Finite Group Databasebeta

← C3×#10593290 #13166660

Subgroups — C3×#10593290 #13166660

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×#64227C212yesmaximal, complemented
2C3×#64221C212yesmaximal, complemented
3C3×#63802C211yesmaximal, characteristic, complemented
4C3×#63797C211yesmaximal, characteristic
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#10593290C311yesmaximal, characteristic, direct product factor
7C3×#3802C2212yes—
8C3×#3797C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11C3×#3775—24no—
12C3×D32⋊C4—24no—
13C3×C8×C3⋊D3C2211yescharacteristic
14C32×Dic12—21no—
15C32×D24—21no—
16C3×#3718—21no—
17C3×#3713—21no—
18#64227C2×C312yescomplemented
19#64221C2×C312yescomplemented
20C3×D3×Q16—61no—
21C3×D3×D8—61no—
22#63802C2×C311yescharacteristic, complemented
23#63797C2×C311yescharacteristic
24C8×#157C2×C311yescharacteristic, complemented
25C2×C3×D32—22no—
26C2×C3×#157D412yes—
27C3×C4×C3⋊D3C2311yescharacteristic
28C32×D12—22no—
29C32×Dic6—22no—
30C3×#627—22no—
31C3×#625—22no—
32C3×#624—22no—
33C3×#622D412yes—
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)D411yescharacteristic
36C3×#495—91nomaximal
37C3×D3×Q8—62no—
38C3×D3×D4—62no—
39#3802C22×C312yes—
40#3797C22×C312yes—
41C4×#157C22×C311yescharacteristic
42#3787C22×C311yescharacteristic
43#3775—24no—
44D32⋊C4—24no—
45C32×Q16—61no—
46C32×D8—61no—
47C8×C3⋊D3C22×C311yescharacteristic
48C3×#216—62no—
49C3×#213—62no—
50C3×Dic12—21no—
51C3×Dic12—21no—
52C3×Dic12—41no—
53C3×D24—21no—
54C3×D24—21no—
55C3×D24—41no—
56C3×D3×C8—61no—
57C3×D3×C8—61no—
58#3718—21no—
59#3713—21no—
60C2×C3×C3⋊D3C2×D411yescharacteristic
61C2×C32×D3—42no—
62C3×D32—44no—
63C3×#157—22no—
64C33×C4C2×D411yescharacteristic
65C3×C3⋊Dic3C2×D411yescharacteristic
66C32×Dic3—42no—
67C2×C3×Q16—91no—
68C2×C3×SD16—92no—
69C2×C3×D8—91no—
70C3×#124—92no—
71D3×Q16—61no—
72D3×D8—61no—
73C3×C4×C8—91no—
74C22×C3×D3—122no—
75C2×D32—22no—
76C2×#157C3×D412yes—
77C32×Q8—62no—
78C32×D4—62no—
79C4×C3⋊D3C23×C311yescharacteristic
80C3×#67—122no—
81C3×D12—22no—
82C3×D12—22no—
83C3×D12—42no—
84C3×C4×D3—61no—
85C3×C4×D3—61no—
86C3×C4×D3—122no—
87C3×Dic6—22no—
88C3×Dic6—22no—
89C3×Dic6—42no—
90C3×Dic6—122no—
91#627—22no—
92#625—22no—
93#624—22no—
94#622C3×D412yes—
95C32×C8—21no—
96C32×C8—21no—
97C32×C8—41no—
98C32×C8—41no—
99C32×C8C3×D411yescharacteristic
100C3⋊(C3⋊C8)C3×D411yescharacteristic
101C3×C3⋊C8—61no—
102C3×C3⋊C8—61no—
103#495—91noSylow
104C2×C33#12711yescharacteristic
105C3×C3⋊D3#13512yes—
106C32×D3—82no—
107C2×C3×Q8—92no—
108C2×C3×D4—92no—
109D3×Q8—62no—
110D3×D4—62no—
111C3×Q16—61no—
112C3×Q16—61no—
113C3×Q16—121no—
114C3×Q16—181no—
115C3×SD16—184no—
116C3×D8—61no—
117C3×D8—61no—
118C3×D8—121no—
119C3×D8—181no—
120C2×C3×C8—91no—
121C2×C3×C8—91no—
122C3×C22⋊C4—184no—
123C3×C42—91no—
124#216—62no—
125#213—62no—
126Dic12—21no—
127D24—21no—
128D3×C8—61no—
129D3×C8—61no—
130C22×C32—122no—
131C2×C3⋊D3C2×C3×D411yescharacteristic
132C2×C3×D3—42no—
133C2×C3×D3—42no—
134C2×C3×D3—61no—
