Finite Group Databasebeta

← #1823×C211 #41009443

Subgroups — #1823×C211 #41009443

Order 27008.

344 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22×C4⋊C4×C211C211yesmaximal, characteristic
2C2×C22⋊C8×C211C211yesmaximal, characteristic
3#380×C211C211yesmaximal, characteristic
4#340×C211C214yesmaximal, complemented
5C23×C4×C211—21no—
6C23×C4×C211C2211yescharacteristic
7C22×C8×C211C2212yes—
8C2×C4⋊C4×C211—21no—
9C2×C4⋊C4×C211—22no—
10C2×C4⋊C4×C211C414yescomplemented
11C2×C4⋊C4×C211C2212yes—
12C2×C4⋊C4×C211C2212yes—
13C2×C22⋊C4×C211—21no—
14C22⋊C8×C211—22no—
15#95×C211—21no—
16C24×C211D411yescharacteristic
17C22×C4×C211—24no—
18C22×C4×C211—24no—
19C22×C4×C211—22no—
20C22×C4×C211—22no—
21C22×C4×C211—42no—
22C22×C4×C211—42no—
23C22×C4×C211C2×C412yes—
24C22×C4×C211D411yescharacteristic
25C22×C4×C211D412yes—
26C22×C4×C211C2311yescharacteristic
27C2×C8×C211—22no—
28C2×C8×C211—24no—
29C4⋊C4×C211—24no—
30C4⋊C4×C211—22no—
31C4⋊C4×C211—24no—
32C4⋊C4×C211C2×C414yes—
33C22⋊C4×C211—42no—
34C23×C211—22no—
35C23×C211—22no—
36C23×C211C22⋊C412yes—
37C23×C211D812yes—
38C23×C211SD1612yes—
39C23×C211C2×D411yescharacteristic
40C2×C4×C211—22no—
41C2×C4×C211—24no—
42C2×C4×C211—24no—
43C2×C4×C211—24no—
44C2×C4×C211—48no—
45C2×C4×C211—42no—
46C2×C4×C211—42no—
47C2×C4×C211—42no—
48C2×C4×C211—42no—
49C2×C4×C211—42no—
50C2×C4×C211C22⋊C412yes—
51C2×C4×C211C22×C411yescharacteristic
52C2×C4×C211C2×D411yescharacteristic
53C2×C4×C211#4114yes—
54C8×C211—42no—
55C22×C211—22no—
56C22×C211—22no—
57C22×C211—24no—
58C22×C211—24no—
59C22×C211#10214yes—
60C22×C211C2×C22⋊C411yescharacteristic
61C22×C211#12312yes—
62C22×C211C2×D811yescharacteristic
63C22×C211C2×SD1611yescharacteristic
64C22×C211#13712yes—
65C4×C211—22no—
66C4×C211—44no—
67C4×C211—82no—
68C4×C211#11712yes—
69C4×C211#12312yes—
70C2×C211—24no—
71C2×C211#38111yescharacteristic
72C2×C211C2×#10211yescharacteristic
73C2×C211#41911yescharacteristic
74C2×C211#48012yes—
75C2×C211#48312yes—
76C211#182311yescharacteristic, direct product factor, Sylow
77#1823C21111yesmaximal, characteristic, direct product factor, Sylow
78C22×C4⋊C4C2×C21111yescharacteristic
79C2×C22⋊C8C2×C21111yescharacteristic
80#380C2×C21111yescharacteristic
81#340C2×C21114yescomplemented
82C23×C4—21no—
83C23×C4C22×C21111yescharacteristic
84C22×C8C22×C21112yes—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4C4×C21114yescomplemented
88C2×C4⋊C4C22×C21112yes—
89C2×C4⋊C4C22×C21112yes—
90C2×C22⋊C4—21no—
91C22⋊C8—22no—
92#95—21no—
93C24D4×C21111yescharacteristic
94C22×C4—24no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4C2×C4×C21112yes—
101C22×C4D4×C21111yescharacteristic
102C22×C4D4×C21112yes—
103C22×C4C23×C21111yescharacteristic
104C2×C8—22no—
105C2×C8—24no—
106C4⋊C4—24no—
107C4⋊C4—22no—
108C4⋊C4—24no—
109C4⋊C4C2×C4×C21114yes—
110C22⋊C4—42no—
111C23—22no—
112C23—22no—
113C23C22⋊C4×C21112yes—
114C23D8×C21112yes—
115C23SD16×C21112yes—
116C23C2×D4×C21111yescharacteristic
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—24no—
120C2×C4—24no—
121C2×C4—48no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4C22⋊C4×C21112yes—
128C2×C4C22×C4×C21111yescharacteristic
129C2×C4C2×D4×C21111yescharacteristic
130C2×C4#41×C21114yes—
131C8—42no—
132C22—22no—
133C22—22no—
134C22—24no—
135C22—24no—
136C22#102×C21114yes—
137C22C2×C22⋊C4×C21111yescharacteristic
138C22#123×C21112yes—
139C22C2×D8×C21111yescharacteristic
140C22C2×SD16×C21111yescharacteristic
141C22#137×C21112yes—
142C4—22no—
143C4—44no—
144C4—82no—
145C4#117×C21112yes—
146C4#123×C21112yes—
147C2—24no—
148C2#381×C21111yescharacteristic
149C2C2×#102×C21111yescharacteristic
150C2#419×C21111yescharacteristic
151C2#480×C21112yes—
152C2#483×C21112yes—
subgroupG/H
center, socleC23×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC24×C211D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
4. #340×C211C2semidirect
10. C2×C4⋊C4×C211C4semidirect
76. C211#1823direct (N × H)
77. #1823C211direct (N × H)
81. #340C2×C211semidirect
87. C2×C4⋊C4C4×C211semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×C2112
C2×C22⋊C8×C2112
#380×C2112
#340×C2112
#1823211