Finite Group Databasebeta

← #13147056

Subgroups — #13147056

Order 1728.

500 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×C3×Q8⋊C9C211yesmaximal, characteristic, complemented
2C2×C3⋊#3689C211yesmaximal, characteristic, complemented
3C3⋊#63421C211yesmaximal, characteristic, complemented
4#10592926—31nomaximal
5#10589523—33nomaximal
6C2×#6233—41nomaximal
7C2×C3×Q8⋊C9—21no—
8C2×C3×Q8⋊C9C2211yescharacteristic
9C3⋊#3689—21no—
10C22×C32×Q8D311yescharacteristic
11C2×#3755—31no—
12C2×#3755—31no—
13#64139—31no—
14C22×Q8⋊C9D313yescomplemented
15C2×#3689—33no—
16#63662—31no—
17#63662—31no—
18#63657—31no—
19#63421—33no—
20C23×C3×C9—41no—
21C22×C3⋊D9—41no—
22#6233—44no—
23C2×C3⋊Dic9—41no—
24C3×Q8⋊C9D411yescharacteristic, complemented
25#5227—93no—
26#5227—91no—
27C2×C32×Q8—21no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C2×C32×Q8C2×D311yescharacteristic
31C22×C32×C4—31no—
32C2×#638—61no—
33C2×C4⋊(C3⋊D3)—31no—
34#3755—61no—
35#3755—62no—
36#3755—61no—
37#3752—61no—
38C4⋊(C3⋊Dic3)—31no—
39C2×C3⋊(C3⋊C8)—31no—
40C2×C3⋊(C3⋊C8)—31no—
41C2×#611—123no—
42C2×Q8⋊C9—23no—
43C2×Q8⋊C9C2×D313yes—
44#3689—63no—
45C22×C3×C9—41no—
46C22×C3×C9—42no—
47C22×C3×C9—81no—
48C22×C3×C9—81no—
49C2×C3⋊D9—42no—
50C2×C3⋊D9—81no—
51C2×C3⋊D9—81no—
52C3⋊Dic9—42no—
53C22×C3×Q8D913yescomplemented
54C22×C3×Q8C3⋊D311yescharacteristic
55C2×#215—93no—
56C2×#215—91no—
57C2×#215—93no—
58C2×#215—91no—
59#930—93no—
60#930—91no—
61#835—93no—
62#835—91no—
63#835—93no—
64#835—91no—
65#830—93no—
66#830—91no—
67C23×C32S411yescharacteristic
68C22×C3⋊D3—61no—
69C32×Q8—31no—
70C32×Q8—32no—
71C32×Q8—61no—
72C32×Q8#6711yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77#638—122no—
78C2×C3⋊Dic3—61no—
79C4⋊(C3⋊D3)—61no—
80C4⋊(C3⋊D3)—61no—
81C23×C9—43no—
82C22×D9—123no—
83C3⋊(C3⋊C8)—61no—
84C3⋊(C3⋊C8)—61no—
85#611—1212no—
86C2×Dic9—123no—
87Q8⋊C9#6713yescomplemented
88#448—271noSylow
89C2×C3×C9—41no—
90C2×C3×C9—41no—
91C2×C3×C9—41no—
92C2×C3×C9—82no—
93C3⋊D9—82no—
94C2×C3×Q8—23no—
95C2×C3×Q8—21no—
96C2×C3×Q8—33no—
97C2×C3×Q8—31no—
98C2×C3×Q8—63no—
99C2×C3×Q8—61no—
100C2×C3×Q8C2×D913yes—
101C2×C3×Q8C2×C3⋊D311yescharacteristic
102C22×C3×C4—33no—
103C22×C3×C4—31no—
104C2×#67—183no—
105C2×#67—181no—
106C2×D12—91no—
107C2×D12—93no—
108#215—183no—
109#215—181no—
110#215—182no—
111#215—186no—
112#215—183no—
113#215—181no—
114(C2×D3)⋊C4—181no—
115(C2×D3)⋊C4—183no—
116C4⋊Dic3—91no—
117C4⋊Dic3—93no—
118C2×C3⋊C8—91no—
119C2×C3⋊C8—91no—
120C2×C3⋊C8—93no—
121C2×C3⋊C8—93no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32GL(2,𝔽3)12yes—
125C22×C32C2×S411yescharacteristic
126C2×C3⋊D3—121no—
127C2×C3⋊D3—121no—
128C2×C3⋊D3—121no—
129C32×C4—31no—
130C32×C4—31no—
131C32×C4—61no—
132C3⋊Dic3—121no—
133C22×C9—43no—
134C22×C9—46no—
135C22×C9—83no—
136C22×C9—83no—
137C2×D9—126no—
138C2×D9—243no—
139C2×D9—243no—
140Dic9—126no—
141C22×Q8C3⋊D911yescharacteristic, complemented
142C2×SD16—271no—
143C2×SD16—271no—
144#121—271no—
145#103—271no—
146#103—271no—
147C22⋊C8—271no—
148C3×C9—41noSylow
149C23×C3(C22⋊C9)⋊C213yes—
150C23×C3C3⋊S411yescharacteristic