135C2×C3×D3—61no—
136C2×C3×D3—82no—
137C2×C3×D3—244no—
138D32—44no—
139#157—22no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4—122no—
145C32×C4C2×C3×D411yescharacteristic
146C3⋊Dic3C2×C3×D411yescharacteristic
147C3×Dic3—42no—
148C3×Dic3—42no—
149C3×Dic3—61no—
150C3×Dic3—61no—
151C3×Dic3—82no—
152C2×Q16—91no—
153C2×SD16—92no—
154C2×D8—91no—
155#124—92no—
156C4×C8—91no—
157C33#49511yescharacteristic, complemented, Sylow
158C23×C3—182no—
159C22×D3—122no—
160C3×Q8—62no—
161C3×Q8—62no—
162C3×Q8—122no—
163C3×Q8—182no—
164C3×D4—62no—
165C3×D4—62no—
166C3×D4—122no—
167C3×D4—182no—
168C2×C3×C4—91no—
169C2×C3×C4—92no—
170C2×C3×C4—182no—
171#67—122no—
172D12—22no—
173C4×D3—61no—
174C4×D3—61no—
175C4×D3—122no—
176Dic6—22no—
177Dic6—122no—
178C3×C8—21no—
179C3×C8—21no—
180C3×C8—41no—
181C3×C8—41no—
182C3×C8—92no—
183C3×C8—91no—
184C3×C8GO+(4,𝔽2)11yescharacteristic
185C3⋊C8—61no—
186C3⋊C8—61no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—242no—
192C2×C32C3×#12711yescharacteristic
193C3⋊D3C3×#13512yes—
194C3×D3—62no—
195C3×D3—62no—
196C3×D3—82no—
197C3×D3—82no—
198C3×D3—162no—
199C2×Q8—92no—
200C2×D4—92no—
201Q16—61no—
202Q16—181no—
203SD16—184no—
204D8—61no—
205D8—181no—
206C2×C8—91no—
207C2×C8—91no—
208C22⋊C4—184no—
209C42—91no—
210C22×C3—91no—
211C22×C3—122no—
212C22×C3—122no—
213C22×C3—242no—
214C22×C3—364no—
215C2×D3—42no—
216C2×D3—61no—
217C2×D3—61no—
218C2×D3—244no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—91no—
224C3×C4—122no—
225C3×C4—122no—
226C3×C4—182no—
227C3×C4—242no—
228C3×C4C2×GO+(4,𝔽2)11yescharacteristic
229Dic3—42no—
230Dic3—61no—
231Dic3—61no—
232C32—21no—
233C32—21no—
234C32—41no—
235C32—41no—
236C32C3×#49511yescharacteristic, complemented
237C23—182no—
238Q8—62no—
239Q8—182no—
240D4—62no—
241D4—182no—
242C2×C4—91no—
243C2×C4—92no—
244C2×C4—182no—
245C8—92no—
246C8—91no—
247C8C3×GO+(4,𝔽2)11yescharacteristic
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—41no—
251C2×C3—41no—
252C2×C3—92no—
253C2×C3—242no—
254C2×C3—242no—
255C2×C3—482no—
256C2×C3#6423111yescharacteristic
257D3—62no—
258D3—62no—
259D3—82no—
260C22—91no—
261C22—122no—
262C22—364no—
263C4—91no—
264C4—122no—
265C4—182no—
266C4C2×C3×GO+(4,𝔽2)11yescharacteristic
267C3—21no—
268C3—21no—
269C3—41no—
270C3—41no—
271C3#1059329011yescharacteristic, direct product factor
272C2—92no—
273C2—242no—
274C2C3×#6423111yescharacteristic
subgroupG/H
centerC2×C3#64231
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#127
prime radicalC3×#63797C2
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×#64227C2semidirect
2. C3×#64221C2semidirect
3. C3×#63802C2semidirect
5. C3×C8×#157C2semidirect
6. #10593290C3direct (N × H)
18. #64227C2×C3semidirect
19. #64221C2×C3semidirect
22. #63802C2×C3semidirect
24. C8×#157C2×C3semidirect
157. C33#495semidirect
236. C32C3×#495semidirect
271. C3#10593290direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642272
C3×#642212
C3×#638022
C3×#637972
C3×C8×#1572
#105932903
C3×#4959