Finite Group Databasebeta

← #41011251

Subgroups — #41011251

Order 27008.

226 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 61 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#27213110C211yesmaximal, characteristic
2#27213009C211yesmaximal, characteristic
3C4×C22⋊C4×C211C211yesmaximal, characteristic
4C2×#16809946—21no—
5C2×C22⋊Dic211—21no—
6C22⋊(C211⋊C8)—21no—
7C22⋊(C211⋊C8)C412yescomplemented
8C211⋊#95—21no—
9C211⋊#95C412yescomplemented
10C23×C4×C211C2211yescharacteristic
11C2×C22⋊C4×C211—21no—
12C2×C42×C211C412yescomplemented
13#95×C211—21no—
14C22×Dic211—22no—
15#16809946—42no—
16C2×C211⋊C8—22no—
17C22⋊Dic211—42no—
18C24×C211D411yescharacteristic
19C22×C4×C211—22no—
20C22×C4×C211—42no—
21C22×C4×C211C2×C411yescharacteristic
22C22×C4×C211C2×C411yescharacteristic
23C22×C4×C211C2×C411yescharacteristic
24C22×C4×C211D411yescharacteristic
25C22×C4×C211D411yescharacteristic
26C22×C4×C211Q811yescharacteristic
27C22⋊C4×C211—24no—
28C42×C211—42no—
29C2×Dic211—42no—
30C2×Dic211—42no—
31C2×Dic211—42no—
32C211⋊C8—42no—
33C23×C211—22no—
34C23×C211—22no—
35C23×C211—41no—
36C23×C211C22⋊C411yescharacteristic
37C23×C211C22⋊C411yescharacteristic
38C23×C211C4⋊C411yescharacteristic
39C2×C4×C211—22no—
40C2×C4×C211—22no—
41C2×C4×C211—24no—
42C2×C4×C211—41no—
43C2×C4×C211—41no—
44C2×C4×C211—41no—
45C2×C4×C211—41no—
46C2×C4×C211—42no—
47C2×C4×C211—42no—
48C2×C4×C211C4211yescharacteristic
49C2×C4×C211C22⋊C411yescharacteristic
50C2×C4×C211C4⋊C411yescharacteristic
51C2×C4×C211C4⋊C411yescharacteristic
52C22×C211—21no—
53C22×C211—21no—
54C22×C211—21no—
55C22×C211—21no—
56C22×C211—21no—
57C22×C211—21no—
58C22×C211—21no—
59C22×C211—44no—
60C22×C211#9511yescharacteristic
61C22×C211#9912yes—
62C22×C211#10412yes—
63C4×C211—21no—
64C4×C211—21no—
65C4×C211—21no—
66C4×C211—21no—
67C4×C211—44no—
68Dic211—82no—
69C2×C211—22no—
70C2×C211—22no—
71C2×C211—41no—
72C2×C211H(3,ℤ/4)11yescharacteristic
73C2×C211#33911yescharacteristic
74C2×C211#34211yescharacteristic
75C211#128411yescharacteristic, complemented, Sylow
76#1284—2111nomaximal, Sylow
77#407—2111no—
78#381—2111no—
79C4×C22⋊C4D21111yescharacteristic
80C23×C4C2×D21111yescharacteristic
81C2×M16—4221no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—4221no—
84C2×C42Dic21112yescomplemented
85C22⋊C8—4221no—
86C22⋊C8—2112no—
87#95—21no—
88#95—4221no—
89#95—2112no—
90C24C22⋊D21111yescharacteristic
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—4222no—
94C22×C4C4×D21111yescharacteristic
95C22×C4C4×D21111yescharacteristic
96C22×C4C2×Dic21111yescharacteristic
97C22×C4C22⋊D21111yescharacteristic
98C22×C4D84411yescharacteristic
99C22×C4Dic42211yescharacteristic
100M16—8442no—
101C2×C8—4222no—
102C22⋊C4—24no—
103C22⋊C4—8442no—
104C42—42no—
105C23—22no—
106C23—22no—
107C23—41no—
108C23(C2×D211)⋊C411yescharacteristic
109C23C22⋊Dic21111yescharacteristic
110C23Dic211⋊C411yescharacteristic
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—24no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—8442no—
121C2×C4—8442no—
122C2×C4—8442no—
123C2×C4C4×Dic21111yescharacteristic
124C2×C4(C2×D211)⋊C411yescharacteristic
125C2×C4Dic211⋊C411yescharacteristic
126C2×C4C4⋊Dic21111yescharacteristic
127C8—8442no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—44no—
136C22C211⋊#9511yescharacteristic
137C22#2026989212yes—
138C22#2026990512yes—
139C4—21no—
140C4—21no—
141C4—21no—
142C4—21no—
143C4—44no—
144C4—16882no—
145C2—22no—
146C2—22no—
147C2—41no—
148C2#2721290211yescharacteristic
149C2#2721290611yescharacteristic
150C2#2721291311yescharacteristic
subgroupG/H
centerC22C211⋊#95
commutatorC2×C4×C211C42
FrattiniC23×C4C2×D211
FittingC4×C22⋊C4×C211C2
socleC22×C211#95
prime radicalC24×C211D4
cosocleC23×C4×C211C22
hypercenterC4×C22⋊C4D211
hypocenterC211#1284
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. C22⋊(C211⋊C8)C4semidirect
9. C211⋊#95C4semidirect
12. C2×C42×C211C4semidirect
75. C211#1284semidirect
84. C2×C42Dic211semidirect

Maximal subgroups

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

subgroupindex
#272131102
#272130092
C4×C22⋊C4×C2112
#1284211