Finite Group Databasebeta

← #10588396⋊C3 #13141554

Subgroups — #10588396⋊C3 #13141554

Order 1728.

392 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 112 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×#3664⋊C3C212yesmaximal
2C2×Dic36⋊C3C212yesmaximal, complemented
3#63437⋊C3C211yesmaximal, characteristic
4#63431⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#97C211yesmaximal, characteristic, complemented
6C3×#4715—31nomaximal
7#10588396C311yesmaximal, characteristic, complemented
8C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
9C2×3−1+2⋊Q8C2211yescharacteristic
10C2×3−1+2⋊Q8C2212yes—
11C2×C8×3−1+2C2212yes—
12C42×3−1+2C2211yescharacteristic
13(C2×AGL(1,ℤ/9))⋊C4—22no—
14C4⋊(3−1+2⋊C4)—22no—
15#3664⋊C3—24no—
16Dic36⋊C3—24no—
17C2×C3×Dic12—32no—
18C2×C3×#204—32no—
19C3×#875—31no—
20C3×#869—31no—
21C32×#97D311yescharacteristic
22C2×#3664C2×C312yes—
23C2×Dic36C2×C312yescomplemented
24#63437C2×C311yescharacteristic
25#63431C2×C311yescharacteristic, complemented
26C9×#97—21no—
27C9×#97C2×C311yescharacteristic, complemented
28C22×AGL(1,ℤ/9)—21no—
29C2×C4×3−1+2D412yes—
30C2×C4×3−1+2C2311yescharacteristic
31C2×3−1+2⋊C4—22no—
32C2×3−1+2⋊C4—21no—
33C4⋊AGL(1,ℤ/9)—22no—
343−1+2⋊Q8—22no—
353−1+2⋊Q8—22no—
363−1+2⋊Q8—22no—
37C8×3−1+2D414yes—
38C3×#497—91no—
39#4715—31no—
40C2×C3×D12—31no—
41C2×C3×Dic6—32no—
42C2×C3×Dic6—31no—
43C2×C32×C8C2×D312yes—
44C32×C42C2×D311yescharacteristic
45C3×(C2×D3)⋊C4—62no—
46C3×C4⋊Dic3—62no—
47C3×Dic12—64no—
48C3×#204—64no—
49C2×D36C22×C311yescharacteristic
50C2×Dic18C22×C312yes—
51C2×Dic18C22×C311yescharacteristic
52C2×C8×C9—22no—
53C2×C8×C9C22×C312yes—
54C42×C9—21no—
55C42×C9C22×C311yescharacteristic
56(C2×D9)⋊C4—22no—
57C4⋊Dic9—22no—
58#3664—24no—
59Dic36—24no—
60C22×3−1+2C2×D411yescharacteristic
61C2×AGL(1,ℤ/9)—41no—
62C2×AGL(1,ℤ/9)—42no—
63C4×3−1+2—22no—
64C4×3−1+2C2×D412yes—
653−1+2⋊C4—41no—
663−1+2⋊C4—42no—
67C2×C3×Q16—92no—
68C2×C3×SD16—92no—
69C3×C4⋊Q8—91no—
70C3×#124—91no—
71C2×Dic12—32no—
72C2×#204—32no—
73#875—31no—
74#869—31no—
75C3×#97—31no—
76C3×#97C3×D311yescharacteristic
77C22×C3×D3—61no—
78C2×C32×C4D1212yes—
79C2×C32×C4C22×D311yescharacteristic
80C2×C3×Dic3—62no—
81C2×C3×Dic3—61no—
82C3×D12—62no—
83C3×Dic6—62no—
84C3×Dic6—62no—
85C3×Dic6—62no—
86C22×D9—21no—
87C32×C8D1214yes—
88C2×C4×C9—22no—
89C2×C4×C9—21no—
90C2×C4×C9C3×D412yes—
91C2×C4×C9C23×C311yescharacteristic
92C2×Dic9—22no—
93C2×Dic9—21no—
94D36—22no—
95Dic18—22no—
96Dic18—22no—
97Dic18—22no—
98C8×C9—24no—
99C8×C9C3×D414yes—
100#497—91noSylow
101C2×3−1+2#12711yescharacteristic
102C2×3−1+2#13712yes—
103AGL(1,ℤ/9)—81no—
104C2×C3×Q8—91no—
105C2×C3×Q8—92no—
106C2×C3×D4—91no—
107C2×D12—31no—
108C2×Dic6—32no—
109C2×Dic6—31no—
110C3×Q16—184no—
111C3×SD16—184no—
112C2×C3×C8—32no—
113C2×C3×C8C2×C3×D312yes—
114C3×C4⋊C4—182no—
115C3×C22⋊C4—182no—
116C3×C42—31no—
