Finite Group Databasebeta

← C9×#5429 #13146793

Subgroups — C9×#5429 #13146793

Order 1728.

256 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C9×#994C211yesmaximal, characteristic, complemented
2C9×#986C211yesmaximal, characteristic, complemented
3C9×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5429C311yesmaximal, characteristic
5C9×#453—31nomaximal
6C9×#41⋊C3—21no—
7C9×#41⋊C3C2211yescharacteristic
8C9×GL(2,𝔽3)—21no—
9C9×#215—41nomaximal
10C3×#994C2×C311yescharacteristic
11C3×#986C2×C311yescharacteristic
12C3×GU(2,𝔽3)C2×C311yescharacteristic
13C9×H(5,𝔽2)D311yescharacteristic, complemented
14#63700—21no—
15C9×AGL(1,ℤ/8)—31no—
16C9×AGL(1,ℤ/8)—31no—
17C9×#127—31no—
18C9×#104—31no—
19C9×#104—31no—
20C9×#100—31no—
21C3×Q8×C9—41no—
22C9×D12—41no—
23C9×SL(2,𝔽3)D411yescharacteristic, complemented
24C9×C3⋊C8—41no—
25#5429C911yescharacteristic, direct product factor
26C3×#453—31no—
27C3×#41⋊C3—21no—
28C3×#41⋊C3C22×C311yescharacteristic
29C3×GL(2,𝔽3)—21no—
30C3×#215—41no—
31C9×#41—21no—
32C9×#41—31no—
33C9×#41C2×D311yescharacteristic
34C2×D4×C9—31no—
35C2×D4×C9—31no—
36C2×D4×C9—61no—
37C2×D4×C9—62no—
38#41⋊C9—22no—
39#41⋊C9—21no—
40C9×SD16—61no—
41C9×SD16—61no—
42C9×D8—61no—
43C9×D8—61no—
44C9×M16—31no—
45C9×M16—31no—
46C42×C9—31no—
47C2×D3×C9—81no—
48C3×C4×C9—41no—
49C3×C4×C9—81no—
50C3×H(5,𝔽2)C3×D311yescharacteristic
51#994—21no—
52#994C2×C911yescharacteristic, complemented
53#986C2×C911yescharacteristic, complemented
54C3×AGL(1,ℤ/8)—31no—
55C3×AGL(1,ℤ/8)—31no—
56C3×#127—31no—
57GU(2,𝔽3)C2×C911yescharacteristic, complemented
58C3×#104—31no—
59C3×#104—31no—
60C3×#100—31no—
61C32×Q8—41no—
62C3×D12—41no—
63C3×SL(2,𝔽3)C3×D411yescharacteristic
64C23×C9—32no—
65C23×C9—61no—
66C3×C3⋊C8—41no—
67Q8×C9—81no—
68Q8×C9#6711yescharacteristic
69Q8×C9S411yescharacteristic
70D4×C9—31no—
71D4×C9—31no—
72D4×C9—61no—
73D4×C9—61no—
74D4×C9—61no—
75D4×C9—61no—
76D4×C9—124no—
77C2×C4×C9—31no—
78C2×C4×C9—61no—
79C2×C4×C9—61no—
80Q8⋊C9—21no—
81C8×C9—61no—
82C8×C9—61no—
83#453—31noSylow
84C2×C3×C9—41no—
85D3×C9—161no—
86C3×#41—21no—
87C3×#41—31no—
88C3×#41C2×C3×D311yescharacteristic
89C2×C3×D4—31no—
90C2×C3×D4—31no—
91C2×C3×D4—61no—
92C2×C3×D4—62no—
93#41⋊C3—22no—
94#41⋊C3—21no—
95#41⋊C3—21no—
96#41⋊C3C22×C911yescharacteristic
97GL(2,𝔽3)—21no—
98C3×SD16—61no—
99C3×SD16—61no—
100C3×D8—61no—
101C3×D8—61no—
102C3×M16—31no—
103C3×M16—31no—
104C3×C42—31no—
105#215—41no—
106C2×C3×D3—81no—
107C32×C4—41no—
108C32×C4—81no—
109C22×C9—31no—
110C22×C9—61no—
111C22×C9—122no—
112C22×C9—122no—
113C22×C9—121no—
114C4×C9—21no—
115C4×C9—31no—
116C4×C9—62no—
117C4×C9—82no—
118C4×C9—81no—
119C4×C9C2×S411yescharacteristic
120H(5,𝔽2)D3×C911yescharacteristic, complemented
