Finite Group Databasebeta

← #13161497

Subgroups — #13161497

Order 1728.

530 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 119 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×#64079C212yesmaximal
2C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3#11722080C212yesmaximal
4#11722079C212yesmaximal
5#10590314—31nomaximal
6#10590304—32nomaximal
7C33×C42C2211yescharacteristic
8C3×C4⋊(C3⋊Dic3)C2214yes—
9C3×#3750—24no—
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12#64079D312yes—
13C3×#870—32no—
14C3×#870—32no—
15C3×#870—62no—
16C4×C3⋊(C3⋊C8)D311yescharacteristic
17C3×C4×C3⋊C8—32no—
18C3×C4×C3⋊C8—61no—
19#63576—32no—
20#63575—34no—
21#63574—32no—
22#63573—34no—
23C2×C33×C4D412yes—
24C2×C33×C4C2311yescharacteristic
25C2×C3×C3⋊Dic3—24no—
26C3×C3⋊(C3⋊C8)Q812yes—
27C3×C3⋊(C3⋊C8)Q812yes—
28#5065—92no—
29#4683—91no—
30C32×C42—21no—
31C32×C42—22no—
32C32×C42—41no—
33C32×C42C2×D311yescharacteristic
34C32×C42C2×D312yes—
35C4⋊(C3⋊Dic3)C2×D314yes—
36#3750—24no—
37C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
38C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
39C3×C4⋊Dic3—34no—
40C3×C4⋊Dic3—34no—
41C3×C4⋊Dic3—64no—
42C3×Dic3⋊C4—68no—
43C3×Dic3⋊C4—64no—
44C3×Dic3⋊C4—64no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—32no—
47C2×C3×C3⋊C8—61no—
48C2×C3×C3⋊C8—61no—
49C22×C33C2×D411yescharacteristic
50C33×C4—22no—
51C33×C4C2×Q811yescharacteristic
52C33×C4C2×Q811yescharacteristic
53C3×C3⋊Dic3—44no—
54C3×#125—92no—
55#870—32no—
56#870—32no—
57#870—62no—
58C3×C4×C8—91no—
59#818—92no—
60#817—92no—
61#808—94no—
62#807—94no—
63C4×C3⋊C8—32no—
64C4×C3⋊C8—61no—
65C2×C32×C4—22no—
66C2×C32×C4—24no—
67C2×C32×C4—21no—
68C2×C32×C4—22no—
69C2×C32×C4—42no—
70C2×C32×C4—41no—
71C2×C32×C4D1212yes—
72C2×C32×C4#6714yes—
73C2×C32×C4C22×D311yescharacteristic
74C2×C32×C4C22×D312yes—
75C2×C3⋊Dic3—24no—
76C2×C3×Dic3—68no—
77C2×C3×Dic3—64no—
78C2×C3×Dic3—64no—
79C3⋊(C3⋊C8)Dic612yes—
80C3⋊(C3⋊C8)Dic612yes—
81C3×C3⋊C8—34no—
82C3×C3⋊C8—34no—
83C3×C3⋊C8—62no—
84C3×C3⋊C8—62no—
85#499—271noSylow
86C2×C33C4⋊Q811yescharacteristic
87C2×C33#13512yes—
88C2×C3×C8—91no—
89C2×C3×C8—91no—
90C3×C4⋊C4—94no—
91C3×C4⋊C4—184no—
92C3×C42—21no—
93C3×C42—22no—
94C3×C42—41no—
95C3×C42D3211yescharacteristic
96C3×C42D3212yes—
97C4⋊Dic3—34no—
98C4⋊Dic3—34no—
99C4⋊Dic3—64no—
100Dic3⋊C4—68no—
101Dic3⋊C4—64no—
102Dic3⋊C4—64no—
103C2×C3⋊C8—32no—
104C2×C3⋊C8—32no—
105C2×C3⋊C8—61no—
106C2×C3⋊C8—61no—
107C22×C32—21no—
108C22×C32—22no—
109C22×C32—41no—
110C22×C32C2×D1211yescharacteristic
111C22×C32C2×#6712yes—
112C32×C4—22no—
113C32×C4—24no—
114C32×C4—24no—
115C32×C4—22no—
116C32×C4—22no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—41no—
120C32×C4—41no—
121C32×C4—44no—
122C32×C4—44no—
123C32×C4C2×Dic611yescharacteristic
124C32×C4C2×Dic611yescharacteristic
125C32×C4D3×Q812yes—
126C32×C4D3×Q812yes—
127C3⋊Dic3—44no—
128C3×Dic3—128no—
129C3×Dic3—124no—
130C3×Dic3—124no—
131#125—92no—
132#107—272no—
133#106—272no—
134C4×C8—91no—
135C33#49911yescharacteristic, complemented, Sylow
136C2×C3×C4—22no—
137C2×C3×C4—24no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—42no—
141C2×C3×C4—41no—
142C2×C3×C4—184no—
143C2×C3×C4#62512yes—
144C2×C3×C4#62614yes—
145C2×C3×C4C2×D3212yes—
146C2×C3×C4C2×D3211yescharacteristic
147C2×Dic3—68no—
148C2×Dic3—64no—
149C2×Dic3—64no—
150C3×C8—92no—
151C3×C8—92no—
152C3⋊C8—34no—
153C3⋊C8—34no—
154C3⋊C8—62no—
155C3⋊C8—62no—
156C2×C32—22no—
157C2×C32—24no—
158C2×C32—22no—
159C2×C32—21no—
160C2×C32—41no—
161C2×C32—42no—
162C2×C32#86911yescharacteristic
163C2×C32#90412yes—
164C2×C32#94412yes—
165C2×C32#95014yes—
166C2×C8—91no—
167C2×C8—91no—
168C4⋊C4—94no—
169C4⋊C4—184no—
170C42#111311yescharacteristic
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—41no—
174C22×C3C2×#62511yescharacteristic
175C22×C3C2×#62612yes—
176C3×C4—24no—
177C3×C4—22no—
178C3×C4—24no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—44no—
184C3×C4—44no—
185C3×C4—41no—
186C3×C4—41no—
187C3×C4—364no—
188C3×C4D3×Dic612yes—
189C3×C4D3×Dic612yes—
190C3×C4#379711yescharacteristic
191C3×C4#379711yescharacteristic
192Dic3—128no—
193Dic3—124no—
194Dic3—124no—
195C32—22no—
196C32—21no—
197C32—41no—
198C32#468311yescharacteristic, complemented
199C32#506512yescomplemented
200C2×C4—184no—
201C2×C4#629512yes—
202C2×C4C2×#111311yescharacteristic
203C8—92no—
204C8—92no—
205C2×C3—24no—
206C2×C3—22no—
207C2×C3—21no—
208C2×C3—22no—
209C2×C3—42no—
210C2×C3—41no—
211C2×C3#6382212yes—
212C2×C3#6382914yes—
213C2×C3#6386612yes—
214C2×C3#6387311yescharacteristic
215C22C2×#629511yescharacteristic
216C4—22no—
217C4—364no—
218C4#8904011yescharacteristic
219C4#8904011yescharacteristic
220C3—21no—
221C3—22no—
222C3—41no—
223C3#1059030412yescomplemented
224C3#1059031411yescharacteristic, complemented
225C2#1172366012yes—
226C2#1172367811yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42#1113
hypocenterC33#499
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
135. C33#499semidirect
198. C32#4683semidirect
199. C32#5065semidirect
223. C3#10590304semidirect
224. C3#10590314semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C3×C4×C3⋊(C3⋊C8)2
#117220802
#117220792
#105903143
#105903043