Finite Group Databasebeta

← #27186523

Subgroups — #27186523

Order 12928.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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#20266113C211yesmaximal, characteristic, complemented
3#20266029C214yesmaximal, complemented
4#20265728C211yesmaximal, characteristic
5C2×C4×#87938—21no—
6C4⋊#11717581—21no—
7#15692312C2212yes—
8#15692310—21no—
9(C2×#87938)⋊C4—21no—
10(C2×#87938)⋊C4C2212yes—
11C2×#13108075—21no—
12#15692252C411yescharacteristic, complemented
13#15692244—21no—
14#15692230—21no—
15#15692230C2212yes—
16#15692217—21no—
17#15692217C412yescomplemented
18C42×D101C2211yescharacteristic
19C4⋊Q8×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21C2×#11717581—22no—
22C4×#87938—24no—
23#13108075—44no—
24C2×D404—22no—
25C2×D404—21no—
26C2×D404—21no—
27C2×D404C2×C412yes—
28C2×C4×D101—22no—
29C2×C4×D101D411yescharacteristic
30C2×C4×D101D411yescharacteristic
31C2×C4×D101C2311yescharacteristic
32#13108051—22no—
33#13108051—22no—
34C4×Dic101C2×C411yescharacteristic
35C4×Dic101D411yescharacteristic
36C4×Dic101D411yescharacteristic
37C2×Q8×C101—21no—
38C4⋊C4×C101—21no—
39C4⋊C4×C101C2×C412yes—
40C42×C101C2×C411yescharacteristic
41C22×D101—22no—
42C22×D101C2×D411yescharacteristic
43C2×#87938—24no—
44C2×#87938—42no—
45C2×#87938—42no—
46Q8×C101—42no—
47C2×C4×C101—22no—
48C2×C4×C101C22⋊C411yescharacteristic
49C2×C4×C101C22⋊C411yescharacteristic
50C2×C4×C101C22×C411yescharacteristic
51C2×Dic101C22⋊C411yescharacteristic
52C2×Dic101C22⋊C411yescharacteristic
53C2×Dic101C2×D411yescharacteristic
54D404—42no—
55D404—42no—
56D404—44no—
57D404—42no—
58C4×D101—22no—
59C4×D101—22no—
60C4×D101—41no—
61C4×D101—44no—
62#11717581—42no—
63C22×C101C2×C22⋊C411yescharacteristic
64C2×D101—21no—
65C2×D101—21no—
66C2×D101—42no—
67C2×D101—42no—
68C2×D101—42no—
69C2×D101AGL(1,ℤ/8)12yes—
70#87938—44no—
71C4×C101—21no—
72C4×C101—21no—
73C4×C101—42no—
74C4×C101#10412yes—
75Dic101—21no—
76Dic101—21no—
77Dic101—21no—
78C2×C101#41011yescharacteristic
79C2×C101#41811yescharacteristic
80C2×C101C2×#10411yescharacteristic
81D101—22no—
82D101—82no—
83#1505—1011nomaximal, Sylow
84C101#150511yescharacteristic, complemented, Sylow
85#532—1011no—
86#432—1011no—
87#376—1011no—
88#327—1014no—
89C2×#41—2021no—
90C4⋊Q8#8793811yescharacteristic, complemented
91#127—1011no—
92#124—2021no—
93#121—1012no—
94#121—2021no—
95C4×D4—1012no—
96C4×D4—2021no—
97C2×C42—1011no—
98C2×C42—2021no—
99C4⋊C8—2021no—
100C22⋊C8—1012no—
101#97—2021no—
102#95—1012no—
103#95—2021no—
104#41—4044no—
105C2×Q8—21no—
106C2×D4—1012no—
107C2×D4—2022no—
108C2×D4—2021no—
109C2×D4—2021no—
110C22×C4—1011no—
111C22×C4—1011no—
112C22×C4—1011no—
113C22×C4—2022no—
114C22×C4—2022no—
115C2×C8—2022no—
116C4⋊C4—21no—
117C4⋊C4C2×#8793812yes—
118C22⋊C4—2022no—
119C22⋊C4—2022no—
120C42—1011no—
121C42—1011no—
122C42—1011no—
123C42—2024no—
124C42C2×#8793811yescharacteristic
125C23—1011no—
126C23—2022no—
127Q8—42no—
128D4—4042no—
129D4—4042no—
130D4—4044no—
131D4—4042no—
132C2×C4—22no—
133C2×C4—1011no—
134C2×C4—1011no—
135C2×C4—1011no—
136C2×C4—4041no—
137C2×C4—4044no—
138C2×C4—4042no—
139C2×C4—4042no—
140C2×C4—2022no—
141C2×C4—2024no—
142C2×C4—2022no—
143C2×C4C22⋊#8793811yescharacteristic
144C2×C4C22⋊#8793811yescharacteristic
145C2×C4C22×#8793811yescharacteristic
146C8—4042no—
147C22—1012no—
148C22—4042no—
149C22—4042no—
150C22—4042no—
151C22—2021no—
152C22—2021no—
153C22C2×C22⋊#8793811yescharacteristic
154C4—21no—
155C4—21no—
156C4—42no—
157C4—4044no—
158C4—2021no—
159C4—2021no—
160C4—2021no—
161C4#1569232012yes—
162C2—8082no—
163C2—2022no—
164C2#2026615811yescharacteristic
165C2#2026616911yescharacteristic
166C2C2×#1569232011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265728C2
cosocleC2×C4×D101C23
hypercenterC4⋊Q8#87938
hypocenterC101#1505
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20266199C2semidirect
2. #20266113C2semidirect
3. #20266029C2semidirect
12. #15692252C4semidirect
17. #15692217C4semidirect
19. C4⋊Q8×C101C4semidirect
84. C101#1505semidirect
90. C4⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202661992
#202661132
#202660292
#202657282
#1505101