Finite Group Databasebeta

← #41013488

Subgroups — #41013488

Order 27008.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
2#27213006C211yesmaximal, characteristic
3#27212908C214yesmaximal, complemented
4C22×C4⋊C4×C211C211yesmaximal, characteristic
5C22×C211⋊C8C2212yes—
6C2×C4⋊Dic211C2212yes—
7C2×C22⋊Dic211—21no—
8C22⋊(C211⋊C8)—22no—
9C211⋊#95—21no—
10C23×C4×C211—21no—
11C23×C4×C211C2211yescharacteristic
12C2×C4⋊C4×C211—21no—
13C2×C4⋊C4×C211—22no—
14C2×C4⋊C4×C211C414yescomplemented
15C2×C4⋊C4×C211C2212yes—
16C22×Dic211—22no—
17C2×C211⋊C8—22no—
18C2×C211⋊C8—24no—
19C4⋊Dic211—24no—
20C22⋊Dic211—42no—
21C24×C211D411yescharacteristic
22C22×C4×C211—24no—
23C22×C4×C211—24no—
24C22×C4×C211—22no—
25C22×C4×C211—44no—
26C22×C4×C211C2×C412yes—
27C22×C4×C211D411yescharacteristic
28C22×C4×C211D412yes—
29C22×C4×C211C2311yescharacteristic
30C4⋊C4×C211—24no—
31C4⋊C4×C211—22no—
32C4⋊C4×C211C2×C414yes—
33C2×Dic211—42no—
34C2×Dic211—42no—
35C2×Dic211—42no—
36C211⋊C8—42no—
37C23×C211—22no—
38C23×C211—22no—
39C23×C211C22⋊C412yes—
40C23×C211SD1612yes—
41C23×C211Q1612yes—
42C23×C211C2×D411yescharacteristic
43C2×C4×C211—22no—
44C2×C4×C211—24no—
45C2×C4×C211—24no—
46C2×C4×C211—24no—
47C2×C4×C211—48no—
48C2×C4×C211—44no—
49C2×C4×C211C22⋊C412yes—
50C2×C4×C211C22×C411yescharacteristic
51C2×C4×C211C2×D411yescharacteristic
52C2×C4×C211#4114yes—
53C22×C211—22no—
54C22×C211—22no—
55C22×C211—24no—
56C22×C211—24no—
57C22×C211#10314yes—
58C22×C211C2×C22⋊C411yescharacteristic
59C22×C211#12312yes—
60C22×C211C2×SD1611yescharacteristic
61C22×C211C2×Q1611yescharacteristic
62C22×C211AGL(1,ℤ/8)12yes—
63C4×C211—22no—
64C4×C211—44no—
65C4×C211#11712yes—
66C4×C211#12312yes—
67Dic211—82no—
68C2×C211—24no—
69C2×C211#38111yescharacteristic
70C2×C211C2×#10311yescharacteristic
71C2×C211#41811yescharacteristic
72C2×C211#48112yes—
73C2×C211#48412yes—
74C211#182411yescharacteristic, complemented, Sylow
75#1824—2111nomaximal, Sylow
76C22×C4⋊C4D21111yescharacteristic
77C2×C22⋊C8—2111no—
78#380—2111no—
79#340—2114no—
80C23×C4—21no—
81C23×C4C2×D21111yescharacteristic
82C22×C8—2112no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—2112no—
86C2×C4⋊C4Dic21114yescomplemented
87C2×C4⋊C4C2×D21112yes—
88C2×C22⋊C4—4221no—
89C22⋊C8—4222no—
90#95—4221no—
91C24C22⋊D21111yescharacteristic
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—44no—
96C22×C4—4222no—
97C22×C4C2×Dic21112yes—
98C22×C4C22⋊D21111yescharacteristic
99C22×C4C22⋊D21112yes—
100C22×C4C22×D21111yescharacteristic
101C2×C8—4222no—
102C2×C8—4224no—
103C4⋊C4—24no—
104C4⋊C4—22no—
105C4⋊C4—4224no—
106C4⋊C4C2×Dic21114yes—
107C22⋊C4—8442no—
108C23—22no—
109C23—22no—
110C23C22⋊Dic21112yes—
111C23#1680994912yes—
112C23#1680995212yes—
113C23C2×C22⋊D21111yescharacteristic
114C2×C4—22no—
115C2×C4—24no—
116C2×C4—24no—
117C2×C4—24no—
118C2×C4—48no—
119C2×C4—44no—
120C2×C4—8442no—
121C2×C4—8442no—
122C2×C4—8442no—
123C2×C4C22⋊Dic21112yes—
124C2×C4C22×Dic21111yescharacteristic
125C2×C4C2×C22⋊D21111yescharacteristic
126C2×C4#1680996214yes—
127C8—8442no—
128C22—22no—
129C22—24no—
130C22—24no—
131C22—22no—
132C22#2026990114yes—
133C22C2×C22⋊Dic21111yescharacteristic
134C22#2026996012yes—
135C22C2×#1680994911yescharacteristic
136C22C2×#1680995211yescharacteristic
137C22#2027000312yes—
138C4—22no—
139C4—44no—
140C4—16882no—
141C4#2026993612yes—
142C4#2026996012yes—
143C2—24no—
144C2#2721300911yescharacteristic
145C2C2×#2026990111yescharacteristic
146C2#2721315111yescharacteristic
147C2#2721346412yes—
148C2#2721348112yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×C4⋊C4×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC22×C4⋊C4D211
hypocenterC211#1824
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #27212908C2semidirect
14. C2×C4⋊C4×C211C4semidirect
74. C211#1824semidirect
86. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C211⋊C8)2
#272130062
#272129082
C22×C4⋊C4×C2112
#1824211