Finite Group Databasebeta

← #13147070

Subgroups — #13147070

Order 1728.

448 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 69 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×C3×#41⋊C9C211yesmaximal, characteristic, complemented
2C2×C3⋊#3689C211yesmaximal, characteristic, complemented
3C3⋊#63421C211yesmaximal, characteristic, complemented
4#10592937—31nomaximal
5#10589529—33nomaximal
6C3×#41⋊C9—21no—
7C2×C3×Q8⋊C9C2211yescharacteristic
8C3⋊#3689—21no—
9#88536—41nomaximal
10C2×C32×#41D311yescharacteristic
11C2×#3755—31no—
12C2×#3753—31no—
13#64139—31no—
14C2×#41⋊C9D313yescomplemented
15C2×#3689—33no—
16#63662—31no—
17#63659—31no—
18#63657—31no—
19#63421—33no—
20C22×C3⋊D9—41no—
21C2×C3×C4×C9—41no—
22C2×C3⋊Dic9—41no—
23C3×Q8⋊C9D411yescharacteristic, complemented
24#5238—93no—
25#5238—91no—
26C32×#41—21no—
27C32×#41—61no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C22×C32×C4—31no—
31C2×#638—61no—
32C2×C4⋊(C3⋊D3)—31no—
33#3755—61no—
34#3755—61no—
35#3753—62no—
36#3752—61no—
37C4⋊(C3⋊Dic3)—31no—
38C2×C3⋊(C3⋊C8)—31no—
39C2×C3⋊(C3⋊C8)—31no—
40#41⋊C9—23no—
41C2×Q8⋊C9C2×D313yes—
42#3689—63no—
43(C2×D9)⋊C4—123no—
44C22×C3×C9—41no—
45C2×C3⋊D9—42no—
46C2×C3⋊D9—81no—
47C2×C3⋊D9—81no—
48C3×C4×C9—81no—
49C3⋊Dic9—81no—
50C2×C3×#41D913yescomplemented
51C2×C3×#41C3⋊D311yescharacteristic
52C2×#215—93no—
53C2×#215—91no—
54C2×#213—93no—
55C2×#213—91no—
56#930—93no—
57#930—91no—
58#835—93no—
59#835—91no—
60#832—93no—
61#832—91no—
62#830—93no—
63#830—91no—
64C23×C32—31no—
65C22×C3⋊D3—61no—
66C32×Q8—31no—
67C32×Q8#6711yescharacteristic
68C32×D4—32no—
69C32×D4—61no—
70C2×C32×C4—31no—
71C2×C32×C4—61no—
72C2×C32×C4—61no—
73C2×C32×C4S411yescharacteristic
74#638—122no—
75C2×C3⋊Dic3—61no—
76C4⋊(C3⋊D3)—61no—
77C4⋊(C3⋊D3)—61no—
78C22×D9—123no—
79C3⋊(C3⋊C8)—61no—
80C3⋊(C3⋊C8)—61no—
81C2×C4×C9—43no—
82C2×Dic9—123no—
83Q8⋊C9#6713yescomplemented
84#449—271noSylow
85C2×C3×C9—41no—
86C2×C3×C9—41no—
87C2×C3×C9—41no—
88C3⋊D9—82no—
89C3×#41—23no—
90C3×#41—21no—
91C3×#41—63no—
92C3×#41—61no—
93C2×C3×Q8C2×D913yes—
94C2×C3×Q8C2×C3⋊D311yescharacteristic
95C2×C3×D4—33no—
96C2×C3×D4—31no—
97C22×C3×C4—33no—
98C22×C3×C4—31no—
99C2×#67—181no—
100C2×#67—183no—
101C2×D12—91no—
102C2×D12—93no—
103#215—183no—
104#215—181no—
105#215—183no—
106#215—181no—
107#213—182no—
108#213—186no—
109(C2×D3)⋊C4—183no—
110(C2×D3)⋊C4—181no—
111C4⋊Dic3—91no—
112C4⋊Dic3—93no—
113C2×C3⋊C8—91no—
114C2×C3⋊C8—93no—
115C2×C3⋊C8—91no—
116C2×C3⋊C8—93no—
117C22×C32—61no—
118C22×C32—61no—
119C22×C32—61no—
120C22×C32C2×S411yescharacteristic
121C2×C3⋊D3—121no—
122C2×C3⋊D3—121no—
123C2×C3⋊D3—121no—
124C32×C4—21no—
125C32×C4—31no—
126C32×C4—31no—
127C3⋊Dic3—121no—
128C22×C9—43no—
129C2×D9—126no—
130C2×D9—243no—
131C2×D9—243no—
132C4×C9—83no—
133Dic9—243no—
134C2×#41C3⋊D911yescharacteristic, complemented
135C2×SD16—271no—
136C2×D8—271no—
137#121—271no—
138#103—271no—
139#102—271no—
140C22⋊C8—271no—
