Finite Group Databasebeta

← #13146224

Subgroups — #13146224

Order 1728.

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

normal subgroups only

This group has 227 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
1C2×C9⋊#226C211yesmaximal, characteristic
2SL(2,𝔽3)×Dic9C211yesmaximal, characteristic
3C9×#859C211yesmaximal, characteristic
4#10593195—31nomaximal
5#10588469—31nomaximal
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7C9⋊#226C412yescomplemented
8#88442—41nomaximal
9C2×C3⋊#226—31no—
10Dic3×SL(2,𝔽3)—31no—
11C3×#859D311yescharacteristic
12C2×#3688—61no—
13Q8×Dic9D311yescharacteristic
14C2×#3674—31no—
15#63449—31no—
16C9×#103—31no—
17#63376—31no—
18#63375—31no—
19#63367—31no—
20C2×C3⋊Dic9—41no—
21C2×C9×Dic3—41no—
22C2×C3×Dic9—41no—
23C9×SL(2,𝔽3)C2×C411yescharacteristic
24#5388—91no—
25#4789—91no—
26C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
27C3⋊#226—32no—
28#3724—121no—
29C2×Q8×C9C2×D311yescharacteristic
30C2×Dic18—31no—
31C2×Q8⋊C9—21no—
32#3688—62no—
33C2×C8×C9—31no—
34C9×C4⋊C4—31no—
35#3674—32no—
36#3674—32no—
37C4⋊Dic9—31no—
38Dic9⋊C4—61no—
39C4×Dic9—31no—
40C4×Dic9—61no—
41C2×C9⋊C8—31no—
42C22×C3×C9—41no—
43C3⋊Dic9—42no—
44C9×Dic3—81no—
45C3×Dic9—81no—
46C2×#226—91no—
47C2×#226—181no—
48Q8×Dic3—31no—
49C2×#216—91no—
50#887—91no—
51C4×SL(2,𝔽3)—91no—
52#859D911yescharacteristic
53C3×#103—31no—
54#816—91no—
55#815—91no—
56#808—91no—
57C2×C3⋊Dic3—121no—
58C2×C3×Dic3—41no—
59C2×C3×Dic3—121no—
60C3×SL(2,𝔽3)C4×D311yescharacteristic
61Q8×C9—31no—
62Q8×C9C4×D311yescharacteristic
63C2×C4×C9—31no—
64C2×C4×C9—61no—
65C2×Dic9—31no—
66C2×Dic9—61no—
67C2×Dic9—241no—
68C2×Dic9S411yescharacteristic
69Dic18—32no—
70Dic18—61no—
71Q8⋊C9—21no—
72C8×C9—61no—
73C9⋊C8—32no—
74#441—271noSylow
75C2×C3×C9—41no—
76C2×C3×C9—41no—
77C2×C3×C9—41no—
78C2×C3×Q8D3211yescharacteristic
79C2×Dic6—91no—
80C2×SL(2,𝔽3)—21no—
81C2×SL(2,𝔽3)C2×D911yescharacteristic
82#226—92no—
83#226—182no—
84C2×C3×C8—31no—
85C3×C4⋊C4—31no—
86#216—92no—
87#216—92no—
88C4⋊Dic3—91no—
89Dic3⋊C4—181no—
90Dic3⋊C4—361no—
91C4×Dic3—91no—
92C4×Dic3—181no—
93C2×C3⋊C8—91no—
94C22×C32—41no—
95C3⋊Dic3—122no—
96C3×Dic3—81no—
97C3×Dic3—241no—
98C22×C9—81no—
99C22×C9C2×S411yescharacteristic
100C4×C9—31no—
101C4×C9—31no—
102C4×C9—121no—
103Dic9—21no—
104Dic9—62no—
105Dic9—61no—
106Dic9—242no—
107C2×Q16—271no—
108C4×Q8—91no—
109C4×Q8—271no—
110#106—271no—
111#103—31no—
112#103—271no—
113#97—271no—
114C3×C9—41noSylow
115C3×Q8—31no—
116C3×Q8#62411yescharacteristic
117C2×C3×C4—31no—
118C2×C3×C4—61no—
119C2×C3×C4—361no—
120C2×Dic3—31no—
121C2×Dic3—41no—
122C2×Dic3—91no—
123C2×Dic3—181no—
124C2×Dic3—361no—
125C2×Dic3—721no—
126Dic6—92no—
127Dic6—181no—
128SL(2,𝔽3)—21no—
129SL(2,𝔽3)C4×D911yescharacteristic
130C3×C8—61no—
131C3⋊C8—92no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C9—81no—
136C2×C9—81no—
137C2×C9—81no—
138C2×C9C4×S411yescharacteristic
139C2×C9#98311yescharacteristic
140C2×C9#98411yescharacteristic
141C2×Q8—271no—
142C2×Q8D3×D911yescharacteristic
143Q16—272no—
144Q16—272no—
145C2×C8—31no—
146C2×C8—271no—
147C4⋊C4—31no—
148C4⋊C4—271no—
149C4⋊C4—541no—
150C42—271no—
151C42—541no—
152C22×C3—41no—
153C22×C3—81no—
154C22×C3D3×S411yescharacteristic
155C3×C4—31no—
156C3×C4—31no—
157C3×C4—121no—
158C3×C4—721no—
159Dic3—61no—
160Dic3—81no—
161Dic3—181no—
162Dic3—182no—
163Dic3—362no—
164Dic3—722no—
165C32—41no—
166C9—81no—
167C9#538811yescharacteristic, complemented
168Q8—31no—
169Q8—272no—
170Q8—541no—
171Q8(C3⋊D9)⋊C411yescharacteristic
172C2×C4—31no—
173C2×C4—61no—
174C2×C4—91no—
175C2×C4—271no—
176C2×C4—541no—
177C8—61no—
178C8—272no—
179C2×C3—41no—
180C2×C3—41no—
181C2×C3—41no—
182C2×C3—81no—
183C2×C3—81no—
184C2×C3—81no—
185C2×C3#6419611yescharacteristic
186C2×C3#6420111yescharacteristic
187C2×C3(C3⋊S4)⋊C411yescharacteristic
188C22D9×S411yescharacteristic
189C4—31no—
190C4—31no—
191C4—121no—
192C4—181no—
193C4—542no—
194C4—541no—
195C3—41no—
196C3—81no—
197C3#1059319511yescharacteristic
198C2#1172223311yescharacteristic
199C2#1172223611yescharacteristic
200C2(C9⋊S4)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, hypocenterC9×SL(2,𝔽3)C2×C4
Frattini, socleC22×C3D3×S4
FittingC2×Q8×C9C2×D3
prime radicalC2×C3×SL(2,𝔽3)C2×D3
cosocleC2×C9×SL(2,𝔽3)C22
LCSG ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
7. C9⋊#226C4semidirect
167. C9#5388semidirect

Maximal subgroups

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

subgroupindex
C2×C9⋊#2262
SL(2,𝔽3)×Dic92
C9×#8592
#105931953
#105884693
#884424