Finite Group Databasebeta

← C3×#10592450 #13162789

Subgroups — C3×#10592450 #13162789

Order 1728.

1052 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C2×C3×#3832C211yesmaximal, characteristic
2C3×M16×C3⋊D3C212yesmaximal, complemented
3C33×#101C211yesmaximal, characteristic, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5C3×#63650C212yesmaximal, complemented
6#10592450C311yesmaximal, characteristic, direct product factor
7C3×#4751—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×#3832—22no—
10C3×#3832—22no—
11C2×C3×C4⋊(C3⋊D3)C412yes—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C412yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C33×M16C2212yes—
16C3×#3748C2212yes—
17C3×#3743—22no—
18C3×C8×C3⋊D3—22no—
19C2×#3832C2×C311yescharacteristic
20C2×C3×#239—34no—
21M16×C3⋊D3C2×C312yescomplemented
22C3×D3×M16—38no—
23C32×#101—24no—
24C32×#101D314yescomplemented
25C32×#101C2×C311yescharacteristic, complemented
26#63663C2×C311yescharacteristic, complemented
27#63650C2×C312yescomplemented
28C3×#836—34no—
29C3×#823—38no—
30C22×C3×C3⋊D3C2×C412yes—
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C33×Q8—22no—
33C2×C33×C4C2×C412yes—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4⋊(C3⋊D3)—24no—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4×C3⋊D3—22no—
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3D412yes—
41C3×C4×C3⋊D3D412yes—
42C33×C8—22no—
43C3×C3⋊(C3⋊C8)—22no—
44C3×#413—91no—
45#4751—34no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C22×C311yescharacteristic
49#3832—22no—
50#3832—22no—
51C2×C4⋊(C3⋊D3)C3×C412yes—
52C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
53C2×C4×C3⋊D3C3×C412yes—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C3×#239—68no—
56C3×#239—68no—
57C2×C3×D12—38no—
58C2×C3×D12—34no—
59C2×C3×C4×D3—34no—
60C2×C3×C4×D3—38no—
61C32×M16—28no—
62C32×M16C2×D318yes—
63C32×M16C22×C312yes—
64#3748C22×C312yes—
65#3743—22no—
66C8×C3⋊D3—22no—
67C3×#208—38no—
68C3×ΓL(1,𝔽25)—68no—
69C3×D3×C8—68no—
70C22×C33C22×C411yescharacteristic
71C2×C3×C3⋊D3—22no—
72C2×C3×C3⋊D3—41no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3C22⋊C412yes—
75C33×C4—22no—
76C33×C4C2×D412yes—
77C3×C3⋊Dic3C22⋊C411yescharacteristic
78C3×C3⋊Dic3C22⋊C411yescharacteristic
79C2×C3×#41—91no—
80C2×#239—34no—
81C2×C3×M16—92no—
82D3×M16—38no—
83C3×#101—24no—
84C3×#101—91no—
85C3×#101C3×D314yescomplemented
86C3×#101C3⋊D311yescharacteristic, complemented
87C3×#100—92no—
88#836—34no—
89#823—38no—
90C22×C3⋊D3C2×C3×C412yes—
91C22×C3⋊D3C2×C3×C411yescharacteristic
92C22×C3×D3—34no—
93C22×C3×D3—38no—
94C32×Q8—22no—
95C32×Q8—28no—
96C32×Q8—48no—
97C2×C32×C4—28no—
98C2×C32×C4—24no—
99C2×C32×C4C4×D318yes—
100C2×C32×C4C2×C3×C412yes—
101C2×C32×C4C22×D314yes—
102C2×C32×C4C23×C311yescharacteristic
103C2×C3⋊Dic3C2×C3×C411yescharacteristic
104C4⋊(C3⋊D3)—24no—
105C4⋊(C3⋊D3)—22no—
106C4×C3⋊D3—22no—
107C4×C3⋊D3—22no—
108C4×C3⋊D3C3×D412yes—
109C4×C3⋊D3C3×D412yes—
110C2×C3×Dic3—34no—
111C3×D12—616no—
112C3×D12—68no—
113C3×C4×D3—38no—
114C3×C4×D3—38no—
115C3×C4×D3—68no—
116C3×C4×D3—68no—
117C32×C8—22no—
118C32×C8—28no—
119C32×C8—48no—
120C3⋊(C3⋊C8)—22no—
121C3×C3⋊C8—68no—
122#413—91noSylow
123C2×C33—21no—
124C2×C33C2×C22⋊C411yescharacteristic
125C3×C3⋊D3—22no—
126C3×C3⋊D3—42no—
127C3×#41—182no—
128C3×#41—182no—
129C2×C3×Q8—24no—
130C2×C3×Q8C2×C3×D314yes—
131C2×C3×Q8C2×C3⋊D311yescharacteristic
132C2×C3×D4—92no—
133C2×C3×D4—91no—
134C22×C3×C4—91no—
135C22×C3×C4—92no—
136#239—68no—
137#239—68no—
138C2×D12—38no—
139C2×D12—34no—
140C2×C4×D3—34no—
141C2×C4×D3—38no—
142C3×M16—28no—
143C3×M16—92no—
144C3×M16—182no—
145C3×M16C2×C3×D318yes—
146C3×M16C2×C3⋊D312yes—
147C2×C3×C8—182no—
148#208—38no—
