Finite Group Databasebeta

← #13146981

Subgroups — #13146981

Order 1728.

306 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 51 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
4#10593659—31nomaximal
5#10589527—32nomaximal
6#10588918—31nomaximal
7C9×#41⋊C3—21no—
8C2×C9×SL(2,𝔽3)C2211yescharacteristic
9C9⋊#226—21no—
10#88534—41nomaximal
11C2×C3×#41⋊C3D311yescharacteristic
12C2×C3⋊#226—31no—
13C3⋊#859—31no—
14C2×C9×#41D311yescharacteristic
15C2×#41⋊C9D312yes—
16C2×#3688—32no—
17C2×#3674—31no—
18C2×#3672—31no—
19#63486—31no—
20#63421—32no—
21#63395—31no—
22#63392—31no—
23#63389—31no—
24C2×C3×C4×C9—41no—
25C2×C3⋊Dic9—41no—
26C2×C3⋊Dic9—41no—
27C9×SL(2,𝔽3)D411yescharacteristic
28#5425—93no—
29#5239—91no—
30C3×#41⋊C3—21no—
31C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
32C3⋊#226—61no—
33#3750—121no—
34C9×#41—21no—
35C9×#41—61no—
36C2×Q8×C9C2×D311yescharacteristic
37C2×D4×C9—31no—
38C22×C4×C9—31no—
39C2×Dic18—31no—
40#41⋊C9—22no—
41C2×Q8⋊C9C2×D312yes—
42#3688—62no—
43#3676—61no—
44#3674—61no—
45#3674—61no—
46#3672—62no—
47C4⋊Dic9—31no—
48Dic9⋊C4—61no—
49Dic9⋊C4—122no—
50C2×C9⋊C8—31no—
51C2×C9⋊C8—31no—
52C22×C3×C9—41no—
53C3×C4×C9—81no—
54C3⋊Dic9—42no—
55C3⋊Dic9—81no—
56C2×C3×#41C3⋊D311yescharacteristic
57C2×#41⋊C3D913yes—
58C2×#226—93no—
59C2×#216—91no—
60C2×#214—91no—
61#924—91no—
62#859—93no—
63#835—91no—
64#832—91no—
65#830—91no—
66C2×C32×C4—41no—
67C2×C3⋊Dic3—121no—
68C2×C3⋊Dic3—121no—
69C3×SL(2,𝔽3)#6711yescharacteristic
70C23×C9—31no—
71Q8×C9—31no—
72Q8×C9#6711yescharacteristic
73D4×C9—32no—
74D4×C9—61no—
75C2×C4×C9—31no—
76C2×C4×C9—42no—
77C2×C4×C9—61no—
78C2×C4×C9—61no—
79C2×C4×C9S411yescharacteristic
80C2×Dic9—61no—
81C2×Dic9—61no—
82C2×Dic9—122no—
83C2×Dic9—122no—
84Dic18—61no—
85Dic18—61no—
86Q8⋊C9#6712yes—
87C9⋊C8—61no—
88C9⋊C8—61no—
89#452—271noSylow
90C2×C3×C9—41no—
91C2×C3×C9—41no—
92C2×C3×C9—41no—
93C3×#41—21no—
94C3×#41—61no—
95C2×C3×Q8C2×C3⋊D311yescharacteristic
96C2×C3×D4—31no—
97C22×C3×C4—31no—
98C2×Dic6—91no—
99#41⋊C3—23no—
100C2×SL(2,𝔽3)C2×D913yes—
101#226—183no—
102#217—181no—
103#216—181no—
104#216—181no—
105#214—182no—
106C4⋊Dic3—91no—
107Dic3⋊C4—181no—
108Dic3⋊C4—363no—
109C2×C3⋊C8—91no—
110C2×C3⋊C8—91no—
111C22×C32—41no—
112C32×C4—81no—
113C3⋊Dic3—122no—
114C3⋊Dic3—241no—
115C22×C9—42no—
116C22×C9—61no—
117C22×C9—61no—
118C22×C9—61no—
119C22×C9C2×S411yescharacteristic
120C4×C9—21no—
121C4×C9—31no—
122C4×C9—31no—
123C4×C9—82no—
124Dic9—124no—
125Dic9—121no—