121AGL(1,ℤ/8)—31no—
122AGL(1,ℤ/8)—31no—
123#127—31no—
124#104—31no—
125#104—31no—
126#100—31no—
127C3×C9—41noSylow
128C23×C3—32no—
129C23×C3—61no—
130C3×Q8—41no—
131C3×Q8—81no—
132C3×Q8C3×#6711yescharacteristic
133C3×Q8C3×S411yescharacteristic
134C3×D4—31no—
135C3×D4—31no—
136C3×D4—61no—
137C3×D4—61no—
138C3×D4—61no—
139C3×D4—61no—
140C3×D4—124no—
141C2×C3×C4—31no—
142C2×C3×C4—61no—
143C2×C3×C4—61no—
144D12—41no—
145SL(2,𝔽3)—21no—
146SL(2,𝔽3)D4×C911yescharacteristic, complemented
147C3×C8—61no—
148C3×C8—61no—
149C3⋊C8—41no—
150C2×C32—41no—
151C3×D3—161no—
152C2×C9—61no—
153C2×C9—81no—
154C2×C9—121no—
155C2×C9—241no—
156C2×C9GL(2,ℤ/4)11yescharacteristic
157#41—21no—
158#41—31no—
159#41C2×D3×C911yescharacteristic
160C2×D4—31no—
161C2×D4—31no—
162C2×D4—61no—
163C2×D4—62no—
164SD16—61no—
165SD16—61no—
166D8—61no—
167D8—61no—
168M16—31no—
169M16—31no—
170C42—31no—
171C22×C3—31no—
172C22×C3—61no—
173C22×C3—122no—
174C22×C3—122no—
175C22×C3—121no—
176C2×D3—81no—
177C3×C4—21no—
178C3×C4—31no—
179C3×C4—41no—
180C3×C4—62no—
181C3×C4—81no—
182C3×C4—82no—
183C3×C4—81no—
184C3×C4C2×C3×S411yescharacteristic
185C32—41no—
186C9—81no—
187C9#542911yescharacteristic, direct product factor
188C23—32no—
189C23—61no—
190Q8C9×#6711yescharacteristic
191Q8C9×S411yescharacteristic
192D4—31no—
193D4—31no—
194D4—61no—
195D4—61no—
196D4—61no—
197D4—61no—
198D4—124no—
199C2×C4—31no—
200C2×C4—61no—
201C2×C4—61no—
202C8—61no—
203C8—61no—
204C2×C3—41no—
205C2×C3—61no—
206C2×C3—81no—
207C2×C3—121no—
208C2×C3—241no—
209C2×C3C3×GL(2,ℤ/4)11yescharacteristic
210D3—161no—
211C22—31no—
212C22—61no—
213C22—122no—
214C22—122no—
215C22—121no—
216C4—21no—
217C4—31no—
218C4—62no—
219C4C2×C9×S411yescharacteristic
220C3—41no—
221C3—81no—
222C3C3×#542911yescharacteristic
223C2—61no—
224C2—121no—
225C2—241no—
226C2C9×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C9GL(2,ℤ/4)
commutator#41⋊C3C22×C9
FrattiniC3×C4C2×C3×S4
FittingC9×H(5,𝔽2)D3
socleC2×C3C3×GL(2,ℤ/4)
prime radicalC3×#5429C3
cosocleC3×#41⋊C3C22×C3
hypercenterQ8×C9S4
hypocenterSL(2,𝔽3)D4×C9
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C9 ◁ C4×C9 ◁ Q8×C9

Complements

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

NH ≅ G/Ntype
1. C9×#994C2semidirect
2. C9×#986C2semidirect
3. C9×GU(2,𝔽3)C2semidirect
13. C9×H(5,𝔽2)D3semidirect
23. C9×SL(2,𝔽3)D4semidirect
25. #5429C9direct (N × H)
52. #994C2×C9semidirect
53. #986C2×C9semidirect
57. GU(2,𝔽3)C2×C9semidirect
120. H(5,𝔽2)D3×C9semidirect
146. SL(2,𝔽3)D4×C9semidirect
187. C9#5429direct (N × H)

Maximal subgroups

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

subgroupindex
C9×#9942
C9×#9862
C9×GU(2,𝔽3)2
C3×#54293
C9×#4533
C9×#2154