Finite Group Databasebeta

← #13173375

Subgroups — #13173375

Order 1728.

460 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 71 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
1D4×C3⋊#157C212yesmaximal, complemented
2C3×D8×C3⋊D3C211yesmaximal, characteristic
3#11723107C212yesmaximal, complemented
4#11723093C211yesmaximal, characteristic, complemented
5#11723092C211yesmaximal, characteristic, complemented
6#10593778—31nomaximal
7C22×C3⋊#157—22no—
8C3×D4×C3⋊D3C2212yes—
9#88994—22no—
10C4⋊(C3⋊#157)C2212yes—
11C4×C3⋊#157C2211yescharacteristic
12#88988C2211yescharacteristic
13C33×D8C411yescharacteristic, complemented
14C3×#3753C412yescomplemented
15C3×#3745C411yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17D4×#157—32no—
18D8×C3⋊D3D311yescharacteristic
19C3×D3×D8—62no—
20#63782—32no—
21#63768—31no—
22#63767—31no—
23C22×C3×C3⋊D3—22no—
24C2×C3⋊#157—22no—
25C2×C3⋊#157—44no—
26C2×C3⋊#157D412yes—
27C33×D4C2×C412yescomplemented
28C3×#638—22no—
29C3×C4⋊(C3⋊D3)C2×C412yescomplemented
30C3×C4×C3⋊D3C2311yescharacteristic
31C3⋊#622—21no—
32C33×C8C2×C411yescharacteristic, complemented
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic, complemented
34#5152—91nomaximal
35C22×#157—62no—
36D4×C3⋊D3C2×D312yes—
37C3×D3×D4—64no—
38C22⋊#157—62no—
39C4⋊#157—32no—
40C4×#157—31no—
41#3787—31no—
42C32×D8—22no—
43C32×D8—42no—
44C32×D8Dic311yescharacteristic, complemented
45#3753Dic312yescomplemented
46#3745Dic311yescharacteristic, complemented
47C8×C3⋊D3C2×D311yescharacteristic
48C3×#213—64no—
49C3×D24—62no—
50C3×D3×C8—62no—
51C22×C33—22no—
52C2×C3×C3⋊D3—22no—
53C2×C3×C3⋊D3—42no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3C2×D411yescharacteristic
56C3⋊#157—22no—
57C3⋊#157—42no—
58C33×C4C22×C411yescharacteristic
59C3×C3⋊Dic3#4111yescharacteristic
60C2×C3×D8—91no—
61D4×Dic3—92no—
62D3×D8—62no—
63#832—92no—
64#817—91no—
65#815—91no—
66C22×C3⋊D3—22no—
67C22×C3×D3—124no—
68C2×#157—32no—
69C2×#157—62no—
70C2×#157—124no—
71C32×D4—24no—
72C32×D4—44no—
73C32×D4C2×Dic312yescomplemented
74#638—22no—
75C4⋊(C3⋊D3)C2×Dic312yescomplemented
76C4×C3⋊D3C22×D311yescharacteristic
77C3×#67—124no—
78C3×D12—64no—
79C3×C4×D3—62no—
80#622—61no—
81C32×C8—22no—
82C32×C8—42no—
83C32×C8C2×Dic311yescharacteristic, complemented
84C3⋊(C3⋊C8)C2×Dic311yescharacteristic, complemented
85C3×C3⋊C8—62no—
86#442—271noSylow
87C2×C33—42no—
88C2×C33C4×D411yescharacteristic
89C3×C3⋊D3—42no—
90C3×C3⋊D3AGL(1,ℤ/8)12yes—
91C2×C3×D4—92no—
92C22×Dic3—182no—
93D3×D4—64no—
94C3×D8—22no—
95C3×D8—42no—
96C3×D8—92no—
97C3×D8—91no—
98C3×D8#15711yescharacteristic, complemented
99C2×C3×C8—91no—
100#217—182no—
101#213—64no—
102C4⋊Dic3—92no—
103C4×Dic3—91no—
104C2×C3⋊C8—91no—
105D24—62no—
106D3×C8—62no—
107C22×C32—22no—
108C22×C32—44no—
109C22×C32—84no—
110C2×C3⋊D3—22no—
111C2×C3⋊D3—42no—
112C2×C3⋊D3—42no—
113C2×C3⋊D3D3×D411yescharacteristic
114C2×C3×D3—62no—
115C2×C3×D3—124no—
116C2×C3×D3—244no—
117C2×C3×D3—244no—
118#157—62no—
119#157—122no—
120C32×C4—22no—