126Dic9—121no—
127Dic9—242no—
128C2×#41C3⋊D911yescharacteristic
129C2×Q16—271no—
130C2×SD16—271no—
131C22⋊Q8—271no—
132#103—271no—
133#102—271no—
134C22⋊C8—271no—
135C3×C9—41noSylow
136C23×C3—31no—
137C3×Q8—31no—
138C3×Q8#63811yescharacteristic
139C3×D4—32no—
140C3×D4—61no—
141C2×C3×C4—31no—
142C2×C3×C4—43no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4C3⋊S411yescharacteristic
146C2×Dic3—181no—
147C2×Dic3—181no—
148C2×Dic3—363no—
149C2×Dic3—363no—
150Dic6—181no—
151Dic6—181no—
152SL(2,𝔽3)#61113yes—
153C3⋊C8—181no—
154C3⋊C8—181no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—41no—
158C2×C9—42no—
159C2×C9—42no—
160C2×C9—42no—
161C2×C9—121no—
162C2×C9#98411yescharacteristic
163C2×C9#98511yescharacteristic
164C2×C9GL(2,ℤ/4)11yescharacteristic
165#41—21no—
166#41—61no—
167C2×Q8—271no—
168C2×Q8C2×C3⋊D911yescharacteristic
169C2×D4—31no—
170C22×C4—31no—
171Q16—541no—
172Q16—541no—
173SD16—542no—
174C2×C8—271no—
175C2×C8—271no—
176C4⋊C4—271no—
177C4⋊C4—541no—
178C22⋊C4—541no—
179C22×C3—43no—
180C22×C3—61no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3C2×C3⋊S411yescharacteristic
184C3×C4—21no—
185C3×C4—31no—
186C3×C4—31no—
187C3×C4—83no—
188Dic3—366no—
189Dic3—361no—
190Dic3—361no—
191Dic3—723no—
192C32—41no—
193C9—42no—
194C9#542511yescharacteristic, complemented
195C23—31no—
196Q8—31no—
197Q8—541no—
198Q8—541no—
199Q8#623311yescharacteristic
200D4—32no—
201D4—61no—
202C2×C4—31no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—541no—
206C2×C4—541no—
207C2×C4C9⋊S411yescharacteristic
208C8—541no—
209C8—541no—
210C2×C3—43no—
211C2×C3—43no—
212C2×C3—43no—
213C2×C3—121no—
214C2×C3#6426511yescharacteristic
215C2×C3#6426611yescharacteristic
216C2×C3#6426911yescharacteristic
217C22—61no—
218C22—61no—
219C22—61no—
220C22C2×C9⋊S411yescharacteristic
221C4—21no—
222C4—31no—
223C4—31no—
224C4—1081no—
225C4—1081no—
226C3—43no—
227C3#1059365911yescharacteristic
228C2—121no—
229C2#1172237211yescharacteristic
230C2#1172237311yescharacteristic
231C2#1172237611yescharacteristic
subgroupG/H
centerC22C2×C9⋊S4
commutator, cosocleC2×C9×SL(2,𝔽3)C22
Frattini, socleC22×C3C2×C3⋊S4
FittingC2×C9×#41D3
prime radicalC2×C3×#41⋊C3D3
hypercenterC2×C4C9⋊S4
hypocenterC9×SL(2,𝔽3)D4
LCSG ▷ C2×C9×SL(2,𝔽3) ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C9⋊#226C2semidirect
3. C9⋊#859C2semidirect
194. C9#5425semidirect

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
#105936593
#105895273
#105889183
#885344