Finite Group Databasebeta

← C3×#10593296 #13166665

Subgroups — C3×#10593296 #13166665

Order 1728.

436 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 66 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×#64228C211yesmaximal, characteristic, complemented
2C3×#64225C211yesmaximal, characteristic
3C3×#64221C212yesmaximal, complemented
4C3×#63796C212yesmaximal, complemented
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#10593296C311yesmaximal, characteristic, direct product factor
7C3×#3802C2212yes—
8C3×#3797C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11C3×#3776—22no—
12C3×#3774—22no—
13C3×D32⋊C4—24no—
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×#204—22no—
16C3×#3715—22no—
17#64228C2×C311yescharacteristic, complemented
18#64225C2×C311yescharacteristic
19#64221C2×C312yescomplemented
20C3×D3×SD16—62no—
21#63796C2×C312yescomplemented
22C8×#157C2×C311yescharacteristic, complemented
23C2×C3×D32—22no—
24C2×C3×#157D412yes—
25C3×C4×C3⋊D3C2311yescharacteristic
26C32×D12—22no—
27C32×Dic6—22no—
28C3×#627—22no—
29C3×#625—22no—
30C3×#624—22no—
31C3×#622D412yes—
32C33×C8D411yescharacteristic
33C3×C3⋊(C3⋊C8)D411yescharacteristic
34C3×#495—91nomaximal
35C3×D3×Q8—62no—
36C3×D3×D4—62no—
37#3802C22×C312yes—
38#3797C22×C312yes—
39C4×#157C22×C311yescharacteristic
40#3787C22×C311yescharacteristic
41#3776—22no—
42#3774—22no—
43D32⋊C4—24no—
44C32×SD16—62no—
45C8×C3⋊D3C22×C311yescharacteristic
46C3×#215—62no—
47C3×#214—62no—
48C3×#204—22no—
49C3×#204—22no—
50C3×#204—42no—
51C3×D3×C8—62no—
52#3715—22no—
53C2×C3×C3⋊D3C2×D411yescharacteristic
54C2×C32×D3—42no—
55C3×D32—44no—
56C3×#157—22no—
57C33×C4C2×D411yescharacteristic
58C3×C3⋊Dic3C2×D411yescharacteristic
59C32×Dic3—42no—
60C2×C3×Q16—91no—
61C2×C3×SD16—92no—
62C2×C3×D8—91no—
63C3×#124—92no—
64D3×SD16—62no—
65C3×C4×C8—91no—
66C22×C3×D3—122no—
67C2×D32—22no—
68C2×#157C3×D412yes—
69C32×Q8—62no—
70C32×D4—62no—
71C4×C3⋊D3C23×C311yescharacteristic
72C3×#67—122no—
73C3×D12—22no—
74C3×D12—22no—
75C3×D12—42no—
76C3×C4×D3—62no—
77C3×C4×D3—122no—
78C3×Dic6—22no—
79C3×Dic6—22no—
80C3×Dic6—42no—
81C3×Dic6—122no—
82#627—22no—
83#625—22no—
84#624—22no—
85#622C3×D412yes—
86C32×C8—22no—
87C32×C8—42no—
88C32×C8C3×D411yescharacteristic
89C3⋊(C3⋊C8)C3×D411yescharacteristic
90C3×C3⋊C8—62no—
91#495—91noSylow
92C2×C33#12711yescharacteristic
93C3×C3⋊D3#13512yes—
94C32×D3—82no—
95C2×C3×Q8—92no—
96C2×C3×D4—92no—
97D3×Q8—62no—
98D3×D4—62no—
99C3×Q16—182no—
100C3×SD16—62no—
101C3×SD16—62no—
102C3×SD16—122no—
103C3×SD16—182no—
104C3×D8—182no—
105C2×C3×C8—91no—
106C2×C3×C8—91no—
107C3×C22⋊C4—184no—
108C3×C42—91no—
109#215—62no—
110#214—62no—
111#204—22no—
112D3×C8—62no—
113C22×C32—122no—
114C2×C3⋊D3C2×C3×D411yescharacteristic
115C2×C3×D3—42no—
116C2×C3×D3—42no—
