Finite Group Databasebeta

← Dic3×#41⋊C9 #13147286

Subgroups — Dic3×#41⋊C9 #13147286

Order 1728.

363 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×C3×#41⋊C9C211yesmaximal, characteristic
2Dic3×Q8⋊C9C212yesmaximal, complemented
3C3×Dic3×#41C311yesmaximal, characteristic
4C4×#41⋊C9—31nomaximal
5C3×#41⋊C9C412yescomplemented
6C2×C3×Q8⋊C9C2211yescharacteristic
7C4×C9×Dic3—41nomaximal
8C2×C32×#41C2×C311yescharacteristic
9C3×Q8×Dic3C2×C312yes—
10C3×D4×Dic3—32no—
11C3×#926—31no—
12C2×C3×C4×Dic3—31no—
13C2×#41⋊C9—21no—
14C2×#41⋊C9D311yescharacteristic
15C4×Q8⋊C9—32no—
16C2×C3×C4×C9—41no—
17C2×C9×Dic3—42no—
18C3×Q8⋊C9C2×C411yescharacteristic, complemented
19C3×C4×#41—31no—
20Dic3×#41C911yescharacteristic, complemented
21C32×#41—32no—
22C32×#41C3×C412yes—
23C2×C32×Q8C22×C311yescharacteristic
24C2×C32×D4—31no—
25C22×C32×C4—31no—
26C22×C3×Dic3—32no—
27C3×#217—32no—
28C3×C4⋊Dic3—32no—
29C3×C4×Dic3—31no—
30C3×C4×Dic3—32no—
31C3×C4×Dic3A411yescharacteristic
32#41⋊C9—22no—
33#41⋊C9Dic312yesdirect product factor
34C2×Q8⋊C9—21no—
35C2×Q8⋊C9C2×D311yescharacteristic
36C42×C9—121no—
37C22×C3×C9—41no—
38C3×C4×C9—42no—
39C9×Dic3—44no—
40C2×C3×#41—21no—
41C2×C3×#41C2×C911yescharacteristic
42C2×C3×#41C3×D311yescharacteristic
43C3×C4×Q8—32no—
44C3×C4×D4—92no—
45C3×#117—91no—
46C2×C3×C42—91no—
47Q8×Dic3C2×C912yes—
48D4×Dic3—32no—
49#926—31no—
50C2×C4×Dic3—31no—
51C23×C32—31no—
52C32×Q8—31no—
53C32×Q8C2×C3×C411yescharacteristic
54C32×D4—32no—
55C32×D4—32no—
56C2×C32×C4—31no—
57C2×C32×C4—32no—
58C2×C32×C4—32no—
59C2×C32×C4C2×A411yescharacteristic
60C2×C3×Dic3—32no—
61C2×C3×Dic3—64no—
62C2×C3×Dic3C2×A412yes—
63C2×C4×C9—41no—
64C2×C4×C9—81no—
65C2×C4×C9—122no—
66Q8⋊C9—21no—
67Q8⋊C9C2×Dic311yescharacteristic, complemented
68C4×#41—31noSylow
69C2×C3×C9—41no—
70C2×C3×C9—42no—
71C3×#41—22no—
72C3×#41—32no—
73C3×#41—32no—
74C3×#41—62no—
75C3×#41C4×C912yescomplemented
76C3×#41C3×Dic312yes—
77C2×C3×Q8—21no—
78C2×C3×Q8C22×C911yescharacteristic
79C2×C3×Q8C2×C3×D311yescharacteristic
80C2×C3×D4—31no—
81C2×C3×D4—31no—
82C2×C3×D4—61no—
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—61no—
86C22×C3×C4—92no—
87C22×Dic3—32no—
88C3×C4⋊C4—92no—
89C3×C22⋊C4—92no—
90C3×C42—31no—
91C3×C42—92no—
92C3×C42—91no—
93#217—32no—
94C4⋊Dic3—32no—
95C4×Dic3—31no—
96C4×Dic3—32no—
97C4×Dic3C22⋊C911yescharacteristic
98C22×C32—32no—
99C22×C32—62no—
100C22×C32C22×A411yescharacteristic
101C32×C4—31no—
102C32×C4—31no—
103C32×C4C4×A412yes—
104C3×Dic3—62no—
105C3×Dic3#41⋊C314yes—
106C22×C9—41no—
107C22×C9—81no—
108C4×C9—42no—
109C4×C9—82no—
110C4×C9—124no—
111C2×#41D3×C911yescharacteristic
112C4×Q8—32no—
113C4×D4—92no—
114#117—91no—
115C2×C42—91no—
116C3×C9—41noSylow