141C3×C9—41noSylow
142C23×C3—31no—
143C23×C3—33no—
144C22×D3—181no—
145C22×D3—183no—
146C3×Q8—33no—
147C3×Q8—31no—
148C3×Q8#61113yes—
149C3×Q8#63811yescharacteristic
150C3×D4—32no—
151C3×D4—36no—
152C3×D4—61no—
153C3×D4—63no—
154C2×C3×C4—31no—
155C2×C3×C4—33no—
156C2×C3×C4—61no—
157C2×C3×C4—63no—
158C2×C3×C4—61no—
159C2×C3×C4—63no—
160C2×C3×C4(C22⋊C9)⋊C213yes—
161C2×C3×C4C3⋊S411yescharacteristic
162#67—362no—
163#67—366no—
164C2×Dic3—181no—
165C2×Dic3—183no—
166D12—181no—
167D12—183no—
168D12—181no—
169D12—183no—
170C3⋊C8—181no—
171C3⋊C8—183no—
172C3⋊C8—181no—
173C3⋊C8—183no—
174C2×C32—121no—
175C2×C32#98511yescharacteristic
176C2×C32#98611yescharacteristic
177C2×C32GL(2,ℤ/4)11yescharacteristic
178C3⋊D3—241no—
179C2×C9—43no—
180C2×C9—43no—
181C2×C9—43no—
182D9—246no—
183#41—21no—
184#41—61no—
185C2×Q8C2×C3⋊D911yescharacteristic
186C2×D4—31no—
187C2×D4—271no—
188C2×D4—541no—
189C22×C4—31no—
190SD16—541no—
191SD16—541no—
192D8—542no—
193C2×C8—271no—
194C2×C8—271no—
195C4⋊C4—271no—
196C22⋊C4—541no—
197C22×C3—61no—
198C22×C3—61no—
199C22×C3—61no—
200C22×C3—63no—
201C22×C3—63no—
202C22×C3—63no—
203C22×C3C2×(C22⋊C9)⋊C213yes—
204C22×C3C2×C3⋊S411yescharacteristic
205C2×D3—361no—
206C2×D3—361no—
207C2×D3—361no—
208C2×D3—363no—
209C2×D3—363no—
210C2×D3—363no—
211C3×C4—21no—
212C3×C4—23no—
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—33no—
216C3×C4—33no—
217Dic3—361no—
218Dic3—363no—
219C32#542711yescharacteristic
220C9—43no—
221C23—31no—
222C23—541no—
223Q8—31no—
224Q8#623311yescharacteristic
225D4—32no—
226D4—61no—
227D4—541no—
228D4—541no—
229D4—1082no—
230C2×C4—31no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—541no—
234C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
235C8—541no—
236C8—541no—
237C2×C3—123no—
238C2×C3—121no—
239C2×C3#6369113yes—
240C2×C3#6369213yes—
241C2×C3#6369413yes—
242C2×C3#6426611yescharacteristic
243C2×C3#6426711yescharacteristic
244C2×C3#6426911yescharacteristic
245D3—723no—
246D3—721no—
247C22—61no—
248C22—61no—
249C22—61no—
250C22—1081no—
251C22—1081no—
252C22—1081no—
253C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
254C4—21no—
255C4—31no—
256C4—31no—
257C4—1081no—
258C3#1058952913yescomplemented
259C3#1059366111yescharacteristic
260C2—121no—
261C2—2161no—
262C2#1172239011yescharacteristic
263C2#1172239111yescharacteristic
264C2#1172239611yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocleC2×C3×Q8⋊C9C22
FrattiniC22×C3C2×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
hypocenterC3×Q8⋊C9D4
LCSG ▷ C2×C3×Q8⋊C9 ▷ C3×Q8⋊C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. C2×C3⋊#3689C2semidirect
3. C3⋊#63421C2semidirect
14. C2×#41⋊C9D3semidirect
23. C3×Q8⋊C9D4semidirect
50. C2×C3×#41D9semidirect
83. Q8⋊C9#67semidirect
134. C2×#41C3⋊D9semidirect
258. C3#10589529semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C2×C3⋊#36892
C3⋊#634212
#105929373
#105895293
#885364