Finite Group Databasebeta

← #44549671

Subgroups — #44549671

Order 32128.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#30632082C211yesmaximal, characteristic, complemented
2#30632046C212yesmaximal, complemented
3#30631427C212yesmaximal, complemented
4#30631427C211yesmaximal, characteristic, complemented
5H(3,𝔽4)×C251C211yesmaximal, characteristic
6C2×#16860800—21no—
7C2×#16860798—21no—
8#22548715C2211yescharacteristic, complemented
9#22548639—22no—
10#22548636—22no—
11#22548626C2212yescomplemented
12#22548626C2211yescharacteristic, complemented
13#124×C251—22no—
14#124×C251—21no—
15#124×C251C2212yes—
16#124×C251C2211yescharacteristic
17#120×C251—22no—
18#120×C251—21no—
19C2×Dic502—21no—
20#16860800—42no—
21#16860798—42no—
22C2×C251⋊C8—21no—
23C2×C251⋊C8—22no—
24C4⋊Dic251—22no—
25C24×C251—21no—
26C2×Q8×C251D412yes—
27C2×Q8×C251D411yescharacteristic
28C2×D4×C251—21no—
29C2×D4×C251—22no—
30C2×D4×C251D412yes—
31C2×D4×C251D411yescharacteristic
32C22⋊C4×C251—22no—
33C22⋊C4×C251—21no—
34C22⋊C4×C251—22no—
35C22⋊C4×C251—22no—
36C22⋊C4×C251—22no—
37C42×C251—21no—
38C42×C251C2311yescharacteristic
39Dic502—41no—
40Dic502—41no—
41C2×Dic251—41no—
42C251⋊C8—22no—
43C251⋊C8—42no—
44C23×C251—21no—
45C23×C251—22no—
46C23×C251—81no—
47C23×C251—82no—
48Q8×C251—42no—
49Q8×C251—41no—
50D4×C251—42no—
51D4×C251—41no—
52D4×C251—81no—
53D4×C251—82no—
54C2×C4×C251—21no—
55C2×C4×C251—22no—
56C2×C4×C251C2×D412yes—
57C2×C4×C251C2×D411yescharacteristic
58C22×C251—41no—
59C22×C251—42no—
60C22×C251—42no—
61C22×C251—42no—
62C22×C251—42no—
63C22×C251—84no—
64C22×C251#12011yescharacteristic
65C4×C251—21no—
66C4×C251—22no—
67C4×C251—41no—
68C4×C251—42no—
69Dic251—81no—
70C2×C251—81no—
71C2×C251—82no—
72C2×C251#45311yescharacteristic
73C2×C251#45512yes—
74C251#165911yescharacteristic, complemented, Sylow
75#1659—2511nomaximal, Sylow
76H(3,𝔽4)D25111yescharacteristic
77#497—2511no—
78#489—2512no—
79#329—2511no—
80#329—2512no—
81C2×Q16—5021no—
82C2×SD16—5021no—
83C4⋊Q8—2511no—
84#124—22no—
85#124—21no—
86#124C2×D25112yes—
87#124C2×D25111yescharacteristic
88#120—22no—
89#120—21no—
90#103—5022no—
91#102—5022no—
92#97—2512no—
93#97—2511no—
94C24—21no—
95C2×Q8—5021no—
96C2×Q8C22⋊D25112yes—
97C2×Q8C22⋊D25111yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D25112yes—
101C2×D4C22⋊D25111yescharacteristic
102Q16—10042no—
103SD16—10042no—
104C2×C8—5022no—
105C2×C8—5021no—
106C4⋊C4—5022no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D25111yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—10041no—
121Q8—10041no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—10041no—
129C2×C4C2×C22⋊D25112yes—
130C2×C4C2×C22⋊D25111yescharacteristic
131C8—10042no—
132C8—5022no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#2254868411yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—20081no—
145C2—81no—
146C2—82no—
147C2#3063185511yescharacteristic
148C2#3063186512yes—
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting, prime radicalH(3,𝔽4)×C251C2
socleC22×C251#120
hypercenterH(3,𝔽4)D251
hypocenterC251#1659
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #30632082C2semidirect
2. #30632046C2semidirect
3. #30631427C2semidirect
4. #30631427C2semidirect
8. #22548715C22semidirect
11. #22548626C22semidirect
12. #22548626C22semidirect
74. C251#1659semidirect

Maximal subgroups

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

subgroupindex
#306320822
#306320462
#306314272
H(3,𝔽4)×C2512
#1659251