Finite Group Databasebeta

← #27186535

Subgroups — #27186535

Order 12928.

242 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 58 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#20266200C211yesmaximal, characteristic, complemented
2C4×C4⋊#87938C211yesmaximal, characteristic, complemented
3#20266030C212yesmaximal, complemented
4#20266029C212yesmaximal
5#20265712C211yesmaximal, characteristic
6C2×C4⋊#87938—21no—
7C2×C4×#87938—21no—
8#15692322—21no—
9#15692312C2212yes—
10#15692310—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—24no—
26C4×#87938—42no—
27C2×Dic202C2×C411yescharacteristic
28C2×D404—21no—
29C2×D404C2×C411yescharacteristic
30C2×C4×D101—21no—
31C2×C4×D101D411yescharacteristic
32C2×C4×D101D411yescharacteristic
33C2×C4×D101C2311yescharacteristic
34C4⋊Dic101—21no—
35#13108053—21no—
36#13108051—21no—
37#13108051—21no—
38#13108051—21no—
39#13108051—21no—
40#13108051—21no—
41C4×Dic101—21no—
42C4×Dic101C2×C411yescharacteristic
43C4×Dic101D411yescharacteristic
44C4×Dic101D411yescharacteristic
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101C2×C411yescharacteristic
48C4⋊C4×C101C2×C411yescharacteristic
49C42×C101C2×C411yescharacteristic
50C22×D101—21no—
51C22×D101C2×D411yescharacteristic
52C2×#87938—24no—
53C2×#87938—42no—
54C2×#87938—42no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101C22⋊C411yescharacteristic
59C2×C4×C101C22×C411yescharacteristic
60C2×Dic101—21no—
61C2×Dic101C22⋊C411yescharacteristic
62C2×Dic101C22⋊C411yescharacteristic
63C2×Dic101C2×D411yescharacteristic
64D404—41no—
65D404—41no—
66D404—41no—
67C4×D101—21no—
68C4×D101—21no—
69C4×D101—22no—
70C4×D101—41no—
71C4×D101—42no—
72Dic202—41no—
73#11717581—42no—
74C22×C101C2×C22⋊C411yescharacteristic
75C2×D101—21no—
76C2×D101—21no—
77C2×D101—41no—
78C2×D101—41no—
79C2×D101—41no—
80C2×D101AGL(1,ℤ/8)11yescharacteristic
81C2×D101#13711yescharacteristic
82#87938—44no—
83C4×C101—21no—
84C4×C101—21no—
85C4×C101—21no—
86C4×C101—41no—
87C4×C101—41no—
88Dic101—21no—
89Dic101—21no—
90Dic101—41no—
91Dic101#10412yes—
92C2×C101#41011yescharacteristic
93C2×C101#41711yescharacteristic
94C2×C101C2×#10411yescharacteristic
95D101—21no—
96D101—21no—
97D101—81no—
98#1509—1011nomaximal, Sylow
99C101#150911yescharacteristic, complemented, Sylow
100#529—1011no—
101#432—1011no—
102C4×C4⋊C4—1011no—
103#328—1012no—
104#327—1012no—
105#126—2021no—
106#125#8793811yescharacteristic, complemented
107#124—1011no—
108#123—2021no—
109C22⋊Q8—1011no—
110#121—1011no—
111C4×Q8—1011no—
112C4×D4—1011no—
113C4×D4—2021no—
114#117—2021no—
115C2×C4⋊C4—2021no—
116C2×C42—1011no—
117C2×C42—2021no—
118C4⋊C8—2021no—
119C22⋊C8—1012no—
120#97—2021no—
121#95—1012no—
122C2×Q8—1011no—
123C2×D4—1011no—
124C2×D4—2021no—
125C22×C4—1011no—
126C22×C4—1011no—
127C22×C4—1011no—
128C22×C4—2021no—
129C22×C4—2022no—
130C2×C8—2022no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4—2024no—
136C4⋊C4C2×#8793811yescharacteristic
137C4⋊C4C2×#8793811yescharacteristic
138C22⋊C4—2021no—
139C22⋊C4—2021no—
140C22⋊C4—2021no—
141C22⋊C4—2021no—
142C22⋊C4—2021no—
143C42—1011no—
144C42—1011no—
145C42—1011no—
146C42—4042no—
147C42—2021no—
148C42C2×#8793811yescharacteristic
149C23—1011no—
150C23—2021no—
151Q8—4041no—
152D4—4041no—
153D4—4041no—
154D4—4041no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—1011no—
158C2×C4—1011no—
159C2×C4—1011no—
160C2×C4—4041no—
161C2×C4—4042no—
162C2×C4—4042no—
163C2×C4—4042no—
164C2×C4—2022no—
165C2×C4—2021no—
166C2×C4—2021no—
167C2×C4—2024no—
168C2×C4—2021no—
169C2×C4C22⋊#8793811yescharacteristic
170C2×C4C22⋊#8793811yescharacteristic
171C2×C4C22×#8793811yescharacteristic
172C8—4042no—
173C22—1011no—
174C22—1011no—
175C22—4041no—
176C22—4041no—
177C22—4041no—
178C22—2021no—
179C22—2021no—
180C22C2×C22⋊#8793811yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—1012no—
187C4—4044no—
188C4—4041no—
189C4—2021no—
190C4—2021no—
191C2—8081no—
192C2—2021no—
193C2—2021no—
194C2#2026615811yescharacteristic
195C2#2026616811yescharacteristic
196C2D101⋊#10411yescharacteristic
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#1509
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. #20266200C2semidirect
2. C4×C4⋊#87938C2semidirect
3. #20266030C2semidirect
13. #15692243C4semidirect
17. #15692220C4semidirect
19. #15692217C4semidirect
22. #125×C101C4semidirect
99. C101#1509semidirect
106. #125#87938semidirect

Maximal subgroups

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

subgroupindex
#202662002
C4×C4⋊#879382
#202660302
#202660292
#202657122
#1509101