Finite Group Databasebeta

← C3×#10592742 #13163081

Subgroups — C3×#10592742 #13163081

Order 1728.

920 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C33×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C3×#63662C212yesmaximal, complemented
4C3×#63657C211yesmaximal, characteristic
5C3×#63635C212yesmaximal, complemented
6#10592742C311yesmaximal, characteristic, direct product factor
7C3×#5043—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3—21no—
11C33×C4⋊C4—21no—
12C33×C4⋊C4C2211yescharacteristic
13C33×C22⋊C4—21no—
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C22⋊Q8—24no—
18C32×C22⋊Q8D314yes—
19C32×C22⋊Q8C2×C311yescharacteristic
20C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—34no—
22#63662C2×C312yescomplemented
23#63657C2×C311yescharacteristic
24#63635C2×C312yescomplemented
25C3×#835—38no—
26C3×#830—34no—
27C3×#808—38no—
28C23×C33D411yescharacteristic
29C33×Q8—22no—
30C2×C33×C4—22no—
31C2×C33×C4—21no—
32C2×C33×C4—21no—
33C2×C33×C4D411yescharacteristic
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—42no—
37C3×C3⋊(C3⋊C8)—22no—
38C3×#483—91no—
39#5043—34no—
40C2×C32×Q8—24no—
41C2×C32×Q8C2×D314yes—
42C2×C32×Q8C22×C311yescharacteristic
43C22×C32×C4—24no—
44C22×C32×C4C2×D314yes—
45C22×C32×C4C22×C311yescharacteristic
46C22×C3⋊Dic3—21no—
47C22×C3×Dic3—64no—
48C32×C4⋊C4—28no—
49C32×C4⋊C4—24no—
50C32×C4⋊C4—24no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4C2×D314yes—
53C32×C4⋊C4C22×C311yescharacteristic
54C32×C22⋊C4—28no—
55C32×C22⋊C4—24no—
56C32×C22⋊C4—21no—
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)C22×C312yes—
59C2×C3⋊(C3⋊C8)C22×C312yes—
60C3×C4⋊Dic3—38no—
61C3×C4⋊Dic3—64no—
62C2×C3×C3⋊C8—38no—
63C22×C33—21no—
64C22×C33—21no—
65C22×C33SD1612yes—
66C22×C33C2×D411yescharacteristic
67C33×C4—21no—
68C33×C4—41no—
69C33×C4—41no—
70C33×C4#4112yes—
71C3×C3⋊Dic3—42no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8C3×D314yes—
74C3×C22⋊Q8C3⋊D311yescharacteristic
75C2×C3×C4⋊C4—91no—
76C2×C4⋊Dic3—34no—
77C3×#106—92no—
78C3×#103—92no—
79C3×C22⋊C8—91no—
80#835—38no—
81#830—34no—
82#808—38no—
83C23×C32—24no—
84C23×C32#6714yes—
85C23×C32C3×D411yescharacteristic
86C32×Q8—22no—
87C32×Q8—28no—
88C32×Q8—48no—
89C2×C32×C4—22no—
90C2×C32×C4—28no—
91C2×C32×C4—28no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—24no—
97C2×C32×C4—24no—
98C2×C32×C4—28no—
99C2×C32×C4—48no—
100C2×C32×C4#6714yes—
101C2×C32×C4C3×D411yescharacteristic
102C2×C32×C4C22×D314yes—
103C2×C32×C4C23×C311yescharacteristic
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—42no—
106C2×C3×Dic3—68no—
107C2×C3×Dic3—128no—
108C3⋊(C3⋊C8)—22no—
109C3×C3⋊C8—68no—
110#483—91noSylow
111C2×C33—22no—
112C2×C33#12311yescharacteristic
113C2×C33C2×SD1611yescharacteristic
114C2×C33#13711yescharacteristic
115C2×C3×Q8—24no—
116C2×C3×Q8C2×C3×D314yes—
117C2×C3×Q8C2×C3⋊D311yescharacteristic
118C22×C3×C4—24no—
119C22×C3×C4—181no—
120C22×C3×C4C2×C3×D314yes—
121C22×C3×C4C2×C3⋊D311yescharacteristic
122C22×Dic3—64no—
123C2×C3×C8—92no—
124C3×C4⋊C4—28no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—92no—
129C3×C4⋊C4—181no—
130C3×C4⋊C4C2×C3×D314yes—
131C3×C4⋊C4C2×C3⋊D311yescharacteristic
132C3×C22⋊C4—28no—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—21no—
135C4⋊Dic3—38no—
136C4⋊Dic3—64no—
137C2×C3⋊C8—38no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—28no—
141C22×C32—24no—
142C22×C32—24no—
143C22×C32—24no—
