Finite Group Databasebeta

← C22×C3×#3689 #13164406

Subgroups — C22×C3×#3689 #13164406

Order 1728.

747 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 747 conjugacy classes of subgroups — too many to draw. Only its 116 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×C3×Q8⋊C9C211yesmaximal, characteristic, complemented
2C2×C3×#3689C216yesmaximal, direct product factor
3C22×C3×#215—31nomaximal
4C22×#3689C311yesmaximal, characteristic, direct product factor
5C23×C3×D9—41nomaximal
6C2×C3×Q8⋊C9C2213yescomplemented
7C3×#3689C2214yesdirect 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×C316yesdirect product factor
16C23×C3×C9—41no—
17C22×C3×D9—46no—
18C22×C3×D9—48no—
19C3×Q8⋊C9C2311yescharacteristic, 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×C314yesdirect product factor
35C22×C3×C9—44no—
36C22×C3×C9—43no—
37C2×C3×D9—44no—
38C2×C3×D9—424no—
39C22×C3×Q8—21no—
40C22×C3×Q8D911yescharacteristic, complemented
41C22×C3×Q8C3×D311yescharacteristic
42C22×C3×D4—91no—
43C22×D12—31no—
44C2×C3×SD16—96no—
45C2×C3×SD16—96no—
46C22×C3×C8—91no—
47C2×#215—36no—
48C2×#215—36no—
49C22×C3⋊C8—31no—
50C23×C32S411yescharacteristic
51C22×C3×D3—66no—
52C22×C3×D3—124no—
53C32×Q8—33no—
54C32×Q8—61no—
55C32×Q8C22×D311yescharacteristic
56C2×C32×C4—33no—
57C2×C32×C4—33no—
58C3×D12—34no—
59C3×D12—66no—
60C23×C9—41no—
61C23×C9—81no—
62C22×D9—46no—
63C22×D9—48no—
64C3×C3⋊C8—34no—
65Q8⋊C9—21no—
66Q8⋊C9C23×C311yescharacteristic, complemented
67C22×SD16—91noSylow
68C2×C3×C9—46no—
69C2×C3×C9—41no—
70C3×D9—48no—
71C24×C3—181no—
72C23×D3—61no—
73C2×C3×Q8—23no—
74C2×C3×Q8—33no—
75C2×C3×Q8—33no—
76C2×C3×Q8—63no—
77C2×C3×Q8C2×D913yescomplemented
78C2×C3×Q8C2×C3×D313yes—
79C2×C3×D4—96no—
80C2×C3×D4—183no—
81C22×C3×C4—31no—
82C22×C3×C4—31no—
83C22×C3×C4—61no—
84C2×D12—36no—
85C2×D12—63no—
86C3×SD16—912no—
87C3×SD16—94no—
88C2×C3×C8—96no—
89#215—312no—
90#215—34no—
91C2×C3⋊C8—36no—
92C22×C32GL(2,𝔽3)14yes—
93C22×C32C2×S413yes—
94C2×C3×D3—64no—
95C2×C3×D3—1212no—
96C32×C4—31no—
97C32×C4—33no—
98C22×C9—43no—
99C22×C9—44no—
100C22×C9—83no—
101C22×C9—84no—
102C2×D9—44no—
103C2×D9—424no—
104C22×Q8C3×D911yescharacteristic, complemented
105C22×D4—91no—
106C2×SD16—96no—
107C2×SD16—96no—
108C22×C8—91no—
109C3×C9—41noSylow
110C23×C3—21no—
111C23×C3—186no—
112C23×C3—364no—
113C23×C3(C22⋊C9)⋊C211yescharacteristic
114C23×C3C3×S411yescharacteristic
115C22×D3—66no—
116C22×D3—124no—
117C3×Q8—21no—
118C3×Q8—33no—
119C3×Q8—33no—
120C3×Q8—63no—
121C3×Q8—61no—
122C3×Q8—61no—
123C3×Q8—62no—
124C3×Q8C22×D911yescharacteristic, complemented
