Finite Group Databasebeta

← #10588735⋊C3 #13142931

Subgroups — #10588735⋊C3 #13142931

Order 1728.

326 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 92 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×#124C211yesmaximal, characteristic
2#63432⋊C3C211yesmaximal, characteristic, complemented
3#63395⋊C3C212yesmaximal, complemented
4#63392⋊C3C212yesmaximal
5C4×3−1+2⋊C8C211yesmaximal, characteristic, complemented
6C3×#5056—31nomaximal
7#10588735C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2211yescharacteristic
9C2×D4×3−1+2C2211yescharacteristic
10C22⋊C4×3−1+2—22no—
11C42×3−1+2C2211yescharacteristic
12C4⋊(3−1+2⋊C4)C2212yes—
13(3−1+2⋊C4)⋊C4—22no—
14C2×3−1+2⋊C8C2212yes—
15C32×#124D311yescharacteristic
16C3×#870—31no—
17C3×#835—32no—
18C3×#832—32no—
19C3×C4×C3⋊C8—31no—
20C9×#124—21no—
21C9×#124C2×C311yescharacteristic
22#63432C2×C311yescharacteristic, complemented
23#63395C2×C312yescomplemented
24#63392C2×C312yes—
25C4×C9⋊C8C2×C311yescharacteristic, complemented
26C23×3−1+2—21no—
27Q8×3−1+2—22no—
28D4×3−1+2—22no—
29C2×C4×3−1+2—21no—
30C2×C4×3−1+2D412yes—
31C2×C4×3−1+2C2311yescharacteristic
32C2×3−1+2⋊C4—22no—
333−1+2⋊C8—22no—
34C3×#488—91no—
35#5056—31no—
36C2×C32×Q8C2×D311yescharacteristic
37C2×C32×D4C2×D311yescharacteristic
38C32×C22⋊C4—22no—
39C32×C42C2×D311yescharacteristic
40C3×C4⋊Dic3—32no—
41C3×Dic3⋊C4—62no—
42C2×C3×C3⋊C8—32no—
43C2×Q8×C9—21no—
44C2×Q8×C9C22×C311yescharacteristic
45C2×D4×C9—21no—
46C2×D4×C9C22×C311yescharacteristic
47C9×C22⋊C4—24no—
48C9×C22⋊C4—22no—
49C42×C9—21no—
50C42×C9C22×C311yescharacteristic
51C4⋊Dic9C22×C312yes—
52Dic9⋊C4—22no—
53C2×C9⋊C8C22×C312yes—
54C22×3−1+2—41no—
55C22×3−1+2—42no—
56C22×3−1+2C2×D411yescharacteristic
57C4×3−1+2—22no—
58C4×3−1+2—41no—
59C4×3−1+2#4112yes—
603−1+2⋊C4—42no—
61C3×#125—91no—
62C3×#124—31no—
63C3×#124C3×D311yescharacteristic
64#870—31no—
65C3×#103—92no—
66C3×#102—92no—
67C3×C4×C8—91no—
68#835—32no—
69#832—32no—
70C4×C3⋊C8—31no—
71C23×C32—21no—
72C32×Q8—22no—
73C32×D4—22no—
74C2×C32×C4—21no—
75C2×C32×C4#6712yes—
76C2×C32×C4C22×D311yescharacteristic
77C2×C3×Dic3—62no—
78C23×C9—22no—
79C23×C9—21no—
80C3×C3⋊C8—62no—
81Q8×C9—22no—
82Q8×C9—42no—
83D4×C9—22no—
84D4×C9—42no—
85C2×C4×C9—22no—
86C2×C4×C9—21no—
87C2×C4×C9—22no—
88C2×C4×C9—21no—
89C2×C4×C9C3×D412yes—
90C2×C4×C9C23×C311yescharacteristic
91C2×Dic9—22no—
92C9⋊C8—22no—
93#488—91noSylow
94C2×3−1+2—81no—
95C2×3−1+2#12411yescharacteristic
96C2×3−1+2#13512yes—
97C2×C3×Q8—31no—
98C2×C3×Q8C2×C3×D311yescharacteristic
99C2×C3×D4—31no—
100C2×C3×D4C2×C3×D311yescharacteristic
101C2×C3×C8—92no—
102C3×C4⋊C4—92no—
103C3×C4⋊C4—182no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—62no—
106C3×C42—31no—