117C3×C42C2×C3×D311yescharacteristic
118(C2×D3)⋊C4—62no—
119C4⋊Dic3—62no—
120Dic12—64no—
121#204—64no—
122C22×C32C2×D1211yescharacteristic
123C2×C3×D3—121no—
124C2×C3×D3—122no—
125C32×C4—22no—
126C32×C4C2×D1212yes—
127C3×Dic3—122no—
128C3×Dic3—121no—
129C22×C9—21no—
130C22×C9C2×C3×D411yescharacteristic
131C2×D9—41no—
132C2×D9—42no—
133C4×C9—22no—
134C4×C9—22no—
135C4×C9—42no—
136C4×C9C2×C3×D412yes—
137Dic9—41no—
138Dic9—42no—
139C2×Q16—92no—
140C2×SD16—92no—
141C4⋊Q8—91no—
142#124—91no—
143#97AGL(1,ℤ/9)11yescharacteristic, complemented
1443−1+2#49711yescharacteristic, complemented, Sylow
145C23×C3—181no—
146C22×D3—61no—
147C3×Q8—182no—
148C3×Q8—182no—
149C3×Q8—182no—
150C3×D4—182no—
151C2×C3×C4—32no—
152C2×C3×C4—31no—
153C2×C3×C4—181no—
154C2×C3×C4—182no—
155C2×C3×C4C3×D1212yes—
156C2×C3×C4C22×C3×D311yescharacteristic
157C2×Dic3—62no—
158C2×Dic3—61no—
159D12—62no—
160Dic6—62no—
161Dic6—62no—
162Dic6—62no—
163C3×C8—34no—
164C3×C8C3×D1214yes—
165C2×C32#87411yescharacteristic
166C2×C32#90912yes—
167C3×D3—241no—
168C2×C9—22no—
169C2×C9—21no—
170C2×C9C3×#12711yescharacteristic
171C2×C9C3×#13712yes—
172D9—81no—
173C2×Q8—91no—
174C2×Q8—92no—
175C2×D4—91no—
176Q16—184no—
177SD16—184no—
178C2×C8C2×AGL(1,ℤ/9)12yes—
179C4⋊C4—182no—
180C22⋊C4—182no—
181C42C2×AGL(1,ℤ/9)11yescharacteristic
182C22×C3—31no—
183C22×C3—361no—
184C22×C3—362no—
185C22×C3C2×C3×D1211yescharacteristic
186C2×D3—122no—
187C2×D3—121no—
188C3×C4—22no—
189C3×C4—32no—
190C3×C4—62no—
191C3×C4—361no—
192C3×C4—362no—
193C3×C4C2×C3×D1212yes—
194Dic3—122no—
195Dic3—121no—
196C32#471511yescharacteristic
197C9—21no—
198C9C3×#49711yescharacteristic, complemented
199C23—181no—
200Q8—182no—
201Q8—182no—
202Q8—182no—
203D4—182no—
204C2×C4—181no—
205C2×C4—182no—
206C2×C4C4⋊AGL(1,ℤ/9)12yes—
207C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
208C8C4⋊AGL(1,ℤ/9)14yes—
209C2×C3—32no—
210C2×C3—31no—
211C2×C3—721no—
212C2×C3C3×#87411yescharacteristic
213C2×C3C3×#90912yes—
214D3—241no—
215C22—362no—
216C22—361no—
217C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
218C4—22no—
219C4—361no—
220C4—362no—
221C4C2×C4⋊AGL(1,ℤ/9)12yes—
222C3—31no—
223C3C3×#471511yescharacteristic
224C2—721no—
225C2#1172139411yescharacteristic
226C2#63471⋊C312yes—
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#97C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenter#97AGL(1,ℤ/9)
hypocenterC9C3×#497
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. C2×Dic36⋊C3C2semidirect
4. #63431⋊C3C2semidirect
5. 3−1+2×#97C2semidirect
7. #10588396C3semidirect
23. C2×Dic36C2×C3semidirect
25. #63431C2×C3semidirect
27. C9×#97C2×C3semidirect
143. #97AGL(1,ℤ/9)semidirect
144. 3−1+2#497semidirect
198. C9C3×#497semidirect

Maximal subgroups

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

subgroupindex
C2×#3664⋊C32
C2×Dic36⋊C32
#63437⋊C32
#63431⋊C32
3−1+2×#972
C3×#47153
#105883963