Finite Group Databasebeta

← D3×C22⋊(Q8⋊C9) #13164656

Subgroups — D3×C22⋊(Q8⋊C9) #13164656

Order 1728.

701 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 701 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C3×C22⋊(Q8⋊C9)C211yesmaximal, characteristic, complemented
2C22×C3×D3×Q8C311yesmaximal, characteristic
3C2×C22⋊(Q8⋊C9)—31nomaximal
4C2×D3×C22⋊C9—41nomaximal
5D3×Q8⋊C9—44nomaximal
6C22×C32×Q8C2×C311yescharacteristic
7C2×C3×D3×Q8—34no—
8C2×C3×D3×Q8—34no—
9C22×C3×C4×D3—31no—
10C22×C3×Dic6—31no—
11C22⋊(Q8⋊C9)—21no—
12C22⋊(Q8⋊C9)D311yescharacteristic, direct product factor
13C2×C3×C22⋊C9—41no—
14D3×C22⋊C9—42no—
15C3×Q8⋊C9—44no—
16C23×C3×Q8—31no—
17C22×D3×Q8C911yescharacteristic, complemented
18C23×C3×D3A411yescharacteristic
19C2×C32×Q8—34no—
20C22×C32×C4—31no—
21C22×C3×Dic3—31no—
22C3×D3×Q8—34no—
23C3×D3×Q8—312no—
24C3×D3×Q8—34no—
25C3×D3×Q8A414yes—
26C2×C3×C4×D3—36no—
27C2×C3×C4×D3—36no—
28C2×C3×Dic6—312no—
29C22×C22⋊C9—121no—
30C2×Q8⋊C9—124no—
31C2×D3×C9—161no—
32C3×C22⋊C9—41no—
33C22×C3×Q8—21no—
34C22×C3×Q8—91no—
35C22×C3×Q8—94no—
36C22×C3×Q8—94no—
37C22×C3×Q8C2×C911yescharacteristic, complemented
38C22×C3×Q8C3×D311yescharacteristic
39C23×C3×C4—91no—
40C2×D3×Q8—34no—
41C2×D3×Q8—34no—
42C22×C4×D3—31no—
43C22×Dic6—31no—
44C23×C32C2×A411yescharacteristic
45C22×C3×D3—31no—
46C22×C3×D3—32no—
47C22×C3×D3—31no—
48C22×C3×D3SL(2,𝔽3)12yes—
49C32×Q8—34no—
50C32×Q8C2×A414yes—
51C2×C32×C4—36no—
52C2×C3×Dic3—36no—
53C3×C4×D3—312no—
54C3×C4×D3—34no—
55C3×Dic6—312no—
56C3×Dic6—34no—
57C2×C22⋊C9—41no—
58C2×C22⋊C9—81no—
59C2×C22⋊C9—122no—
60Q8⋊C9—44no—
61Q8⋊C9—84no—
62C23×Q8—31noSylow
63C2×C3×C9—161no—
64D3×C9—162no—
65C24×C3—31no—
66C23×D3C22⋊C911yescharacteristic
67C2×C3×Q8—34no—
68C2×C3×Q8—34no—
69C2×C3×Q8—34no—
70C2×C3×Q8—64no—
71C2×C3×Q8—912no—
72C2×C3×Q8—94no—
73C2×C3×Q8—912no—
74C2×C3×Q8—94no—
75C22×C3×C4—31no—
76C22×C3×C4—31no—
77C22×C3×C4—61no—
78C22×C3×C4—96no—
79C22×C3×C4—96no—
80C22×C3×C4—91no—
81C22×Dic3—31no—
82D3×Q8—34no—
83D3×Q8—312no—
84D3×Q8—34no—
85D3×Q8C22⋊C914yescomplemented
86C2×C4×D3—36no—
87C2×C4×D3—36no—
88C2×Dic6—312no—
89C22×C32—31no—
90C22×C32—31no—
91C22×C32C2×SL(2,𝔽3)11yescharacteristic
92C2×C3×D3—32no—
93C2×C3×D3—31no—
94C2×C3×D3—32no—
95C2×C3×D3—32no—
96C2×C3×D3—32no—
97C2×C3×D3C22⋊A411yescharacteristic
98C32×C4—34no—
99C3×Dic3—34no—
100C22×C9—481no—
101C22⋊C9—41no—
102C22⋊C9—81no—
103C22×Q8—91no—
104C22×Q8—94no—
105C22×Q8—94no—
106C22×Q8D3×C911yescharacteristic, complemented
107C23×C4—91no—
108C3×C9—161noSylow
109C23×C3—21no—
110C23×C3—32no—
111C23×C3—91no—
112C23×C3—91no—
113C23×C3—92no—
114C23×C3C2×C22⋊C911yescharacteristic
115C23×C3D3×A411yescharacteristic
116C22×D3—31no—
117C22×D3—31no—
118C22×D3—32no—
119C22×D3Q8⋊C912yescomplemented
120C3×Q8—24no—
121C3×Q8—34no—
122C3×Q8—34no—
123C3×Q8—64no—