107C3×C42C2×C3×D311yescharacteristic
108C4⋊Dic3—32no—
109Dic3⋊C4—62no—
110C2×C3⋊C8—32no—
111C22×C32—41no—
112C22×C32—42no—
113C22×C32C2×#6711yescharacteristic
114C32×C4—22no—
115C32×C4—41no—
116C32×C4#23712yes—
117C3×Dic3—122no—
118C22×C9—21no—
119C22×C9—42no—
120C22×C9—44no—
121C22×C9—41no—
122C22×C9—42no—
123C22×C9C2×C3×D411yescharacteristic
124C4×C9—22no—
125C4×C9—22no—
126C4×C9—42no—
127C4×C9—42no—
128C4×C9—41no—
129C4×C9C3×#4112yes—
130Dic9—42no—
131#125—91no—
132#124AGL(1,ℤ/9)11yescharacteristic
133#103—92no—
134#102—92no—
135C4×C8—91no—
1363−1+2#48811yescharacteristic, complemented, Sylow
137C23×C3—21no—
138C23×C3—61no—
139C3×Q8—22no—
140C3×Q8—62no—
141C3×D4—22no—
142C3×D4—62no—
143C2×C3×C4—21no—
144C2×C3×C4—32no—
145C2×C3×C4—31no—
146C2×C3×C4—61no—
147C2×C3×C4—182no—
148C2×C3×C4C3×#6712yes—
149C2×C3×C4C22×C3×D311yescharacteristic
150C2×Dic3—62no—
151C3×C8—182no—
152C3⋊C8—62no—
153C2×C32—81no—
154C2×C32#93611yescharacteristic
155C2×C32#95012yes—
156C2×C9—21no—
157C2×C9—22no—
158C2×C9—82no—
159C2×C9—81no—
160C2×C9C3×#12411yescharacteristic
161C2×C9C3×#13512yes—
162C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
163C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
164C2×C8—92no—
165C4⋊C4—92no—
166C4⋊C4—182no—
167C22⋊C4—22no—
168C42C2×AGL(1,ℤ/9)11yescharacteristic
169C22×C3—31no—
170C22×C3—41no—
171C22×C3—42no—
172C22×C3—121no—
173C22×C3—122no—
174C22×C3C2×C3×#6711yescharacteristic
175C3×C4—22no—
176C3×C4—32no—
177C3×C4—41no—
178C3×C4—62no—
179C3×C4—121no—
180C3×C4—362no—
181C3×C4C3×#23712yes—
182Dic3—122no—
183C32#505611yescharacteristic
184C9—21no—
185C9C3×#48811yescharacteristic, complemented
186C23—21no—
187Q8—22no—
188D4—22no—
189C2×C4—21no—
190C2×C4—182no—
191C2×C4#611⋊C312yes—
192C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
193C8—182no—
194C2×C3—31no—
195C2×C3—32no—
196C2×C3—81no—
197C2×C3—241no—
198C2×C3C3×#93611yescharacteristic
199C2×C3C3×#95012yes—
200C22—41no—
201C22—42no—
202C22C2×#611⋊C311yescharacteristic
203C4—22no—
204C4—41no—
205C4—362no—
206C4#3699⋊C312yes—
207C3—31no—
208C3C3×#505611yescharacteristic
209C2—81no—
210C2#63498⋊C311yescharacteristic
211C2#63513⋊C312yes—
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#124C2
socleC22×C3C2×C3×#67
prime radicalC2×C32×D4C2×D3
hypercenter#124AGL(1,ℤ/9)
hypocenterC9C3×#488
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #63432⋊C3C2semidirect
3. #63395⋊C3C2semidirect
5. C4×3−1+2⋊C8C2semidirect
7. #10588735C3semidirect
22. #63432C2×C3semidirect
23. #63395C2×C3semidirect
25. C4×C9⋊C8C2×C3semidirect
136. 3−1+2#488semidirect
185. C9C3×#488semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1242
#63432⋊C32
#63395⋊C32
#63392⋊C32
C4×3−1+2⋊C82
C3×#50563
#105887353