149ΓL(1,𝔽25)—68no—
150D3×C8—68no—
151C22×C32—24no—
152C22×C32C2×C4×D314yes—
153C22×C32C22×C3×C411yescharacteristic
154C2×C3⋊D3—22no—
155C2×C3⋊D3—41no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3C3×C22⋊C412yes—
158C2×C3×D3—38no—
159C2×C3×D3—68no—
160C2×C3×D3—124no—
161C2×C3×D3—128no—
162C32×C4—22no—
163C32×C4—28no—
164C32×C4—28no—
165C32×C4—48no—
166C32×C4D3×D418yes—
167C32×C4C2×C3×D412yes—
168C3⋊Dic3C3×C22⋊C411yescharacteristic
169C3⋊Dic3C3×C22⋊C411yescharacteristic
170C3×Dic3—34no—
171C3×Dic3—34no—
172C2×#41—91no—
173C2×M16—92no—
174#101—91no—
175#101C3×C3⋊D311yescharacteristic, complemented
176#100—92no—
177C33#41311yescharacteristic, complemented, Sylow
178C23×C3—91no—
179C23×C3—92no—
180C22×D3—34no—
181C22×D3—38no—
182C3×Q8—28no—
183C3×Q8—22no—
184C3×Q8—48no—
185C3×D4—184no—
186C3×D4—182no—
187C2×C3×C4—28no—
188C2×C3×C4—24no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4C3×C4×D318yes—
195C2×C3×C4C4×C3⋊D312yes—
196C2×C3×C4C22×C3×D314yes—
197C2×C3×C4C22×C3⋊D311yescharacteristic
198C2×Dic3—34no—
199D12—616no—
200D12—68no—
201C4×D3—38no—
202C4×D3—38no—
203C4×D3—68no—
204C4×D3—68no—
205C3×C8—28no—
206C3×C8—22no—
207C3×C8—48no—
208C3×C8—182no—
209C3⋊C8—68no—
210C2×C32—21no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32—44no—
214C2×C32D3×C22⋊C414yes—
215C2×C32C2×C3×C22⋊C411yescharacteristic
216C3⋊D3—22no—
217C3⋊D3—42no—
218C3×D3—68no—
219C3×D3—128no—
220#41—182no—
221#41—182no—
222C2×Q8C2×C3×C3⋊D311yescharacteristic
223C2×D4—92no—
224C2×D4—91no—
225C22×C4—91no—
226C22×C4—92no—
227M16—92no—
228M16—182no—
229M16C2×C3×C3⋊D312yes—
230C2×C8—182no—
231C22×C3—24no—
232C22×C3—92no—
233C22×C3—182no—
234C22×C3—361no—
235C22×C3—362no—
236C22×C3C2×C3×C4×D314yes—
237C22×C3C2×C4×C3⋊D311yescharacteristic
238C2×D3—38no—
239C2×D3—68no—
240C2×D3—124no—
241C2×D3—128no—
242C3×C4—28no—
243C3×C4—22no—
244C3×C4—28no—
245C3×C4—48no—
246C3×C4—91no—
247C3×C4—91no—
248C3×C4C3×D3×D418yes—
249C3×C4D4×C3⋊D312yes—
250Dic3—34no—
251Dic3—34no—
252C32—24no—
253C32#475114yescomplemented
254C32C3×#41311yescharacteristic, complemented
255C23—91no—
256C23—92no—
257Q8—22no—
258D4—184no—
259D4—182no—
260C2×C4—91no—
261C2×C4—92no—
262C2×C4—92no—
263C2×C4—182no—
264C2×C4—182no—
265C2×C4C3×C4×C3⋊D312yes—
266C2×C4C22×C3×C3⋊D311yescharacteristic
267C8—22no—
268C8—182no—
269C2×C3—24no—
270C2×C3—21no—
271C2×C3—24no—
272C2×C3—44no—
273C2×C3—182no—
274C2×C3—362no—
275C2×C3C3×D3×C22⋊C414yes—
276C2×C3C22⋊C4×C3⋊D311yescharacteristic
277D3—68no—
278D3—128no—
279C22—92no—
280C22—182no—
281C22—361no—
282C22—362no—
283C22C2×C3×C4×C3⋊D311yescharacteristic
284C4—22no—
285C4—91no—
286C4—91no—
287C4C3×D4×C3⋊D312yes—
288C3—24no—
289C3C3×#475114yescomplemented
290C3#1059245011yescharacteristic, direct product factor
291C2—21no—
292C2—182no—
293C2—362no—
294C2C3×C22⋊C4×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3C22⋊C4×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#101C2
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3832C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#101C3⋊D3
hypocenterC32C3×#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. C33×#101C2semidirect
4. C3×#63663C2semidirect
5. C3×#63650C2semidirect
6. #10592450C3direct (N × H)
21. M16×C3⋊D3C2×C3semidirect
24. C32×#101D3semidirect
25. C32×#101C2×C3semidirect
26. #63663C2×C3semidirect
27. #63650C2×C3semidirect
85. C3×#101C3×D3semidirect
86. C3×#101C3⋊D3semidirect
175. #101C3×C3⋊D3semidirect
177. C33#413semidirect
253. C32#4751semidirect
254. C32C3×#413semidirect
289. C3C3×#4751semidirect
290. C3#10592450direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C3×M16×C3⋊D32
C33×#1012
C3×#636632
C3×#636502
#105924503
C3×#47513