Finite Group Databasebeta

← C3×Q8×S4 #10596560

Subgroups — C3×Q8×S4 #10596560

Order 576.

324 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C3×Q8×A4C211yesmaximal, characteristic, complemented
2C3×C4×S4C213yesmaximal
3C3×A4⋊Q8C213yesmaximal, complemented
4Q8×S4C311yesmaximal, characteristic, direct product factor
5C3×D4×Q8—31nomaximal
6C2×C3×S4C2211yescharacteristic
7C3×D3×Q8—41nomaximal
8C3×C4×A4C2213yes—
9C3×SL(2,ℤ/4)C2213yes—
10C22×C3×Q8—31no—
11C22×C3×Q8D311yescharacteristic, complemented
12Q8×A4—21no—
13Q8×A4C2×C311yescharacteristic, complemented
14C4×S4C2×C313yes—
15A4⋊Q8C2×C313yescomplemented
16C3×C4⋊Q8—33no—
17C3×C22⋊Q8—33no—
18C3×C22⋊Q8—33no—
19C3×C4×Q8—31no—
20C3×C4×D4—33no—
21C2×C3×A4C2311yescharacteristic
22C3×S4Q812yesdirect product factor
23C32×Q8—41no—
24C3×C4×D3—43no—
25C3×Dic6—43no—
26D4×Q8—31noSylow
27C2×S4C22×C311yescharacteristic
28C2×C3×Q8—31no—
29C2×C3×Q8—33no—
30C2×C3×Q8—33no—
31C2×C3×Q8—63no—
32C2×C3×Q8—61no—
33C2×C3×D4—31no—
34C22×C3×C4—33no—
35C22×C3×C4C2×D313yes—
36D3×Q8—41no—
37C4×A4—23no—
38C4×A4C22×C313yes—
39SL(2,ℤ/4)C22×C313yes—
40C3×C4⋊C4—33no—
41C3×C4⋊C4—33no—
42C3×C4⋊C4—36no—
43C3×C22⋊C4—33no—
44C3×C22⋊C4—33no—
45C3×C42—33no—
46C2×C3×D3—41no—
47C3×A4C2×Q811yescharacteristic, complemented
48C32×C4—43no—
49C3×Dic3—43no—
50C22×Q8—31no—
51C22×Q8C3×D311yescharacteristic, complemented
52C4⋊Q8—33no—
53C22⋊Q8—33no—
54C22⋊Q8—33no—
55C4×Q8—31no—
56C4×D4—33no—
57C23×C3—31no—
58C23×C3C22×D311yescharacteristic
59C2×A4—21no—
60C2×A4C23×C311yescharacteristic
61S4C3×Q812yesdirect product factor
62C3×Q8—33no—
63C3×Q8—41no—
64C3×Q8—61no—
65C3×Q8—63no—
66C3×Q8—63no—
67C3×Q8—81no—
68C3×Q8S411yescharacteristic, direct product factor
69C3×D4—32no—
70C3×D4—32no—
71C2×C3×C4—31no—
72C2×C3×C4—33no—
73C2×C3×C4—33no—
74C2×C3×C4—33no—
75C2×C3×C4—33no—
76C2×C3×C4—63no—
77C2×C3×C4—63no—
78C4×D3—43no—
79Dic6—43no—
80C2×C32—41no—
81C3×D3—42no—
82C2×Q8—31no—
83C2×Q8—33no—
84C2×Q8—33no—
85C2×Q8—63no—
86C2×Q8—61no—
87C2×D4—31no—
88C22×C4—33no—
89C22×C4C2×C3×D313yes—
90C4⋊C4—33no—
91C4⋊C4—33no—
92C4⋊C4—36no—
93C22⋊C4—33no—
94C22⋊C4—33no—
95C42—33no—
96C22×C3—31no—
97C22×C3—31no—
98C22×C3—32no—
99C22×C3—61no—
100C22×C3—61no—
101C22×C3D3×Q811yescharacteristic, complemented
102C2×D3—41no—
103A4—21no—
104A4C2×C3×Q811yescharacteristic, complemented
105C3×C4—32no—
106C3×C4—33no—
107C3×C4—43no—
108C3×C4—63no—
109C3×C4—63no—
110C3×C4—83no—
111C3×C4C2×S413yes—
112Dic3—43no—
113C32—41noSylow
114C23—31no—
115C23C22×C3×D311yescharacteristic
116Q8—33no—
117Q8—61no—
118Q8—63no—
119Q8—63no—
120Q8C3×S411yescharacteristic, direct product factor
121D4—32no—
122D4—32no—
123C2×C4—31no—
124C2×C4—33no—
125C2×C4—33no—
126C2×C4—33no—
127C2×C4—33no—
128C2×C4—63no—
129C2×C4—63no—
130C2×C3—31no—
131C2×C3—31no—
132C2×C3—41no—
133C2×C3—62no—
134C2×C3—81no—
135C2×C3C22×S411yescharacteristic
136D3—42no—
137C22—31no—
138C22—31no—
139C22—32no—
140C22—61no—
141C22—61no—
142C22C3×D3×Q811yescharacteristic, complemented
143C4—32no—
144C4—33no—
145C4—63no—
146C4—63no—
147C4C2×C3×S413yes—
148C3—41no—
149C3—81no—
150C3Q8×S411yescharacteristic, direct product factor
151C2—31no—
152C2—31no—
153C2—62no—
154C2C22×C3×S411yescharacteristic
subgroupG/H
centerC2×C3C22×S4
commutator, cosocleC2×A4C23×C3
FrattiniC2C22×C3×S4
FittingC22×C3×Q8D3
socleC23×C3C22×D3
prime radicalC2×C3×S4C22
hypercenterC3×Q8S4
hypocenterA4C2×C3×Q8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×A4C2semidirect
3. C3×A4⋊Q8C2semidirect
4. Q8×S4C3direct (N × H)
11. C22×C3×Q8D3semidirect
13. Q8×A4C2×C3semidirect
15. A4⋊Q8C2×C3semidirect
22. C3×S4Q8direct (N × H)
47. C3×A4C2×Q8semidirect
51. C22×Q8C3×D3semidirect
61. S4C3×Q8direct (N × H)
68. C3×Q8S4direct (N × H)
101. C22×C3D3×Q8semidirect
104. A4C2×C3×Q8semidirect
120. Q8C3×S4direct (N × H)
142. C22C3×D3×Q8semidirect
150. C3Q8×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×A42
C3×C4×S42
C3×A4⋊Q82
Q8×S43
C3×D4×Q83
C3×D3×Q84