Finite Group Databasebeta

← #44549669

Subgroups — #44549669

Order 32128.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#30632429C211yesmaximal, characteristic
2#30632049C212yesmaximal, complemented
3#30631427C211yesmaximal, characteristic, complemented
4#30631426C212yesmaximal, complemented
5#497×C251C211yesmaximal, characteristic, complemented
6#22548704—21no—
7#22548703C2212yes—
8#22548701—22no—
9#22548685—22no—
10#22548683—21no—
11#22548638—22no—
12#22548635—22no—
13#22548626C2212yescomplemented
14C2×Q16×C251—21no—
15C2×SD16×C251—21no—
16C4⋊Q8×C251C2211yescharacteristic, complemented
17#124×C251C2211yescharacteristic
18#97×C251C2211yescharacteristic, complemented
19C23×D251—21no—
20C2×Dic502D412yes—
21C2×D1004—21no—
22C2×D1004D412yes—
23C2×C22⋊D251—22no—
24C2×C251⋊C8—22no—
25C22⋊Dic251—21no—
26(C2×D251)⋊C4—22no—
27(C2×D251)⋊C4—22no—
28(C2×D251)⋊C4—22no—
29C4×Dic251—21no—
30C2×Q8×C251—21no—
31C2×Q8×C251D411yescharacteristic
32C2×D4×C251D411yescharacteristic
33Q16×C251—42no—
34SD16×C251—42no—
35C2×C8×C251—21no—
36C4⋊C4×C251—22no—
37C22⋊C4×C251—22no—
38C42×C251C2311yescharacteristic
39C22×D251—22no—
40C22×D251—81no—
41C22×D251—82no—
42Dic502—42no—
43D1004—42no—
44D1004—81no—
45C22⋊D251—82no—
46C2×Dic251—22no—
47C251⋊C8—42no—
48C23×C251—21no—
49Q8×C251—41no—
50Q8×C251—41no—
51Q8×C251—41no—
52D4×C251—41no—
53C2×C4×C251—21no—
54C2×C4×C251—41no—
55C2×C4×C251C2×D412yes—
56C2×C4×C251C2×D411yescharacteristic
57C8×C251—22no—
58C22×C251—41no—
59C22×C251—42no—
60C22×C251#12011yescharacteristic
61C2×D251—42no—
62C2×D251—42no—
63C2×D251—42no—
64C2×D251—84no—
65C4×C251—21no—
66C4×C251—22no—
67C4×C251—41no—
68C4×C251—81no—
69Dic251—42no—
70C2×C251—81no—
71C2×C251#45311yescharacteristic
72C2×C251#45512yes—
73D251—82no—
74C251#165911yescharacteristic, complemented, Sylow
75#1659—2511nomaximal, Sylow
76H(3,𝔽4)—2511no—
77#497D25111yescharacteristic, complemented
78#489—2512no—
79#329—2511no—
80#329—2512no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D25111yescharacteristic, complemented
84#124—5022no—
85#124—5021no—
86#124—2512no—
87#124C2×D25111yescharacteristic
88#120—5022no—
89#120—5021no—
90#103—5022no—
91#102—5022no—
92#97—2512no—
93#97C2×D25111yescharacteristic, complemented
94C24—5021no—
95C2×Q8—21no—
96C2×Q8—2512no—
97C2×Q8C22⋊D25111yescharacteristic
98C2×D4—5021no—
99C2×D4—5022no—
100C2×D4—2512no—
101C2×D4C22⋊D25111yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—5022no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—5022no—
109C22⋊C4—5021no—
110C22⋊C4—5022no—
111C22⋊C4—5022no—
112C42—5021no—
113C42C22×D25111yescharacteristic
114C23—21no—
115C23—20081no—
116C23—20082no—
117C23—5022no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—10042no—
122D4—41no—
123D4—20081no—
124D4—20082no—
125D4—10042no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—5022no—
129C2×C4D4×D25112yes—
130C2×C4C2×C22⋊D25111yescharacteristic
131C8—22no—
132C8—10042no—
133C22—41no—
134C22—42no—
135C22—20084no—
136C22—10042no—
137C22—10042no—
138C22—10042no—
139C22#2254868311yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—10042no—
145C2—81no—
146C2—20082no—
147C2#3063185411yescharacteristic
148C2#3063186312yes—
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#497×C251C2
socleC22×C251#120
prime radical#30632429C2
hypercenter#497D251
hypocenterC251#1659
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #30632049C2semidirect
3. #30631427C2semidirect
4. #30631426C2semidirect
5. #497×C251C2semidirect
13. #22548626C22semidirect
16. C4⋊Q8×C251C22semidirect
18. #97×C251C22semidirect
74. C251#1659semidirect
77. #497D251semidirect
83. C4⋊Q8C2×D251semidirect
93. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324292
#306320492
#306314272
#306314262
#497×C2512
#1659251