Finite Group Databasebeta

← #44549099

Subgroups — #44549099

Order 32128.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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#30632238C211yesmaximal, characteristic
2#30631709C212yesmaximal, complemented
3#30631436C211yesmaximal, characteristic, complemented
4#30631430C212yesmaximal, complemented
5#333×C251C211yesmaximal, characteristic, complemented
6C2×#16860811C412yes—
7C2×#16860793—22no—
8#22548712C2211yescharacteristic
9#22548704C412yes—
10#22548690—22no—
11#22548677—22no—
12C42×D251C2211yescharacteristic
13C4⋊(C251⋊C8)C2212yes—
14Dic251⋊C8—22no—
15C4⋊Q8×C251C2211yescharacteristic
16C4⋊C8×C251C2212yes—
17#16860811—44no—
18C2×D1004—22no—
19C2×D1004—22no—
20C2×D1004C2×C412yes—
21C2×C4×D251—22no—
22C2×C4×D251C2×C411yescharacteristic
23C2×C4×D251D412yes—
24#16860793—44no—
25C2×C251⋊C8—22no—
26(C2×D251)⋊C4—24no—
27C4×Dic251C2×C411yescharacteristic
28C4×Dic251D412yes—
29C2×Q8×C251C2×C412yes—
30C2×C8×C251—22no—
31C4⋊C4×C251—22no—
32C42×C251C2311yescharacteristic
33C22×D251—22no—
34C22×D251C22⋊C411yescharacteristic
35D1004—42no—
36D1004—44no—
37D1004—42no—
38D1004—42no—
39C2×Dic251C22⋊C412yes—
40C2×Dic251C22⋊C411yescharacteristic
41C4×D251—22no—
42C4×D251—22no—
43C4×D251—22no—
44C4×D251—44no—
45C251⋊C8—42no—
46Q8×C251—42no—
47C2×C4×C251—22no—
48C2×C4×C251C22×C411yescharacteristic
49C2×C4×C251C2×D412yes—
50C8×C251—42no—
51C22×C251C2×C22⋊C411yescharacteristic
52C2×D251—21no—
53C2×D251—22no—
54C2×D251—42no—
55C2×D251—44no—
56C4×C251—21no—
57C4×C251—42no—
58C4×C251AGL(1,ℤ/8)12yes—
59C4×C251#13712yes—
60Dic251—22no—
61Dic251#10012yes—
62C2×C251C2×#10011yescharacteristic
63C2×C251#41712yes—
64D251—41no—
65D251—82no—
66C251#156011yescharacteristic, complemented, Sylow
67#1560—2511nomaximal, Sylow
68#532—2511no—
69#423—2512no—
70#333—2511no—
71#333D25111yescharacteristic, complemented
72#331—2512no—
73C2×#41—2512no—
74C2×M16—5022no—
75C4⋊Q8C2×D25111yescharacteristic
76#127—2511no—
77#124—2512no—
78#121—5022no—
79C4×D4—5022no—
80C2×C42—2511no—
81C4⋊C8—5022no—
82C4⋊C8—2512no—
83C4⋊C8C2×D25112yes—
84#41—10044no—
85C2×Q8C4×D25112yes—
86C2×D4—5022no—
87C2×D4—5022no—
88C2×D4—2512no—
89C22×C4—5022no—
90C22×C4—2511no—
91C22×C4—2512no—
92M16—10044no—
93C2×C8—22no—
94C2×C8—5022no—
95C4⋊C4—22no—
96C22⋊C4—5024no—
97C42—2511no—
98C42—2512no—
99C42C22×D25111yescharacteristic
100C23—5022no—
101C23—2511no—
102Q8—42no—
103D4—10042no—
104D4—10044no—
105D4—10042no—
106D4—10042no—
107C2×C4—22no—
108C2×C4—10044no—
109C2×C4—5022no—
110C2×C4—5022no—
111C2×C4—5022no—
112C2×C4—2512no—
113C2×C4—2511no—
114C2×C4C2×C4×D25111yescharacteristic
115C2×C4D4×D25112yes—
116C8—42no—
117C8—10042no—
118C22—10044no—
119C22—10042no—
120C22—5022no—
121C22—5021no—
122C22C22⋊C4×D25111yescharacteristic
123C4—21no—
124C4—42no—
125C4—5022no—
126C4—2512no—
127C4#2254874212yes—
128C4#2254874712yes—
129C2—20082no—
130C2—10041no—
131C2#100×D25111yescharacteristic
132C2#3063168712yes—
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
Fitting#333×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632238C2
cosocleC42×C251C23
hypercenter#333D251
hypocenterC251#1560
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #30631709C2semidirect
3. #30631436C2semidirect
4. #30631430C2semidirect
5. #333×C251C2semidirect
66. C251#1560semidirect
71. #333D251semidirect

Maximal subgroups

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

subgroupindex
#306322382
#306317092
#306314362
#306314302
#333×C2512
#1560251