Finite Group Databasebeta

← #13158903

Subgroups — #13158903

Order 1728.

700 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C3×#64078C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic
3#11722173C211yesmaximal, characteristic
4#11722056C212yesmaximal
5#11722053C212yesmaximal
6#10592763—31nomaximal
7#10590315—34nomaximal
8C2×C3×#634—21no—
9C33×C42C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C32×C4⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2211yescharacteristic
15C32×Dic3⋊C4—21no—
16C32×Dic3⋊C4—21no—
17C32×#125—31no—
18#64078D311yescharacteristic
19C3×#870D314yes—
20C3×#869—34no—
21#63635—32no—
22#63634—32no—
23#63630—31no—
24#63630—34no—
25#63630—64no—
26#63576—38no—
27#63574—38no—
28C2×C33×C4D411yescharacteristic
29C2×C33×C4D411yescharacteristic
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C3×#634—42no—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C3⋊(C3⋊(C3⋊C8))Q814yescomplemented
36#5081—91no—
37#5064—94no—
38C2×#634—21no—
39C2×C3×Dic6—64no—
40C32×C4⋊C4—31no—
41C32×C4⋊C4—31no—
42C32×C4⋊C4—61no—
43C32×C4⋊C4—61no—
44C32×C42—24no—
45C32×C42C2×D314yes—
46C32×C42C2×D311yescharacteristic
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)C2×D312yes—
49C2×C3⋊(C3⋊C8)—32no—
50C2×C3⋊(C3⋊C8)—38no—
51C2×C3⋊(C3⋊C8)—68no—
52C3×C4⋊Dic3—38no—
53C3×C4⋊Dic3—64no—
54C3×C4⋊Dic3C2×D314yes—
55C3×C4⋊Dic3C2×D314yes—
56C3×Dic3⋊C4—24no—
57C3×Dic3⋊C4—24no—
58C22×C33C2×D411yescharacteristic
59C33×C4—21no—
60C33×C4—21no—
61C33×C4C2×Q812yes—
62C3×C3⋊Dic3—42no—
63C32×Dic3—41no—
64C32×Dic3—41no—
65C3×C4⋊Q8—91no—
66C3×#125—34no—
67#870C3⋊D311yescharacteristic
68#869—34no—
69#808—98no—
70#808—92no—
71#807—98no—
72#807—92no—
73#803—94no—
74#803—91no—
75#803—184no—
76C2×C32×C4—24no—
77C2×C32×C4—24no—
78C2×C32×C4—24no—
79C2×C32×C4—61no—
80C2×C32×C4—61no—
81C2×C32×C4#6714yes—
82C2×C32×C4#6714yes—
83C2×C32×C4#6711yescharacteristic
84C2×C32×C4#6711yescharacteristic
85C2×C32×C4C22×D314yes—
86C2×C32×C4C22×D311yescharacteristic
87C2×C3⋊Dic3—22no—
88#634—42no—
89C2×C3×Dic3—24no—
90C2×C3×Dic3—24no—
91C2×C3×Dic3—68no—
92C3×Dic6—128no—
93C3⋊(C3⋊C8)—34no—
94C3⋊(C3⋊C8)—316no—
95C3⋊(C3⋊C8)—616no—
96#501—271noSylow
97C2×C33C4⋊Q811yescharacteristic
98C2×C33AGL(1,ℤ/8)11yescharacteristic
99C2×C33#13711yescharacteristic
100C2×C3×Q8—181no—
101C2×Dic6—64no—
102C3×C4⋊C4—34no—
103C3×C4⋊C4—34no—
104C3×C4⋊C4—64no—
105C3×C4⋊C4—64no—
106C3×C4⋊C4—92no—
107C3×C4⋊C4—181no—
108C3×C42—24no—
109C3×C42D3214yes—
110C3×C42C2×C3⋊D311yescharacteristic
111C4⋊Dic3—38no—
112C4⋊Dic3—64no—
113C4⋊Dic3C2×C3⋊D311yescharacteristic
114C4⋊Dic3C2×C3⋊D311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C2×C3⋊C8—98no—
118C2×C3⋊C8—92no—
119C2×C3⋊C8—188no—
120C22×C32—24no—
121C22×C32C2×#6714yes—
