Finite Group Databasebeta

← #13146221

Subgroups — #13146221

Order 1728.

223 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 41 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#10593198—31nomaximal
5#10588864—31nomaximal
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7C9×#226C412yescomplemented
8Dic3⋊Dic9—41nomaximal
9C2×C3×#226D311yescharacteristic
10Dic3×SL(2,𝔽3)—31no—
11C3⋊#859—31no—
12C2×C9×Q16—31no—
13Q8×Dic9—31no—
14Q8×Dic9D311yescharacteristic
15#63421—61no—
16#63395—31no—
17#63395—31no—
18#63377—31no—
19#63375—31no—
20C2×C3⋊Dic9—41no—
21C2×C9×Dic3—41no—
22C2×C3×Dic9—41no—
23C9×SL(2,𝔽3)C2×C411yescharacteristic
24#5388—91no—
25#5184—91no—
26C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
27C3×#226Dic312yes—
28Dic3⋊Dic3—121no—
29C2×Q8×C9—31no—
30C2×Q8×C9C2×D311yescharacteristic
31C2×Q8⋊C9—21no—
32C9×Q16—32no—
33C9×Q16—32no—
34C2×C8×C9—31no—
35C4⋊Dic9—31no—
36C4⋊Dic9—31no—
37C4⋊Dic9—61no—
38C4×Dic9—31no—
39C4×Dic9—61no—
40C2×C9⋊C8—31no—
41C22×C3×C9—41no—
42C3⋊Dic9—81no—
43C9×Dic3—42no—
44C3×Dic9—81no—
45C2×#226D911yescharacteristic
46C2×C3×Q16—31no—
47Q8×Dic3—31no—
48Q8×Dic3—91no—
49C4×SL(2,𝔽3)—91no—
50#859—91no—
51#859—181no—
52#835—91no—
53#835—91no—
54#817—91no—
55#815—91no—
56C2×C3⋊Dic3—121no—
57C2×C3×Dic3—41no—
58C2×C3×Dic3—121no—
59C3×SL(2,𝔽3)C2×Dic311yescharacteristic
60Q8×C9—31no—
61Q8×C9—32no—
62Q8×C9—61no—
63Q8×C9C4×D311yescharacteristic
64C2×C4×C9—31no—
65C2×C4×C9—61no—
66C2×Dic9—31no—
67C2×Dic9—61no—
68C2×Dic9—241no—
69C2×Dic9S411yescharacteristic
70Q8⋊C9—21no—
71C8×C9—32no—
72C9⋊C8—61no—
73#441—271noSylow
74C2×C3×C9—41no—
75C2×C3×C9—41no—
76C2×C3×C9—41no—
77C2×C3×Q8—31no—
78C2×C3×Q8D3211yescharacteristic
79C2×SL(2,𝔽3)—21no—
80C2×SL(2,𝔽3)C2×D911yescharacteristic
81#226Dic912yescomplemented
82C3×Q16—32no—
83C3×Q16—32no—
84C2×C3×C8—31no—
85C4⋊Dic3—91no—
86C4⋊Dic3—91no—
87C4⋊Dic3—181no—
88Dic3⋊C4—361no—
89C4×Dic3—91no—
90C4×Dic3—181no—
91C2×C3⋊C8—91no—
92C22×C32—41no—
93C3⋊Dic3—241no—
94C3×Dic3—42no—
95C3×Dic3—241no—
96C22×C9—81no—
97C22×C9C2×S411yescharacteristic
98C4×C9—31no—
99C4×C9—31no—
100C4×C9—62no—
101Dic9—21no—
102Dic9—61no—
103Dic9—121no—
104Dic9—242no—
105C2×Q16—31no—
106C4×Q8—91no—
107C4×Q8—271no—
108#106—271no—
109#103—271no—
110#103—271no—
111#97—271no—
112C3×C9—41noSylow
113C3×Q8—31no—
114C3×Q8—32no—
115C3×Q8—61no—
116C3×Q8D3×Dic311yescharacteristic
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—
126SL(2,𝔽3)—21no—
127SL(2,𝔽3)C2×Dic911yescharacteristic
128C3×C8—32no—
129C3⋊C8—181no—
130C2×C32—41no—
131C2×C32—41no—
132C2×C32—41no—
133C2×C9—81no—
134C2×C9—81no—
135C2×C9—81no—
136C2×C9C4×S411yescharacteristic
137C2×C9#98311yescharacteristic
138C2×C9#98411yescharacteristic
139C2×Q8—31no—
140C2×Q8D3×D911yescharacteristic
141Q16—32no—
142Q16—32no—
143C2×C8—31no—
144C2×C8—271no—
145C4⋊C4—271no—
146C4⋊C4—271no—
147C4⋊C4—541no—
148C42—271no—
149C42—541no—
150C22×C3—41no—
151C22×C3—81no—
152C22×C3D3×S411yescharacteristic
153C3×C4—31no—
154C3×C4—31no—
155C3×C4—62no—
156C3×C4—721no—
157Dic3—42no—
158Dic3—61no—
159Dic3—181no—
160Dic3—361no—
161Dic3—721no—
162Dic3—722no—
163C32—41no—
164C9—81no—
165C9#538811yescharacteristic, complemented
166Q8—31no—
167Q8—32no—
168Q8—61no—
169Q8D3×Dic911yescharacteristic
170C2×C4—31no—
171C2×C4—61no—
172C2×C4—91no—
173C2×C4—271no—
174C2×C4—541no—
175C8—32no—
176C8—541no—
177C2×C3—41no—
178C2×C3—41no—
179C2×C3—41no—
180C2×C3—81no—
181C2×C3—81no—
182C2×C3—81no—
183C2×C3#6419611yescharacteristic
184C2×C3#6420211yescharacteristic
185C2×C3Dic3×S411yescharacteristic
186C22D9×S411yescharacteristic
187C4—31no—
188C4—31no—
189C4—62no—
190C4—181no—
191C4—541no—
192C4—1081no—
193C3—41no—
194C3—81no—
195C3#1059319811yescharacteristic
196C2#1172223411yescharacteristic
197C2#1172223611yescharacteristic
198C2S4×Dic911yescharacteristic
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
81. #226Dic9semidirect
165. 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
#105931983
#105888643
Dic3⋊Dic94