Finite Group Databasebeta

← C9×#5425 #13146789

Subgroups — C9×#5425 #13146789

Order 1728.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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×C9×#41⋊C3C211yesmaximal, characteristic
2C2×C9×#226C211yesmaximal, characteristic, complemented
3C9×#859C211yesmaximal, characteristic, complemented
4C3×#5425C311yesmaximal, characteristic
5C9×#452—31nomaximal
6C9×#41⋊C3—21no—
7C2×C9×SL(2,𝔽3)C2211yescharacteristic
8C9×#226—21no—
9C9×Dic3⋊C4—41nomaximal
10C2×C3×#41⋊C3C2×C311yescharacteristic
11C2×C3×#226C2×C311yescharacteristic
12C3×#859C2×C311yescharacteristic
13C2×C9×#41D311yescharacteristic
14C2×#41⋊C9—21no—
15C2×C9×Q16—31no—
16C2×C9×SD16—31no—
17C9×C22⋊Q8—31no—
18C9×#103—31no—
19C9×#102—31no—
20C9×C22⋊C8—31no—
21C2×C3×C4×C9—41no—
22C2×C9×Dic3—41no—
23C2×C9×Dic3—41no—
24C9×SL(2,𝔽3)D411yescharacteristic
25#5425C911yescharacteristic, direct product factor
26C3×#452—31no—
27C3×#41⋊C3—21no—
28C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
29C3×#226—21no—
30C3×Dic3⋊C4—41no—
31C9×#41—21no—
32C9×#41—61no—
33C2×Q8×C9—31no—
34C2×Q8×C9C2×D311yescharacteristic
35C2×D4×C9—31no—
36C22×C4×C9—31no—
37#41⋊C9—22no—
38C2×Q8⋊C9—21no—
39C9×Q16—61no—
40C9×Q16—61no—
41C9×SD16—62no—
42C2×C8×C9—31no—
43C2×C8×C9—31no—
44C9×C4⋊C4—31no—
45C9×C4⋊C4—61no—
46C9×C22⋊C4—61no—
47C22×C3×C9—41no—
48C3×C4×C9—81no—
49C9×Dic3—42no—
50C9×Dic3—81no—
51C2×C3×#41C3×D311yescharacteristic
52C2×#41⋊C3—21no—
53C2×#41⋊C3C2×C911yescharacteristic
54C2×#226C2×C911yescharacteristic, complemented
55C2×C3×Q16—31no—
56C2×C3×SD16—31no—
57C3×C22⋊Q8—31no—
58#859C2×C911yescharacteristic, complemented
59C3×#103—31no—
60C3×#102—31no—
61C3×C22⋊C8—31no—
62C2×C32×C4—41no—
63C2×C3×Dic3—41no—
64C2×C3×Dic3—41no—
65C3×SL(2,𝔽3)C3×D411yescharacteristic
66C23×C9—31no—
67Q8×C9—31no—
68Q8×C9—61no—
69Q8×C9—61no—
70Q8×C9#6711yescharacteristic
71D4×C9—32no—
72D4×C9—61no—
73C2×C4×C9—31no—
74C2×C4×C9—61no—
75C2×C4×C9—61no—
76C2×C4×C9—61no—
77C2×C4×C9—61no—
78C2×C4×C9—81no—
79C2×C4×C9S411yescharacteristic
80Q8⋊C9—21no—
81C8×C9—61no—
82C8×C9—61no—
83#452—31noSylow
84C2×C3×C9—41no—
85C2×C3×C9—41no—
86C2×C3×C9—41no—
87C3×#41—21no—
88C3×#41—61no—
89C2×C3×Q8—31no—
90C2×C3×Q8C2×C3×D311yescharacteristic
91C2×C3×D4—31no—
92C22×C3×C4—31no—
93#41⋊C3—22no—
94#41⋊C3—21no—
95C2×SL(2,𝔽3)—21no—
96C2×SL(2,𝔽3)C22×C911yescharacteristic
97#226—21no—
98C3×Q16—61no—
99C3×Q16—61no—
100C3×SD16—62no—
101C2×C3×C8—31no—
102C2×C3×C8—31no—
103C3×C4⋊C4—31no—
104C3×C4⋊C4—61no—
105C3×C22⋊C4—61no—
106Dic3⋊C4—41no—
107C22×C32—41no—
108C32×C4—81no—
109C3×Dic3—42no—
110C3×Dic3—81no—
111C22×C9—61no—
112C22×C9—61no—
113C22×C9—61no—
114C22×C9—81no—
115C22×C9C2×S411yescharacteristic
116C4×C9—21no—
117C4×C9—31no—
118C4×C9—31no—
