Finite Group Databasebeta

← C3×C3⋊D3×#101 #13162788

Subgroups — C3×C3⋊D3×#101 #13162788

Order 1728.

1028 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 202 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×Q8×C3⋊D3C211yesmaximal, characteristic
2C3×M16×C3⋊D3C212yesmaximal
3C33×#101C211yesmaximal, characteristic, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5C3×#63651C212yesmaximal, complemented
6C3⋊D3×#101C311yesmaximal, characteristic, direct product factor
7C3×D3×#101—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×Q8×C3⋊D3—22no—
10C3×Q8×C3⋊D3—22no—
11C2×C3×C4×C3⋊D3C412yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C412yes—
14C2×C3×#634C2211yescharacteristic
15C33×M16C2212yes—
16C3×#3748C2212yes—
17C3×#3743—22no—
18C3×C8×C3⋊D3—22no—
19C2×Q8×C3⋊D3C2×C311yescharacteristic
20C2×C3×D3×Q8—34no—
21M16×C3⋊D3C2×C312yes—
22C3×D3×M16—38no—
23C32×#101—24no—
24C32×#101D314yescomplemented
25C32×#101C2×C311yescharacteristic, complemented
26#63663C2×C311yescharacteristic, complemented
27#63651C2×C312yescomplemented
28C3×#836—34no—
29C3×#824—38no—
30C22×C3×C3⋊D3C2×C411yescharacteristic
31C33×Q8—22no—
32C2×C33×C4C2×C412yes—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3—22no—
38C3×C4×C3⋊D3D412yes—
39C3×C4×C3⋊D3D412yes—
40C3×#634—22no—
41C3×#634—24no—
42C33×C8—22no—
43C3×C3⋊(C3⋊C8)—22no—
44C2×C3×#101—91no—
45D3×#101—34no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C22×C311yescharacteristic
49Q8×C3⋊D3—22no—
50Q8×C3⋊D3—22no—
51C2×C4×C3⋊D3C3×C412yes—
52C2×C4×C3⋊D3C22×C311yescharacteristic
53C2×#634C3×C412yes—
54C2×#634C22×C311yescharacteristic
55C3×D3×Q8—68no—
56C3×D3×Q8—68no—
57C2×C3×C4×D3—34no—
58C2×C3×C4×D3—38no—
59C2×C3×Dic6—38no—
60C2×C3×Dic6—34no—
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⋊D3C22⋊C411yescharacteristic
73C2×C3×C3⋊D3C22⋊C411yescharacteristic
74C33×C4—22no—
75C33×C4C2×D412yes—
76C3×C3⋊Dic3—22no—
77C3×C3⋊Dic3C22⋊C412yes—
78C22×C3×Q8—91no—
79C2×D3×Q8—34no—
80C2×C3×M16—92no—
81D3×M16—38no—
82C3×#101—24no—
83C3×#101—91no—
84C3×#101—92no—
85C3×#101C3×D314yescomplemented
86C3×#101C3⋊D311yescharacteristic, direct product factor
87#836—34no—
88#824—38no—
89C22×C3⋊D3C2×C3×C411yescharacteristic
90C22×C3×D3—34no—
91C32×Q8—22no—
92C32×Q8—28no—
93C32×Q8—48no—
94C2×C32×C4—28no—
95C2×C32×C4—24no—
96C2×C32×C4C4×D318yes—
97C2×C32×C4C2×C3×C412yes—
98C2×C32×C4C22×D314yes—
99C2×C32×C4C23×C311yescharacteristic
100C2×C3⋊Dic3C2×C3×C412yes—
101C2×C3⋊Dic3C2×C3×C411yescharacteristic
102C4×C3⋊D3—22no—
103C4×C3⋊D3—22no—
104C4×C3⋊D3C3×D412yes—
105C4×C3⋊D3C3×D412yes—
106#634—22no—
107#634—24no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—38no—
110C3×C4×D3—38no—
111C3×C4×D3—38no—
112C3×C4×D3—68no—
113C3×C4×D3—68no—
114C3×Dic6—616no—
115C3×Dic6—68no—
116C32×C8—22no—
117C32×C8—28no—
118C32×C8—48no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—68no—
121C2×#101—91noSylow
122C2×C33—21no—
123C2×C33C2×C22⋊C411yescharacteristic
124C3×C3⋊D3—21no—
125C3×C3⋊D3#10112yesdirect product factor
126C2×C3×Q8—24no—
127C2×C3×Q8—92no—
128C2×C3×Q8—91no—
129C2×C3×Q8—182no—
130C2×C3×Q8—182no—
131C2×C3×Q8C2×C3×D314yes—
132C2×C3×Q8C2×C3⋊D311yescharacteristic
133C22×C3×C4—91no—
134C22×C3×C4—92no—
135D3×Q8—68no—
136D3×Q8—68no—
137C2×C4×D3—34no—
138C2×C4×D3—38no—
139C2×Dic6—38no—
140C2×Dic6—34no—
141C3×M16—28no—
142C3×M16—92no—
143C3×M16—182no—
144C3×M16C2×C3×D318yes—
145C3×M16C2×C3⋊D312yes—
146C2×C3×C8—182no—
147#208—38no—
148ΓL(1,𝔽25)—68no—
149D3×C8—68no—
150C22×C32—24no—
151C22×C32C2×C4×D314yes—