125C3×Q8C22×C3×D311yescharacteristic
126C3×D4—94no—
127C3×D4—186no—
128C2×C3×C4—33no—
129C2×C3×C4—33no—
130C2×C3×C4—33no—
131C2×C3×C4—33no—
132C2×C3×C4—63no—
133C2×C3×C4—63no—
134D12—34no—
135D12—66no—
136C3×C8—94no—
137C3⋊C8—34no—
138C2×C32C2×GL(2,𝔽3)16yes—
139C2×C32C22×S411yescharacteristic
140C3×D3—124no—
141C2×C9—41no—
142C2×C9—46no—
143C2×C9—81no—
144C2×C9—86no—
145D9—48no—
146C24—181no—
147C2×Q8—33no—
148C2×Q8C2×C3×D913yescomplemented
149C2×D4—96no—
150C2×D4—183no—
151C22×C4—31no—
152SD16—912no—
153SD16—94no—
154C2×C8—96no—
155C22×C3—24no—
156C22×C3—23no—
157C22×C3—184no—
158C22×C3—3612no—
159C22×C3#368914yesdirect product factor
160C22×C3C2×(C22⋊C9)⋊C213yes—
161C22×C3C3×GL(2,𝔽3)14yes—
162C22×C3C2×C3×S413yes—
163C2×D3—64no—
164C2×D3—1212no—
165C3×C4—31no—
166C3×C4—33no—
167C3×C4—31no—
168C3×C4—33no—
169C3×C4—61no—
170C3×C4—63no—
171C32C22×GL(2,𝔽3)11yescharacteristic
172C9—41no—
173C9—81no—
174C23—186no—
175C23—364no—
176C23C3×(C22⋊C9)⋊C211yescharacteristic
177Q8—33no—
178Q8—61no—
179Q8C22×C3×D911yescharacteristic, complemented
180D4—94no—
181D4—186no—
182C2×C4—33no—
183C2×C4—33no—
184C8—94no—
185C2×C3—21no—
186C2×C3—26no—
187C2×C3—364no—
188C2×C3C2×#368916yesdirect product factor
189C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
190C2×C3C2×C3×GL(2,𝔽3)16yes—
191C2×C3C22×C3×S411yescharacteristic
192D3—124no—
193C22—184no—
194C22—3612no—
195C22C3×#368914yesdirect product factor
196C22C2×C3×(C22⋊C9)⋊C213yes—
197C4—31no—
198C4—33no—
199C3—21no—
200C3C22×#368911yescharacteristic, direct product factor
201C3C22×C3×GL(2,𝔽3)11yescharacteristic
202C2—364no—
203C2C2×C3×#368916yesdirect product factor
204C2C22×C3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC23×C3(C22⋊C9)⋊C2
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC23×C32S4
LCSG ▷ Q8⋊C9
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8⋊C9C2semidirect
2. C2×C3×#3689C2direct (N × H)
4. C22×#3689C3direct (N × H)
6. C2×C3×Q8⋊C9C22semidirect
7. C3×#3689C22direct (N × H)
14. C22×Q8⋊C9C2×C3semidirect
15. C2×#3689C2×C3direct (N × H)
19. C3×Q8⋊C9C23semidirect
33. C2×Q8⋊C9C22×C3semidirect
34. #3689C22×C3direct (N × H)
40. C22×C3×Q8D9semidirect
66. Q8⋊C9C23×C3semidirect
77. C2×C3×Q8C2×D9semidirect
104. C22×Q8C3×D9semidirect
124. C3×Q8C22×D9semidirect
148. C2×Q8C2×C3×D9semidirect
159. C22×C3#3689direct (N × H)
179. Q8C22×C3×D9semidirect
188. C2×C3C2×#3689direct (N × H)
195. C22C3×#3689direct (N × H)
200. C3C22×#3689direct (N × H)
203. C2C2×C3×#3689direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
C2×C3×#36892
C22×C3×#2153
C22×#36893
C23×C3×D94