117C23×C3—31no—
118C23×C3—31no—
119C23×C3—61no—
120C3×Q8—21no—
121C3×Q8—31no—
122C3×Q8—31no—
123C3×Q8—61no—
124C3×Q8C2×C4×C911yescharacteristic
125C3×Q8C2×C3×Dic311yescharacteristic
126C3×D4—32no—
127C3×D4—32no—
128C3×D4—32no—
129C3×D4—32no—
130C3×D4—62no—
131C3×D4—62no—
132C2×C3×C4—21no—
133C2×C3×C4—32no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—62no—
141C2×C3×C4—62no—
142C2×C3×C4—61no—
143C2×C3×C4—92no—
144C2×C3×C4—184no—
145C2×C3×C4C2×C22⋊C911yescharacteristic
146C2×C3×C4D3×A411yescharacteristic
147C2×Dic3—32no—
148C2×Dic3—64no—
149C2×Dic3C2×C22⋊C912yes—
150C2×C32—62no—
151C2×C32C2×C4×A411yescharacteristic
152C2×C32C2×#41⋊C312yes—
153C2×C9—42no—
154C2×C9—41no—
155C2×C9—82no—
156C2×C9—81no—
157#41—32no—
158#41C9×Dic312yescomplemented
159C2×Q8C2×D3×C911yescharacteristic
160C2×D4—31no—
161C22×C4—31no—
162C22×C4—92no—
163C4⋊C4—92no—
164C22⋊C4—92no—
165C42—31no—
166C42—92no—
167C42—91no—
168C22×C3—21no—
169C22×C3—32no—
170C22×C3—32no—
171C22×C3—62no—
172C22×C3—62no—
173C22×C3—62no—
174C22×C3—122no—
175C22×C3C22×C22⋊C911yescharacteristic
176C22×C3C2×D3×A411yescharacteristic
177C3×C4—22no—
178C3×C4—34no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—61no—
184C3×C4—61no—
185C3×C4—182no—
186C3×C4C4×C22⋊C912yes—
187C3×C4Dic3×A412yes—
188Dic3—62no—
189Dic3#41⋊C914yesdirect product factor
190C32C4×#41⋊C311yescharacteristic
191C9—41no—
192C9—81no—
193C23—31no—
194Q8—31no—
195Q8C2×C9×Dic311yescharacteristic
196D4—32no—
197D4—32no—
198C2×C4—32no—
199C2×C4—31no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—92no—
203C2×C4—184no—
204C2×C4D3×C22⋊C911yescharacteristic
205C2×C3—22no—
206C2×C3—21no—
207C2×C3—62no—
208C2×C3—62no—
209C2×C3—122no—
210C2×C3C2×C4×C22⋊C911yescharacteristic
211C2×C3C2×#41⋊C912yes—
212C2×C3D3×#41⋊C312yes—
213C2×C3C2×Dic3×A411yescharacteristic
214C22—32no—
215C22—62no—
216C22C2×D3×C22⋊C911yescharacteristic
217C4—34no—
218C4—31no—
219C4—31no—
220C4—182no—
221C4Dic3×C22⋊C912yes—
222C3—21no—
223C3C4×#41⋊C911yescharacteristic, complemented
224C3Dic3×#41⋊C311yescharacteristic
225C2—62no—
226C2D3×#41⋊C912yes—
227C2C2×Dic3×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
FrattiniC22×C3C2×D3×A4
Fitting, prime radicalC2×C32×#41C2×C3
socleC22×C32C22×A4
cosocleC2×C32×Q8C22×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. Dic3×Q8⋊C9C2semidirect
5. C3×#41⋊C9C4semidirect
18. C3×Q8⋊C9C2×C4semidirect
20. Dic3×#41C9semidirect
33. #41⋊C9Dic3direct (N × H)
67. Q8⋊C9C2×Dic3semidirect
75. C3×#41C4×C9semidirect
158. #41C9×Dic3semidirect
189. Dic3#41⋊C9direct (N × H)
223. C3C4×#41⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
Dic3×Q8⋊C92
C3×Dic3×#413
C4×#41⋊C93
C4×C9×Dic34