Finite Group Databasebeta

← C22×D3×Q8⋊C9 #13164616

Subgroups — C22×D3×Q8⋊C9 #13164616

Order 1728.

1015 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 229 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×D3×Q8⋊C9C216yesmaximal, direct product factor
3C22×C3×D3×Q8C311yesmaximal, characteristic
4C23×Q8⋊C9—31nomaximal
5C23×D3×C9—41nomaximal
6C2×C3×Q8⋊C9C2213yescomplemented
7D3×Q8⋊C9C2214yesdirect product factor
8C22×C32×Q8C2×C311yescharacteristic
9C2×C3×D3×Q8—36no—
10C2×C3×D3×Q8C2×C316yes—
11C22×C3×C4×D3—31no—
12C22×C3×Dic6—31no—
13C22×Q8⋊C9—21no—
14C22×Q8⋊C9—36no—
15C22×Q8⋊C9D311yescharacteristic, direct product factor
16C23×C3×C9—41no—
17C22×D3×C9—48no—
18C22×D3×C9—46no—
19C3×Q8⋊C9C2311yescharacteristic, complemented
20C23×C3×Q8—31no—
21C22×D3×Q8C911yescharacteristic, complemented
22C23×C3×D3A411yescharacteristic
23C2×C32×Q8—33no—
24C2×C32×Q8C22×C313yes—
25C22×C32×C4—31no—
26C22×C3×Dic3—31no—
27C3×D3×Q8—38no—
28C3×D3×Q8—312no—
29C3×D3×Q8C22×C314yes—
30C2×C3×C4×D3—36no—
31C2×C3×C4×D3—36no—
32C2×C3×Dic6—36no—
33C2×C3×Dic6—36no—
34C24×C9—121no—
35C2×Q8⋊C9—23no—
36C2×Q8⋊C9—34no—
37C2×Q8⋊C9C2×D313yesdirect product factor
38C22×C3×C9—44no—
39C22×C3×C9—43no—
40C2×D3×C9—424no—
41C2×D3×C9—44no—
42C22×C3×Q8—21no—
43C22×C3×Q8—36no—
44C22×C3×Q8—91no—
45C22×C3×Q8—96no—
46C22×C3×Q8C2×C911yescharacteristic, complemented
47C22×C3×Q8C3×D311yescharacteristic
48C23×C3×C4—91no—
49C2×D3×Q8—36no—
50C2×D3×Q8C2×C916yescomplemented
51C22×C4×D3—31no—
52C22×Dic6—31no—
53C23×C32C2×A411yescharacteristic
54C22×C3×D3SL(2,𝔽3)18yes—
55C22×C3×D3C2×A416yes—
56C32×Q8—33no—
57C32×Q8—32no—
58C32×Q8C23×C311yescharacteristic
59C2×C32×C4—33no—
60C2×C32×C4—33no—
61C2×C3×Dic3—36no—
62C3×C4×D3—312no—
63C3×C4×D3—34no—
64C3×Dic6—312no—
65C3×Dic6—34no—
66C23×C9—41no—
67C23×C9—81no—
68C23×C9—128no—
69C23×C9—126no—
70Q8⋊C9—21no—
71Q8⋊C9C22×D311yescharacteristic, direct product factor
72C23×Q8—31noSylow
73C2×C3×C9—46no—
74C2×C3×C9—41no—
75D3×C9—48no—
76C24×C3—31no—
77C23×D3C22⋊C911yescharacteristic
78C2×C3×Q8—23no—
79C2×C3×Q8—33no—
80C2×C3×Q8—33no—
81C2×C3×Q8—34no—
82C2×C3×Q8—63no—
83C2×C3×Q8—96no—
84C2×C3×Q8—96no—
85C2×C3×Q8—98no—
86C2×C3×Q8—912no—
87C2×C3×Q8C22×C913yescomplemented
88C2×C3×Q8C2×C3×D313yes—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—61no—
92C22×C3×C4—96no—
93C22×C3×C4—96no—
94C22×C3×C4—91no—
95C22×Dic3—31no—
96D3×Q8—38no—
97D3×Q8—312no—
98D3×Q8C22×C914yescomplemented
99C2×C4×D3—36no—
100C2×C4×D3—36no—
101C2×Dic6—36no—
102C2×Dic6—36no—
103C22×C32C2×SL(2,𝔽3)14yes—
104C22×C32C22×A413yes—
105C2×C3×D3C2×SL(2,𝔽3)124yes—
106C2×C3×D3C22×A414yes—
107C32×C4—33no—
108C32×C4—31no—
109C3×Dic3—34no—
110C22×C9—43no—
111C22×C9—44no—
112C22×C9—83no—
113C22×C9—84no—
114C22×C9—124no—
115C22×C9—1224no—
116C22×Q8—36no—
117C22×Q8—91no—
118C22×Q8—96no—
119C22×Q8D3×C911yescharacteristic, complemented
120C23×C4—91no—
121C3×C9—41noSylow
122C23×C3—21no—
123C23×C3—36no—
124C23×C3—38no—
125C23×C3C2×C22⋊C911yescharacteristic
126C23×C3D3×A411yescharacteristic
