Finite Group Databasebeta

← D3×#395⋊C3 #11921179

Subgroups — D3×#395⋊C3 #11921179

Order 1152.

396 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 32 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
1D3×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#395⋊C3C211yesmaximal, characteristic, complemented
3C3⋊(#395⋊C3)C211yesmaximal, characteristic, complemented
4D3×#395C311yesmaximal, characteristic, complemented
5C2×#395⋊C3—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7C4×D3×A4—41nomaximal
8C22×D3×Q8C2×C311yescharacteristic
9C2×C22⋊SL(2,𝔽3)—31no—
10C2×D3×C4⋊C4—31no—
11C3×#395C2×C311yescharacteristic, complemented
12#5230C2×C311yescharacteristic, complemented
13#4979—31no—
14D3×#95—31no—
15#4656—31no—
16#395⋊C3—21no—
17#395⋊C3—31no—
18#395⋊C3D311yescharacteristic, direct product factor
19C2×D3×A4—41no—
20C3×C4×A4—41no—
21Dic3×A4—41no—
22D3×SL(2,𝔽3)—82no—
23C2×#395—31noSylow
24C22×C3×Q8C22×C311yescharacteristic
25C2×D3×Q8—62no—
26C2×D3×Q8—62no—
27C22×C4×D3—31no—
28C22×C4×D3—31no—
29C22×C4×D3A411yescharacteristic
30C22×Dic6—31no—
31C22⋊SL(2,𝔽3)—21no—
32C22⋊SL(2,𝔽3)C2×D311yescharacteristic
33C2×C4×A4—121no—
34C2×C3×C4⋊C4—31no—
35C2×C4⋊Dic3—31no—
36C2×Dic3⋊C4—31no—
37C2×Dic3⋊C4—31no—
38D3×C4⋊C4—62no—
39D3×C4⋊C4—62no—
40C3×#95—31no—
41C3⋊#95—32no—
42C3⋊#95—31no—
43C2×C3×A4—41no—
44D3×A4—42no—
45C3×C4×D3—161no—
46C3×SL(2,𝔽3)—82no—
47C23×Q8—31no—
48C22×C4⋊C4—91no—
49#395—31no—
50#395—91no—
51#395—91no—
52#395C3×D311yescharacteristic, complemented
53C2×#95—91no—
54C23×D3C2×A411yescharacteristic
55C22×A4—121no—
56C2×C3×Q8—62no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C22×C3×C4C2×A411yescharacteristic
60C22×Dic3—31no—
61C22×Dic3—31no—
62C22×Dic3C2×A411yescharacteristic
63D3×Q8—22no—
64D3×Q8—62no—
65D3×Q8—64no—
66D3×Q8—62no—
67D3×Q8—62no—
68C2×C4×D3—32no—
69C2×C4×D3—32no—
70C2×C4×D3—34no—
71C2×C4×D3—61no—
72C2×C4×D3—61no—
73C2×C4×D3—61no—
74C2×C4×D3—62no—
75C2×C4×D3—62no—
76C2×C4×D3—62no—
77C2×C4×D3—61no—
78C2×Dic6—64no—
79C2×Dic6—62no—
80C2×SL(2,𝔽3)—242no—
81C4×A4—41no—
82C4×A4—81no—
83C4×A4—121no—
84C3×C4⋊C4—62no—
85C4⋊Dic3—62no—
86Dic3⋊C4—62no—
87Dic3⋊C4—62no—
88C2×C3×D3—161no—
89C3×A4—41no—
90C32×C4—161no—
91C3×Dic3—161no—
92C22×Q8—91no—
93C22×Q8—182no—
94C22×Q8—182no—
95C22×Q8C2×C3×D311yescharacteristic
96C23×C4—31no—
97C23×C4—91no—
98C23×C4—91no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—91no—
101C2×C4⋊C4—91no—
102C2×C4⋊C4—91no—
103C2×C4⋊C4—182no—
104C2×C4⋊C4—182no—
105#95—31no—
106#95—91no—
107#95—92no—
108C23×C3C22×A411yescharacteristic
109C22×D3—31no—
110C22×D3—31no—
111C22×D3—32no—
112C22×D3#41⋊C312yes—
113C2×A4—41no—
114C2×A4—81no—
115C2×A4—122no—
116C3×Q8—22no—
117C3×Q8—62no—
118C2×C3×C4—32no—
119C2×C3×C4—32no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×C3×C4—62no—
123C2×C3×C4—61no—
