Finite Group Databasebeta

← #44549444

Subgroups — #44549444

Order 32128.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#30632417C211yesmaximal, characteristic, complemented
2#30632061C211yesmaximal, characteristic
3#30631888C212yesmaximal
4#30631417C212yesmaximal, complemented
5#333×C251C211yesmaximal, characteristic, complemented
6C2×#16860799—22no—
7C2×#16860797—22no—
8#22548714—22no—
9#22548713—22no—
10#22548712C2211yescharacteristic
11C4×Dic502C2212yes—
12#22548680—22no—
13#22548637—22no—
14C4×C251⋊C8C2211yescharacteristic
15C4⋊Q8×C251C2211yescharacteristic
16C4⋊C8×C251C2212yes—
17C2×Dic502—22no—
18C2×Dic502D412yes—
19C2×D1004—22no—
20C2×D1004—21no—
21#16860799—44no—
22#16860797—44no—
23C2×C251⋊C8—21no—
24C4⋊Dic251D412yes—
25Dic251⋊C4—24no—
26Dic251⋊C4—22no—
27C4×Dic251—22no—
28C4×Dic251—22no—
29C2×Q8×C251D412yes—
30C2×C8×C251—22no—
31C4⋊C4×C251—22no—
32C42×C251C2311yescharacteristic
33C22×D251—41no—
34Dic502—22no—
35Dic502—82no—
36Dic502SD1614yescomplemented
37D1004—42no—
38D1004—42no—
39D1004—42no—
40C2×Dic251—22no—
41C2×Dic251—22no—
42C251⋊C8—22no—
43Q8×C251—42no—
44C2×C4×C251—22no—
45C2×C4×C251C2×D412yes—
46C2×C4×C251C2×D411yescharacteristic
47C8×C251—42no—
48C22×C251#12011yescharacteristic
49C2×D251—82no—
50C2×D251—81no—
51C4×C251—21no—
52C4×C251—42no—
53C4×C251C2×SD1612yes—
54C4×C251#13712yes—
55Dic251—24no—
56Dic251—42no—
57C2×C251#44812yes—
58C2×C251#45311yescharacteristic
59D251—161no—
60C251#162311yescharacteristic, complemented, Sylow
61#1623—2511nomaximal, Sylow
62Q82—2511no—
63#492—2511no—
64#460—2512no—
65#333D25111yescharacteristic, complemented
66Q8⋊C8—2512no—
67C2×SD16—5022no—
68C2×SD16—5022no—
69C4⋊Q8—5022no—
70C4⋊Q8—5022no—
71C4⋊Q8C2×D25111yescharacteristic
72#127—2511no—
73C4×Q8—5022no—
74C4×Q8—2512no—
75C4⋊C8C2×D25112yes—
76#102—5022no—
77C4×C8—2511no—
78C2×Q8—5022no—
79C2×Q8—2512no—
80C2×Q8D100412yes—
81C2×D4—5022no—
82C2×D4—5021no—
83SD16—10044no—
84SD16—10044no—
85C2×C8—22no—
86C2×C8—5021no—
87C4⋊C4—22no—
88C4⋊C4—5024no—
89C4⋊C4—5022no—
90C4⋊C4—2512no—
91C42—5022no—
92C42—5022no—
93C42C22×D25111yescharacteristic
94C23—10041no—
95Q8—42no—
96Q8—20082no—
97Q8—5022no—
98Q8—2514no—
99D4—10042no—
100D4—10042no—
101D4—10042no—
102C2×C4—22no—
103C2×C4—5022no—
104C2×C4—5022no—
105C2×C4D4×D25112yes—
106C2×C4C2×D100411yescharacteristic
107C8—42no—
108C8—5022no—
109C22—20082no—
110C22—20081no—
111C22#2254868511yescharacteristic
112C4—21no—
113C4—42no—
114C4—10042no—
115C4—5024no—
116C4SD16×D25112yes—
117C4#2254874712yes—
118C2—40161no—
119C2#3063182712yes—
120C2#3063185711yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#333×C251C2
socleC22×C251#120
prime radical#22548712C22
hypercenter#333D251
hypocenterC251#1623
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #30632417C2semidirect
4. #30631417C2semidirect
5. #333×C251C2semidirect
36. Dic502SD16semidirect
60. C251#1623semidirect
65. #333D251semidirect

Maximal subgroups

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

subgroupindex
#306324172
#306320612
#306318882
#306314172
#333×C2512
#1623251