Finite Group Databasebeta

← #13146218

Subgroups — #13146218

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
2C2×C9×#226C211yesmaximal, characteristic
3SL(2,𝔽3)×Dic9C211yesmaximal, characteristic
4#10593201—31nomaximal
5#10588862—31nomaximal
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7C9⋊#226—21no—
8C9×#226—21no—
9Dic9⋊Dic3—41nomaximal
10C2×C3⋊#226—31no—
11C2×C3×#226D311yescharacteristic
12Dic3×SL(2,𝔽3)—31no—
13C2×#3688—61no—
14C2×C9×Q16—31no—
15Q8×Dic9D311yescharacteristic
16#63506—31no—
17C2×#3674—31no—
18#63395—31no—
19#63376—31no—
20Dic9⋊C8—31no—
21C2×C3⋊Dic9—41no—
22C2×C9×Dic3—41no—
23C2×C3×Dic9—41no—
24C9×SL(2,𝔽3)D411yescharacteristic
25C4⋊#226—91no—
26#5182—91no—
27C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
28C3⋊#226—61no—
29C3×#226—21no—
30Dic3⋊Dic3—121no—
31C2×Q8×C9—31no—
32C2×Q8×C9C2×D311yescharacteristic
33C2×Dic18—31no—
34C2×Q8⋊C9—21no—
35#3688—62no—
36C9×Q16—61no—
37C9×Q16—61no—
38C2×C8×C9—31no—
39#3674—61no—
40#3674—61no—
41C4⋊Dic9—31no—
42Dic9⋊C4—61no—
43Dic9⋊C4—61no—
44C4×Dic9—31no—
45C2×C9⋊C8—31no—
46C22×C3×C9—41no—
47C3⋊Dic9—81no—
48C9×Dic3—81no—
49C3×Dic9—42no—
50C2×#226—91no—
51C2×#226—181no—
52C2×#226D911yescharacteristic
53C2×C3×Q16—31no—
54Q8×Dic3—31no—
55#944—91no—
56C2×#216—91no—
57C4×SL(2,𝔽3)—91no—
58#835—91no—
59#816—91no—
60Dic3⋊C8—91no—
61C2×C3⋊Dic3—121no—
62C2×C3×Dic3—41no—
63C2×C3×Dic3—121no—
64C3×SL(2,𝔽3)#6711yescharacteristic
65Q8×C9—31no—
66Q8×C9—61no—
67Q8×C9—61no—
68Q8×C9D1211yescharacteristic
69C2×C4×C9—31no—
70C2×C4×C9—61no—
71C2×Dic9—31no—
72C2×Dic9—61no—
73C2×Dic9—241no—
74C2×Dic9S411yescharacteristic
75Dic18—61no—
76Dic18—61no—
77Q8⋊C9—21no—
78C8×C9—61no—
79C9⋊C8—61no—
80#462—271noSylow
81C2×C3×C9—41no—
82C2×C3×C9—41no—
83C2×C3×C9—41no—
84C2×C3×Q8—31no—
85C2×C3×Q8D3211yescharacteristic
86C2×Dic6—91no—
87C2×SL(2,𝔽3)—21no—
88C2×SL(2,𝔽3)C2×D911yescharacteristic
89#226—21no—
90#226—182no—
91#226—181no—
92C3×Q16—61no—
93C3×Q16—61no—
94C2×C3×C8—31no—
95#216—181no—
96#216—181no—
97C4⋊Dic3—91no—
98C4⋊Dic3—361no—
99Dic3⋊C4—181no—
100Dic3⋊C4—181no—
101C4×Dic3—91no—
102C2×C3⋊C8—91no—
103C22×C32—41no—
104C3⋊Dic3—241no—
105C3×Dic3—81no—
106C3×Dic3—122no—
107C22×C9—81no—
108C22×C9C2×S411yescharacteristic
109C4×C9—31no—
110C4×C9—31no—
111C4×C9—121no—
112Dic9—61no—
113Dic9—121no—
114Dic9—242no—
115Dic9#22612yescomplemented
116C2×Q16—31no—
117C2×Q16—271no—
118C4⋊Q8—271no—
119C4×Q8—91no—
120C4⋊C8—271no—
121#103—271no—
122#103—271no—
123C3×C9—41noSylow
124C3×Q8—31no—
125C3×Q8—61no—
126C3×Q8—61no—
127C3×Q8#62611yescharacteristic
128C2×C3×C4—31no—
129C2×C3×C4—61no—
130C2×C3×C4—361no—
131C2×Dic3—31no—
132C2×Dic3—41no—
133C2×Dic3—91no—
134C2×Dic3—181no—
135C2×Dic3—361no—
136C2×Dic3—721no—
137Dic6—181no—
138Dic6—181no—
139SL(2,𝔽3)—21no—
140SL(2,𝔽3)#61111yescharacteristic
141C3×C8—61no—
142C3⋊C8—181no—
143C2×C32—41no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C9—81no—
147C2×C9—81no—
148C2×C9—81no—
149C2×C9C4⋊S411yescharacteristic
150C2×C9C2×#22611yescharacteristic
151C2×C9#98411yescharacteristic
152C2×Q8—31no—
153C2×Q8—271no—
154C2×Q8D3×D911yescharacteristic
155Q16—61no—
156Q16—61no—
157Q16—541no—
158Q16—541no—
159C2×C8—31no—
160C2×C8—271no—
161C4⋊C4—271no—
162C4⋊C4—541no—
163C4⋊C4—541no—
164C42—271no—
165C22×C3—41no—
166C22×C3—81no—
167C22×C3D3×S411yescharacteristic
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—121no—
171C3×C4—362no—
172Dic3—32no—
173Dic3—81no—
174Dic3—181no—
175Dic3—361no—
176Dic3—721no—
177Dic3—722no—
178C32—41no—
179C9—81no—
180C9C4⋊#22611yescharacteristic, complemented
181Q8—31no—
182Q8—61no—
183Q8—61no—
184Q8—541no—
185Q8—541no—
186Q8#619511yescharacteristic
187C2×C4—31no—
188C2×C4—61no—
189C2×C4—91no—
190C2×C4—271no—
191C2×C4—541no—
192C8—61no—
193C8—541no—
194C2×C3—41no—
195C2×C3—41no—
196C2×C3—41no—
197C2×C3—81no—
198C2×C3—81no—
199C2×C3—81no—
200C2×C3#6419611yescharacteristic
201C2×C3D3×#22611yescharacteristic
202C2×C3#6420711yescharacteristic
203C22D9×S411yescharacteristic
204C4—31no—
205C4—31no—
206C4—92no—
207C4—121no—
208C4—541no—
209C4—1081no—
210C3—41no—
211C3—81no—
212C3#1059320111yescharacteristic
213C2D9×#22611yescharacteristic
214C2#1172223611yescharacteristic
215C2#1172224311yescharacteristic
subgroupG/H
center, hypercenterC22D9×S4
commutator, cosocleC2×C9×SL(2,𝔽3)C22
Frattini, socleC22×C3D3×S4
FittingC2×Q8×C9C2×D3
prime radicalC2×C3×SL(2,𝔽3)C2×D3
hypocenterC9×SL(2,𝔽3)D4
LCSG ▷ C2×C9×SL(2,𝔽3) ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
115. Dic9#226semidirect
180. C9C4⋊#226semidirect

Maximal subgroups

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

subgroupindex
C2×C9⋊#2262
C2×C9×#2262
SL(2,𝔽3)×Dic92
#105932013
#105888623
Dic9⋊Dic34