Finite Group Databasebeta

← C22×#3689⋊C3 #13164407

Subgroups — C22×#3689⋊C3 #13164407

Order 1728.

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

normal subgroups only

This group has 726 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, complemented
2C2×#3689⋊C3C216yesmaximal, direct product factor
3C22×C3×#215—31nomaximal
4C22×#3689C311yesmaximal, characteristic, complemented
5C23×AGL(1,ℤ/9)—41nomaximal
6C2×#6204C2213yescomplemented
7#3689⋊C3C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic
9C22×C3×D12—31no—
10C2×C3×#215—36no—
11C2×C3×#215—36no—
12C22×C3×C3⋊C8—31no—
13C22×Q8⋊C9—21no—
14C22×Q8⋊C9C2×C311yescharacteristic, complemented
15C2×#3689C2×C316yescomplemented
16C23×3−1+2—41no—
17C22×AGL(1,ℤ/9)—46no—
18C22×AGL(1,ℤ/9)—48no—
19#6204C2311yescharacteristic, complemented
20C22×C3×SD16—91no—
21C22×#215—31no—
22C23×C3×D3—61no—
23C2×C32×Q8—33no—
24C2×C32×Q8C2×D313yes—
25C22×C32×C4—31no—
26C2×C3×D12—36no—
27C2×C3×D12—63no—
28C23×D9—41no—
29C3×#215—312no—
30C3×#215—34no—
31C2×C3×C3⋊C8—36no—
32C2×Q8⋊C9—23no—
33C2×Q8⋊C9C22×C313yescomplemented
34#3689C22×C314yescomplemented
35C22×3−1+2—44no—
36C22×3−1+2—43no—
37C2×AGL(1,ℤ/9)—44no—
38C2×AGL(1,ℤ/9)—424no—
39C22×C3×Q8—31no—
40C22×C3×Q8C3×D311yescharacteristic
41C22×C3×D4—91no—
42C22×D12—31no—
43C2×C3×SD16—96no—
44C2×C3×SD16—96no—
45C22×C3×C8—91no—
46C2×#215—36no—
47C2×#215—36no—
48C22×C3⋊C8—31no—
49C23×C32S411yescharacteristic
50C22×C3×D3—66no—
51C22×C3×D3—124no—
52C32×Q8—33no—
53C32×Q8—61no—
54C32×Q8C22×D311yescharacteristic
55C2×C32×C4—33no—
56C2×C32×C4—33no—
57C3×D12—34no—
58C3×D12—66no—
59C23×C9—41no—
60C23×C9—81no—
61C22×D9—46no—
62C22×D9—48no—
63C3×C3⋊C8—34no—
64Q8⋊C9—21no—
65Q8⋊C9C23×C311yescharacteristic, complemented
66C22×SD16—91noSylow
67C2×3−1+2—46no—
68C2×3−1+2—41no—
69AGL(1,ℤ/9)—48no—
70C24×C3—181no—
71C23×D3—61no—
72C2×C3×Q8—33no—
73C2×C3×Q8—33no—
74C2×C3×Q8—33no—
75C2×C3×Q8—63no—
76C2×C3×Q8C2×C3×D313yes—
77C2×C3×D4—96no—
78C2×C3×D4—183no—
79C22×C3×C4—31no—
80C22×C3×C4—31no—
81C22×C3×C4—61no—
82C2×D12—36no—
83C2×D12—63no—
84C3×SD16—912no—
85C3×SD16—94no—
86C2×C3×C8—96no—
87#215—312no—
88#215—34no—
89C2×C3⋊C8—36no—
90C22×C32GL(2,𝔽3)14yes—
91C22×C32C2×S413yes—
92C2×C3×D3—64no—
93C2×C3×D3—1212no—
94C32×C4—31no—
95C32×C4—33no—
96C22×C9—43no—
97C22×C9—44no—
98C22×C9—83no—
99C22×C9—84no—
100C2×D9—44no—
101C2×D9—424no—
102C22×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
