Finite Group Databasebeta

← C22×#3688⋊C3 #13164405

Subgroups — C22×#3688⋊C3 #13164405

Order 1728.

566 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 95 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×#6204C211yesmaximal, characteristic
2C2×#3688⋊C3C216yesmaximal, direct product factor
3C22×C3×#216—31nomaximal
4C22×#3688C311yesmaximal, characteristic, complemented
5C22×3−1+2⋊C4—41nomaximal
6C2×#6204C2213yes—
7#3688⋊C3C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic
9C22×C3×Dic6—31no—
10C2×C3×#216—36no—
11C2×C3×#216—36no—
12C22×C3×C3⋊C8—31no—
13C22×Q8⋊C9—21no—
14C22×Q8⋊C9C2×C311yescharacteristic
15C2×#3688C2×C316yescomplemented
16C23×3−1+2—41no—
17C2×3−1+2⋊C4—46no—
18#6204C2311yescharacteristic
19C22×C3×Q16—91no—
20C22×#216—31no—
21C2×C32×Q8—33no—
22C2×C32×Q8C2×D313yes—
23C22×C32×C4—31no—
24C22×C3×Dic3—61no—
25C2×C3×Dic6—36no—
26C2×C3×Dic6—63no—
27C3×#216—312no—
28C3×#216—34no—
29C2×C3×C3⋊C8—36no—
30C22×Dic9—41no—
31C2×Q8⋊C9—23no—
32C2×Q8⋊C9C22×C313yes—
33#3688C22×C314yescomplemented
34C22×3−1+2—44no—
35C22×3−1+2—43no—
363−1+2⋊C4—44no—
37C22×C3×Q8—31no—
38C22×C3×Q8—91no—
39C22×C3×Q8C3×D311yescharacteristic
40C22×Dic6—31no—
41C2×C3×Q16—96no—
42C2×C3×Q16—96no—
43C22×C3×C8—91no—
44C2×#216—36no—
45C2×#216—36no—
46C22×C3⋊C8—31no—
47C23×C32S411yescharacteristic
48C32×Q8—33no—
49C32×Q8—61no—
50C32×Q8C22×D311yescharacteristic
51C2×C32×C4—33no—
52C2×C32×C4—33no—
53C2×C3×Dic3—66no—
54C3×Dic6—34no—
55C3×Dic6—66no—
56C23×C9—41no—
57C23×C9—81no—
58C3×C3⋊C8—34no—
59C2×Dic9—46no—
60Q8⋊C9—21no—
61Q8⋊C9C23×C311yescharacteristic
62C22×Q16—91noSylow
63C2×3−1+2—46no—
64C2×3−1+2—41no—
65C2×C3×Q8—33no—
66C2×C3×Q8—33no—
67C2×C3×Q8—33no—
68C2×C3×Q8—63no—
69C2×C3×Q8—96no—
70C2×C3×Q8—183no—
71C2×C3×Q8C2×C3×D313yes—
72C22×C3×C4—31no—
73C22×C3×C4—31no—
74C22×C3×C4—61no—
75C22×C3×C4—181no—
76C22×Dic3—61no—
77C2×Dic6—36no—
78C2×Dic6—63no—
79C3×Q16—912no—
80C3×Q16—94no—
81C2×C3×C8—96no—
82#216—312no—
83#216—34no—
84C2×C3⋊C8—36no—
85C22×C32#22614yes—
86C22×C32C2×S413yes—
87C32×C4—31no—
88C32×C4—33no—
89C3×Dic3—64no—
90C22×C9—43no—
91C22×C9—44no—
92C22×C9—83no—
93C22×C9—84no—
94Dic9—44no—
95C22×Q8—91no—
96C22×Q8AGL(1,ℤ/9)11yescharacteristic
97C2×Q16—96no—
98C2×Q16—96no—
99C22×C8—91no—
1003−1+2—41noSylow
101C23×C3—31no—
102C23×C3C3×S411yescharacteristic
103C3×Q8—33no—
104C3×Q8—33no—
105C3×Q8—31no—
106C3×Q8—63no—
107C3×Q8—61no—
108C3×Q8—63no—
109C3×Q8—94no—
110C3×Q8—186no—
111C3×Q8C22×C3×D311yescharacteristic
112C2×C3×C4—33no—
113C2×C3×C4—33no—
114C2×C3×C4—33no—
115C2×C3×C4—33no—
116C2×C3×C4—63no—
117C2×C3×C4—63no—
118C2×C3×C4—186no—
119C2×Dic3—66no—
120Dic6—34no—
121Dic6—66no—
122C3×C8—94no—
123C3⋊C8—34no—
124C2×C32C2×#22616yes—
125C2×C32C22×S411yescharacteristic
126C2×C9—41no—
127C2×C9—46no—
128C2×C9—81no—
129C2×C9—86no—
130C2×Q8—33no—
131C2×Q8—96no—
132C2×Q8—183no—
133C2×Q8C2×AGL(1,ℤ/9)13yes—
134C22×C4—31no—
135C22×C4—181no—
136Q16—912no—
137Q16—94no—
138C2×C8—96no—
139C22×C3—34no—
140C22×C3—33no—
141C22×C3C3×#22614yes—
142C22×C3C2×C3×S413yes—
143C3×C4—31no—
144C3×C4—33no—
145C3×C4—31no—
146C3×C4—33no—
147C3×C4—61no—
148C3×C4—63no—
149C3×C4—184no—
150Dic3—64no—
151C32C22×#22611yescharacteristic
152C9—41no—
153C9—81no—
154C23((C22⋊C9)⋊C2)⋊C311yescharacteristic
155Q8—33no—
156Q8—61no—
157Q8—94no—
158Q8—186no—
159Q8C22×AGL(1,ℤ/9)11yescharacteristic
160C2×C4—33no—
161C2×C4—33no—
162C2×C4—186no—
163C8—94no—
164C2×C3—31no—
165C2×C3—36no—
166C2×C3C2×C3×#22616yes—
167C2×C3C22×C3×S411yescharacteristic
168C22#3688⋊C314yesdirect product factor
169C22C2×((C22⋊C9)⋊C2)⋊C313yes—
170C4—31no—
171C4—33no—
172C4—184no—
173C3—31no—
174C3C22×C3×#22611yescharacteristic
175C2C2×#3688⋊C316yesdirect product factor
176C2C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC23((C22⋊C9)⋊C2)⋊C3
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×S4
prime radicalC23×C32S4
LCSG ▷ Q8⋊C9
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#3688⋊C3C2direct (N × H)
4. C22×#3688C3semidirect
7. #3688⋊C3C22direct (N × H)
15. C2×#3688C2×C3semidirect
33. #3688C22×C3semidirect
168. C22#3688⋊C3direct (N × H)
175. C2C2×#3688⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62042
C2×#3688⋊C32
C22×C3×#2163
C22×#36883
C22×3−1+2⋊C44