Finite Group Databasebeta

← #13147054

Subgroups — #13147054

Order 1728.

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

normal subgroups only

This group has 430 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
2C2×C3⋊#3688C211yesmaximal, characteristic, complemented
3C3⋊#63421C211yesmaximal, characteristic, complemented
4#10592927—31nomaximal
5#10589521—33nomaximal
6C2×C3×Q8⋊C9—21no—
7C2×C3×Q8⋊C9C2211yescharacteristic
8C3⋊#3688—21no—
9#88551—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×#3756—31no—
12C2×#3756—31no—
13#64133—31no—
14C22×Q8⋊C9D313yes—
15C2×#3688—33no—
16#63662—31no—
17#63662—31no—
18#63657—31no—
19#63421—33no—
20C23×C3×C9—41no—
21C2×C3⋊Dic9—41no—
22C2×C3⋊Dic9—41no—
23C3×Q8⋊C9D411yescharacteristic
24#5228—93no—
25#5228—91no—
26C2×C32×Q8—21no—
27C2×C32×Q8—31no—
28C2×C32×Q8—61no—
29C2×C32×Q8C2×D311yescharacteristic
30C22×C32×C4—31no—
31C2×#634—31no—
32#3757—61no—
33#3756—61no—
34#3756—62no—
35#3756—61no—
36C4⋊(C3⋊Dic3)—31no—
37#3750—61no—
38C2×C3⋊(C3⋊C8)—31no—
39C2×C3⋊(C3⋊C8)—31no—
40C2×Q8⋊C9—23no—
41C2×Q8⋊C9C2×D313yes—
42#3688—63no—
43#3676—123no—
44C22×C3×C9—41no—
45C22×C3×C9—42no—
46C22×C3×C9—81no—
47C22×C3×C9—81no—
48C3⋊Dic9—81no—
49C3⋊Dic9—81no—
50C22×C3×Q8D913yes—
51C22×C3×Q8C3⋊D311yescharacteristic
52C2×#216—93no—
53C2×#216—91no—
54C2×#216—93no—
55C2×#216—91no—
56#924—93no—
57#924—91no—
58#835—93no—
59#835—91no—
60#835—93no—
61#835—91no—
62#830—93no—
63#830—91no—
64C23×C32S411yescharacteristic
65C32×Q8—31no—
66C32×Q8—32no—
67C32×Q8—61no—
68C32×Q8#6711yescharacteristic
69C2×C32×C4—31no—
70C2×C32×C4—31no—
71C2×C32×C4—61no—
72C2×C32×C4—61no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—61no—
75#634—61no—
76#634—61no—
77C23×C9—43no—
78C3⋊(C3⋊C8)—61no—
79C3⋊(C3⋊C8)—61no—
80C2×Dic9—123no—
81C2×Dic9—123no—
82Q8⋊C9#6713yes—
83#451—271noSylow
84C2×C3×C9—41no—
85C2×C3×C9—41no—
86C2×C3×C9—41no—
87C2×C3×C9—82no—
88C2×C3×Q8—23no—
89C2×C3×Q8—21no—
90C2×C3×Q8—33no—
91C2×C3×Q8—31no—
92C2×C3×Q8—63no—
93C2×C3×Q8—61no—
94C2×C3×Q8C2×D913yes—
95C2×C3×Q8C2×C3⋊D311yescharacteristic
96C22×C3×C4—33no—
97C22×C3×C4—31no—
98C2×Dic6—91no—
99C2×Dic6—93no—
100#217—183no—
101#217—181no—
102#216—183no—
103#216—181no—
104#216—182no—
105#216—186no—
106#216—183no—
107#216—181no—
108C4⋊Dic3—91no—
109C4⋊Dic3—93no—
110Dic3⋊C4—181no—
111Dic3⋊C4—183no—
112C2×C3⋊C8—91no—
113C2×C3⋊C8—91no—
114C2×C3⋊C8—93no—
115C2×C3⋊C8—93no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32#22612yes—
119C22×C32C2×S411yescharacteristic
120C32×C4—31no—
121C32×C4—31no—
122C32×C4—61no—
123C3⋊Dic3—121no—
124C3⋊Dic3—121no—
125C22×C9—43no—
126C22×C9—46no—
127C22×C9—83no—
128C22×C9—83no—
129Dic9—243no—
130Dic9—243no—
131C22×Q8C3⋊D911yescharacteristic
132C2×Q16—271no—