144C22×C32—44no—
145C22×C32—44no—
146C22×C32#21518yes—
147C22×C32C3×SD1612yes—
148C22×C32C2×#6714yes—
149C22×C32C2×C3×D411yescharacteristic
150C32×C4—21no—
151C32×C4—28no—
152C32×C4—24no—
153C32×C4—44no—
154C32×C4—48no—
155C32×C4—48no—
156C32×C4—44no—
157C32×C4—44no—
158C32×C4—41no—
159C32×C4—41no—
160C32×C4#23718yes—
161C32×C4C3×#4112yes—
162C3⋊Dic3—42no—
163C3×Dic3—128no—
164C22⋊Q8C3×C3⋊D311yescharacteristic
165C2×C4⋊C4—91no—
166#106—92no—
167#103—92no—
168C22⋊C8—91no—
169C33#48311yescharacteristic, complemented, Sylow
170C23×C3—24no—
171C23×C3C3×#6714yes—
172C23×C3#63811yescharacteristic
173C3×Q8—28no—
174C3×Q8—22no—
175C3×Q8—48no—
176C2×C3×C4—28no—
177C2×C3×C4—22no—
178C2×C3×C4—28no—
179C2×C3×C4—28no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—24no—
185C2×C3×C4—24no—
186C2×C3×C4—48no—
187C2×C3×C4—182no—
188C2×C3×C4—362no—
189C2×C3×C4C3×#6714yes—
190C2×C3×C4#63811yescharacteristic
191C2×C3×C4C22×C3×D314yes—
192C2×C3×C4C22×C3⋊D311yescharacteristic
193C2×Dic3—68no—
194C2×Dic3—128no—
195C3×C8—182no—
196C3⋊C8—68no—
197C2×C32—22no—
198C2×C32—24no—
199C2×C32—24no—
200C2×C32—24no—
201C2×C32—28no—
202C2×C32—48no—
203C2×C32#93514yes—
204C2×C32C2×#21514yes—
205C2×C32#95114yes—
206C2×C32C3×#12311yescharacteristic
207C2×C32C2×C3×SD1611yescharacteristic
208C2×C32C3×#13711yescharacteristic
209C2×Q8C2×C3×C3⋊D311yescharacteristic
210C22×C4—181no—
211C22×C4C2×C3×C3⋊D311yescharacteristic
212C2×C8—92no—
213C4⋊C4—21no—
214C4⋊C4—92no—
215C4⋊C4—181no—
216C4⋊C4C2×C3×C3⋊D311yescharacteristic
217C22⋊C4—21no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—28no—
223C22×C3—24no—
224C22×C3—44no—
225C22×C3—44no—
226C22×C3C3×#21518yes—
227C22×C3#375512yes—
228C22×C3C2×C3×#6714yes—
229C22×C3C2×#63811yescharacteristic
230C3×C4—24no—
231C3×C4—21no—
232C3×C4—28no—
233C3×C4—44no—
234C3×C4—48no—
235C3×C4—48no—
236C3×C4—44no—
237C3×C4—44no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—362no—
241C3×C4C3×#23718yes—
242C3×C4#383012yes—
243Dic3—128no—
244C32—24no—
245C32#504314yescomplemented
246C32C3×#48311yescharacteristic, complemented
247C23C3×#63811yescharacteristic
248Q8—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—182no—
253C2×C4—362no—
254C2×C4C3×#63811yescharacteristic
255C2×C4C22×C3×C3⋊D311yescharacteristic
256C8—182no—
257C2×C3—28no—
258C2×C3—22no—
259C2×C3—24no—
260C2×C3—24no—
261C2×C3—24no—
262C2×C3—48no—
263C2×C3C3×#93514yes—
264C2×C3C2×C3×#21514yes—
265C2×C3C3×#95114yes—
266C2×C3#6414411yescharacteristic
267C2×C3C2×#375511yescharacteristic
268C2×C3#6416011yescharacteristic
269C22—21no—
270C22—21no—
271C22C3×#375512yes—
272C22C2×C3×#63811yescharacteristic
273C4—21no—
274C4—41no—
275C4—41no—
276C4—362no—
277C4C3×#383012yes—
278C3—24no—
279C3C3×#504314yescomplemented
280C3#1059274211yescharacteristic, direct product factor
281C2—22no—
282C2C3×#6414411yescharacteristic
283C2C2×C3×#375511yescharacteristic
284C2C3×#6416011yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterC32C3×#483
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63662C2semidirect
5. C3×#63635C2semidirect
6. #10592742C3direct (N × H)
22. #63662C2×C3semidirect
24. #63635C2×C3semidirect
169. C33#483semidirect
245. C32#5043semidirect
246. C32C3×#483semidirect
279. C3C3×#5043semidirect
280. C3#10592742direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
C2×C3×C4⋊(C3⋊Dic3)2
C3×#636622
C3×#636572
C3×#636352
#105927423
C3×#50433