Finite Group Databasebeta

← #41012520

Subgroups — #41012520

Order 27008.

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

normal subgroups only

This group has 218 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#27213882C211yesmaximal, characteristic
2#27213518C211yesmaximal, characteristic
3#27213442C212yesmaximal, complemented
4#27212876C212yesmaximal, complemented
5#331×C211C211yesmaximal, characteristic
6C2×#16809950—22no—
7#20269977C2211yescharacteristic
8#20269964—22no—
9#20269958—22no—
10C4×Dic422C2212yes—
11D4×Dic211—22no—
12#20269911—22no—
13#20269902—22no—
14C4×C211⋊C8C2211yescharacteristic
15#127×C211C2211yescharacteristic
16C4⋊C8×C211C2212yes—
17C2×Dic422—21no—
18C2×Dic422D412yes—
19C22×Dic211—22no—
20#16809950—44no—
21C2×C211⋊C8—21no—
22C4⋊Dic211—22no—
23C4⋊Dic211D412yes—
24Dic211⋊C4—22no—
25C22⋊Dic211—24no—
26C22⋊Dic211—22no—
27C4×Dic211—22no—
28C2×D4×C211—22no—
29C2×D4×C211D412yes—
30C2×C8×C211—22no—
31C42×C211C2311yescharacteristic
32Dic422—22no—
33Dic422—22no—
34Dic422—42no—
35C2×Dic211—22no—
36C2×Dic211—22no—
37C2×Dic211—41no—
38C2×Dic211—82no—
39C211⋊C8—22no—
40C23×C211—22no—
41D4×C211—42no—
42D4×C211—42no—
43D4×C211—42no—
44C2×C4×C211C2×D412yes—
45C2×C4×C211C2×D411yescharacteristic
46C8×C211—42no—
47C22×C211—44no—
48C22×C211—42no—
49C22×C211#12011yescharacteristic
50C4×C211—21no—
51C4×C211#13512yes—
52C4×C211#13712yes—
53Dic211—42no—
54Dic211—42no—
55Dic211—81no—
56C2×C211—82no—
57C2×C211#45212yes—
58C2×C211#45311yescharacteristic
59C211#163011yescharacteristic, complemented, Sylow
60#1630—2111nomaximal, Sylow
61#559—2111no—
62#492—2111no—
63#477—2112no—
64#331D21111yescharacteristic
65Q8⋊C8—2112no—
66C2×SD16—4222no—
67C4⋊Q8—2111no—
68#127C2×D21111yescharacteristic
69#124—4222no—
70#123—4222no—
71C4×Q8—2112no—
72C4×D4—4222no—
73#107—4222no—
74C4⋊C8C2×D21112yes—
75#103—4222no—
76C4×C8—2111no—
77C2×Q8—4221no—
78C2×Q8—2112no—
79C2×D4—22no—
80C2×D4D84412yes—
81C22×C4—4222no—
82SD16—8444no—
83C2×C8—22no—
84C2×C8—4221no—
85C4⋊C4—4222no—
86C4⋊C4—4222no—
87C4⋊C4—2112no—
88C22⋊C4—4224no—
89C22⋊C4—4222no—
90C42—4222no—
91C42C22×D21111yescharacteristic
92C23—22no—
93Q8—8442no—
94Q8—4222no—
95Q8—4222no—
96D4—42no—
97D4—42no—
98D4—42no—
99C2×C4—8441no—
100C2×C4—16882no—
101C2×C4—4222no—
102C2×C4—4222no—
103C2×C4D4×D21112yes—
104C2×C4C2×D84411yescharacteristic
105C8—42no—
106C8—4222no—
107C22—42no—
108C22—44no—
109C22#2026994711yescharacteristic
110C4—21no—
111C4—8442no—
112C4—8442no—
113C4—16881no—
114C4#2026999612yes—
115C4#2027000612yes—
116C2—82no—
117C2#2721330912yes—
118C2#2721331611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211#120
prime radical#127×C211C22
hypercenter#331D211
hypocenterC211#1630
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #27213442C2semidirect
4. #27212876C2semidirect
59. C211#1630semidirect

Maximal subgroups

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

subgroupindex
#272138822
#272135182
#272134422
#272128762
#331×C2112
#1630211