Finite Group Databasebeta

← #46835448

Subgroups — #46835448

Order 34688.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#31774613C211yesmaximal, characteristic, complemented
2#31774271C211yesmaximal, characteristic, complemented
3#31774093C212yesmaximal, complemented
4#31773624C212yesmaximal, complemented
5#333×C271C211yesmaximal, characteristic, complemented
6C2×#16864641—22no—
7C2×D2168—22no—
8#22576157C2211yescharacteristic
9#22576149—22no—
10#22576135—22no—
11C4×D1084C2212yes—
12#22576118—22no—
13#22576082—22no—
14C4×C271⋊C8C2211yescharacteristic
15C4⋊Q8×C271C2211yescharacteristic
16C4⋊C8×C271C2212yes—
17C2×D1084—22no—
18C2×D1084—22no—
19C2×D1084—21no—
20C2×D1084D412yes—
21C2×C4×D271—22no—
22#16864641—44no—
23D2168—44no—
24C2×C271⋊C8—21no—
25C4⋊Dic271D412yes—
26(C2×D271)⋊C4—24no—
27(C2×D271)⋊C4—22no—
28C4×Dic271—22no—
29C2×Q8×C271D412yes—
30C2×C8×C271—22no—
31C4⋊C4×C271—22no—
32C42×C271C2311yescharacteristic
33C22×D271—22no—
34C22×D271—41no—
35D1084—22no—
36D1084—22no—
37D1084—42no—
38D1084—42no—
39D1084—42no—
40D1084—82no—
41C2×Dic271—22no—
42C4×D271—42no—
43C4×D271—42no—
44C271⋊C8—22no—
45Q8×C271—42no—
46C2×C4×C271—22no—
47C2×C4×C271C2×D412yes—
48C2×C4×C271C2×D411yescharacteristic
49C8×C271—42no—
50C22×C271#12011yescharacteristic
51C2×D271—42no—
52C2×D271—42no—
53C2×D271—42no—
54C2×D271—82no—
55C2×D271—81no—
56C4×C271—21no—
57C4×C271—42no—
58C4×C271#13512yes—
59C4×C271AGL(1,ℤ/8)12yes—
60Dic271—42no—
61C2×C271#44912yes—
62C2×C271#45311yescharacteristic
63D271—82no—
64D271—161no—
65C271#162411yescharacteristic, complemented, Sylow
66#1624—2711nomaximal, Sylow
67#555—2711no—
68#492—2711no—
69#459—2712no—
70#333D27111yescharacteristic, complemented
71D4⋊C8—2712no—
72C2×SD16—5422no—
73C2×D8—5422no—
74C4⋊Q8C2×D27111yescharacteristic
75#127—2711no—
76#124—5422no—
77#121—5422no—
78C4×D4—2712no—
79#117—5422no—
80C4⋊C8C2×D27112yes—
81#102—5422no—
82C4×C8—2711no—
83C2×Q8D108412yes—
84C2×D4—2712no—
85C2×D4—5422no—
86C2×D4—5422no—
87C2×D4—5421no—
88C22×C4—5422no—
89SD16—10844no—
90D8—10844no—
91C2×C8—22no—
92C2×C8—5421no—
93C4⋊C4—22no—
94C4⋊C4—2712no—
95C22⋊C4—5424no—
96C22⋊C4—5422no—
97C42—5422no—
98C42C22×D27111yescharacteristic
99C23—5422no—
100C23—10841no—
101Q8—42no—
102D4—5422no—
103D4—5422no—
104D4—10842no—
105D4—10842no—
106D4—10842no—
107D4—21682no—
108C2×C4—22no—
109C2×C4—5422no—
110C2×C4—10842no—
111C2×C4—10842no—
112C2×C4D4×D27112yes—
113C2×C4C2×D108411yescharacteristic
114C8—42no—
115C8—5422no—
116C22—10842no—
117C22—10842no—
118C22—10842no—
119C22—21682no—
120C22—21681no—
121C22#2257613011yescharacteristic
122C4—21no—
123C4—42no—
124C4—10842no—
125C4#2257618312yes—
126C4#2257618712yes—
127C2—21682no—
128C2—43361no—
129C2#3177404512yes—
130C2#3177406711yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#333×C271C2
socleC22×C271#120
hypercenter#333D271
hypocenterC271#1624
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #31774613C2semidirect
2. #31774271C2semidirect
3. #31774093C2semidirect
4. #31773624C2semidirect
5. #333×C271C2semidirect
65. C271#1624semidirect
70. #333D271semidirect

Maximal subgroups

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

subgroupindex
#317746132
#317742712
#317740932
#317736242
#333×C2712
#1624271