Finite Group Databasebeta

← C22×C3⋊(#41⋊C9) #13164615

Subgroups — C22×C3⋊(#41⋊C9) #13164615

Order 1728.

855 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 855 conjugacy classes of subgroups — too many to draw. Only its 149 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⋊(#41⋊C9)C216yesmaximal, direct product factor
3C22×C3×#239C311yesmaximal, characteristic
4C22×#41⋊C9—31nomaximal
5C22×C9×Dic3—41nomaximal
6C2×C3×Q8⋊C9C2213yescomplemented
7C3⋊(#41⋊C9)C2214yesdirect product factor
8C22×C32×Q8C2×C311yescharacteristic
9C2×C3×#239—36no—
10C2×C3×#239C2×C316yes—
11C22×C3×D12—31no—
12C22×C3×C4×D3—31no—
13C2×#41⋊C9—36no—
14C22×Q8⋊C9—21no—
15C22×Q8⋊C9D311yescharacteristic, complemented
16C23×C3×C9—41no—
17C2×C9×Dic3—46no—
18C3×Q8⋊C9C2311yescharacteristic, complemented
19C22×C3×#41—31no—
20C22×#239C911yescharacteristic, complemented
21C23×C3×D3—31no—
22C2×C32×Q8—33no—
23C2×C32×Q8C22×C313yes—
24C22×C32×C4—31no—
25C22×C3×Dic3A411yescharacteristic
26C3×#239—38no—
27C3×#239—312no—
28C3×#239C22×C314yes—
29C2×C3×D12—36no—
30C2×C3×D12—36no—
31C2×C3×C4×D3—36no—
32C2×C3×C4×D3—36no—
33C22×C4×C9—121no—
34#41⋊C9—34no—
35C2×Q8⋊C9—23no—
36C2×Q8⋊C9C2×D313yescomplemented
37C22×C3×C9—44no—
38C22×C3×C9—43no—
39C9×Dic3—44no—
40C2×C3×#41—36no—
41C2×C3×#41—96no—
42C22×C3×Q8—21no—
43C22×C3×Q8C2×C911yescharacteristic
44C22×C3×Q8C3×D311yescharacteristic
45C22×C3×D4—91no—
46C23×C3×C4—91no—
47C2×#239—36no—
48C2×#239C2×C916yescomplemented
49C22×D12—31no—
50C22×C4×D3—31no—
51C23×C32C2×A411yescharacteristic
52C22×C3×D3—36no—
53C22×C3×D3—64no—
54C32×Q8—33no—
55C32×Q8—32no—
56C32×Q8C23×C311yescharacteristic
57C2×C32×C4—33no—
58C2×C32×C4—33no—
59C2×C3×Dic3C2×A416yes—
60C3×D12—34no—
61C3×D12—312no—
62C3×C4×D3—312no—
63C3×C4×D3—34no—
64C23×C9—41no—
65C23×C9—81no—
66C2×C4×C9—126no—
67Q8⋊C9—21no—
68Q8⋊C9C22×D311yescharacteristic, complemented
69C22×#41—31noSylow
70C2×C3×C9—46no—
71C2×C3×C9—41no—
72C24×C3—91no—
73C23×D3—31no—
74C3×#41—34no—
75C3×#41—98no—
76C3×#41—912no—
77C2×C3×Q8—23no—
78C2×C3×Q8—33no—
79C2×C3×Q8—33no—
80C2×C3×Q8—63no—
81C2×C3×Q8C22×C913yes—
82C2×C3×Q8C2×C3×D313yes—
83C2×C3×D4—96no—
84C2×C3×D4—96no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—61no—
89C22×C3×C4—96no—
90C22×C3×C4—96no—
91C22×Dic3C22⋊C911yescharacteristic
92#239—38no—
93#239—312no—
94#239C22×C914yescomplemented
95C2×D12—36no—
96C2×D12—36no—
97C2×C4×D3—36no—
98C2×C4×D3—36no—
99C22×C32#41⋊C314yes—
100C22×C32C22×A413yes—
101C2×C3×D3—34no—
102C2×C3×D3—612no—
103C32×C4—33no—
104C32×C4—31no—
105C3×Dic3C22×A414yes—
106C22×C9—43no—
107C22×C9—44no—
108C22×C9—83no—
109C22×C9—84no—
110C4×C9—124no—
111C2×#41—36no—
112C2×#41—96no—
113C22×Q8D3×C911yescharacteristic
114C22×D4—91no—
115C23×C4—91no—
116C3×C9—41noSylow
117C23×C3—21no—
118C23×C3—96no—
119C23×C3—184no—
