Finite Group Databasebeta

← C2×C3×#356 #72846

Subgroups — C2×C3×#356 #72846

Order 384.

240 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 88 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×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#101C212yesmaximal
3C3×#356C214yesmaximal, direct product factor
4C2×#356C311yesmaximal, characteristic, direct product factor, Sylow
5C22×C3×Q8C411yescharacteristic
6C22×C3×Q8C2211yescharacteristic
7C2×C3×M16—22no—
8C3×C4⋊Q8—22no—
9C3×C4⋊Q8C412yes—
10C3×C4⋊Q8C2212yes—
11C2×C3×C4⋊C4—22no—
12C2×C3×C42C411yescharacteristic
13C3×#101—22no—
14C3×#101C2214yes—
15C2×C4⋊Q8C2×C311yescharacteristic
16C2×#101C2×C312yes—
17#356C2×C314yesdirect product factor
18C2×C3×Q8—21no—
19C2×C3×Q8—41no—
20C2×C3×Q8—41no—
21C2×C3×Q8—41no—
22C2×C3×Q8—41no—
23C2×C3×Q8C2×C412yes—
24C2×C3×Q8C2×C411yescharacteristic
25C2×C3×Q8D412yes—
26C2×C3×Q8C2311yescharacteristic
27C22×C3×C4—21no—
28C22×C3×C4—21no—
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4D412yes—
31C3×M16—24no—
32C3×M16—42no—
33C2×C3×C8—42no—
34C3×C4⋊C4—24no—
35C3×C4⋊C4—24no—
36C3×C42—21no—
37C3×C42C2×C412yes—
38C22×Q8C3×C411yescharacteristic
39C22×Q8C22×C311yescharacteristic
40C2×M16—22no—
41C4⋊Q8—22no—
42C4⋊Q8C3×C412yes—
43C4⋊Q8C22×C312yes—
44C2×C4⋊C4—22no—
45C2×C42C3×C411yescharacteristic
46#101—22no—
47#101C22×C314yes—
48C23×C3C22⋊C411yescharacteristic
49C3×Q8—41no—
50C3×Q8—42no—
51C3×Q8—41no—
52C3×Q8—42no—
53C3×Q8—42no—
54C2×C3×C4—24no—
55C2×C3×C4—22no—
56C2×C3×C4—21no—
57C2×C3×C4—22no—
58C2×C3×C4—21no—
59C2×C3×C4—22no—
60C2×C3×C4—22no—
61C2×C3×C4—41no—
62C2×C3×C4—41no—
63C2×C3×C4C22⋊C411yescharacteristic
64C2×C3×C4C22⋊C412yes—
65C2×C3×C4C22×C411yescharacteristic
66C2×C3×C4C2×D412yes—
67C3×C8—44no—
68C2×Q8—21no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×Q8C2×C3×C412yes—
74C2×Q8C2×C3×C411yescharacteristic
75C2×Q8C3×D412yes—
76C2×Q8C23×C311yescharacteristic
77C22×C4—21no—
78C22×C4—21no—
79C22×C4C2×C3×C411yescharacteristic
80C22×C4C3×D412yes—
81M16—24no—
82M16—42no—
83C2×C8—42no—
84C4⋊C4—24no—
85C4⋊C4—24no—
86C42—21no—
87C42C2×C3×C412yes—
88C22×C3—22no—
89C22×C3#9911yescharacteristic
90C22×C3#9911yescharacteristic
91C22×C3C2×C22⋊C411yescharacteristic
92C3×C4—24no—
93C3×C4—22no—
94C3×C4—22no—
95C3×C4—21no—
96C3×C4—21no—
97C3×C4—42no—
98C23C3×C22⋊C411yescharacteristic
99Q8—41no—
100Q8—42no—
101Q8—41no—
102Q8—42no—
103Q8—42no—
104C2×C4—24no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4C3×C22⋊C411yescharacteristic
114C2×C4C3×C22⋊C412yes—
115C2×C4C22×C3×C411yescharacteristic
116C2×C4C2×C3×D412yes—
117C8—44no—
118C2×C3—21no—
119C2×C3—21no—
120C2×C3#35612yesdirect product factor
121C2×C3C2×#9911yescharacteristic
122C22—22no—
123C22C3×#9911yescharacteristic
124C22C3×#9911yescharacteristic
125C22C2×C3×C22⋊C411yescharacteristic
126C4—24no—
127C4—22no—
128C4—21no—
129C4—22no—
130C4—21no—
131C4—42no—
132C3C2×#35611yescharacteristic, direct product factor, Sylow
133C2—21no—
134C2—21no—
135C2C3×#35612yesdirect product factor
136C2C2×C3×#9911yescharacteristic
subgroupG/H
center, socleC22×C3#99
commutatorC2×C4C22×C3×C4
Frattini, cosocleC2×Q8C23×C3
prime radicalC23×C3C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#356C2direct (N × H)
4. C2×#356C3direct (N × H)
17. #356C2×C3direct (N × H)
120. C2×C3#356direct (N × H)
132. C3C2×#356direct (N × H)
135. C2C3×#356direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
C2×C3×#1012
C3×#3562
C2×#3563