Finite Group Databasebeta

← #10588761⋊C3 #13143057

Subgroups — #10588761⋊C3 #13143057

Order 1728.

304 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 100 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
13−1+2×C4⋊Q8C211yesmaximal, characteristic
2#63432⋊C3C211yesmaximal, characteristic
3#63367⋊C3C212yesmaximal
4#63366⋊C3C212yesmaximal
5#63362⋊C3C211yesmaximal, characteristic
6C3×#5081—31nomaximal
7#10588761C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2—21no—
9C4⋊C4×3−1+2—21no—
10C4⋊C4×3−1+2C2212yes—
11C42×3−1+2C2211yescharacteristic
12C4⋊(3−1+2⋊C4)C2211yescharacteristic
13C4⋊(3−1+2⋊C4)C2211yescharacteristic
14(3−1+2⋊C4)⋊C4—21no—
15(3−1+2⋊C4)⋊C4—21no—
16C2×3−1+2⋊C8C2212yes—
17C32×C4⋊Q8D311yescharacteristic
18C3×#870—31no—
19C3×#808—32no—
20C3×#807—32no—
21C3×#803—31no—
22C9×C4⋊Q8—21no—
23C9×C4⋊Q8C2×C311yescharacteristic
24#63432C2×C311yescharacteristic
25#63367C2×C312yes—
26#63366C2×C312yes—
27#63362C2×C311yescharacteristic
28Q8×3−1+2—42no—
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2D411yescharacteristic
31C2×C4×3−1+2D411yescharacteristic
32C2×C4×3−1+2C2311yescharacteristic
33C2×3−1+2⋊C4—21no—
34C2×3−1+2⋊C4—21no—
353−1+2⋊C8Q814yescomplemented
36C3×#501—91no—
37#5081—31no—
38C2×C32×Q8—21no—
39C32×C4⋊C4—21no—
40C32×C4⋊C4C2×D312yes—
41C32×C42C2×D311yescharacteristic
42C3×C4⋊Dic3—31no—
43C3×C4⋊Dic3—31no—
44C3×Dic3⋊C4—61no—
45C3×Dic3⋊C4—61no—
46C2×C3×C3⋊C8—32no—
47C2×Q8×C9—22no—
48C2×Q8×C9—21no—
49C9×C4⋊C4—22no—
50C9×C4⋊C4—22no—
51C9×C4⋊C4—21no—
52C9×C4⋊C4C22×C312yes—
53C42×C9—21no—
54C42×C9C22×C311yescharacteristic
55C4⋊Dic9C22×C311yescharacteristic
56C4⋊Dic9C22×C311yescharacteristic
57Dic9⋊C4—21no—
58Dic9⋊C4—21no—
59C2×C9⋊C8C22×C312yes—
60C22×3−1+2C2×D411yescharacteristic
61C4×3−1+2—21no—
62C4×3−1+2—21no—
63C4×3−1+2—42no—
64C4×3−1+2C2×Q812yes—
653−1+2⋊C4—41no—
663−1+2⋊C4—41no—
67C3×C4⋊Q8—31no—
68C3×C4⋊Q8C3×D311yescharacteristic
69C3×#125—91no—
70#870—31no—
71C3×#107—92no—
72C3×#106—92no—
73C3×#97—91no—
74#808—32no—
75#807—32no—
76#803—31no—
77C32×Q8—42no—
78C2×C32×C4—22no—
79C2×C32×C4#6711yescharacteristic
80C2×C32×C4#6711yescharacteristic
81C2×C32×C4C22×D311yescharacteristic
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C3×C3⋊C8—34no—
85Q8×C9—44no—
86Q8×C9—42no—
87C2×C4×C9—24no—
88C2×C4×C9—22no—
89C2×C4×C9—21no—
90C2×C4×C9—21no—
91C2×C4×C9—21no—
92C2×C4×C9C3×D411yescharacteristic
93C2×C4×C9C3×D411yescharacteristic
94C2×C4×C9C23×C311yescharacteristic
95C2×Dic9—21no—
96C2×Dic9—21no—
97C9⋊C8C3×Q814yescomplemented
98#501—91noSylow
99C2×3−1+2C4⋊Q811yescharacteristic
100C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
101C2×3−1+2#13711yescharacteristic
102C2×C3×Q8—21no—
103C2×C3×Q8—61no—
104C2×C3×C8—92no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—32no—
107C3×C4⋊C4—61no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—91no—
110C3×C4⋊C4—181no—
111C3×C4⋊C4—181no—
