Finite Group Databasebeta

← #41011551

Subgroups — #41011551

Order 27008.

106 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 23 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#27213328C211yesmaximal, characteristic
2#27212895C211yesmaximal, characteristic, complemented
3#356×C211C211yesmaximal, characteristic, complemented
4#20270025C411yescharacteristic
5#20270008—21no—
6C4⋊(C211⋊C8)—21no—
7#20269905—21no—
8#20269896C411yescharacteristic
9C4⋊Q8×C211C2211yescharacteristic
10#101×C211—21no—
11#16809963—41no—
12#16809962—21no—
13#16809962—41no—
14Q8×D211—41no—
15#16809952—41no—
16#16809949—41no—
17#16809946—21no—
18C2×C211⋊C8—41no—
19C2×Q8×C211C2×C411yescharacteristic
20C2×Q8×C211D411yescharacteristic
21M16×C211—41no—
22C4⋊C4×C211—41no—
23C42×C211D411yescharacteristic
24Dic422—21no—
25Dic422—41no—
26D844—21no—
27D844—41no—
28C22⋊D211—41no—
29C4×D211—41no—
30C4×D211—42no—
31C211⋊C8—41no—
32C211⋊C8—81no—
33Q8×C211—41no—
34Q8×C211—41no—
35C2×C4×C211—21no—
36C2×C4×C211—21no—
37C2×C4×C211—21no—
38C2×C4×C211C22⋊C411yescharacteristic
39C8×C211—81no—
40C22×C211#9911yescharacteristic
41C2×D211—41no—
42C4×C211—21no—
43C4×C211—21no—
44C4×C211—21no—
45C4×C211—41no—
46C4×C211—41no—
47Dic211—41no—
48C2×C211—21no—
49C2×C211AΣL(1,𝔽16)11yescharacteristic
50D211—81no—
51C211#140511yescharacteristic, complemented, Sylow
52#1405—2111nomaximal, Sylow
53#456—2111no—
54#356D21111yescharacteristic, complemented
55#333—2111no—
562−1+4—2111no—
57#137—4221no—
58C4⋊Q8C2×D21111yescharacteristic
59C4⋊C8—4221no—
60#104—4221no—
61#101—21no—
62#101—2111no—
63#41—8441no—
64#41—8441no—
65#41—4221no—
66C2×Q8—8441no—
67C2×Q8C4×D21111yescharacteristic
68C2×Q8C22⋊D21111yescharacteristic
69Q16—8441no—
70SD16—8441no—
71M16—41no—
72M16—4221no—
73C2×C8—8441no—
74C4⋊C4—41no—
75C42D84411yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—8441no—
79Q8—4221no—
80D4—8441no—
81D4—8441no—
82D4—4221no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—8441no—
87C2×C4—8442no—
88C2×C4(C2×D211)⋊C411yescharacteristic
89C8—81no—
90C8—8441no—
91C8—16881no—
92C22—8441no—
93C22#2026989111yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—8441no—
100C2—21no—
101C2—16881no—
102C2#2721293211yescharacteristic
subgroupG/H
centerC2#27212932
commutatorC2×Q8×C211C2×C4
FrattiniC4⋊Q8C2×D211
Fitting#356×C211C2
socleC2×C211AΣL(1,𝔽16)
prime radical#20270025C4
cosocleC4⋊Q8×C211C22
hypercenter#356D211
hypocenterC211#1405
LCSG ▷ C2×Q8×C211 ▷ C2×C4×C211 ▷ C22×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #27212895C2semidirect
3. #356×C211C2semidirect
51. C211#1405semidirect
54. #356D211semidirect

Maximal subgroups

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

subgroupindex
#272133282
#272128952
#356×C2112
#1405211