Finite Group Databasebeta

← C3×#10593299 #13166668

Subgroups — C3×#10593299 #13166668

Order 1728.

376 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 74 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×#64223C211yesmaximal, characteristic
2C3×#63805C212yesmaximal, complemented
3C3×C8×#157C211yesmaximal, characteristic, complemented
4C3×#63733C211yesmaximal, characteristic, complemented
5C3×#63731C212yesmaximal, complemented
6#10593299C311yesmaximal, characteristic, direct product factor
7C3×#3798C2212yes—
8C3×C4×#157C2211yescharacteristic
9C3×#3787C2211yescharacteristic
10C3×#3776C411yescharacteristic, complemented
11C3×#3775C412yescomplemented
12C3×#3774C411yescharacteristic, complemented
13C3×C8×C3⋊D3C2211yescharacteristic
14C32×ΓL(1,𝔽25)—22no—
15C3×#3712C2212yes—
16C3×D3×C3⋊C8—22no—
17#64223C2×C311yescharacteristic
18C3×#907—62no—
19#63805C2×C312yescomplemented
20C8×#157C2×C311yescharacteristic, complemented
21#63733C2×C311yescharacteristic, complemented
22#63731C2×C312yescomplemented
23C2×C3×#157—21no—
24C3×C4×C3⋊D3C2311yescharacteristic
25C32×C4×D3—22no—
26C3×#627C2×C412yes—
27C3×#626—22no—
28C3×#625C2×C412yes—
29C3×#622C2×C411yescharacteristic
30C3×#622C2×C411yescharacteristic
31C33×C8D411yescharacteristic
32C3×C3⋊(C3⋊C8)D411yescharacteristic
33C32×C3⋊C8—22no—
34C3×#443—91nomaximal
35C3×#235—62no—
36#3798C22×C312yes—
37C4×#157C22×C311yescharacteristic
38#3787C22×C311yescharacteristic
39#3776C3×C411yescharacteristic, complemented
40#3775C3×C412yescomplemented
41#3774C3×C411yescharacteristic, complemented
42C32×M16—62no—
43C8×C3⋊D3C22×C311yescharacteristic
44C2×C3×C3⋊C8—62no—
45C3×ΓL(1,𝔽25)—22no—
46C3×ΓL(1,𝔽25)—22no—
47C3×ΓL(1,𝔽25)—42no—
48C3×ΓL(1,𝔽25)—62no—
49C3×D3×C8—62no—
50C3×D3×C8—62no—
51#3712C22×C312yes—
52D3×C3⋊C8—22no—
53C2×C3×C3⋊D3#4111yescharacteristic
54C2×C32×D3—22no—
55C3×#157—22no—
56C33×C4C2×D411yescharacteristic
57C3×C3⋊Dic3C22×C411yescharacteristic
58C32×Dic3—22no—
59C3×#135—91no—
60C3×#131—92no—
61#907—62no—
62C3×#108—91no—
63C3×#104—92no—
64C3×C4×C8—91no—
65C2×#157—21no—
66C2×C32×C4—62no—
67C4×C3⋊D3C23×C311yescharacteristic
68C3×#67—62no—
69C3×#67—62no—
70C3×D12—62no—
71C3×C4×D3—22no—
72C3×C4×D3—22no—
73C3×C4×D3—42no—
74C3×C4×D3—62no—
75C3×Dic6—62no—
76#627C2×C3×C412yes—
77#626—22no—
78#625C2×C3×C412yes—
79#622C2×C3×C411yescharacteristic
80#622C2×C3×C411yescharacteristic
81C32×C8—22no—
82C32×C8—42no—
83C32×C8—62no—
84C32×C8C3×D411yescharacteristic
85C3⋊(C3⋊C8)C3×D411yescharacteristic
86C3×C3⋊C8—22no—
87C3×C3⋊C8—22no—
88C3×C3⋊C8—42no—
89C3×C3⋊C8—62no—
90#443—91noSylow
91C2×C33C4×D411yescharacteristic
92C3×C3⋊D3—21no—
93C32×D3—42no—
94C3×#41—92no—
95#235—62no—
96C3×Q16—91no—
97C3×SD16—92no—
98C3×D8—91no—
99C3×M16—62no—
100C3×M16—62no—
101C3×M16—92no—
102C3×M16—122no—
103C2×C3×C8—91no—
104C2×C3×C8—91no—
105C2×C3×C8—182no—
106C3×C42—91no—
107C2×C3⋊C8—62no—
108ΓL(1,𝔽25)—22no—
109ΓL(1,𝔽25)—62no—
110D3×C8—62no—
111D3×C8—62no—