120C23×C3C2×C22⋊C911yescharacteristic
121C23×C3D3×A411yescharacteristic
122C22×D3—36no—
123C22×D3—64no—
124C3×Q8—21no—
125C3×Q8—33no—
126C3×Q8—33no—
127C3×Q8—32no—
128C3×Q8—32no—
129C3×Q8—63no—
130C3×Q8—62no—
131C3×Q8C23×C911yescharacteristic
132C3×Q8C22×C3×D311yescharacteristic
133C3×D4—94no—
134C3×D4—912no—
135C2×C3×C4—36no—
136C2×C3×C4—33no—
137C2×C3×C4—33no—
138C2×C3×C4—33no—
139C2×C3×C4—33no—
140C2×C3×C4—63no—
141C2×C3×C4—63no—
142C2×C3×C4—94no—
143C2×C3×C4—912no—
144C2×Dic3C2×C22⋊C916yes—
145D12—34no—
146D12—312no—
147C4×D3—312no—
148C4×D3—34no—
149C2×C32C2×#41⋊C316yes—
150C2×C32C23×A411yescharacteristic
151C3×D3—64no—
152C2×C9—46no—
153C2×C9—41no—
154C2×C9—86no—
155C2×C9—81no—
156C24—91no—
157#41—34no—
158#41—98no—
159#41—912no—
160C2×Q8—33no—
161C2×Q8C2×D3×C913yes—
162C2×D4—96no—
163C2×D4—96no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—96no—
167C22×C4—96no—
168C22×C3—24no—
169C22×C3—23no—
170C22×C3—94no—
171C22×C3—1812no—
172C22×C3#41⋊C914yescomplemented
173C22×C3C22×C22⋊C913yes—
174C22×C3C3⋊(#41⋊C3)14yes—
175C22×C3C2×D3×A413yes—
176C2×D3—34no—
177C2×D3—612no—
178C3×C4—31no—
179C3×C4—33no—
180C3×C4—34no—
181C3×C4—31no—
182C3×C4—33no—
183C3×C4—61no—
184C3×C4—63no—
185Dic3C22×C22⋊C914yes—
186C32C22×#41⋊C311yescharacteristic
187C9—41no—
188C9—81no—
189C23—96no—
190C23—184no—
191C23D3×C22⋊C911yescharacteristic
192Q8—33no—
193Q8—32no—
194Q8C22×D3×C911yescharacteristic
195D4—94no—
196D4—912no—
197C2×C4—33no—
198C2×C4—33no—
199C2×C4—36no—
200C2×C4—94no—
201C2×C4—912no—
202C2×C3—21no—
203C2×C3—26no—
204C2×C3—184no—
205C2×C3C2×#41⋊C916yescomplemented
206C2×C3C23×C22⋊C911yescharacteristic
207C2×C3C2×C3⋊(#41⋊C3)16yes—
208C2×C3C22×D3×A411yescharacteristic
209D3—64no—
210C22—94no—
211C22—1812no—
212C22C3⋊(#41⋊C9)14yesdirect product factor
213C22C2×D3×C22⋊C913yes—
214C4—31no—
215C4—33no—
216C4—34no—
217C3—21no—
218C3C22×#41⋊C911yescharacteristic, complemented
219C3C22×C3⋊(#41⋊C3)11yescharacteristic
220C2—184no—
221C2C2×C3⋊(#41⋊C9)16yesdirect product factor
222C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×A4
commutator, hypocenterC3×Q8C23×C9
FrattiniC2×C3C22×D3×A4
FittingC22×C32×Q8C2×C3
socleC23×C32C2×A4
prime radicalC22×C3×#239C3
cosocleC32×Q8C23×C3
LCSG ▷ C3×Q8
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⋊(#41⋊C9)C2direct (N × H)
6. C2×C3×Q8⋊C9C22semidirect
7. C3⋊(#41⋊C9)C22direct (N × H)
15. C22×Q8⋊C9D3semidirect
18. C3×Q8⋊C9C23semidirect
20. C22×#239C9semidirect
36. C2×Q8⋊C9C2×D3semidirect
48. C2×#239C2×C9semidirect
68. Q8⋊C9C22×D3semidirect
94. #239C22×C9semidirect
172. C22×C3#41⋊C9semidirect
205. C2×C3C2×#41⋊C9semidirect
212. C22C3⋊(#41⋊C9)direct (N × H)
218. C3C22×#41⋊C9semidirect
221. C2C2×C3⋊(#41⋊C9)direct (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⋊(#41⋊C9)2
C22×C3×#2393
C22×#41⋊C93
C22×C9×Dic34