Finite Group Databasebeta

← #13180423

Subgroups — #13180423

Order 1728.

298 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 49 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×#3777C211yesmaximal, characteristic, complemented
2#11724355C211yesmaximal, characteristic
3#11724243C211yesmaximal, characteristic, complemented
4#11724103C212yesmaximal, complemented
5#11724101C212yesmaximal, complemented
6#10596412—31nomaximal
7C22×C3×#157—21no—
8C22×C3×#157C2211yescharacteristic
9C2×#157⋊D3C2211yescharacteristic
10C2×C3⋊AGL(1,𝔽9)C2212yes—
11#89049—21no—
12C3×#3777—21no—
13C3×#3777C2212yes—
14#88940C2211yescharacteristic
15#88939—21no—
16C2×#3777D311yescharacteristic, complemented
17#64242—31no—
18C22⋊AGL(1,𝔽9)—31no—
19AGL(1,𝔽9)⋊C4—32no—
20GO+(4,𝔽2)⋊C4—32no—
21C22×C3×C3⋊D3D411yescharacteristic
22C2×#1113—21no—
23C2×C3×#157—22no—
24C2×C3×#157—22no—
25C2×C3×#157—42no—
26C2×C3×#157D411yescharacteristic
27C2×C3×#157C2311yescharacteristic
28#157⋊D3—22no—
29C3⋊AGL(1,𝔽9)—22no—
30C3×#638—41no—
31#6295—41no—
32#6294—21no—
33#4968—91nomaximal
34C22×#157—21no—
35C22×#157C2×D311yescharacteristic
36C2×GO+(4,𝔽2)—31no—
37C2×AGL(1,𝔽9)—32no—
38#3811—61no—
39D3×#67—121no—
40#3777—21no—
41#3777C2×D312yes—
42#3773—31no—
43D32⋊C4—61no—
44C22×C33SD1611yescharacteristic
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3—41no—
48C2×C3×C3⋊D3SD1611yescharacteristic
49C2×C3×C3⋊D3C2×D411yescharacteristic
50#1113—41no—
51#1113—41no—
52C3×#157—21no—
53C3×#157—42no—
54C3×#157#4112yes—
55C3×C3⋊Dic3—41no—
56C2×C3×C4⋊C4—91no—
57#930—91no—
58#830—91no—
59#809—92no—
60#808—92no—
61C22×C3⋊D3#6711yescharacteristic
62C22×C3×D3—121no—
63C2×D32—61no—
64C2×D32—121no—
65C2×#157—22no—
66C2×#157—22no—
67C2×#157—42no—
68C2×#157#6711yescharacteristic
69C2×#157C22×D311yescharacteristic
70GO+(4,𝔽2)—62no—
71AGL(1,𝔽9)—62no—
72#638—41no—
73C3×#67—121no—
74C3×#67—121no—
75C3×#67—241no—
76#626—121no—
77#626—121no—
78#625—121no—
79#624—61no—
80D3×Dic3—121no—
81#481—271noSylow
82C2×C33—21no—
83C2×C33C2×SD1611yescharacteristic
84C3×C3⋊D3—21no—
85C3×C3⋊D3—81no—
86C3×C3⋊D3#12311yescharacteristic
87C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
88C22×C3×C4—91no—
89C22×C3×C4—181no—
90C2×#67—181no—
91D3×D4—361no—
92C2×D12—91no—
93C3×C4⋊C4—92no—
94C3×C4⋊C4—181no—
95(C2×D3)⋊C4—181no—
96C4⋊Dic3—91no—
97C2×C3⋊C8—92no—
98C22×C32—41no—
99C22×C32—81no—
100C22×C32#21511yescharacteristic
101C2×C3⋊D3—21no—
102C2×C3⋊D3—21no—
103C2×C3⋊D3—41no—
104C2×C3⋊D3#21511yescharacteristic
105C2×C3⋊D3C2×#6711yescharacteristic
106C2×C3×D3—121no—
107C2×C3×D3—121no—
108C2×C3×D3—121no—
109C2×C3×D3—121no—
110C2×C3×D3—241no—
111C2×C3×D3—241no—
112C2×C3×D3—241no—
113D32—121no—
114D32—121no—
115D32—241no—
116D32—241no—
117#157—21no—
118#157—42no—
119#157#23512yes—
120C3⋊Dic3—41no—
121C3×Dic3—121no—
122C3×Dic3—121no—