119C4×C9—82no—
120C4×C9—121no—
121C4×C9—121no—
122C2×#41D3×C911yescharacteristic
123C2×Q16—31no—
124C2×SD16—31no—
125C22⋊Q8—31no—
126#103—31no—
127#102—31no—
128C22⋊C8—31no—
129C3×C9—41noSylow
130C23×C3—31no—
131C3×Q8—31no—
132C3×Q8—61no—
133C3×Q8—61no—
134C3×Q8C3×#6711yescharacteristic
135C3×D4—32no—
136C3×D4—61no—
137C2×C3×C4—31no—
138C2×C3×C4—41no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—61no—
143C2×C3×C4—81no—
144C2×C3×C4C3×S411yescharacteristic
145C2×Dic3—41no—
146C2×Dic3—41no—
147SL(2,𝔽3)—21no—
148SL(2,𝔽3)D4×C911yescharacteristic
149C3×C8—61no—
150C3×C8—61no—
151C2×C32—41no—
152C2×C32—41no—
153C2×C32—41no—
154C2×C9—81no—
155C2×C9—81no—
156C2×C9—81no—
157C2×C9—121no—
158C2×C9#98411yescharacteristic
159C2×C9#98511yescharacteristic
160C2×C9GL(2,ℤ/4)11yescharacteristic
161#41—21no—
162#41—61no—
163C2×Q8—31no—
164C2×Q8C2×D3×C911yescharacteristic
165C2×D4—31no—
166C22×C4—31no—
167Q16—61no—
168Q16—61no—
169SD16—62no—
170C2×C8—31no—
171C2×C8—31no—
172C4⋊C4—31no—
173C4⋊C4—61no—
174C22⋊C4—61no—
175C22×C3—41no—
176C22×C3—61no—
177C22×C3—61no—
178C22×C3—61no—
179C22×C3—81no—
180C22×C3C2×C3×S411yescharacteristic
181C3×C4—21no—
182C3×C4—31no—
183C3×C4—31no—
184C3×C4—81no—
185C3×C4—82no—
186C3×C4—121no—
187C3×C4—121no—
188Dic3—42no—
189Dic3—81no—
190C32—41no—
191C9—81no—
192C9#542511yescharacteristic, direct product factor
193C23—31no—
194Q8—31no—
195Q8—61no—
196Q8—61no—
197Q8C9×#6711yescharacteristic
198D4—32no—
199D4—61no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4C9×S411yescharacteristic
206C8—61no—
207C8—61no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—81no—
212C2×C3—81no—
213C2×C3—81no—
214C2×C3—121no—
215C2×C3C3×#98411yescharacteristic
216C2×C3C3×#98511yescharacteristic
217C2×C3C3×GL(2,ℤ/4)11yescharacteristic
218C22—61no—
219C22—61no—
220C22—61no—
221C22C2×C9×S411yescharacteristic
222C4—21no—
223C4—31no—
224C4—31no—
225C4—121no—
226C4—121no—
227C3—41no—
228C3—81no—
229C3C3×#542511yescharacteristic
230C2—121no—
231C2C9×#98411yescharacteristic
232C2C9×#98511yescharacteristic
233C2C9×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C9C2×S4
commutatorC2×SL(2,𝔽3)C22×C9
Frattini, socleC22×C3C2×C3×S4
FittingC2×C9×#41D3
prime radicalC2×C3×#41⋊C3C2×C3
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC2×C4×C9S4
hypocenterSL(2,𝔽3)D4×C9
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C9 ◁ C2×C4×C9

Complements

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

NH ≅ G/Ntype
2. C2×C9×#226C2semidirect
3. C9×#859C2semidirect
25. #5425C9direct (N × H)
54. C2×#226C2×C9semidirect
58. #859C2×C9semidirect
192. C9#5425direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C9×#41⋊C32
C2×C9×#2262
C9×#8592
C3×#54253
C9×#4523
C9×Dic3⋊C44