Finite Group Databasebeta

← #13166254

Subgroups — #13166254

Order 1728.

460 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#11724259C212yesmaximal, complemented
2C3⋊D3×D24C211yesmaximal, characteristic
3#11723092C211yesmaximal, characteristic, complemented
4C3×#63768C211yesmaximal, characteristic, complemented
5#11722706C212yesmaximal, complemented
6#10593778—31nomaximal
7C2×D3×#157—22no—
8C3⋊D3×D12C2212yes—
9C4×C3⋊#157C2211yescharacteristic
10#88988C2211yescharacteristic
11C3×C4⋊#157C2212yes—
12(C2×D3)⋊#157—22no—
13#88852C411yescharacteristic, complemented
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×D24C411yescharacteristic, complemented
16#88790C412yescomplemented
17D4×#157—32no—
18D8×C3⋊D3—31no—
19D3×D24—62no—
20#63782—32no—
21#63768D311yescharacteristic, complemented
22#63767—31no—
23C2×D3×C3⋊D3—22no—
24C2×C3⋊#157D412yes—
25C2×C3×#157—22no—
26D3×#157—44no—
27C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
28C3×C4×C3⋊D3C2311yescharacteristic
29C32×D12C2×C412yescomplemented
30#6291—22no—
31C3⋊#622—21no—
32C33×C8C2×C411yescharacteristic, complemented
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic, complemented
34#4869—91nomaximal
35C22×#157—62no—
36D4×C3⋊D3—32no—
37D3×D12—64no—
38C22⋊#157—62no—
39C4⋊#157C2×D312yes—
40C4×#157—31no—
41#3787—31no—
42C32×D8—31no—
43#3753—32no—
44#3745—31no—
45#3745—62no—
46#3745—122no—
47C8×C3⋊D3C2×D311yescharacteristic
48C3×D24—22no—
49C3×D3×C8—62no—
50#3714—64no—
51C2×C3⋊(C3⋊D3)—22no—
52C2×C3×C3⋊D3C2×D411yescharacteristic
53C2×C32×D3—22no—
54D3×C3⋊D3—42no—
55D3×C3⋊D3—42no—
56C3⋊#157—22no—
57C3×#157—42no—
58C33×C4C22×C411yescharacteristic
59C3×C3⋊Dic3#4111yescharacteristic
60D3×D8—182no—
61C2×D24—91no—
62#893—92no—
63C3×#106—91no—
64#815—91no—
65#809—92no—
66C22×C3⋊D3—62no—
67C2×D32—124no—
68C2×#157—22no—
69C2×#157—32no—
70C2×#157—124no—
71C32×D4—32no—
72#638—62no—
73C4⋊(C3⋊D3)—32no—
74C4⋊(C3⋊D3)—64no—
75C4⋊(C3⋊D3)—124no—
76C4×C3⋊D3C22×D311yescharacteristic
77C3×D12—24no—
78C3×C4×D3—62no—
79#626—124no—
80#622—61no—
81C32×C8—22no—
82C32×C8—42no—
83C32×C8C4×D311yescharacteristic, complemented
84C3⋊(C3⋊C8)C4×D311yescharacteristic, complemented
85C3×C3⋊C8—62no—
86#442—271noSylow
87C2×C33C4×D411yescharacteristic
88C3⋊(C3⋊D3)—42no—
89C3×C3⋊D3AGL(1,ℤ/8)12yes—
90C32×D3—42no—
91D3×D4—184no—
92C2×D12—92no—
93C2×C4×D3—182no—
94C3×D8—62no—
95C2×C3×C8—91no—
96C3×C4⋊C4—92no—
97#213—184no—
98(C2×D3)⋊C4—182no—
99C4×Dic3—91no—
100C2×C3⋊C8—91no—
101D24—92no—
102D24—91no—
103D24—182no—
104D24—362no—
105D24#15711yescharacteristic, complemented
106D3×C8—62no—
107C22×C32—62no—
108C2×C3⋊D3—62no—
109C2×C3⋊D3—124no—
110C2×C3⋊D3—122no—
111C2×C3⋊D3—122no—
112C2×C3⋊D3—244no—
113C2×C3⋊D3D3×D411yescharacteristic
114C2×C3×D3—44no—
115C2×C3×D3—62no—
116D32—244no—
117D32—244no—
118#157—42no—
119#157—62no—
120C32×C4—22no—
