Finite Group Databasebeta

← #27186521

Subgroups — #27186521

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#20266028C214yesmaximal, complemented
4#20265725C211yesmaximal, characteristic
5C2×C4×#87938—21no—
6C4⋊#11717581—21no—
7#15692312C2212yes—
8#15692310—21no—
9(C2×#87938)⋊C4—21no—
10(C2×#87938)⋊C4C2212yes—
11C2×#13108073—21no—
12#15692255C411yescharacteristic, complemented
13#15692242—21no—
14#15692226—21no—
15#15692226C2212yes—
16D4×Dic101—21no—
17D4×Dic101C412yescomplemented
18C42×D101C2211yescharacteristic
19#127×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21C2×#11717581—22no—
22C4×#87938—24no—
23#13108073—44no—
24C2×Dic202—21no—
25C2×C22⋊D101—22no—
26C22×Dic101—22no—
27C2×C4×D101D411yescharacteristic
28C2×C4×D101D411yescharacteristic
29C2×C4×D101C2311yescharacteristic
30C4⋊Dic101—21no—
31C4⋊Dic101C2×C412yes—
32C22⋊Dic101—22no—
33C22⋊Dic101—22no—
34C4×Dic101C2×C411yescharacteristic
35C4×Dic101D411yescharacteristic
36C4×Dic101D411yescharacteristic
37C2×D4×C101—21no—
38C2×D4×C101—21no—
39C2×D4×C101C2×C412yes—
40C42×C101C2×C411yescharacteristic
41C23×C101—22no—
42C22×D101C2×D411yescharacteristic
43C2×#87938—24no—
44C2×#87938—42no—
45C2×#87938—42no—
46D4×C101—42no—
47D4×C101—42no—
48D4×C101—42no—
49C2×C4×C101C22⋊C411yescharacteristic
50C2×C4×C101C22⋊C411yescharacteristic
51C2×C4×C101C22×C411yescharacteristic
52C22⋊D101—44no—
53C2×Dic101—22no—
54C2×Dic101—44no—
55C2×Dic101C22⋊C411yescharacteristic
56C2×Dic101C22⋊C411yescharacteristic
57C2×Dic101C2×D411yescharacteristic
58C4×D101—22no—
59C4×D101—22no—
60C4×D101—41no—
61Dic202—42no—
62#11717581—42no—
63C22×C101—42no—
64C22×C101—42no—
65C22×C101—42no—
66C22×C101C2×C22⋊C411yescharacteristic
67C2×D101—21no—
68C2×D101—21no—
69C2×D101AGL(1,ℤ/8)12yes—
70#87938—44no—
71C4×C101—21no—
72C4×C101—21no—
73C4×C101#10412yes—
74Dic101—21no—
75Dic101—21no—
76Dic101—21no—
77Dic101—42no—
78C2×C101—82no—
79C2×C101#41011yescharacteristic
80C2×C101#41811yescharacteristic
81C2×C101C2×#10411yescharacteristic
82D101—22no—
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—1011no—
91#127#8793811yescharacteristic, complemented
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—2021no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—2022no—
109C2×D4C2×#8793812yes—
110C22×C4—1011no—
111C22×C4—1011no—
112C22×C4—1011no—
113C22×C4—2022no—
114C22×C4—2022no—
115C2×C8—2022no—
116C4⋊C4—1012no—
117C4⋊C4—2021no—
118C22⋊C4—2022no—
119C22⋊C4—2022no—
120C42—1011no—
121C42—1011no—
122C42—1011no—
123C42—2024no—
124C42C2×#8793811yescharacteristic
125C23—22no—
126C23—1011no—
127Q8—4042no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—4044no—
132C2×C4—1011no—
133C2×C4—1011no—
134C2×C4—1011no—
135C2×C4—4041no—
136C2×C4—4042no—
137C2×C4—4042no—
138C2×C4—4044no—
139C2×C4—2022no—
140C2×C4—2024no—
141C2×C4—2022no—
142C2×C4—2022no—
143C2×C4C22⋊#8793811yescharacteristic
144C2×C4C22⋊#8793811yescharacteristic
145C2×C4C22×#8793811yescharacteristic
146C8—4042no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22—1012no—
151C22—2021no—
152C22—2021no—
153C22C2×C22⋊#8793811yescharacteristic
154C4—21no—
155C4—21no—
156C4—4044no—
157C4—4042no—
158C4—2021no—
159C4—2021no—
160C4—2021no—
161C4#1569231912yes—
162C2—82no—
163C2—2022no—
164C2#2026615711yescharacteristic
165C2#2026616911yescharacteristic
166C2C2×#1569231911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#127×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265725C2
cosocleC2×C4×D101C23
hypercenter#127#87938
hypocenterC101#1505
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20266199C2semidirect
2. #20266113C2semidirect
3. #20266028C2semidirect
12. #15692255C4semidirect
17. D4×Dic101C4semidirect
19. #127×C101C4semidirect
84. C101#1505semidirect
91. #127#87938semidirect

Maximal subgroups

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

subgroupindex
#202661992
#202661132
#202660282
#202657252
#1505101