
102#328—1012no—
103#327—1012no—
104#126—2021no—
105#125#8793811yescharacteristic, complemented
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—2021no—
115C2×C42—1011no—
116C4⋊C8—2021no—
117C22⋊C8—1012no—
118#97—2021no—
119#95—1012no—
120#95—2021no—
121C2×Q8—1011no—
122C2×D4—1011no—
123C2×D4—2021no—
124C22×C4—1011no—
125C22×C4—1011no—
126C22×C4—1011no—
127C22×C4—2021no—
128C22×C4—2022no—
129C2×C8—2022no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—4042no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4C2×#8793811yescharacteristic
136C4⋊C4C2×#8793811yescharacteristic
137C22⋊C4—2021no—
138C22⋊C4—2021no—
139C22⋊C4—2021no—
140C22⋊C4—2021no—
141C22⋊C4—2021no—
142C42—1011no—
143C42—1011no—
144C42—1011no—
145C42—2021no—
146C42C2×#8793811yescharacteristic
147C23—1011no—
148C23—2021no—
149Q8—4041no—
150D4—4041no—
151D4—4041no—
152D4—4041no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—1011no—
156C2×C4—1011no—
157C2×C4—1011no—
158C2×C4—4041no—
159C2×C4—4041no—
160C2×C4—4042no—
161C2×C4—4042no—
162C2×C4—4042no—
163C2×C4—4042no—
164C2×C4—2021no—
165C2×C4—2021no—
166C2×C4—2021no—
167C2×C4C22⋊#8793811yescharacteristic
168C2×C4C22⋊#8793811yescharacteristic
169C2×C4C22×#8793811yescharacteristic
170C8—4042no—
171C22—1011no—
172C22—1011no—
173C22—4041no—
174C22—4041no—
175C22—4041no—
176C22—2021no—
177C22—2021no—
178C22C2×C22⋊#8793811yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—8082no—
185C4—4041no—
186C4—2021no—
187C4—2021no—
188C4—2021no—
189C2—8081no—
190C2—2021no—
191C2—2021no—
192C2#2026615811yescharacteristic
193C2#2026616811yescharacteristic
194C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#125×C101C4
socleC22×C101C2×C22⋊C4
prime radical#15692230C22
cosocleC2×C4×D101C23
hypercenter#125#87938
hypocenterC101#1512
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #20266199C2semidirect
2. #20266121C2semidirect
3. #20266030C2semidirect
13. #15692243C4semidirect
17. #15692220C4semidirect
19. #15692217C4semidirect
22. #125×C101C4semidirect
98. C101#1512semidirect
105. #125#87938semidirect

Maximal subgroups

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

subgroupindex
#202661992
#202661212
#202660302
#202660292
#202657122
#1512101