123C3×Dic3—241no—
124#121—271no—
125C2×C4⋊C4—91no—
126#106—272no—
127#102—272no—
128C22⋊C8—271no—
129C33#48111yescharacteristic, complemented, Sylow
130C23×C3—91no—
131C22×D3—121no—
132C22×D3—181no—
133C22×D3—361no—
134C3×D4—361no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—182no—
138C2×C3×C4—182no—
139C2×C3×C4—362no—
140#67—121no—
141#67—121no—
142#67—241no—
143#67—361no—
144#67—361no—
145C2×Dic3—181no—
146D12—182no—
147D12—361no—
148C4×D3—361no—
149C3⋊C8—182no—
150C2×C32—21no—
151C2×C32—41no—
152C2×C32—81no—
153C2×C32—81no—
154C2×C32—161no—
155C2×C32C2×#21511yescharacteristic
156C3⋊D3—21no—
157C3⋊D3—81no—
158C3⋊D3#92911yescharacteristic
159C3⋊D3#93211yescharacteristic
160C3×D3—121no—
161C3×D3—121no—
162C3×D3—241no—
163C3×D3—241no—
164C3×D3—241no—
165C3×D3—481no—
166C2×D4—271no—
167C2×D4—541no—
168C22×C4—91no—
169C22×C4—181no—
170C2×C8—272no—
171C4⋊C4—92no—
172C4⋊C4—181no—
173C4⋊C4—271no—
174C22⋊C4—541no—
175C22×C3—41no—
176C22×C3—81no—
177C22×C3—91no—
178C22×C3—91no—
179C22×C3—181no—
180C22×C3—181no—
181C22×C3—361no—
182C22×C3AΓL(1,𝔽9)11yescharacteristic
183C2×D3—121no—
184C2×D3—121no—
185C2×D3—121no—
186C2×D3—121no—
187C2×D3—241no—
188C2×D3—241no—
189C2×D3—241no—
190C2×D3—361no—
191C2×D3—361no—
192C2×D3—721no—
193C2×D3—721no—
194C3×C4—92no—
195C3×C4—181no—
196C3×C4—361no—
197C3×C4—362no—
198Dic3—121no—
199Dic3—121no—
200Dic3—241no—
201C32—41no—
202C32—81no—
203C32#496811yescharacteristic, complemented
204C23—91no—
205C23—541no—
206D4—361no—
207D4—542no—
208D4—1081no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—182no—
212C2×C4—182no—
213C2×C4—362no—
214C2×C4—541no—
215C8—542no—
216C2×C3—21no—
217C2×C3—41no—
218C2×C3—81no—
219C2×C3—81no—
220C2×C3—91no—
221C2×C3—91no—
222C2×C3—161no—
223C2×C3—181no—
224C2×C3—721no—
225C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
226D3—121no—
227D3—121no—
228D3—241no—
229D3—241no—
230D3—241no—
231D3—481no—
232C22—91no—
233C22—91no—
234C22—181no—
235C22—181no—
236C22—361no—
237C22—1081no—
238C22—1081no—
239C22#8909211yescharacteristic
240C4—92no—
241C4—181no—
242C4—362no—
243C4—361no—
244C3—41no—
245C3—81no—
246C3#1059641211yescharacteristic, complemented
247C2—21no—
248C2—91no—
249C2—91no—
250C2—181no—
251C2—721no—
252C2C2×#8909211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#89092
commutator, cosocleC2×C3×#157C23
FittingC22×C33SD16
socleC2×C33C2×SD16
prime radical#11724355C2
hypercenterC22#89092
hypocenterC33#481
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3777C2semidirect
3. #11724243C2semidirect
4. #11724103C2semidirect
5. #11724101C2semidirect
16. C2×#3777D3semidirect
129. C33#481semidirect
203. C32#4968semidirect
246. C3#10596412semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37772
#117243552
#117242432
#117241032
#117241012
#105964123
#49689