Finite Group Databasebeta

← #10588553⋊C3 #13142239

Subgroups — #10588553⋊C3 #13142239

Order 1728.

348 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 116 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×Dic36⋊C3C211yesmaximal, characteristic
2#63449⋊C3C212yesmaximal
33−1+2×#107C211yesmaximal, characteristic
4C8×3−1+2⋊C4C211yesmaximal, characteristic
5#63368⋊C3C212yesmaximal
6C3×#4873—31nomaximal
7#10588553C311yesmaximal, characteristic, complemented
8C2×3−1+2⋊Q8C2212yes—
9C2×C8×3−1+2C2211yescharacteristic
10C4⋊C4×3−1+2C2212yes—
11(3−1+2⋊C4)⋊C4—22no—
12C4×3−1+2⋊C4—22no—
13C4×3−1+2⋊C4C2211yescharacteristic
14C2×3−1+2⋊C8C2211yescharacteristic
15Dic36⋊C3C414yescomplemented
16C2×C3×Dic12—31no—
17C3×#887—32no—
18C32×#107D311yescharacteristic
19C3×C8×Dic3—31no—
20C3×#810—32no—
21C2×Dic36C2×C311yescharacteristic
22#63449C2×C312yes—
23C9×#107—21no—
24C9×#107C2×C311yescharacteristic
25C8×Dic9C2×C311yescharacteristic
26#63368C2×C312yes—
27C2×C4×3−1+2—22no—
28C2×C4×3−1+2C2311yescharacteristic
29C2×3−1+2⋊C4—22no—
30C2×3−1+2⋊C4D411yescharacteristic
31C2×3−1+2⋊C4D411yescharacteristic
323−1+2⋊Q8—22no—
333−1+2⋊Q8C2×C414yes—
34C8×3−1+2C2×C412yes—
353−1+2⋊C8—21no—
36C3×C4×Q16—91no—
37#4873—31no—
38C2×C3×Dic6—32no—
39C2×C32×C8C2×D311yescharacteristic
40C32×C4⋊C4C2×D312yes—
41C3×Dic3⋊C4—62no—
42C3×C4×Dic3—31no—
43C3×C4×Dic3—62no—
44C2×C3×C3⋊C8—31no—
45C3×Dic12—34no—
46C2×Dic18C22×C312yes—
47C2×C8×C9—21no—
48C2×C8×C9C22×C311yescharacteristic
49C9×C4⋊C4—22no—
50C9×C4⋊C4C22×C312yes—
51Dic9⋊C4—22no—
52C4×Dic9—22no—
53C4×Dic9C22×C311yescharacteristic
54C2×C9⋊C8C22×C311yescharacteristic
55Dic36C3×C414yescomplemented
56C22×3−1+2C2×D411yescharacteristic
57C4×3−1+2—42no—
58C4×3−1+2C22×C411yescharacteristic
59C4×3−1+2#4111yescharacteristic
603−1+2⋊C4—24no—
613−1+2⋊C4—21no—
623−1+2⋊C4Q1612yescomplemented
63C2×C3×Q16—91no—
64C3×C4×Q8—92no—
65C2×Dic12—31no—
66#887—32no—
67C3×#107—31no—
68C3×#107C3×D311yescharacteristic
69C3×#103—92no—
70C3×C4×C8—91no—
71C8×Dic3—31no—
72#810—32no—
73C2×C32×C4—22no—
74C2×C32×C4C22×D311yescharacteristic
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—62no—
78C3×Dic6—34no—
79C3×Dic6—62no—
80C32×C8C4×D312yes—
81C3×C3⋊C8—61no—
82C2×C4×C9—22no—
83C2×C4×C9—21no—
84C2×C4×C9—42no—
85C2×C4×C9C23×C311yescharacteristic
86C2×Dic9—22no—
87C2×Dic9C3×D411yescharacteristic
88C2×Dic9C3×D411yescharacteristic
89Dic18—22no—
90Dic18C2×C3×C414yes—
91C8×C9—22no—
92C8×C9C2×C3×C412yes—
93C9⋊C8—21no—
94C4×Q16—91noSylow
95C2×3−1+2C4×D411yescharacteristic
96C2×3−1+2C2×Q1611yescharacteristic
97C2×3−1+2#13511yescharacteristic
98C2×C3×Q8—92no—
99C2×Dic6—32no—
100C3×Q16—94no—
101C2×C3×C8—31no—
102C2×C3×C8—91no—
103C2×C3×C8C2×C3×D311yescharacteristic
104C3×C4⋊C4—32no—
105C3×C4⋊C4—182no—
106C3×C4⋊C4C2×C3×D312yes—
107C3×C42—91no—
108C3×C42—182no—
109Dic3⋊C4—62no—
110C4×Dic3—31no—
111C4×Dic3—62no—
112C2×C3⋊C8—31no—
