Finite Group Databasebeta

← #10589506⋊C3 #13146853

Subgroups — #10589506⋊C3 #13146853

Order 1728.

233 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 37 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#63691⋊C3C211yesmaximal, characteristic
2#11721359C211yesmaximal, characteristic, complemented
3#63422⋊C3C211yesmaximal, characteristic, complemented
4C3×#5144—31nomaximal
5#10589506C311yesmaximal, characteristic, complemented
6#88691C2211yescharacteristic
7#3689⋊C3C411yescharacteristic, complemented
8#3688⋊C3C411yescharacteristic, complemented
9C8×AGL(1,ℤ/9)—41nomaximal
10C32×#131D311yescharacteristic
11C3×#950—31no—
12C3×#901—31no—
13#63691C2×C311yescharacteristic
14C3×#837—31no—
15C3×#825—31no—
16C3×#822—31no—
17C3×C8×Dic3—31no—
18#131⋊C9—21no—
19#131⋊C9C2×C311yescharacteristic, complemented
20#63422C2×C311yescharacteristic, complemented
21C4×AGL(1,ℤ/9)—41no—
22#6204C2×C411yescharacteristic
23C8×3−1+2—41no—
243−1+2⋊C8—41no—
25C3×#443—91no—
26#5144—31no—
27C32×#41C2×D311yescharacteristic
28C3×#235—31no—
29C32×M16—31no—
30C2×C32×C8—31no—
31C3×#216—31no—
32C3×#215—31no—
33C3×#214—31no—
34C3×#213—31no—
35C3×C4×Dic3—31no—
36C3×#208—31no—
37C2×C3×C3⋊C8—31no—
38C3×ΓL(1,𝔽25)—61no—
39C3×D3×C8—61no—
40#41⋊C9—21no—
41#41⋊C9C22×C311yescharacteristic
42#3689C3×C411yescharacteristic, complemented
43#3688C3×C411yescharacteristic, complemented
44C8×D9—41no—
45C2×AGL(1,ℤ/9)—41no—
46C4×3−1+2—41no—
473−1+2⋊C4—41no—
48C3×#135—91no—
49C3×#131—31no—
50C3×#131—91no—
51C3×#131C3×D311yescharacteristic
52#950—31no—
53#901—31no—
54C3×#108—91no—
55C3×#104—91no—
56C3×#104—91no—
57C3×C4×C8—91no—
58#837—31no—
59#825—31no—
60#822—31no—
61C8×Dic3—31no—
62C32×Q8C4×D311yescharacteristic
63C32×D4—31no—
64C2×C32×C4—31no—
65C3×#67—61no—
66C2×C3×Dic3—61no—
67C3×D12—31no—
68C3×C4×D3—61no—
69C3×Dic6—31no—
70C32×C8—31no—
71C32×C8S411yescharacteristic
72C3×C3⋊C8—31no—
73C3×C3⋊C8—31no—
74C3×C3⋊C8—61no—
75C4×D9—41no—
76Q8⋊C9—21no—
77Q8⋊C9C2×C3×C411yescharacteristic
78C8×C9—41no—
79C8×C9—81no—
80C9⋊C8—41no—
81#443—91noSylow
82C2×3−1+2—41no—
83AGL(1,ℤ/9)—42no—
84C3×#41—31no—
85C3×#41—91no—
86C3×#41C2×C3×D311yescharacteristic
87#235—31no—
88C3×Q16—91no—
89C3×SD16—91no—
90C3×SD16—91no—
91C3×D8—91no—
92C3×M16—31no—
93C3×M16—31no—
94C3×M16—61no—
95C3×M16—91no—
96C3×M16—181no—
97C2×C3×C8—31no—
98C2×C3×C8—31no—
99C2×C3×C8—61no—
100C2×C3×C8—91no—
101C2×C3×C8—181no—
102C3×C42—91no—
103#216—31no—
104#215—31no—
105#214—31no—
106#213—31no—
107C4×Dic3—31no—
108#208—31no—
109C2×C3⋊C8—31no—
110ΓL(1,𝔽25)—61no—
111D3×C8—61no—
112C22×C32—31no—
113C2×C3×D3—61no—
114C32×C4—31no—