124C2×C3×C4—481no—
125C2×Dic3—32no—
126C2×Dic3—32no—
127C2×Dic3—61no—
128C2×Dic3—61no—
129C2×Dic3—62no—
130C2×Dic3—61no—
131C4×D3—41no—
132C4×D3—64no—
133C4×D3—64no—
134C4×D3—64no—
135C4×D3—62no—
136C4×D3—62no—
137C4×D3—121no—
138Dic6—64no—
139Dic6—62no—
140Dic6—62no—
141SL(2,𝔽3)—82no—
142SL(2,𝔽3)—162no—
143C2×C32—161no—
144C3×D3—162no—
145C24—31no—
146C2×Q8—62no—
147C2×Q8—62no—
148C2×Q8—182no—
149C2×Q8—184no—
150C2×Q8—182no—
151C2×Q8—182no—
152C2×Q8—184no—
153C2×Q8—182no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—92no—
158C22×C4—92no—
159C22×C4—94no—
160C22×C4—91no—
161C22×C4—91no—
162C22×C4—181no—
163C22×C4—182no—
164C22×C4—181no—
165C22×C4—181no—
166C22×C4—182no—
167C22×C4—182no—
168C22×C4—181no—
169C22×C4D3×A411yescharacteristic
170C4⋊C4—62no—
171C4⋊C4—182no—
172C4⋊C4—182no—
173C4⋊C4—182no—
174C22×C3—31no—
175C22×C3—31no—
176C22×C3—481no—
177C22×C3C2×#41⋊C311yescharacteristic
178C2×D3—32no—
179C2×D3—31no—
180C2×D3—32no—
181C2×D3—32no—
182C2×D3—32no—
183C2×D3#120⋊C311yescharacteristic
184A4—41no—
185A4—81no—
186C3×C4—41no—
187C3×C4—62no—
188C3×C4—62no—
189C3×C4—161no—
190C3×C4—321no—
191C3×C4—481no—
192Dic3—41no—
193Dic3—62no—
194Dic3—62no—
195C32—161noSylow
196C23—32no—
197C23—91no—
198C23—91no—
199C23—92no—
200C23C2×D3×A411yescharacteristic
201Q8—22no—
202Q8—62no—
203Q8—182no—
204Q8—182no—
205Q8—184no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—92no—
213C2×C4—92no—
214C2×C4—121no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4—184no—
218C2×C4—184no—
219C2×C4—184no—
220C2×C4—181no—
221C2×C4—181no—
222C2×C4—182no—
223C2×C4—181no—
224C2×C4—361no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—161no—
228C2×C3—321no—
229C2×C3—482no—
230C2×C3C2×#120⋊C311yescharacteristic
231D3—32no—
232D3#395⋊C312yesdirect product factor
233C22—31no—
234C22—31no—
235C22—31no—
236C22—92no—
237C22—92no—
238C22—92no—
239C22—92no—
240C22—91no—
241C22D3×#41⋊C311yescharacteristic
242C4—41no—
243C4—62no—
244C4—62no—
245C4—121no—
246C4—182no—
247C4—182no—
248C3—161no—
249C3—321no—
250C3C2×#395⋊C311yescharacteristic, complemented
251C2—31no—
252C2—31no—
253C2—32no—
254C2—92no—
255C2D3×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2D3×#120⋊C3
commutator, cosocle, hypocenterC22×C3×Q8C22×C3
FrattiniC23C2×D3×A4
FittingC3×#395C2×C3
socleC23×C3C22×A4
prime radicalD3×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22×C3×Q8
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C3×#395⋊C3C2semidirect
3. C3⋊(#395⋊C3)C2semidirect
4. D3×#395C3semidirect
11. C3×#395C2×C3semidirect
12. #5230C2×C3semidirect
18. #395⋊C3D3direct (N × H)
52. #395C3×D3semidirect
232. D3#395⋊C3direct (N × H)
250. C3C2×#395⋊C3semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊SL(2,𝔽3)2
C3×#395⋊C32
C3⋊(#395⋊C3)2
D3×#3953
C2×#395⋊C33
C4×D3×A44