124C3×Q8—94no—
125C3×Q8—912no—
126C3×Q8C2×C22⋊C914yescomplemented
127C3×Q8D3×A414yes—
128C2×C3×C4—36no—
129C2×C3×C4—36no—
130C2×C3×C4—66no—
131C2×C3×C4—94no—
132C2×C3×C4—912no—
133C2×C3×C4—96no—
134C2×Dic3—36no—
135C4×D3—312no—
136C4×D3—34no—
137Dic6—312no—
138Dic6—34no—
139C2×C32—31no—
140C2×C32—31no—
141C2×C32C2×C22⋊A411yescharacteristic
142C3×D3—32no—
143C3×D3C22⋊SL(2,𝔽3)12yes—
144C2×C9—161no—
145C2×C9—321no—
146C2×C9—482no—
147C24—31no—
148C2×Q8—34no—
149C2×Q8—34no—
150C2×Q8—912no—
151C2×Q8—94no—
152C2×Q8—912no—
153C2×Q8—94no—
154C22×C4—31no—
155C22×C4—96no—
156C22×C4—96no—
157C22×C4—91no—
158C22×C3—21no—
159C22×C3—31no—
160C22×C3—31no—
161C22×C3—31no—
162C22×C3—31no—
163C22×C3—31no—
164C22×C3—61no—
165C22×C3—61no—
166C22×C3—92no—
167C22×C3—92no—
168C22×C3—92no—
169C22×C3—92no—
170C22×C3—91no—
171C22×C3C2×Q8⋊C911yescharacteristic, complemented
172C22×C3D3×SL(2,𝔽3)11yescharacteristic
173C2×D3—32no—
174C2×D3—31no—
175C2×D3—32no—
176C2×D3—32no—
177C2×D3—32no—
178C2×D3C22⋊(C22⋊C9)11yescharacteristic
179C3×C4—34no—
180C3×C4—34no—
181C3×C4—64no—
182C3×C4—94no—
183Dic3—34no—
184C32C2×C22⋊SL(2,𝔽3)11yescharacteristic
185C9—161no—
186C9—321no—
187C23—32no—
188C23—91no—
189C23—91no—
190C23—92no—
191C23D3×C22⋊C911yescharacteristic
192Q8—34no—
193Q8—94no—
194Q8—912no—
195Q8D3×C22⋊C914yescomplemented
196C2×C4—36no—
197C2×C4—94no—
198C2×C4—912no—
199C2×C4—96no—
200C2×C3—21no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—32no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—92no—
209C2×C3C2×C22⋊(C22⋊C9)11yescharacteristic
210C2×C3D3×C22⋊A411yescharacteristic
211D3—32no—
212D3C22⋊(Q8⋊C9)12yesdirect product factor
213C22—31no—
214C22—31no—
215C22—31no—
216C22—92no—
217C22—92no—
218C22—92no—
219C22—92no—
220C22—91no—
221C22D3×Q8⋊C911yescharacteristic, complemented
222C4—34no—
223C4—94no—
224C3—21no—
225C3C2×C22⋊(Q8⋊C9)11yescharacteristic, complemented
226C3D3×C22⋊SL(2,𝔽3)11yescharacteristic
227C2—31no—
228C2—31no—
229C2—32no—
230C2—92no—
231C2D3×C22⋊(C22⋊C9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C3D3×C22⋊A4
commutator, hypocenterC22×C3×Q8C2×C9
Fitting, cosocleC22×C32×Q8C2×C3
socleC23×C32C2×A4
prime radicalC23×C3×D3A4
LCSG ▷ C22×C3×Q8
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×C22⋊(Q8⋊C9)C2semidirect
12. C22⋊(Q8⋊C9)D3direct (N × H)
17. C22×D3×Q8C9semidirect
37. C22×C3×Q8C2×C9semidirect
85. D3×Q8C22⋊C9semidirect
106. C22×Q8D3×C9semidirect
119. C22×D3Q8⋊C9semidirect
126. C3×Q8C2×C22⋊C9semidirect
171. C22×C3C2×Q8⋊C9semidirect
195. Q8D3×C22⋊C9semidirect
212. D3C22⋊(Q8⋊C9)direct (N × H)
221. C22D3×Q8⋊C9semidirect
225. C3C2×C22⋊(Q8⋊C9)semidirect

Maximal subgroups

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

subgroupindex
C3×C22⋊(Q8⋊C9)2
C22×C3×D3×Q83
C2×C22⋊(Q8⋊C9)3
C2×D3×C22⋊C94
D3×Q8⋊C94