Finite Group Databasebeta

← #27186553

Subgroups — #27186553

Order 12928.

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

normal subgroups only

This group has 252 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#20266200C211yesmaximal, characteristic, complemented
2#20266122C211yesmaximal, characteristic, complemented
3#20266029C214yesmaximal, complemented
4#20265728C211yesmaximal, characteristic
5#15692322—21no—
6#15692312C2212yes—
7#15692310—21no—
8(C2×#87938)⋊C4—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—
22#13108075—44no—
23C2×D404—22no—
24C2×D404—21no—
25C2×D404—21no—
26C2×D404C2×C412yes—
27C2×C4×D101—22no—
28C2×C4×D101D411yescharacteristic
29C2×C4×D101D411yescharacteristic
30C2×C4×D101C2311yescharacteristic
31#13108051—22no—
32#13108051—22no—
33C4×Dic101C2×C411yescharacteristic
34C4×Dic101D411yescharacteristic
35C4×Dic101D411yescharacteristic
36C2×Q8×C101—21no—
37C4⋊C4×C101—21no—
38C4⋊C4×C101C2×C412yes—
39C42×C101C2×C411yescharacteristic
40C22×D101—22no—
41C22×D101C2×D411yescharacteristic
42C2×#87938—42no—
43C2×#87938—42no—
44C2×#87938—42no—
45Q8×C101—42no—
46C2×C4×C101—22no—
47C2×C4×C101C22⋊C411yescharacteristic
48C2×C4×C101C22⋊C411yescharacteristic
49C2×C4×C101C22×C411yescharacteristic
50C2×Dic101C22⋊C411yescharacteristic
51C2×Dic101C22⋊C411yescharacteristic
52C2×Dic101C2×D411yescharacteristic
53D404—42no—
54D404—42no—
55D404—42no—
56D404—44no—
57C4×D101—22no—
58C4×D101—41no—
59C4×D101—41no—
60C4×D101—44no—
61#11717581—42no—
62C22×C101C2×C22⋊C411yescharacteristic
63C2×D101—21no—
64C2×D101—21no—
65C2×D101—42no—
66C2×D101—42no—
67C2×D101—42no—
68C2×D101AGL(1,ℤ/8)12yes—
69#87938—82no—
70C4×C101—21no—
71C4×C101—21no—
72C4×C101—21no—
73C4×C101—42no—
74Dic101—21no—
75Dic101—21no—
76Dic101—21no—
77C2×C101#41011yescharacteristic
78C2×C101#41811yescharacteristic
79C2×C101#42111yescharacteristic
80D101—22no—
81D101—82no—
82#1514—1011nomaximal, Sylow
83C101#151411yescharacteristic, complemented, Sylow
84#532—1011no—
85#432—1011no—
86#383—1011no—
87#327—1014no—
88C2×#41—2021no—
89C4⋊Q8#8793811yescharacteristic, complemented
90#127—1011no—
91#124—2021no—
92#121—1012no—
93#121—2021no—
94C4×D4—1012no—
95C4×D4—2021no—
96C2×C42—1011no—
97C4⋊C8—2021no—
98C22⋊C8—1012no—
99#97—2021no—
100#95—1012no—
101#95—2021no—
102#95—2021no—
103#41—4044no—
104C2×Q8—21no—
105C2×D4—1012no—
106C2×D4—2022no—
107C2×D4—2021no—
108C2×D4—2021no—
109C22×C4—1011no—
110C22×C4—1011no—
111C22×C4—1011no—
112C22×C4—2022no—
113C22×C4—2022no—
114C2×C8—2022no—
115C4⋊C4—21no—
116C4⋊C4C2×#8793812yes—
117C22⋊C4—2022no—
118C22⋊C4—2022no—
119C42—1011no—
120C42—1011no—
121C42—1011no—
122C42C2×#8793811yescharacteristic
123C23—1011no—
124C23—2022no—
125Q8—42no—
126D4—4042no—
127D4—4042no—
128D4—4042no—
129D4—4044no—
130C2×C4—22no—
131C2×C4—1011no—
132C2×C4—1011no—
133C2×C4—1011no—
134C2×C4—4041no—
135C2×C4—4041no—
136C2×C4—4044no—
137C2×C4—4042no—
138C2×C4—4042no—
139C2×C4—4042no—
140C2×C4—2022no—
141C2×C4C22⋊#8793811yescharacteristic
142C2×C4C22⋊#8793811yescharacteristic
143C2×C4C22×#8793811yescharacteristic
144C8—4042no—
145C22—1012no—
146C22—4042no—
147C22—4042no—
148C22—4042no—
149C22—2021no—
150C22—2021no—
151C22C2×C22⋊#8793811yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—8082no—
157C4—2021no—
158C4—2021no—
159C4—2021no—
160C2—8082no—
161C2—2022no—
162C2#2026615811yescharacteristic
163C2#2026616911yescharacteristic
164C2#2026617611yescharacteristic
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#1514
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. #20266200C2semidirect
2. #20266122C2semidirect
3. #20266029C2semidirect
12. #15692252C4semidirect
17. #15692217C4semidirect
19. C4⋊Q8×C101C4semidirect
83. C101#1514semidirect
89. C4⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
#202662002
#202661222
#202660292
#202657282
#1514101