113Dic12—34no—
114C22×C32D3×D411yescharacteristic
115C32×C4—42no—
116C32×C4C2×C4×D311yescharacteristic
117C32×C4#23911yescharacteristic
118C3×Dic3—32no—
119C3×Dic3—64no—
120C3×Dic3—61no—
121C22×C9—21no—
122C22×C9C2×C3×D411yescharacteristic
123C4×C9—21no—
124C4×C9—21no—
125C4×C9—42no—
126C4×C9—82no—
127C4×C9C22×C3×C411yescharacteristic
128C4×C9C3×#4111yescharacteristic
129Dic9—24no—
130Dic9—21no—
131Dic9C3×Q1612yescomplemented
132C2×Q16—91no—
133C4×Q8—92no—
134#107AGL(1,ℤ/9)11yescharacteristic
135#103—92no—
136C4×C8—91no—
1373−1+2C4×Q1611yescharacteristic, complemented, Sylow
138C3×Q8—94no—
139C3×Q8—182no—
140C2×C3×C4—22no—
141C2×C3×C4—31no—
142C2×C3×C4—62no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—182no—
146C2×C3×C4C22×C3×D311yescharacteristic
147C2×Dic3—31no—
148C2×Dic3—31no—
149C2×Dic3—62no—
150Dic6—34no—
151Dic6—62no—
152C3×C8—32no—
153C3×C8—181no—
154C3×C8C3×C4×D312yes—
155C3⋊C8—61no—
156C2×C32#89311yescharacteristic
157C2×C32#91211yescharacteristic
158C2×C32D3×Q1611yescharacteristic
159C2×C9—21no—
160C2×C9—21no—
161C2×C9—21no—
162C2×C9C3×C4×D411yescharacteristic
163C2×C9C2×C3×Q1611yescharacteristic
164C2×C9C3×#13511yescharacteristic
165C2×Q8—92no—
166Q16—94no—
167C2×C8—91no—
168C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
169C4⋊C4—182no—
170C4⋊C4C2×AGL(1,ℤ/9)12yes—
171C42—91no—
172C42—182no—
173C22×C3—31no—
174C22×C3C3×D3×D411yescharacteristic
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—42no—
178C3×C4—92no—
179C3×C4—122no—
180C3×C4—181no—
181C3×C4—184no—
182C3×C4C2×C3×C4×D311yescharacteristic
183C3×C4C3×#23911yescharacteristic
184Dic3—32no—
185Dic3—64no—
186Dic3—61no—
187C32#487311yescharacteristic
188C9—21no—
189C9C3×C4×Q1611yescharacteristic, complemented
190Q8—94no—
191Q8—182no—
192C2×C4—22no—
193C2×C4—91no—
194C2×C4—91no—
195C2×C4—182no—
196C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
197C8—181no—
198C8C4×AGL(1,ℤ/9)12yes—
199C2×C3—31no—
200C2×C3—31no—
201C2×C3—31no—
202C2×C3C3×#89311yescharacteristic
203C2×C3C3×#91211yescharacteristic
204C2×C3C3×D3×Q1611yescharacteristic
205C22D4×AGL(1,ℤ/9)11yescharacteristic
206C4—42no—
207C4—92no—
208C4—184no—
209C4—181no—
210C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
211C4#8872411yescharacteristic
212C3—31no—
213C3C3×#487311yescharacteristic
214C2#63455⋊C311yescharacteristic
215C2#63474⋊C311yescharacteristic
216C2Q16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
cosocleC2×C4×C9C23×C3
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×C4×Q16
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
7. #10588553C3semidirect
15. Dic36⋊C3C4semidirect
55. Dic36C3×C4semidirect
62. 3−1+2⋊C4Q16semidirect
131. Dic9C3×Q16semidirect
137. 3−1+2C4×Q16semidirect
189. C9C3×C4×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×Dic36⋊C32
#63449⋊C32
3−1+2×#1072
C8×3−1+2⋊C42
#63368⋊C32
C3×#48733
#105885533