Finite Group Databasebeta

← C3×#10593279 #13166647

Subgroups — C3×#10593279 #13166647

Order 1728.

324 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 54 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×#63809C211yesmaximal, characteristic
2C3×#63802C211yesmaximal, characteristic, complemented
3C3×#63765C211yesmaximal, characteristic, complemented
4C3×#63736C212yesmaximal, complemented
5C3×#63735C212yesmaximal, complemented
6#10593279C311yesmaximal, characteristic, direct product factor
7C3×#3797—21no—
8C3×#3796—21no—
9C3×C8×C3⋊D3C2211yescharacteristic
10C32×Dic12—21no—
11C32×#204—21no—
12C3×#3718C2212yes—
13C3×#3718C2211yescharacteristic
14C3×#3713C2211yescharacteristic
15C3×#3708C2212yes—
16C3×D3×Q16—61no—
17C3×#916—61no—
18#63809C2×C311yescharacteristic
19#63802C2×C311yescharacteristic, complemented
20#63765C2×C311yescharacteristic, complemented
21#63736C2×C312yescomplemented
22#63735C2×C312yescomplemented
23C3×C4×C3⋊D3D411yescharacteristic
24C32×D12—21no—
25C32×Dic6—21no—
26C32×Dic6—22no—
27C3×#627—41no—
28C3×#626—41no—
29C3×#624—41no—
30C3×D3×Dic3—41no—
31C33×C8D411yescharacteristic
32C3×C3⋊(C3⋊C8)C2311yescharacteristic
33C3×#510—91nomaximal
34C3×#239—61no—
35C3×D3×Q8—62no—
36C3×#237—61no—
37#3797—21no—
38#3796—21no—
39C32×Q16—61no—
40C32×SD16—61no—
41C8×C3⋊D3C22×C311yescharacteristic
42C3×#216—61no—
43C3×#216—62no—
44C3×#213—61no—
45C3×Dic12—21no—
46C3×Dic12—21no—
47C3×Dic12—41no—
48C3×#204—21no—
49C3×#204—21no—
50C3×#204—41no—
51C3×D3×C8—61no—
52C3×D3×C8—61no—
53#3718C22×C312yes—
54#3718C22×C311yescharacteristic
55#3713C22×C311yescharacteristic
56#3708C22×C312yes—
57C2×C3×C3⋊D3D811yescharacteristic
58C2×C32×D3—41no—
59C33×C4C2×D411yescharacteristic
60C3×C3⋊Dic3D811yescharacteristic
61C32×Dic3—41no—
62C32×Dic3—42no—
63C3×#135—91no—
64C2×C3×Q16—91no—
65D3×Q16—61no—
66#916—61no—
67C3×Q32—92no—
68C3×SD32—92no—
69C3×M32—91no—
70C32×Q8—61no—
71C32×Q8—62no—
72C32×D4—61no—
73C4×C3⋊D3C3×D411yescharacteristic
74C3×#67—121no—
75C2×C3×Dic3—121no—
76C3×D12—21no—
77C3×D12—21no—
78C3×D12—41no—
79C3×D12—121no—
80C3×C4×D3—61no—
81C3×C4×D3—61no—
82C3×C4×D3—122no—
83C3×C4×D3—121no—
84C3×Dic6—21no—
85C3×Dic6—22no—
86C3×Dic6—21no—
87C3×Dic6—22no—
88C3×Dic6—41no—
89C3×Dic6—42no—
90C3×Dic6—122no—
91#627—41no—
92#626—41no—
93#624—41no—
94D3×Dic3—41no—
95C32×C8—21no—
96C32×C8—21no—
97C32×C8—41no—
98C32×C8—41no—
99C32×C8C3×D411yescharacteristic
100C3⋊(C3⋊C8)C23×C311yescharacteristic
101C3×C3⋊C8—61no—
102C3×C3⋊C8—61no—
103#510—91noSylow
104C2×C33C2×D811yescharacteristic
105C3×C3⋊D3—21no—
106C32×D3—81no—
107C3×#41—181no—
108C2×C3×Q8—181no—
109#239—61no—
110D3×Q8—62no—
111#237—61no—
112C3×Q16—61no—
113C3×Q16—61no—
114C3×Q16—92no—
115C3×Q16—91no—
116C3×Q16—121no—
117C3×SD16—61no—
118C3×SD16—61no—
119C3×SD16—121no—
120C3×D8—91no—
121C2×C3×C8—91no—
122#216—61no—
123#216—62no—
124#213—61no—
125Dic12—21no—
126#204—21no—
127D3×C8—61no—
128D3×C8—61no—
129C3×C16—92no—
130C22×C32—121no—
131C2×C3⋊D3C3×D811yescharacteristic
132C2×C3×D3—41no—
133C2×C3×D3—41no—
134C2×C3×D3—61no—
135C2×C3×D3—61no—