103C22×D4—91no—
104C2×SD16—96no—
105C2×SD16—96no—
106C22×C8—91no—
1073−1+2—41noSylow
108C23×C3—31no—
109C23×C3—186no—
110C23×C3—364no—
111C23×C3C3×S411yescharacteristic
112C22×D3—66no—
113C22×D3—124no—
114C3×Q8—33no—
115C3×Q8—33no—
116C3×Q8—31no—
117C3×Q8—63no—
118C3×Q8—61no—
119C3×Q8—63no—
120C3×Q8C22×C3×D311yescharacteristic
121C3×D4—94no—
122C3×D4—186no—
123C2×C3×C4—33no—
124C2×C3×C4—33no—
125C2×C3×C4—33no—
126C2×C3×C4—33no—
127C2×C3×C4—63no—
128C2×C3×C4—63no—
129D12—34no—
130D12—66no—
131C3×C8—94no—
132C3⋊C8—34no—
133C2×C32C2×GL(2,𝔽3)16yes—
134C2×C32C22×S411yescharacteristic
135C3×D3—124no—
136C2×C9—41no—
137C2×C9—46no—
138C2×C9—81no—
139C2×C9—86no—
140D9—48no—
141C24—181no—
142C2×Q8—33no—
143C2×Q8C2×AGL(1,ℤ/9)13yescomplemented
144C2×D4—96no—
145C2×D4—183no—
146C22×C4—31no—
147SD16—912no—
148SD16—94no—
149C2×C8—96no—
150C22×C3—34no—
151C22×C3—33no—
152C22×C3—184no—
153C22×C3—3612no—
154C22×C3C3×GL(2,𝔽3)14yes—
155C22×C3C2×C3×S413yes—
156C2×D3—64no—
157C2×D3—1212no—
158C3×C4—31no—
159C3×C4—33no—
160C3×C4—31no—
161C3×C4—33no—
162C3×C4—61no—
163C3×C4—63no—
164C32C22×GL(2,𝔽3)11yescharacteristic
165C9—41no—
166C9—81no—
167C23—186no—
168C23—364no—
169C23((C22⋊C9)⋊C2)⋊C311yescharacteristic
170Q8—33no—
171Q8—61no—
172Q8C22×AGL(1,ℤ/9)11yescharacteristic, complemented
173D4—94no—
174D4—186no—
175C2×C4—33no—
176C2×C4—33no—
177C8—94no—
178C2×C3—31no—
179C2×C3—36no—
180C2×C3—364no—
181C2×C3C2×C3×GL(2,𝔽3)16yes—
182C2×C3C22×C3×S411yescharacteristic
183D3—124no—
184C22—184no—
185C22—3612no—
186C22#3689⋊C314yesdirect product factor
187C22C2×((C22⋊C9)⋊C2)⋊C313yes—
188C4—31no—
189C4—33no—
190C3—31no—
191C3C22×C3×GL(2,𝔽3)11yescharacteristic
192C2—364no—
193C2C2×#3689⋊C316yesdirect product factor
194C2C22×((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
LCSG ▷ Q8⋊C9
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6204C2semidirect
2. C2×#3689⋊C3C2direct (N × H)
4. C22×#3689C3semidirect
6. C2×#6204C22semidirect
7. #3689⋊C3C22direct (N × H)
14. C22×Q8⋊C9C2×C3semidirect
15. C2×#3689C2×C3semidirect
19. #6204C23semidirect
33. C2×Q8⋊C9C22×C3semidirect
34. #3689C22×C3semidirect
65. Q8⋊C9C23×C3semidirect
102. C22×Q8AGL(1,ℤ/9)semidirect
143. C2×Q8C2×AGL(1,ℤ/9)semidirect
172. Q8C22×AGL(1,ℤ/9)semidirect
186. C22#3689⋊C3direct (N × H)
193. C2C2×#3689⋊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×#3689⋊C32
C22×C3×#2153
C22×#36893
C23×AGL(1,ℤ/9)4