Finite Group Databasebeta

← C3×#10593278 #13166648

Subgroups — C3×#10593278 #13166648

Order 1728.

356 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 58 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×#63802C211yesmaximal, characteristic, complemented
2C3×#63797C211yesmaximal, characteristic
3C3×#63764C211yesmaximal, characteristic, complemented
4C3×#63735C214yesmaximal, complemented
5#10593278C311yesmaximal, characteristic, direct product factor
6C3×#3802—21no—
7C3×#3797—21no—
8C3×C8×C3⋊D3C2211yescharacteristic
9C32×Dic12—21no—
10C32×D24—21no—
11C3×#3718C2212yescomplemented
12C3×#3713C2212yes—
13C3×#3708C2212yescomplemented
14C3×D3×Q16—61no—
15C3×D3×D8—61no—
16#63802C2×C311yescharacteristic, complemented
17#63797C2×C311yescharacteristic
18#63764C2×C311yescharacteristic, complemented
19#63735C2×C314yescomplemented
20C2×C3×D32—41no—
21C3×C4×C3⋊D3D411yescharacteristic
22C32×D12—22no—
23C32×Dic6—22no—
24C3×#627—41no—
25C3×#625—41no—
26C3×#624—41no—
27C33×C8D411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29C2×C3×SD32—91nomaximal
30C3×D3×Q8—62no—
31C3×D3×D4—62no—
32#3802—21no—
33#3797—21no—
34C32×Q16—61no—
35C32×D8—61no—
36C8×C3⋊D3C22×C311yescharacteristic
37C3×#216—62no—
38C3×#213—62no—
39C3×Dic12—21no—
40C3×Dic12—21no—
41C3×Dic12—41no—
42C3×D24—21no—
43C3×D24—21no—
44C3×D24—41no—
45C3×D3×C8—61no—
46C3×D3×C8—61no—
47#3718C22×C312yescomplemented
48#3713C22×C312yes—
49#3708C22×C312yescomplemented
50C2×C3×C3⋊D3D811yescharacteristic
51C2×C32×D3—42no—
52C3×D32—82no—
53C33×C4C2×D411yescharacteristic
54C3×C3⋊Dic3D811yescharacteristic
55C32×Dic3—42no—
56C2×C3×Q16—91no—
57C2×C3×D8—91no—
58D3×Q16—61no—
59D3×D8—61no—
60C3×SD32—94no—
61C2×C3×C16—91no—
62C22×C3×D3—122no—
63C2×D32—41no—
64C32×Q8—62no—
65C32×D4—62no—
66C4×C3⋊D3C3×D411yescharacteristic
67C3×#67—122no—
68C3×D12—22no—
69C3×D12—22no—
70C3×D12—42no—
71C3×C4×D3—61no—
72C3×C4×D3—61no—
73C3×C4×D3—122no—
74C3×Dic6—22no—
75C3×Dic6—22no—
76C3×Dic6—42no—
77C3×Dic6—122no—
78#627—41no—
79#625—41no—
80#624—41no—
81C32×C8—21no—
82C32×C8—21no—
83C32×C8—41no—
84C32×C8—41no—
85C32×C8C3×D411yescharacteristic
86C3⋊(C3⋊C8)C23×C311yescharacteristic
87C3×C3⋊C8—61no—
88C3×C3⋊C8—61no—
89C2×SD32—91noSylow
90C2×C33C2×D811yescharacteristic
91C3×C3⋊D3SD3212yescomplemented
92C32×D3—82no—
93C2×C3×Q8—181no—
94C2×C3×D4—181no—
95D3×Q8—62no—
96D3×D4—62no—
97C3×Q16—61no—
98C3×Q16—61no—
99C3×Q16—92no—
100C3×Q16—121no—
101C3×D8—61no—
102C3×D8—61no—
103C3×D8—92no—
104C3×D8—121no—
105C2×C3×C8—91no—
106#216—62no—
107#213—62no—
108Dic12—21no—
109D24—21no—
110D3×C8—61no—
111D3×C8—61no—
112C3×C16—92no—
113C22×C32—122no—
114C2×C3⋊D3C3×D811yescharacteristic
115C2×C3×D3—42no—
116C2×C3×D3—42no—
117C2×C3×D3—61no—
118C2×C3×D3—61no—
119C2×C3×D3—82no—
120C2×C3×D3—244no—
121D32—82no—
122C32×C4—21no—
123C32×C4—21no—
124C32×C4—41no—
125C32×C4—41no—
126C32×C4—122no—
127C32×C4C2×C3×D411yescharacteristic
128C3⋊Dic3C3×D811yescharacteristic