136C2×C3×D3—81no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4—41no—
140C32×C4—41no—
141C32×C4—122no—
142C32×C4—121no—
143C32×C4C2×C3×D411yescharacteristic
144C3⋊Dic3C3×D811yescharacteristic
145C3×Dic3—42no—
146C3×Dic3—41no—
147C3×Dic3—41no—
148C3×Dic3—42no—
149C3×Dic3—61no—
150C3×Dic3—61no—
151C3×Dic3—82no—
152C3×Dic3—81no—
153#135—91no—
154C2×Q16—91no—
155Q32—92no—
156SD32—92no—
157M32—91no—
158C33#51011yescharacteristic, complemented, Sylow
159C3×Q8—61no—
160C3×Q8—62no—
161C3×Q8—61no—
162C3×Q8—62no—
163C3×Q8—121no—
164C3×Q8—122no—
165C3×Q8—361no—
166C3×D4—61no—
167C3×D4—61no—
168C3×D4—121no—
169C3×D4—361no—
170C2×C3×C4—91no—
171C2×C3×C4—361no—
172C2×C3×C4—361no—
173#67—121no—
174C2×Dic3—121no—
175D12—21no—
176D12—121no—
177C4×D3—61no—
178C4×D3—61no—
179C4×D3—122no—
180C4×D3—121no—
181Dic6—21no—
182Dic6—22no—
183Dic6—122no—
184C3×C8—21no—
185C3×C8—21no—
186C3×C8—41no—
187C3×C8—41no—
188C3×C8—91no—
189C3×C8GO+(4,𝔽2)11yescharacteristic
190C3⋊C8—61no—
191C3⋊C8—61no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—241no—
197C2×C32C2×C3×D811yescharacteristic
198C3⋊D3—21no—
199C3×D3—62no—
200C3×D3—81no—
201C3×D3—81no—
202C3×D3—121no—
203C3×D3—161no—
204#41—181no—
205C2×Q8—181no—
206Q16—61no—
207Q16—92no—
208Q16—91no—
209SD16—61no—
210D8—91no—
211C2×C8—91no—
212C16—92no—
213C22×C3—91no—
214C22×C3—121no—
215C22×C3—121no—
216C22×C3—241no—
217C2×D3—41no—
218C2×D3—61no—
219C2×D3—61no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—91no—
225C3×C4—122no—
226C3×C4—121no—
227C3×C4—121no—
228C3×C4—122no—
229C3×C4—241no—
230C3×C4—242no—
231C3×C4C2×GO+(4,𝔽2)11yescharacteristic
232Dic3—42no—
233Dic3—41no—
234Dic3—61no—
235Dic3—61no—
236C32—21no—
237C32—21no—
238C32—41no—
239C32—41no—
240C32C3×#51011yescharacteristic, complemented
241Q8—61no—
242Q8—62no—
243Q8—361no—
244D4—61no—
245D4—361no—
246C2×C4—91no—
247C2×C4—361no—
248C2×C4—361no—
249C8—91no—
250C8C3×GO+(4,𝔽2)11yescharacteristic
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—41no—
254C2×C3—41no—
255C2×C3—181no—
256C2×C3—241no—
257C2×C3—241no—
258C2×C3—481no—
259C2×C3#6422511yescharacteristic
260D3—62no—
261D3—81no—
262D3—121no—
263C22—91no—
264C22—121no—
265C4—91no—
266C4—121no—
267C4—122no—
268C4C2×C3×GO+(4,𝔽2)11yescharacteristic
269C3—21no—
270C3—21no—
271C3—41no—
272C3—41no—
273C3#1059327911yescharacteristic, direct product factor
274C2—181no—
275C2—241no—
276C2C3×#6422511yescharacteristic
subgroupG/H
centerC2×C3#64225
commutator, cosocleC3⋊(C3⋊C8)C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33C2×D8
prime radicalC3×#63809C2
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#510
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
2. C3×#63802C2semidirect
3. C3×#63765C2semidirect
4. C3×#63736C2semidirect
5. C3×#63735C2semidirect
6. #10593279C3direct (N × H)
19. #63802C2×C3semidirect
20. #63765C2×C3semidirect
21. #63736C2×C3semidirect
22. #63735C2×C3semidirect
158. C33#510semidirect
240. C32C3×#510semidirect
273. C3#10593279direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638092
C3×#638022
C3×#637652
C3×#637362
C3×#637352
#105932793
C3×#5109