122C22×C32C2×#6711yescharacteristic
123C32×C4—21no—
124C32×C4—21no—
125C32×C4—28no—
126C32×C4—24no—
127C32×C4—24no—
128C32×C4—44no—
129C32×C4—44no—
130C32×C4—121no—
131C32×C4—121no—
132C32×C4D3×Q818yes—
133C32×C4D3×Q812yes—
134C3⋊Dic3—42no—
135C3×Dic3—44no—
136C3×Dic3—44no—
137C3×Dic3—128no—
138C4⋊Q8—91no—
139#125—31no—
140#107—272no—
141#106—272no—
142#97—271no—
143C33#50111yescharacteristic, complemented, Sylow
144C3×Q8—362no—
145C2×C3×C4—24no—
146C2×C3×C4—24no—
147C2×C3×C4—24no—
148C2×C3×C4—64no—
149C2×C3×C4—64no—
150C2×C3×C4—182no—
151C2×C3×C4#62514yes—
152C2×C3×C4#62514yes—
153C2×C3×C4#63811yescharacteristic
154C2×C3×C4#63811yescharacteristic
155C2×C3×C4C2×D3214yes—
156C2×C3×C4C22×C3⋊D311yescharacteristic
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—68no—
160Dic6—128no—
161C3⋊C8—916no—
162C3⋊C8—94no—
163C3⋊C8—1816no—
164C2×C32—24no—
165C2×C32—24no—
166C2×C32—24no—
167C2×C32#93211yescharacteristic
168C2×C32#94211yescharacteristic
169C2×C32#94414yes—
170C2×C32#94411yescharacteristic
171C2×C32#94914yes—
172C2×C32#95114yes—
173C2×Q8—181no—
174C2×C8—272no—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—92no—
180C4⋊C4—181no—
181C42D3×C3⋊D311yescharacteristic
182C22×C3—24no—
183C22×C3C2×#62514yes—
184C22×C3C2×#63811yescharacteristic
185C3×C4—24no—
186C3×C4—24no—
187C3×C4—21no—
188C3×C4—21no—
189C3×C4—28no—
190C3×C4—44no—
191C3×C4—44no—
192C3×C4—124no—
193C3×C4—124no—
194C3×C4—362no—
195C3×C4#379718yes—
196C3×C4Q8×C3⋊D312yes—
197Dic3—41no—
198Dic3—41no—
199Dic3—128no—
200C32—24no—
201C32#506414yescomplemented
202C32#508111yescharacteristic, complemented
203Q8—362no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—182no—
207C2×C4#629011yescharacteristic
208C2×C4#629011yescharacteristic
209C2×C4C2×D3×C3⋊D311yescharacteristic
210C8—274no—
211C2×C3—24no—
212C2×C3—24no—
213C2×C3—24no—
214C2×C3#6382314yes—
215C2×C3#6382714yes—
216C2×C3#6387314yes—
217C2×C3#6415311yescharacteristic
218C2×C3#6415811yescharacteristic
219C2×C3#6416011yescharacteristic
220C22C2×#629011yescharacteristic
221C4—21no—
222C4—21no—
223C4—121no—
224C4—121no—
225C4—362no—
226C4#8901912yes—
227C3—24no—
228C3#1059031514yescomplemented
229C3#1059276311yescharacteristic, complemented
230C2#1172336511yescharacteristic
231C2#1172337311yescharacteristic
232C2#1172345111yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42D3×C3⋊D3
hypocenterC33#501
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
35. C3⋊(C3⋊(C3⋊C8))Q8semidirect
143. C33#501semidirect
201. C32#5064semidirect
202. C32#5081semidirect
228. C3#10590315semidirect
229. C3#10592763semidirect

Maximal subgroups

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

subgroupindex
C3×#640782
C32×#8702
#117221732
#117220562
#117220532
#105927633
#105903153