129C3×Dic3—42no—
130C3×Dic3—42no—
131C3×Dic3—61no—
132C3×Dic3—61no—
133C3×Dic3—82no—
134C2×Q16—91no—
135C2×D8—91no—
136SD32—94no—
137C2×C16—91no—
138C33C2×SD3211yescharacteristic, complemented, Sylow
139C23×C3—361no—
140C22×D3—122no—
141C3×Q8—62no—
142C3×Q8—62no—
143C3×Q8—122no—
144C3×Q8—361no—
145C3×D4—62no—
146C3×D4—62no—
147C3×D4—122no—
148C3×D4—361no—
149C2×C3×C4—91no—
150C2×C3×C4—361no—
151#67—122no—
152D12—22no—
153C4×D3—61no—
154C4×D3—61no—
155C4×D3—122no—
156Dic6—22no—
157Dic6—122no—
158C3×C8—21no—
159C3×C8—21no—
160C3×C8—41no—
161C3×C8—41no—
162C3×C8—91no—
163C3×C8GO+(4,𝔽2)11yescharacteristic
164C3⋊C8—61no—
165C3⋊C8—61no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—41no—
169C2×C32—41no—
170C2×C32—242no—
171C2×C32C2×C3×D811yescharacteristic
172C3⋊D3C3×SD3212yescomplemented
173C3×D3—62no—
174C3×D3—62no—
175C3×D3—82no—
176C3×D3—82no—
177C3×D3—162no—
178C2×Q8—181no—
179C2×D4—181no—
180Q16—61no—
181Q16—92no—
182D8—61no—
183D8—92no—
184C2×C8—91no—
185C16—92no—
186C22×C3—91no—
187C22×C3—122no—
188C22×C3—122no—
189C22×C3—242no—
190C22×C3—722no—
191C2×D3—42no—
192C2×D3—61no—
193C2×D3—61no—
194C2×D3—244no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—41no—
198C3×C4—41no—
199C3×C4—91no—
200C3×C4—122no—
201C3×C4—122no—
202C3×C4—242no—
203C3×C4C2×GO+(4,𝔽2)11yescharacteristic
204Dic3—42no—
205Dic3—61no—
206Dic3—61no—
207C32—21no—
208C32—21no—
209C32—41no—
210C32—41no—
211C32C2×C3×SD3211yescharacteristic, complemented
212C23—361no—
213Q8—62no—
214Q8—361no—
215D4—62no—
216D4—361no—
217C2×C4—91no—
218C2×C4—361no—
219C8—91no—
220C8C3×GO+(4,𝔽2)11yescharacteristic
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—92no—
226C2×C3—242no—
227C2×C3—242no—
228C2×C3—482no—
229C2×C3#6422511yescharacteristic
230D3—62no—
231D3—62no—
232D3—82no—
233C22—91no—
234C22—122no—
235C22—722no—
236C4—91no—
237C4—122no—
238C4C2×C3×GO+(4,𝔽2)11yescharacteristic
239C3—21no—
240C3—21no—
241C3—41no—
242C3—41no—
243C3#1059327811yescharacteristic, direct product factor
244C2—92no—
245C2—242no—
246C2C3×#6422511yescharacteristic
subgroupG/H
centerC2×C3#64225
commutator, cosocleC3⋊(C3⋊C8)C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33C2×D8
prime radicalC3×#63797C2
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C2×C3×SD32
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ 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×#63802C2semidirect
3. C3×#63764C2semidirect
4. C3×#63735C2semidirect
5. #10593278C3direct (N × H)
11. C3×#3718C22semidirect
13. C3×#3708C22semidirect
16. #63802C2×C3semidirect
18. #63764C2×C3semidirect
19. #63735C2×C3semidirect
47. #3718C22×C3semidirect
49. #3708C22×C3semidirect
91. C3×C3⋊D3SD32semidirect
138. C33C2×SD32semidirect
172. C3⋊D3C3×SD32semidirect
211. C32C2×C3×SD32semidirect
243. C3#10593278direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638022
C3×#637972
C3×#637642
C3×#637352
#105932783
C2×C3×SD329