Finite Group Databasebeta

← D3×#6226 #13048642

Subgroups — D3×#6226 #13048642

Order 1296.

470 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 124 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
1#10619985C211yesmaximal, characteristic, complemented
2C3×#6226C211yesmaximal, characteristic, complemented
3C3×C4×D3×C9C211yesmaximal, characteristic
4D3×C3⋊Dic9C212yesmaximal
5#10619868C212yesmaximal, complemented
6D3×#634—31nomaximal
7C2×#6226—31nomaximal
8D3×Dic18—33nomaximal
9C2×C3×D3×C9C2211yescharacteristic
10C32×C4×C9C2211yescharacteristic
11C3⋊(C3⋊Dic9)C2212yes—
12C3×C3⋊Dic9C2212yes—
13C3×C9×Dic3C2211yescharacteristic
14#6308—31no—
15C3×#634—31no—
16C32×C4×D3D311yescharacteristic
17#6293—32no—
18D3×C3⋊Dic3—32no—
19C2×C3×C4×C9—31no—
20C2×C3⋊Dic9—32no—
21#6226—31no—
22#6226—33no—
23#6226—32no—
24#6226—63no—
25#6226—61no—
26#6226D311yescharacteristic, direct product factor
27C4×D3×C9D313yes—
28C3×Dic18—33no—
29D3×Dic9—36no—
30#6189—36no—
31C2×C32×C9C2311yescharacteristic
32C3×D3×C9Q812yescomplemented
33C2×#634—91no—
34D3×Dic6—93no—
35D3×Dic6—91no—
36C2×Dic18—93no—
37C2×C32×D3C2×D311yescharacteristic
38C33×C4C2×D311yescharacteristic
39C3⋊(C3⋊Dic3)—32no—
40C3×C3⋊Dic3—32no—
41C32×Dic3C2×D311yescharacteristic
42C22×C3×C9—31no—
43C2×D3×C9C2×D313yes—
44C3×C4×C9—21no—
45C3×C4×C9—23no—
46C3×C4×C9—31no—
47C3×C4×C9C2×D313yes—
48C3×C4×C9C2×D311yescharacteristic
49C3⋊Dic9—32no—
50C3⋊Dic9—36no—
51C3⋊Dic9—66no—
52C3⋊Dic9—62no—
53C3⋊Dic9C2×D312yes—
54C9×Dic3C2×D313yes—
55C3×Dic9—36no—
56C32×C9C2×Q811yescharacteristic, complemented, Sylow
57C2×C32×C4—31no—
58C2×C3⋊Dic3—92no—
59#634—31no—
60#634—92no—
61#634—91no—
62#634—93no—
63#634—91no—
64#634—181no—
65#634—183no—
66C3×C4×D3D913yes—
67C3×C4×D3C3⋊D311yescharacteristic
68C3×Dic6—93no—
69C3×Dic6—91no—
70#627—96no—
71#627—92no—
72D3×Dic3—96no—
73D3×Dic3—92no—
74C2×C4×C9—33no—
75C2×Dic9—96no—
76Dic18—33no—
77Dic18—96no—
78Dic18—93no—
79Dic18—183no—
80C2×C33C22×D311yescharacteristic
81C32×D3Dic612yes—
82C2×C3×C9—21no—
83C2×C3×C9—23no—
84C2×C3×C9—32no—
85C2×C3×C9C22×D313yes—
86C2×C3×C9C22×D311yescharacteristic
87D3×C9Dic616yescomplemented
88D3×Q8—271no—
89C2×Dic6—271no—
90C2×Dic6—273no—
91C22×C32—31no—
92C2×C3×D3C2×D913yes—
93C2×C3×D3C2×C3⋊D311yescharacteristic
94C32×C4—23no—
95C32×C4—21no—
96C32×C4—31no—
97C32×C4C2×D913yes—
98C32×C4D3211yescharacteristic
99C32×C4C2×C3⋊D311yescharacteristic
100C3⋊Dic3—32no—
101C3⋊Dic3—92no—
102C3⋊Dic3—96no—
103C3⋊Dic3—92no—
104C3⋊Dic3—182no—
105C3⋊Dic3—186no—
106C3×Dic3—96no—
107C3×Dic3—92no—
108C3×Dic3C2×D913yes—
109C3×Dic3C2×C3⋊D311yescharacteristic
110C22×C9—33no—
111C4×C9—23no—
112C4×C9—33no—
113C4×C9D3213yes—
114Dic9—36no—