127C22×D3Q8⋊C918yesdirect product factor
128C22×D3C2×C22⋊C916yes—
129C3×Q8—21no—
130C3×Q8—33no—
131C3×Q8—33no—
132C3×Q8—32no—
133C3×Q8—32no—
134C3×Q8—63no—
135C3×Q8—62no—
136C3×Q8—94no—
137C3×Q8—912no—
138C3×Q8C23×C911yescharacteristic, complemented
139C3×Q8C22×C3×D311yescharacteristic
140C2×C3×C4—33no—
141C2×C3×C4—33no—
142C2×C3×C4—33no—
143C2×C3×C4—33no—
144C2×C3×C4—63no—
145C2×C3×C4—63no—
146C2×C3×C4—94no—
147C2×C3×C4—912no—
148C2×C3×C4—96no—
149C2×Dic3—36no—
150C4×D3—312no—
151C4×D3—34no—
152Dic6—312no—
153Dic6—34no—
154C2×C32C22×SL(2,𝔽3)16yes—
155C2×C32C23×A411yescharacteristic
156C3×D3C22×SL(2,𝔽3)18yes—
157C2×C9—46no—
158C2×C9—41no—
159C2×C9—86no—
160C2×C9—81no—
161C2×C9—128no—
162C24—31no—
163C2×Q8—33no—
164C2×Q8—34no—
165C2×Q8—96no—
166C2×Q8—96no—
167C2×Q8—98no—
168C2×Q8—912no—
169C2×Q8C2×D3×C913yescomplemented
170C22×C4—31no—
171C22×C4—96no—
172C22×C4—96no—
173C22×C4—91no—
174C22×C3—24no—
175C22×C3—23no—
176C22×C3—324no—
177C22×C3—34no—
178C22×C3C2×Q8⋊C914yescomplemented
179C22×C3C22×C22⋊C913yes—
180C22×C3D3×SL(2,𝔽3)14yes—
181C22×C3C2×D3×A413yes—
182C2×D3C2×Q8⋊C9124yesdirect product factor
183C2×D3C22×C22⋊C914yes—
184C3×C4—31no—
185C3×C4—33no—
186C3×C4—31no—
187C3×C4—33no—
188C3×C4—61no—
189C3×C4—63no—
190C3×C4—94no—
191Dic3—34no—
192C32C23×SL(2,𝔽3)11yescharacteristic
193C9—41no—
194C9—81no—
195C23—38no—
196C23—36no—
197C23D3×C22⋊C911yescharacteristic
198Q8—33no—
199Q8—32no—
200Q8—94no—
201Q8—912no—
202Q8C22×D3×C911yescharacteristic, complemented
203C2×C4—33no—
204C2×C4—33no—
205C2×C4—94no—
206C2×C4—912no—
207C2×C4—96no—
208C2×C3—21no—
209C2×C3—26no—
210C2×C3—38no—
211C2×C3C22×Q8⋊C916yescomplemented
212C2×C3C23×C22⋊C911yescharacteristic
213C2×C3C2×D3×SL(2,𝔽3)16yes—
214C2×C3C22×D3×A411yescharacteristic
215D3C22×Q8⋊C918yesdirect product factor
216C22—34no—
217C22—324no—
218C22D3×Q8⋊C914yesdirect product factor
219C22C2×D3×C22⋊C913yes—
220C4—31no—
221C4—33no—
222C4—94no—
223C3—21no—
224C3C23×Q8⋊C911yescharacteristic, complemented
225C3C22×D3×SL(2,𝔽3)11yescharacteristic
226C2—38no—
227C2C2×D3×Q8⋊C916yesdirect product factor
228C2C22×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 radicalC23×C3×D3A4
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×D3×Q8⋊C9C2direct (N × H)
6. C2×C3×Q8⋊C9C22semidirect
7. D3×Q8⋊C9C22direct (N × H)
15. C22×Q8⋊C9D3direct (N × H)
19. C3×Q8⋊C9C23semidirect
21. C22×D3×Q8C9semidirect
37. C2×Q8⋊C9C2×D3direct (N × H)
46. C22×C3×Q8C2×C9semidirect
50. C2×D3×Q8C2×C9semidirect
71. Q8⋊C9C22×D3direct (N × H)
87. C2×C3×Q8C22×C9semidirect
98. D3×Q8C22×C9semidirect
119. C22×Q8D3×C9semidirect
127. C22×D3Q8⋊C9direct (N × H)
138. C3×Q8C23×C9semidirect
169. C2×Q8C2×D3×C9semidirect
178. C22×C3C2×Q8⋊C9semidirect
182. C2×D3C2×Q8⋊C9direct (N × H)
202. Q8C22×D3×C9semidirect
211. C2×C3C22×Q8⋊C9semidirect
215. D3C22×Q8⋊C9direct (N × H)
218. C22D3×Q8⋊C9direct (N × H)
224. C3C23×Q8⋊C9semidirect
227. C2C2×D3×Q8⋊C9direct (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×D3×Q8⋊C92
C22×C3×D3×Q83
C23×Q8⋊C93
C23×D3×C94