Finite Group Databasebeta

← #41003024

Subgroups — #41003024

Order 26944.

142 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 39 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#27212413C211yesmaximal, characteristic
2#27212399C211yesmaximal, characteristic, complemented
3#27212396C211yesmaximal, characteristic, complemented
4#27212366C211yesmaximal, characteristic, complemented
5#27212294C211yesmaximal, characteristic, complemented
6#27212284C211yesmaximal, characteristic, complemented
7#104×C421C211yesmaximal, characteristic, complemented
8#20269771—21no—
9#20269771C2211yescharacteristic
10#20269770—22no—
11#20269770—21no—
12C2×Dic842—21no—
13C2×Dic842C2211yescharacteristic
14C2×Dic842C2211yescharacteristic
15Q8×D421—21no—
16Dic1684—21no—
17#20269761—21no—
18#20269760—21no—
19#20269759—21no—
20#20269755C2211yescharacteristic
21C4⋊Dic421—22no—
22#41×C421C2211yescharacteristic
23M16×C421C2211yescharacteristic
24C42×C421C2211yescharacteristic
25Dic842—22no—
26Dic842—21no—
27Dic842—21no—
28Dic842—21no—
29Dic842—21no—
30Dic842D411yescharacteristic
31D1684D411yescharacteristic
32C22⋊D421—22no—
33C22⋊D421—21no—
34C2×Dic421—22no—
35C2×Dic421—21no—
36C2×Dic421D412yes—
37C4×D421—22no—
38C4×D421—21no—
39#16809890—21no—
40Q8×C421D411yescharacteristic
41D4×C421—21no—
42D4×C421D411yescharacteristic
43C2×C4×C421—21no—
44C2×C4×C421—21no—
45C2×C4×C421C2311yescharacteristic
46C8×C421—21no—
47C22×C421—21no—
48C22×C421C2×D411yescharacteristic
49C2×D421—21no—
50C4×C421—22no—
51C4×C421—21no—
52C4×C421C2×D411yescharacteristic
53C4×C421C2×D411yescharacteristic
54Dic421—21no—
55Dic421—22no—
56Dic421—41no—
57C2×C421—21no—
58C2×C421—41no—
59C2×C421#12011yescharacteristic
60D421—41no—
61C421#45611yescharacteristic, complemented, Sylow
62#456—4211nomaximal, Sylow
632−1+4—4211no—
64#137—4211no—
65#137—4211no—
66C4⋊Q8—4211no—
67#104—4211no—
68#104D42111yescharacteristic, complemented
69#101—4211no—
70#41—8421no—
71#41—8421no—
72#41—8422no—
73#41—4211no—
74#41C2×D42111yescharacteristic
75C2×Q8—8421no—
76C2×Q8—8421no—
77C2×Q8—4211no—
78C2×Q8—4211no—
79Q16—8421no—
80Q16—8421no—
81SD16—8421no—
82SD16—8421no—
83M16—4211no—
84M16C2×D42111yescharacteristic
85C4⋊C4—8422no—
86C42C2×D42111yescharacteristic
87Q8—8422no—
88Q8—8421no—
89Q8—8421no—
90Q8—8421no—
91Q8—8421no—
92Q8—4211no—
93Q8D168411yescharacteristic
94D4—21no—
95D4—8422no—
96D4—8421no—
97D4—4211no—
98D4D168411yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—8422no—
102C2×C4—8422no—
103C2×C4—8421no—
104C2×C4—8421no—
105C2×C4—4212no—
106C2×C4C22×D42111yescharacteristic
107C8—21no—
108C8—8421no—
109C22—21no—
110C22—8421no—
111C22D4×D42111yescharacteristic
112C4—22no—
113C4—21no—
114C4—8421no—
115C4—8422no—
116C4—16841no—
117C4D4×D42111yescharacteristic
118C4C2×D168411yescharacteristic
119C2—21no—
120C2—41no—
121C2—16841no—
122C2#2721233611yescharacteristic
subgroupG/H
centerC2#27212336
commutator, cosocleC2×C4×C421C23
FrattiniC2×C4C22×D421
Fitting#104×C421C2
socleC2×C421#120
prime radical#27212413C2
hypercenter#104D421
hypocenterC421#456
LCSG ▷ C2×C4×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #27212399C2semidirect
3. #27212396C2semidirect
4. #27212366C2semidirect
5. #27212294C2semidirect
6. #27212284C2semidirect
7. #104×C421C2semidirect
61. C421#456semidirect
68. #104D421semidirect

Maximal subgroups

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

subgroupindex
#272124132
#272123992
#272123962
#272123662
#272122942
#272122842
#104×C4212
#456421