115Dic9—96no—
116Dic9—186no—
117C33C2×Dic611yescharacteristic
118C3×C9—21no—
119C3×C9—23no—
120C3×C9C2×Dic613yescomplemented
121C3×C9D3×Q811yescharacteristic, complemented
122C3×Q8—271no—
123C2×C3×C4—33no—
124C2×C3×C4—31no—
125C2×Dic3—272no—
126C2×Dic3—276no—
127C4×D3—272no—
128C4×D3C3⋊D911yescharacteristic
129Dic6—93no—
130Dic6—91no—
131Dic6—272no—
132Dic6—276no—
133Dic6—273no—
134Dic6—271no—
135Dic6—271no—
136Dic6—272no—
137Dic6—543no—
138Dic6—541no—
139C2×C32—23no—
140C2×C32—21no—
141C2×C32—32no—
142C2×C32C22×D913yes—
143C2×C32C2×D3211yescharacteristic
144C2×C32C22×C3⋊D311yescharacteristic
145C3×D3Dic1816yescomplemented
146C3×D3#63412yes—
147C2×C9—23no—
148C2×C9—36no—
149C2×C9C2×D3213yes—
150C2×Q8—811noSylow
151C22×C3—33no—
152C22×C3—31no—
153C2×D3C2×C3⋊D911yescharacteristic
154C3×C4—23no—
155C3×C4—21no—
156C3×C4—33no—
157C3×C4—31no—
158C3×C4—272no—
159C3×C4D3×D913yes—
160C3×C4C2×C3⋊D911yescharacteristic
161C3×C4D3×C3⋊D311yescharacteristic
162Dic3—96no—
163Dic3—92no—
164Dic3—276no—
165Dic3—272no—
166Dic3—272no—
167Dic3—546no—
168Dic3—542no—
169Dic3C2×C3⋊D911yescharacteristic
170C32—23no—
171C32—21no—
172C32C2×Dic1813yescomplemented
173C32D3×Dic611yescharacteristic
174C32C2×#63411yescharacteristic
175C9—23no—
176C9D3×Dic613yescomplemented
177Q8—271no—
178Q8—812no—
179Q8—811no—
180C2×C4—31no—
181C2×C4—812no—
182C2×C3—23no—
183C2×C3—21no—
184C2×C3—36no—
185C2×C3—32no—
186C2×C3C2×D3×D913yes—
187C2×C3C22×C3⋊D911yescharacteristic
188C2×C3C2×D3×C3⋊D311yescharacteristic
189D3#622612yesdirect product factor
190C22—31no—
191C4—31no—
192C4—272no—
193C4—812no—
194C4D3×C3⋊D911yescharacteristic
195C3—23no—
196C3—21no—
197C3D3×Dic1813yescomplemented
198C3C2×#622611yescharacteristic, complemented
199C3D3×#63411yescharacteristic
200C2—32no—
201C2C2×D3×C3⋊D911yescharacteristic
subgroupG/H
centerC2C2×D3×C3⋊D9
commutator, cosocleC2×C32×C9C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC32×C4×C9C22
socleC2×C33C22×D3
prime radicalC2×C32×D3C2×D3
hypercenterC4D3×C3⋊D9
hypocenterC32×C9C2×Q8
LCSG ▷ C2×C32×C9 ▷ C32×C9
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10619985C2semidirect
2. C3×#6226C2semidirect
5. #10619868C2semidirect
26. #6226D3direct (N × H)
32. C3×D3×C9Q8semidirect
56. C32×C9C2×Q8semidirect
87. D3×C9Dic6semidirect
120. C3×C9C2×Dic6semidirect
121. C3×C9D3×Q8semidirect
145. C3×D3Dic18semidirect
172. C32C2×Dic18semidirect
176. C9D3×Dic6semidirect
189. D3#6226direct (N × H)
197. C3D3×Dic18semidirect
198. C3C2×#6226semidirect

Maximal subgroups

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

subgroupindex
#106199852
C3×#62262
C3×C4×D3×C92
D3×C3⋊Dic92
#106198682
D3×#6343
C2×#62263
D3×Dic183