152C22×C32C22×C3×C411yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3C3×C22⋊C411yescharacteristic
155C2×C3⋊D3C3×C22⋊C411yescharacteristic
156C2×C3×D3—34no—
157C2×C3×D3—34no—
158C2×C3×D3—68no—
159C32×C4—22no—
160C32×C4—28no—
161C32×C4—28no—
162C32×C4—48no—
163C32×C4D3×D418yes—
164C32×C4C2×C3×D412yes—
165C3⋊Dic3—22no—
166C3⋊Dic3C3×C22⋊C412yes—
167C3×Dic3—38no—
168C3×Dic3—68no—
169C22×Q8—91no—
170C2×M16—92no—
171#101—91no—
172#101—92no—
173#101C3×C3⋊D311yescharacteristic, direct product factor
174C33C2×#10111yescharacteristic, complemented, Sylow
175C23×C3—91no—
176C22×D3—34no—
177C3×Q8—28no—
178C3×Q8—22no—
179C3×Q8—48no—
180C3×Q8—184no—
181C3×Q8—182no—
182C2×C3×C4—28no—
183C2×C3×C4—24no—
184C2×C3×C4—91no—
185C2×C3×C4—92no—
186C2×C3×C4—92no—
187C2×C3×C4—92no—
188C2×C3×C4—182no—
189C2×C3×C4—182no—
190C2×C3×C4C3×C4×D318yes—
191C2×C3×C4C4×C3⋊D312yes—
192C2×C3×C4C22×C3×D314yes—
193C2×C3×C4C22×C3⋊D311yescharacteristic
194C2×Dic3—34no—
195C2×Dic3—38no—
196C4×D3—38no—
197C4×D3—38no—
198C4×D3—68no—
199C4×D3—68no—
200Dic6—616no—
201Dic6—68no—
202C3×C8—28no—
203C3×C8—22no—
204C3×C8—48no—
205C3×C8—182no—
206C3⋊C8—68no—
207C2×C32—21no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32—44no—
211C2×C32D3×C22⋊C414yes—
212C2×C32C2×C3×C22⋊C411yescharacteristic
213C3⋊D3—21no—
214C3⋊D3C3×#10112yesdirect product factor
215C3×D3—38no—
216C3×D3—64no—
217C2×Q8—92no—
218C2×Q8—91no—
219C2×Q8—182no—
220C2×Q8—182no—
221C2×Q8C2×C3×C3⋊D311yescharacteristic
222C22×C4—91no—
223C22×C4—92no—
224M16—92no—
225M16—182no—
226M16C2×C3×C3⋊D312yes—
227C2×C8—182no—
228C22×C3—24no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—182no—
232C22×C3C2×C3×C4×D314yes—
233C22×C3C2×C4×C3⋊D311yescharacteristic
234C2×D3—34no—
235C2×D3—34no—
236C2×D3—68no—
237C3×C4—28no—
238C3×C4—22no—
239C3×C4—28no—
240C3×C4—48no—
241C3×C4—92no—
242C3×C4—182no—
243C3×C4C3×D3×D418yes—
244C3×C4D4×C3⋊D312yes—
245Dic3—38no—
246Dic3—68no—
247C32—24no—
248C32D3×#10114yescomplemented
249C32C2×C3×#10111yescharacteristic, complemented
250C23—91no—
251Q8—22no—
252Q8—184no—
253Q8—182no—
254C2×C4—91no—
255C2×C4—92no—
256C2×C4—92no—
257C2×C4—92no—
258C2×C4—182no—
259C2×C4—182no—
260C2×C4C3×C4×C3⋊D312yes—
261C2×C4C22×C3×C3⋊D311yescharacteristic
262C8—22no—
263C8—182no—
264C2×C3—24no—
265C2×C3—21no—
266C2×C3—24no—
267C2×C3—44no—
268C2×C3—92no—
269C2×C3—181no—
270C2×C3C3×D3×C22⋊C414yes—
271C2×C3C22⋊C4×C3⋊D311yescharacteristic
272D3—38no—
273D3—64no—
274C22—91no—
275C22—91no—
276C22—182no—
277C22C2×C3×C4×C3⋊D311yescharacteristic
278C4—22no—
279C4—92no—
280C4—182no—
281C4C3×D4×C3⋊D312yes—
282C3—24no—
283C3C3×D3×#10114yescomplemented
284C3C3⋊D3×#10111yescharacteristic, direct product factor
285C2—21no—
286C2—92no—
287C2—181no—
288C2C3×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 radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×#101C3⋊D3
hypocenterC32C2×C3×#101
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
3. C33×#101C2semidirect
4. C3×#63663C2semidirect
5. C3×#63651C2semidirect
6. C3⋊D3×#101C3direct (N × H)
24. C32×#101D3semidirect
25. C32×#101C2×C3semidirect
26. #63663C2×C3semidirect
27. #63651C2×C3semidirect
85. C3×#101C3×D3semidirect
86. C3×#101C3⋊D3direct (N × H)
125. C3×C3⋊D3#101direct (N × H)
173. #101C3×C3⋊D3direct (N × H)
174. C33C2×#101semidirect
214. C3⋊D3C3×#101direct (N × H)
248. C32D3×#101semidirect
249. C32C2×C3×#101semidirect
283. C3C3×D3×#101semidirect
284. C3C3⋊D3×#101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C3×M16×C3⋊D32
C33×#1012
C3×#636632
C3×#636512
C3⋊D3×#1013
C3×D3×#1013