112C22×C32—62no—
113C2×C3⋊D3C3×#4111yescharacteristic
114C2×C3×D3—22no—
115C2×C3×D3—22no—
116C2×C3×D3—42no—
117C2×C3×D3—62no—
118#157—22no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—62no—
122C32×C4C2×C3×D411yescharacteristic
123C3⋊Dic3C22×C3×C411yescharacteristic
124C3×Dic3—22no—
125C3×Dic3—22no—
126C3×Dic3—42no—
127C3×Dic3—62no—
128#135—91no—
129#131—92no—
130#108—91no—
131#104—92no—
132C4×C8—91no—
133C33#44311yescharacteristic, complemented, Sylow
134C3×Q8—92no—
135C3×D4—92no—
136C3×D4—182no—
137C2×C3×C4—62no—
138C2×C3×C4—62no—
139C2×C3×C4—91no—
140C2×C3×C4—122no—
141C2×C3×C4—181no—
142#67—62no—
143#67—62no—
144D12—62no—
145C4×D3—22no—
146C4×D3—62no—
147Dic6—62no—
148C3×C8—22no—
149C3×C8—42no—
150C3×C8—62no—
151C3×C8—62no—
152C3×C8—91no—
153C3×C8—91no—
154C3×C8—91no—
155C3×C8—122no—
156C3×C8GO+(4,𝔽2)11yescharacteristic
157C3⋊C8—22no—
158C3⋊C8—62no—
159C2×C32—22no—
160C2×C32—42no—
161C2×C32—122no—
162C2×C32C3×C4×D411yescharacteristic
163C3⋊D3—21no—
164C3×D3—42no—
165C3×D3—42no—
166C3×D3—82no—
167C3×D3—122no—
168#41—92no—
169Q16—91no—
170SD16—92no—
171D8—91no—
172M16—62no—
173M16—92no—
174C2×C8—91no—
175C2×C8—91no—
176C2×C8—182no—
177C42—91no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3—91no—
181C22×C3—122no—
182C2×D3—22no—
183C2×D3—62no—
184C3×C4—22no—
185C3×C4—42no—
186C3×C4—62no—
187C3×C4—62no—
188C3×C4—91no—
189C3×C4—122no—
190C3×C4—182no—
191C3×C4C2×GO+(4,𝔽2)11yescharacteristic
192Dic3—22no—
193Dic3—62no—
194C32—22no—
195C32—42no—
196C32C3×#44311yescharacteristic, complemented
197Q8—92no—
198D4—92no—
199D4—182no—
200C2×C4—62no—
201C2×C4—91no—
202C2×C4—181no—
203C8—62no—
204C8—91no—
205C8—91no—
206C8—91no—
207C8C3×GO+(4,𝔽2)11yescharacteristic
208C2×C3—22no—
209C2×C3—42no—
210C2×C3—122no—
211C2×C3—122no—
212C2×C3—181no—
213C2×C3—242no—
214C2×C3C4×GO+(4,𝔽2)11yescharacteristic
215D3—42no—
216D3—122no—
217C22—62no—
218C22—91no—
219C4—62no—
220C4—91no—
221C4—182no—
222C4C2×C3×GO+(4,𝔽2)11yescharacteristic
223C3—22no—
224C3—42no—
225C3#1059329911yescharacteristic, direct product factor
226C2—122no—
227C2—181no—
228C2C3×C4×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC3×C4C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33C4×D4
prime radicalC3×#64223C2
cosocleC4×C3⋊D3C23×C3
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#443
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63805C2semidirect
3. C3×C8×#157C2semidirect
4. C3×#63733C2semidirect
5. C3×#63731C2semidirect
6. #10593299C3direct (N × H)
10. C3×#3776C4semidirect
11. C3×#3775C4semidirect
12. C3×#3774C4semidirect
19. #63805C2×C3semidirect
20. C8×#157C2×C3semidirect
21. #63733C2×C3semidirect
22. #63731C2×C3semidirect
39. #3776C3×C4semidirect
40. #3775C3×C4semidirect
41. #3774C3×C4semidirect
133. C33#443semidirect
196. C32C3×#443semidirect
225. C3#10593299direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642232
C3×#638052
C3×C8×#1572
C3×#637332
C3×#637312
#105932993
C3×#4439