151C22×D3—181no—
152C22×D3—183no—
153C3×Q8—33no—
154C3×Q8—31no—
155C3×Q8—32no—
156C3×Q8—36no—
157C3×Q8—61no—
158C3×Q8—63no—
159C3×Q8#61113yescomplemented
160C3×Q8#63811yescharacteristic
161C2×C3×C4—31no—
162C2×C3×C4—33no—
163C2×C3×C4—31no—
164C2×C3×C4—33no—
165C2×C3×C4—61no—
166C2×C3×C4—63no—
167C2×C3×C4—61no—
168C2×C3×C4—63no—
169#67—366no—
170#67—362no—
171C2×Dic3—181no—
172C2×Dic3—183no—
173D12—181no—
174D12—181no—
175D12—183no—
176D12—183no—
177C3⋊C8—181no—
178C3⋊C8—181no—
179C3⋊C8—183no—
180C3⋊C8—183no—
181C2×C32—22no—
182C2×C32C2×GL(2,𝔽3)11yescharacteristic
183C2×C32#98311yescharacteristic
184C2×C32GL(2,ℤ/4)11yescharacteristic
185C3⋊D3—241no—
186C2×C9—43no—
187C2×C9—43no—
188C2×C9—43no—
189C2×C9—86no—
190D9—246no—
191C2×Q8—21no—
192C2×Q8—31no—
193C2×Q8—61no—
194C2×Q8C2×C3⋊D911yescharacteristic
195C2×D4—271no—
196C2×D4—541no—
197C22×C4—31no—
198SD16—541no—
199SD16—542no—
200SD16—541no—
201C2×C8—271no—
202C2×C8—271no—
203C4⋊C4—271no—
204C22⋊C4—541no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—23no—
208C22×C3—23no—
209C22×C3#368916yescomplemented
210C22×C3C2×(C22⋊C9)⋊C213yes—
211C22×C3C3⋊GL(2,𝔽3)12yes—
212C22×C3C2×C3⋊S411yescharacteristic
213C2×D3—361no—
214C2×D3—361no—
215C2×D3—361no—
216C2×D3—363no—
217C2×D3—363no—
218C2×D3—363no—
219C3×C4—31no—
220C3×C4—31no—
221C3×C4—33no—
222C3×C4—33no—
223C3×C4—61no—
224C3×C4—63no—
225Dic3—361no—
226Dic3—363no—
227C32#542111yescharacteristic
228C9—43no—
229C23—541no—
230C23C3⋊((C22⋊C9)⋊C2)11yescharacteristic
231Q8—31no—
232Q8—32no—
233Q8—61no—
234Q8#623311yescharacteristic, complemented
235D4—541no—
236D4—541no—
237D4—1082no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—541no—
243C8—541no—
244C8—541no—
245C2×C3—26no—
246C2×C3—22no—
247C2×C3C2×#368913yes—
248C2×C3#6368913yes—
249C2×C3#6369413yes—
250C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
251C2×C3#6426411yescharacteristic
252C2×C3#6426911yescharacteristic
253D3—723no—
254D3—721no—
255C22—21no—
256C22—21no—
257C22—1081no—
258C22—1081no—
259C22—1081no—
260C22C3⋊#368912yescomplemented
261C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
262C4—31no—
263C4—31no—
264C4—61no—
265C4—1081no—
266C3#1058952313yescomplemented
267C3#1059365511yescharacteristic
268C2—22no—
269C2—2161no—
270C2C2×C3⋊#368911yescharacteristic
271C2#1172238511yescharacteristic
272C2#1172239611yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocleC2×C3×Q8⋊C9C22
FrattiniC22×C3C2×C3⋊S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
hypercenterC23C3⋊((C22⋊C9)⋊C2)
hypocenterC3×Q8⋊C9D4
LCSG ▷ C2×C3×Q8⋊C9 ▷ C3×Q8⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8⋊C9C2semidirect
2. C2×C3⋊#3689C2semidirect
3. C3⋊#63421C2semidirect
14. C22×Q8⋊C9D3semidirect
24. C3×Q8⋊C9D4semidirect
53. C22×C3×Q8D9semidirect
87. Q8⋊C9#67semidirect
141. C22×Q8C3⋊D9semidirect
159. C3×Q8#611semidirect
209. C22×C3#3689semidirect
234. Q8#6233semidirect
260. C22C3⋊#3689semidirect
266. C3#10589523semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
C2×C3⋊#36892
C3⋊#634212
#105929263
#105895233
C2×#62334