121C32×C4—42no—
122C32×C4C22×Dic311yescharacteristic
123C3⋊Dic3#23711yescharacteristic
124C3×Dic3—62no—
125C2×D8—91no—
126C4×D4—272no—
127#106—271no—
128#102—272no—
129#97—271no—
130C33#44211yescharacteristic, complemented, Sylow
131C23×C3—182no—
132C22×D3—124no—
133C3×D4—24no—
134C3×D4—44no—
135C3×D4—92no—
136C3×D4—182no—
137C3×D4C2×#15712yescomplemented
138C2×C3×C4—91no—
139#67—124no—
140C2×Dic3—92no—
141C2×Dic3—182no—
142C2×Dic3—364no—
143D12—64no—
144C4×D3—62no—
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—42no—
154C2×C32—84no—
155C2×C32—164no—
156C2×C32D4×Dic311yescharacteristic
157C3⋊D3—42no—
158C3⋊D3#91112yes—
159C3×D3—64no—
160C3×D3—244no—
161C2×D4—92no—
162C22×C4—542no—
163D8—92no—
164D8—91no—
165D8C3⋊#15711yescharacteristic, complemented
166C2×C8—91no—
167C2×C8—271no—
168C4⋊C4—272no—
169C22⋊C4—542no—
170C42—271no—
171C22×C3—22no—
172C22×C3—44no—
173C22×C3—84no—
174C22×C3—91no—
175C22×C3—182no—
176C22×C3—362no—
177C22×C3—362no—
178C2×D3—62no—
179C2×D3—124no—
180C2×D3—244no—
181C2×D3—244no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—91no—
185C3×C4C22×#15711yescharacteristic
186Dic3—62no—
187Dic3—182no—
188Dic3—362no—
189C32—22no—
190C32—42no—
191C32#515211yescharacteristic, complemented
192C23—182no—
193D4—92no—
194D4—182no—
195D4C2×C3⋊#15712yescomplemented
196C2×C4—91no—
197C2×C4—272no—
198C2×C4—542no—
199C2×C4—1084no—
200C8—91no—
201C8—541no—
202C8C2×C3⋊#15711yescharacteristic, complemented
203C2×C3—22no—
204C2×C3—42no—
205C2×C3—42no—
206C2×C3—84no—
207C2×C3—92no—
208C2×C3—164no—
209C2×C3—362no—
210C2×C3D4×#15711yescharacteristic
211D3—64no—
212D3—244no—
213C22—22no—
214C22—91no—
215C22—182no—
216C22—362no—
217C22—362no—
218C4—91no—
219C4—542no—
220C4—1082no—
221C4C22×C3⋊#15711yescharacteristic
222C3—22no—
223C3—42no—
224C3#1059377811yescharacteristic, complemented
225C2—42no—
226C2—92no—
227C2—362no—
228C2D4×C3⋊#15711yescharacteristic
subgroupG/H
centerC2D4×C3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC33×D8C4
socleC2×C33C4×D4
prime radicalC3×D8×C3⋊D3C2
cosocleC3×C4×C3⋊D3C23
hypercenterD8C3⋊#157
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊#157C2semidirect
3. #11723107C2semidirect
4. #11723093C2semidirect
5. #11723092C2semidirect
13. C33×D8C4semidirect
14. C3×#3753C4semidirect
15. C3×#3745C4semidirect
27. C33×D4C2×C4semidirect
29. C3×C4⋊(C3⋊D3)C2×C4semidirect
32. C33×C8C2×C4semidirect
33. C3×C3⋊(C3⋊C8)C2×C4semidirect
44. C32×D8Dic3semidirect
45. #3753Dic3semidirect
46. #3745Dic3semidirect
73. C32×D4C2×Dic3semidirect
75. C4⋊(C3⋊D3)C2×Dic3semidirect
83. C32×C8C2×Dic3semidirect
84. C3⋊(C3⋊C8)C2×Dic3semidirect
98. C3×D8#157semidirect
130. C33#442semidirect
137. C3×D4C2×#157semidirect
148. C3×C8C2×#157semidirect
165. D8C3⋊#157semidirect
191. C32#5152semidirect
195. D4C2×C3⋊#157semidirect
202. C8C2×C3⋊#157semidirect
224. C3#10593778semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊#1572
C3×D8×C3⋊D32
#117231072
#117230932
#117230922
#105937783
#51529