Finite Group Databasebeta

← Q8×C3⋊S4 #10596581

Subgroups — Q8×C3⋊S4 #10596581

Order 576.

388 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 61 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
2C4×C3⋊S4C213yesmaximal
3#64259C213yesmaximal, complemented
4Q8×S4—33nomaximal
5Q8×#67—31nomaximal
6C2×C3⋊S4C2211yescharacteristic
7Q8×C3⋊D3—41nomaximal
8C3×C4×A4C2213yes—
9C3⋊SL(2,ℤ/4)C2213yes—
10C22×C3×Q8D311yescharacteristic, complemented
11C2×D3×Q8—31no—
12Q8×A4D313yescomplemented
13C4×S4—39no—
14A4⋊Q8—39no—
15#946—33no—
16Q8×Dic3—31no—
17#944—33no—
18C4×#67—33no—
19#924—33no—
20C2×C3×A4C2311yescharacteristic
21C3⋊S4Q812yesdirect product factor
22C32×Q8—41no—
23C4×C3⋊D3—43no—
24#634—43no—
25D4×Q8—91noSylow
26C2×S4—33no—
27C2×C3×Q8—31no—
28C2×C3×Q8—33no—
29C22×C3×C4C2×D313yes—
30C2×#67—31no—
31D3×Q8—63no—
32D3×Q8—61no—
33D3×Q8—123no—
34C2×C4×D3—33no—
35C2×Dic6—33no—
36C4×A4C2×D319yes—
37SL(2,ℤ/4)—39no—
38#217—33no—
39(C2×D3)⋊C4—33no—
40C4⋊Dic3—33no—
41Dic3⋊C4—36no—
42Dic3⋊C4—33no—
43C4×Dic3—33no—
44C2×C3⋊D3—41no—
45C3×A4C2×Q811yescharacteristic, complemented
46C32×C4—43no—
47C3⋊Dic3—43no—
48C22×Q8—91no—
49C22×Q8C3⋊D311yescharacteristic, complemented
50C4⋊Q8—93no—
51C22⋊Q8—93no—
52C22⋊Q8—93no—
53C4×Q8—91no—
54C4×D4—93no—
55C23×C3C22×D311yescharacteristic
56C22×D3—31no—
57C2×A4C22×D313yes—
58S4—36no—
59C3×Q8—33no—
60C3×Q8—43no—
61C3×Q8—61no—
62C3×Q8S411yescharacteristic, complemented
63C2×C3×C4—33no—
64C2×C3×C4—33no—
65#67—32no—
66#67—32no—
67C2×Dic3—31no—
68C2×Dic3—33no—
69C2×Dic3—33no—
70C4×D3—63no—
71C4×D3—63no—
72C4×D3—129no—
73Dic6—63no—
74Dic6—63no—
75Dic6—129no—
76C2×C32—41no—
77C3⋊D3—42no—
78C2×Q8—31no—
79C2×Q8—33no—
80C2×Q8—93no—
81C2×Q8—183no—
82C2×Q8—181no—
83C2×D4—91no—
84C22×C4—93no—
85C22×C4C2×C3⋊D313yes—
86C4⋊C4—93no—
87C4⋊C4—93no—
88C4⋊C4—96no—
89C22⋊C4—93no—
90C22⋊C4—93no—
91C42—93no—
92C22×C3—31no—
93C22×C3—31no—
94C22×C3D3×Q811yescharacteristic, complemented
95C2×D3—32no—
96C2×D3—61no—
97C2×D3—61no—
98C2×D3—123no—
99A4D3×Q813yescomplemented
100C3×C4—33no—
101C3×C4—49no—
102C3×C4C2×S413yes—
103Dic3—32no—
104Dic3—63no—
105Dic3—63no—
106Dic3—129no—
107C32—41noSylow
108C23—91no—
109C23C22×C3⋊D311yescharacteristic
110Q8—33no—
111Q8—61no—
112Q8—183no—
113Q8—183no—
114Q8C3⋊S411yescharacteristic, direct product factor
115D4—92no—
116D4—92no—
117C2×C4—33no—
118C2×C4—33no—
119C2×C4—91no—
120C2×C4—93no—
121C2×C4—93no—
122C2×C4—183no—
123C2×C4—183no—
124C2×C3—31no—
125C2×C3—31no—
126C2×C3—43no—
127C2×C3C22×S411yescharacteristic
128D3—62no—
129D3—126no—
130C22—31no—
131C22—31no—
132C22—92no—
133C22—181no—
134C22—181no—
135C22Q8×C3⋊D311yescharacteristic, complemented
136C4—33no—
137C4—92no—
138C4—183no—
139C4—183no—
140C4C2×C3⋊S413yes—
141C3—43no—
142C3Q8×S411yescharacteristic, complemented
143C2—31no—
144C2—31no—
145C2—182no—
146C2C22×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C22×C3⋊S4
commutator, cosocleC2×C3×A4C23
FittingC22×C3×Q8D3
socleC23×C3C22×D3
prime radicalC2×C3⋊S4C22
hypercenterQ8C3⋊S4
hypocenterC3×A4C2×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×A4C2semidirect
3. #64259C2semidirect
10. C22×C3×Q8D3semidirect
12. Q8×A4D3semidirect
21. C3⋊S4Q8direct (N × H)
45. C3×A4C2×Q8semidirect
49. C22×Q8C3⋊D3semidirect
62. C3×Q8S4semidirect
94. C22×C3D3×Q8semidirect
99. A4D3×Q8semidirect
114. Q8C3⋊S4direct (N × H)
135. C22Q8×C3⋊D3semidirect
142. C3Q8×S4semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×A42
C4×C3⋊S42
#642592
Q8×S43
Q8×#673
Q8×C3⋊D34