133C2×Q16—271no—
134C22⋊Q8—271no—
135#103—271no—
136#103—271no—
137C22⋊C8—271no—
138C3×C9—41noSylow
139C23×C3(C22⋊C9)⋊C213yes—
140C23×C3C3⋊S411yescharacteristic
141C3×Q8—33no—
142C3×Q8—31no—
143C3×Q8—32no—
144C3×Q8—36no—
145C3×Q8—61no—
146C3×Q8—63no—
147C3×Q8#61113yes—
148C3×Q8#63811yescharacteristic
149C2×C3×C4—31no—
150C2×C3×C4—33no—
151C2×C3×C4—31no—
152C2×C3×C4—33no—
153C2×C3×C4—61no—
154C2×C3×C4—63no—
155C2×C3×C4—61no—
156C2×C3×C4—63no—
157C2×Dic3—181no—
158C2×Dic3—181no—
159C2×Dic3—183no—
160C2×Dic3—183no—
161Dic6—181no—
162Dic6—181no—
163Dic6—183no—
164Dic6—183no—
165C3⋊C8—181no—
166C3⋊C8—181no—
167C3⋊C8—183no—
168C3⋊C8—183no—
169C2×C32—22no—
170C2×C32C2×#22611yescharacteristic
171C2×C32#98311yescharacteristic
172C2×C32GL(2,ℤ/4)11yescharacteristic
173C2×C9—43no—
174C2×C9—43no—
175C2×C9—43no—
176C2×C9—86no—
177C2×Q8—21no—
178C2×Q8—31no—
179C2×Q8—61no—
180C2×Q8—271no—
181C2×Q8C2×C3⋊D911yescharacteristic
182C22×C4—31no—
183Q16—541no—
184Q16—542no—
185Q16—541no—
186C2×C8—271no—
187C2×C8—271no—
188C4⋊C4—271no—
189C4⋊C4—541no—
190C22⋊C4—541no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—23no—
194C22×C3—23no—
195C22×C3#368816yescomplemented
196C22×C3C2×(C22⋊C9)⋊C213yes—
197C22×C3C3⋊#22612yes—
198C22×C3C2×C3⋊S411yescharacteristic
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—33no—
202C3×C4—33no—
203C3×C4—61no—
204C3×C4—63no—
205Dic3—361no—
206Dic3—361no—
207Dic3—363no—
208Dic3—363no—
209C32#541911yescharacteristic
210C9—43no—
211C23C3⋊((C22⋊C9)⋊C2)11yescharacteristic
212Q8—31no—
213Q8—32no—
214Q8—61no—
215Q8—541no—
216Q8—541no—
217Q8#623311yescharacteristic
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—541no—
223C2×C4—541no—
224C8—541no—
225C8—541no—
226C2×C3—26no—
227C2×C3—22no—
228C2×C3C2×#368813yes—
229C2×C3#6368913yes—
230C2×C3#6369413yes—
231C2×C3C2×C3⋊#22611yescharacteristic
232C2×C3#6426411yescharacteristic
233C2×C3#6426911yescharacteristic
234C22—21no—
235C22—21no—
236C22C3⋊#368812yescomplemented
237C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
238C4—31no—
239C4—31no—
240C4—61no—
241C4—1081no—
242C4—1081no—
243C3#1058952113yescomplemented
244C3#1059365311yescharacteristic
245C2—22no—
246C2C2×C3⋊#368811yescharacteristic
247C2#1172238511yescharacteristic
248C2#1172239611yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocleC2×C3×Q8⋊C9C22
FrattiniC22×C3C2×C3⋊S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC23×C32S4
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
2. C2×C3⋊#3688C2semidirect
3. C3⋊#63421C2semidirect
195. C22×C3#3688semidirect
236. C22C3⋊#3688semidirect
243. C3#10589521semidirect

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⋊#36882
C3⋊#634212
#105929273
#105895213
#885514