Finite Group Databasebeta

← C3×#2026 #72741

Subgroups — C3×#2026 #72741

Order 384.

258 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 100 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×C3×#125C211yesmaximal, characteristic
2C2×C3×#103C212yesmaximal
3C3×#391C211yesmaximal, characteristic
4C3×#340C212yesmaximal
5C3×#336C211yesmaximal, characteristic
6#2026C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q8C2211yescharacteristic
8C22×C3×C8C2212yes—
9C3×#125—24no—
10C2×C3×C4⋊C4—22no—
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C4⋊C4C2212yes—
13C2×C3×C42C2211yescharacteristic
14C3×#103—24no—
15C3×#95—22no—
16C2×#125C2×C311yescharacteristic
17C2×#103C2×C312yes—
18#391C2×C311yescharacteristic
19#340C2×C312yes—
20#336C2×C311yescharacteristic
21C2×C3×Q8—21no—
22C2×C3×Q8—22no—
23C2×C3×Q8—22no—
24C2×C3×Q8Q812yes—
25C22×C3×C4—21no—
26C22×C3×C4—22no—
27C22×C3×C4—21no—
28C22×C3×C4D412yes—
29C22×C3×C4C2311yescharacteristic
30C2×C3×C8—22no—
31C2×C3×C8—22no—
32C2×C3×C8—22no—
33C3×C4⋊C4—28no—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—22no—
36C3×C4⋊C4D414yes—
37C3×C42—22no—
38C22×Q8C22×C311yescharacteristic
39C22×C8C22×C312yes—
40#125—24no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4C22×C311yescharacteristic
43C2×C4⋊C4C22×C312yes—
44C2×C42C22×C311yescharacteristic
45#103—24no—
46#95—22no—
47C23×C3C2×D411yescharacteristic
48C3×Q8—24no—
49C3×Q8—42no—
50C2×C3×C4—22no—
51C2×C3×C4—24no—
52C2×C3×C4—22no—
53C2×C3×C4—22no—
54C2×C3×C4—22no—
55C2×C3×C4—41no—
56C2×C3×C4—41no—
57C2×C3×C4—42no—
58C2×C3×C4—42no—
59C2×C3×C4—41no—
60C2×C3×C4—42no—
61C2×C3×C4C2×D412yes—
62C2×C3×C4C2×Q811yescharacteristic
63C2×C3×C4#4112yes—
64C2×C3×C4#4111yescharacteristic
65C3×C8—42no—
66C2×Q8—21no—
67C2×Q8—22no—
68C2×Q8—22no—
69C2×Q8C3×Q812yes—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4C3×D412yes—
74C22×C4C23×C311yescharacteristic
75C2×C8—22no—
76C2×C8—22no—
77C2×C8—22no—
78C4⋊C4—28no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4C3×D414yes—
82C42—22no—
83C22×C3#12011yescharacteristic
84C22×C3C22⋊Q811yescharacteristic
85C22×C3#12411yescharacteristic
86C22×C3#13512yes—
87C22×C3#13712yes—
88C3×C4—42no—
89C3×C4—42no—
90C3×C4—44no—
91C3×C4—81no—
92C3×C4C22⋊Q812yes—
93C3×C4#12412yes—
94C23C2×C3×D411yescharacteristic
95Q8—24no—
96Q8—42no—
97C2×C4—22no—
98C2×C4—24no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4C2×C3×D412yes—
109C2×C4C2×C3×Q811yescharacteristic
110C2×C4C3×#4112yes—
111C2×C4C3×#4111yescharacteristic
112C8—42no—
113C2×C3#39311yescharacteristic
114C2×C3#45212yes—
115C2×C3#47912yes—
116C2×C3#48811yescharacteristic
117C2×C3#49111yescharacteristic
118C22C3×#12011yescharacteristic
119C22C3×C22⋊Q811yescharacteristic
120C22C3×#12411yescharacteristic
121C22C3×#13512yes—
122C22C3×#13712yes—
123C4—42no—
124C4—42no—
125C4—44no—
126C4—81no—
127C4C3×C22⋊Q812yes—
128C4C3×#12412yes—
129C3#202611yescharacteristic, direct product factor, Sylow
130C2C3×#39311yescharacteristic
131C2C3×#45212yes—
132C2C3×#47912yes—
133C2C3×#48811yescharacteristic
134C2C3×#49111yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
6. #2026C3direct (N × H)
129. C3#2026direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1252
C2×C3×#1032
C3×#3912
C3×#3402
C3×#3362
#20263