117C2×C3×D3—62no—
118C2×C3×D3—82no—
119C2×C3×D3—244no—
120D32—44no—
121#157—22no—
122C32×C4—22no—
123C32×C4—42no—
124C32×C4—122no—
125C32×C4C2×C3×D411yescharacteristic
126C3⋊Dic3C2×C3×D411yescharacteristic
127C3×Dic3—42no—
128C3×Dic3—42no—
129C3×Dic3—62no—
130C3×Dic3—82no—
131C2×Q16—91no—
132C2×SD16—92no—
133C2×D8—91no—
134#124—92no—
135C4×C8—91no—
136C33#49511yescharacteristic, complemented, Sylow
137C23×C3—182no—
138C22×D3—122no—
139C3×Q8—62no—
140C3×Q8—62no—
141C3×Q8—122no—
142C3×Q8—182no—
143C3×D4—62no—
144C3×D4—62no—
145C3×D4—122no—
146C3×D4—182no—
147C2×C3×C4—91no—
148C2×C3×C4—92no—
149C2×C3×C4—182no—
150#67—122no—
151D12—22no—
152C4×D3—62no—
153C4×D3—122no—
154Dic6—22no—
155Dic6—122no—
156C3×C8—22no—
157C3×C8—42no—
158C3×C8—92no—
159C3×C8—91no—
160C3×C8GO+(4,𝔽2)11yescharacteristic
161C3⋊C8—62no—
162C2×C32—22no—
163C2×C32—42no—
164C2×C32—242no—
165C2×C32C3×#12711yescharacteristic
166C3⋊D3C3×#13512yes—
167C3×D3—64no—
168C3×D3—82no—
169C3×D3—82no—
170C3×D3—162no—
171C2×Q8—92no—
172C2×D4—92no—
173Q16—182no—
174SD16—62no—
175SD16—182no—
176D8—182no—
177C2×C8—91no—
178C2×C8—91no—
179C22⋊C4—184no—
180C42—91no—
181C22×C3—91no—
182C22×C3—122no—
183C22×C3—122no—
184C22×C3—242no—
185C22×C3—364no—
186C2×D3—42no—
187C2×D3—62no—
188C2×D3—244no—
189C3×C4—22no—
190C3×C4—42no—
191C3×C4—91no—
192C3×C4—122no—
193C3×C4—122no—
194C3×C4—182no—
195C3×C4—242no—
196C3×C4C2×GO+(4,𝔽2)11yescharacteristic
197Dic3—42no—
198Dic3—62no—
199C32—22no—
200C32—42no—
201C32C3×#49511yescharacteristic, complemented
202C23—182no—
203Q8—62no—
204Q8—182no—
205D4—62no—
206D4—182no—
207C2×C4—91no—
208C2×C4—92no—
209C2×C4—182no—
210C8—92no—
211C8—91no—
212C8C3×GO+(4,𝔽2)11yescharacteristic
213C2×C3—22no—
214C2×C3—42no—
215C2×C3—92no—
216C2×C3—242no—
217C2×C3—242no—
218C2×C3—482no—
219C2×C3#6423111yescharacteristic
220D3—64no—
221D3—82no—
222C22—91no—
223C22—122no—
224C22—364no—
225C4—91no—
226C4—122no—
227C4—182no—
228C4C2×C3×GO+(4,𝔽2)11yescharacteristic
229C3—22no—
230C3—42no—
231C3#1059329611yescharacteristic, direct product factor
232C2—92no—
233C2—242no—
234C2C3×#6423111yescharacteristic
subgroupG/H
centerC2×C3#64231
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#127
prime radicalC3×#64225C2
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#495
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#64228C2semidirect
3. C3×#64221C2semidirect
4. C3×#63796C2semidirect
5. C3×C8×#157C2semidirect
6. #10593296C3direct (N × H)
17. #64228C2×C3semidirect
19. #64221C2×C3semidirect
21. #63796C2×C3semidirect
22. C8×#157C2×C3semidirect
136. C33#495semidirect
201. C32C3×#495semidirect
231. C3#10593296direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642282
C3×#642252
C3×#642212
C3×#637962
C3×C8×#1572
#105932963
C3×#4959