121C32×C4—42no—
122C32×C4C2×C4×D311yescharacteristic
123C3⋊Dic3#23911yescharacteristic
124C3×Dic3—62no—
125C2×D8—271no—
126C4×D4—272no—
127#106—91no—
128#102—272no—
129#97—271no—
130C33#44211yescharacteristic, complemented, Sylow
131C22×D3—182no—
132C22×D3—364no—
133C3×D4—64no—
134C2×C3×C4—91no—
135C2×C3×C4—182no—
136#67—364no—
137C2×Dic3—92no—
138D12—92no—
139D12—184no—
140D12—182no—
141D12—364no—
142D12C2×#15712yescomplemented
143C4×D3—62no—
144C4×D3—364no—
145C3×C8—22no—
146C3×C8—42no—
147C3×C8—91no—
148C3×C8C2×#15711yescharacteristic, complemented
149C3⋊C8—62no—
150C3⋊C8—181no—
151C2×C32—22no—
152C2×C32—42no—
153C2×C32—122no—
154C2×C32#89311yescharacteristic
155C3⋊D3—122no—
156C3⋊D3—244no—
157C3⋊D3—484no—
158C3⋊D3#91412yes—
159C3×D3—64no—
160C3×D3—84no—
161C2×D4—272no—
162C22×C4—542no—
163D8—31no—
164D8—272no—
165D8—271no—
166C2×C8—91no—
167C2×C8—271no—
168C4⋊C4—92no—
169C22⋊C4—542no—
170C42—271no—
171C22×C3—91no—
172C22×C3—124no—
173C2×D3—22no—
174C2×D3—62no—
175C2×D3—182no—
176C2×D3—364no—
177C2×D3—362no—
178C2×D3—362no—
179C2×D3—724no—
180C2×D3—724no—
181C2×D3—724no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—91no—
185C3×C4—362no—
186C3×C4C22×#15711yescharacteristic
187Dic3—62no—
188Dic3—182no—
189C32—22no—
190C32—42no—
191C32#486911yescharacteristic, complemented
192C23—542no—
193D4—32no—
194D4—272no—
195D4—542no—
196C2×C4—91no—
197C2×C4—182no—
198C2×C4—272no—
199C2×C4—1084no—
200C8—91no—
201C8—541no—
202C8D3×#15711yescharacteristic, complemented
203C2×C3—22no—
204C2×C3—42no—
205C2×C3—92no—
206C2×C3—244no—
207C2×C3D4×#15711yescharacteristic
208D3—42no—
209D3—64no—
210D3—362no—
211D3—724no—
212D3—1444no—
213C22—62no—
214C22—91no—
215C22—542no—
216C22—1082no—
217C22—1082no—
218C4—91no—
219C4—362no—
220C4—542no—
221C4C2×D3×#15711yescharacteristic
222C3—22no—
223C3—42no—
224C3#1059377811yescharacteristic, complemented
225C2—92no—
226C2—122no—
227C2—1082no—
228C2#1172425911yescharacteristic
subgroupG/H
centerC2#11724259
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radicalC3⋊D3×D24C2
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724259C2semidirect
3. #11723092C2semidirect
4. C3×#63768C2semidirect
5. #11722706C2semidirect
13. #88852C4semidirect
15. C32×D24C4semidirect
16. #88790C4semidirect
21. #63768D3semidirect
27. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
29. C32×D12C2×C4semidirect
32. C33×C8C2×C4semidirect
33. C3×C3⋊(C3⋊C8)C2×C4semidirect
83. C32×C8C4×D3semidirect
84. C3⋊(C3⋊C8)C4×D3semidirect
105. D24#157semidirect
130. C33#442semidirect
142. D12C2×#157semidirect
148. C3×C8C2×#157semidirect
191. C32#4869semidirect
202. C8D3×#157semidirect
224. C3#10593778semidirect

Maximal subgroups

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

subgroupindex
#117242592
C3⋊D3×D242
#117230922
C3×#637682
#117227062
#105937783
#48699