115C32×C4C2×S411yescharacteristic
116C3×Dic3—62no—
117C3×Dic3—61no—
118C2×D9—41no—
119C4×C9—41no—
120C4×C9—81no—
121Dic9—41no—
122#135—91no—
123#131—91no—
124#131AGL(1,ℤ/9)11yescharacteristic, complemented
125#108—91no—
126#104—91no—
127#104—91no—
128C4×C8—91no—
1293−1+2—41noSylow
130C3×Q8—31no—
131C3×Q8—91no—
132C3×Q8C3×C4×D311yescharacteristic
133C3×D4—31no—
134C3×D4—31no—
135C3×D4—61no—
136C3×D4—91no—
137C3×D4—181no—
138C2×C3×C4—31no—
139C2×C3×C4—31no—
140C2×C3×C4—61no—
141C2×C3×C4—181no—
142C2×C3×C4—181no—
143#67—61no—
144C2×Dic3—61no—
145D12—31no—
146C4×D3—61no—
147Dic6—31no—
148C3×C8—31no—
149C3×C8—31no—
150C3×C8—31no—
151C3×C8—61no—
152C3×C8—91no—
153C3×C8—91no—
154C3×C8—181no—
155C3×C8C3×S411yescharacteristic
156C3⋊C8—31no—
157C3⋊C8—31no—
158C3⋊C8—61no—
159C2×C32—61no—
160C2×C32C4×S411yescharacteristic
161C3×D3—121no—
162C2×C9—41no—
163C2×C9—81no—
164D9—42no—
165#41—91no—
166#41C2×AGL(1,ℤ/9)11yescharacteristic
167Q16—91no—
168SD16—91no—
169SD16—91no—
170D8—91no—
171M16—31no—
172M16—91no—
173M16—181no—
174C2×C8—31no—
175C2×C8—91no—
176C2×C8—181no—
177C42—91no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—61no—
181C22×C3—181no—
182C2×D3—61no—
183C3×C4—31no—
184C3×C4—31no—
185C3×C4—31no—
186C3×C4—61no—
187C3×C4—181no—
188C3×C4—182no—
189C3×C4C2×C3×S411yescharacteristic
190Dic3—62no—
191Dic3—61no—
192C32#540411yescharacteristic
193C9—41no—
194C9—81no—
195Q8—91no—
196Q8C4×AGL(1,ℤ/9)11yescharacteristic
197D4—31no—
198D4—91no—
199D4—181no—
200C2×C4—31no—
201C2×C4—181no—
202C2×C4—181no—
203C8—31no—
204C8—91no—
205C8—91no—
206C8—181no—
207C8((C22⋊C9)⋊C2)⋊C311yescharacteristic
208C2×C3—31no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3—121no—
212C2×C3—361no—
213C2×C3C3×C4×S411yescharacteristic
214D3—121no—
215C22—31no—
216C22—181no—
217C4—31no—
218C4—182no—
219C4—181no—
220C4C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
221C3—31no—
222C3C3×#540411yescharacteristic
223C2—61no—
224C2—361no—
225C2C4×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC8((C22⋊C9)⋊C2)⋊C3
commutator, hypocenterQ8⋊C9C2×C3×C4
FrattiniC3×C4C2×C3×S4
FittingC32×#131D3
socleC2×C3C3×C4×S4
prime radical#63691⋊C3C2
cosocle#41⋊C9C22×C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11721359C2semidirect
3. #63422⋊C3C2semidirect
5. #10589506C3semidirect
7. #3689⋊C3C4semidirect
8. #3688⋊C3C4semidirect
19. #131⋊C9C2×C3semidirect
20. #63422C2×C3semidirect
42. #3689C3×C4semidirect
43. #3688C3×C4semidirect
124. #131AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
#63691⋊C32
#117213592
#63422⋊C32
C3×#51443
#105895063
C8×AGL(1,ℤ/9)4