112C3×C4⋊C4C2×C3×D312yes—
113C3×C42—31no—
114C3×C42C2×C3×D311yescharacteristic
115C4⋊Dic3—31no—
116C4⋊Dic3—31no—
117Dic3⋊C4—61no—
118Dic3⋊C4—61no—
119C2×C3⋊C8—32no—
120C22×C32C2×#6711yescharacteristic
121C32×C4—21no—
122C32×C4—21no—
123C32×C4—42no—
124C32×C4D3×Q812yes—
125C3×Dic3—121no—
126C3×Dic3—121no—
127C22×C9—21no—
128C22×C9C2×C3×D411yescharacteristic
129C4×C9—22no—
130C4×C9—22no—
131C4×C9—21no—
132C4×C9—21no—
133C4×C9—22no—
134C4×C9—44no—
135C4×C9—42no—
136C4×C9C2×C3×Q812yes—
137Dic9—41no—
138Dic9—41no—
139C4⋊Q8AGL(1,ℤ/9)11yescharacteristic
140#125—91no—
141#107—92no—
142#106—92no—
143#97—91no—
1443−1+2#50111yescharacteristic, complemented, Sylow
145C3×Q8—42no—
146C3×Q8—122no—
147C2×C3×C4—22no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—62no—
152C2×C3×C4—181no—
153C2×C3×C4—181no—
154C2×C3×C4C3×#6711yescharacteristic
155C2×C3×C4C3×#6711yescharacteristic
156C2×C3×C4C22×C3×D311yescharacteristic
157C2×Dic3—61no—
158C2×Dic3—61no—
159C3×C8—94no—
160C3⋊C8—34no—
161C2×C32#93211yescharacteristic
162C2×C32#94211yescharacteristic
163C2×C32#94411yescharacteristic
164C2×C9—21no—
165C2×C9—21no—
166C2×C9—21no—
167C2×C9C3×C4⋊Q811yescharacteristic
168C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
169C2×C9C3×#13711yescharacteristic
170C2×Q8—21no—
171C2×C8—92no—
172C4⋊C4—21no—
173C4⋊C4—91no—
174C4⋊C4—91no—
175C4⋊C4—181no—
176C4⋊C4—181no—
177C4⋊C4C2×AGL(1,ℤ/9)12yes—
178C42C2×AGL(1,ℤ/9)11yescharacteristic
179C22×C3—31no—
180C22×C3C2×C3×#6711yescharacteristic
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—32no—
184C3×C4—42no—
185C3×C4—61no—
186C3×C4—61no—
187C3×C4—122no—
188C3×C4—361no—
189C3×C4—361no—
190C3×C4C3×D3×Q812yes—
191Dic3—121no—
192Dic3—121no—
193C32#508111yescharacteristic
194C9—21no—
195C9C3×#50111yescharacteristic, complemented
196Q8—42no—
197C2×C4—22no—
198C2×C4—181no—
199C2×C4—181no—
200C2×C4#611⋊C311yescharacteristic
201C2×C4#611⋊C311yescharacteristic
202C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
203C8—94no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—31no—
207C2×C3C3×#93211yescharacteristic
208C2×C3C3×#94211yescharacteristic
209C2×C3C3×#94411yescharacteristic
210C22C2×#611⋊C311yescharacteristic
211C4—21no—
212C4—21no—
213C4—42no—
214C4—361no—
215C4—361no—
216C4Q8×AGL(1,ℤ/9)12yes—
217C3—31no—
218C3C3×#508111yescharacteristic
219C2#63495⋊C311yescharacteristic
220C2#63505⋊C311yescharacteristic
221C2#63506⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C4⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC22×C32C2×#67
hypercenterC4⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#501
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
7. #10588761C3semidirect
35. 3−1+2⋊C8Q8semidirect
97. C9⋊C8C3×Q8semidirect
144. 3−1+2#501semidirect
195. C9C3×#501semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C4⋊Q82
#63432⋊C32
#63367⋊C32
#63366⋊C32
#63362⋊C32
C3×#50813
#105887613