Finite Group Databasebeta

← C3×#10592887 #13163227

Subgroups — C3×#10592887 #13163227

Order 1728.

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

normal subgroups only

This group has 924 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×C33×Q16C211yesmaximal, characteristic, complemented
2C3×#64155C212yesmaximal
3C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
4C3×#63662C212yesmaximal, complemented
5C3×#63645C211yesmaximal, characteristic, complemented
6#10592887C311yesmaximal, characteristic, direct product factor
7C3×#5188—34nomaximal
8C2×C33×Q8C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C33×Q16—22no—
11C2×C33×C8C2211yescharacteristic
12C3×#3752—22no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×#3750—22no—
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×C8×C3⋊D3—22no—
17C2×C32×Q16—24no—
18C2×C32×Q16D314yescomplemented
19C2×C32×Q16C2×C311yescharacteristic, complemented
20#64155C2×C312yes—
21C2×C8×C3⋊D3C2×C311yescharacteristic
22C3×#946—38no—
23C2×C3×D3×C8—34no—
24#63662C2×C312yescomplemented
25#63645C2×C311yescharacteristic, complemented
26C3×#835—38no—
27C3×#818—34no—
28C22×C3×C3⋊D3D411yescharacteristic
29C33×Q8—22no—
30C33×Q8—22no—
31C2×C33×C4—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3—21no—
37C33×C8D412yes—
38C3×C3⋊(C3⋊C8)—21no—
39C3×#467—91no—
40#5188—34no—
41C2×C32×Q8—28no—
42C2×C32×Q8C2×D318yes—
43C2×C32×Q8C22×C312yes—
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C2×C3×C4×D3—34no—
46C32×Q16—216no—
47C32×Q16—28no—
48C32×Q16—22no—
49C2×C32×C8—24no—
50C2×C32×C8C2×D314yes—
51C2×C32×C8C22×C311yescharacteristic
52#3752—22no—
53C4⋊(C3⋊Dic3)C22×C312yes—
54#3750—22no—
55C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
56C8×C3⋊D3—22no—
57C3×(C2×D3)⋊C4—68no—
58C3×C4⋊Dic3—38no—
59C3×Dic3⋊C4—68no—
60C2×C3×C3⋊C8—34no—
61C3×D3×C8—68no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—21no—
65C2×C3×C3⋊D3Q1612yescomplemented
66C33×C4—42no—
67C33×C4C2×D411yescharacteristic
68C33×C4#4111yescharacteristic
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3—42no—
71C2×C3×Q16—24no—
72C2×C3×Q16C3×D314yescomplemented
73C2×C3×Q16C3⋊D311yescharacteristic, complemented
74C22×C3×C8—91no—
75C3×C22⋊Q8—92no—
76#946—38no—
77C2×D3×C8—34no—
78C3×#107—91no—
79C3×#103—92no—
80#835—38no—
81#818—34no—
82C22×C3⋊D3C3×D411yescharacteristic
83C22×C3×D3—34no—
84C32×Q8—216no—
85C32×Q8—28no—
86C32×Q8—22no—
87C32×Q8—22no—
88C32×Q8—28no—
89C32×Q8—48no—
90C2×C32×C4—22no—
91C2×C32×C4—24no—
92C2×C32×C4—28no—
93C2×C32×C4—48no—
94C2×C32×C4C22×D314yes—
95C2×C32×C4C23×C311yescharacteristic
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3C3×D411yescharacteristic
98C4×C3⋊D3—21no—
99C4×C3⋊D3—21no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—68no—
102C3×C4×D3—64no—
103C3×C4×D3—64no—
104C32×C8—28no—
105C32×C8#6718yes—
106C32×C8C3×D412yes—
107C3⋊(C3⋊C8)—21no—
108C3×C3⋊C8—64no—
109#467—91noSylow
110C2×C33#12111yescharacteristic
111C2×C33C2×Q1611yescharacteristic
112C2×C33#13511yescharacteristic
113C3×C3⋊D3—22no—
114C2×C3×Q8—28no—
115C2×C3×Q8C2×C3×D318yes—
116C2×C3×Q8C2×C3⋊D312yes—
117C22×C3×C4—91no—
118C2×C4×D3—34no—
119C3×Q16—216no—
120C3×Q16—28no—
121C3×Q16—22no—
122C2×C3×C8—24no—
123C2×C3×C8—91no—
124C2×C3×C8—182no—
125C2×C3×C8C2×C3×D314yes—
126C2×C3×C8C2×C3⋊D311yescharacteristic
127C3×C4⋊C4—92no—
128C3×C4⋊C4—182no—
129C3×C22⋊C4—182no—
130(C2×D3)⋊C4—68no—
131C4⋊Dic3—38no—
132Dic3⋊C4—68no—
133C2×C3⋊C8—34no—
134D3×C8—68no—
135C22×C32—24no—
136C22×C32D3×D414yes—
137C22×C32C2×C3×D411yescharacteristic
138C2×C3⋊D3—21no—
139C2×C3⋊D3—21no—
140C2×C3⋊D3C3×Q1612yescomplemented
141C2×C3×D3—38no—
142C2×C3×D3—64no—
143C2×C3×D3—64no—
144C32×C4—24no—
145C32×C4—24no—
146C32×C4—416no—
147C32×C4—48no—
148C32×C4—42no—
149C32×C4#23714yes—
150C32×C4C2×#6714yes—
151C32×C4C2×C3×D411yescharacteristic
152C32×C4C3×#4111yescharacteristic
153C3⋊Dic3—21no—
154C3⋊Dic3—42no—
155C3×Dic3—64no—
156C3×Dic3—128no—
157C2×Q16C3×C3⋊D311yescharacteristic, complemented
158C22×C8—91no—
159C22⋊Q8—92no—
160#107—91no—
161#103—92no—
162C33#46711yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C22×D3—34no—
165C3×Q8—216no—
166C3×Q8—28no—
167C3×Q8—28no—
168C3×Q8—22no—
169C3×Q8—22no—
170C3×Q8—48no—
171C2×C3×C4—28no—
172C2×C3×C4—22no—
173C2×C3×C4—24no—
174C2×C3×C4—48no—
175C2×C3×C4—91no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4—182no—
179C2×C3×C4C22×C3×D314yes—
180C2×C3×C4C22×C3⋊D311yescharacteristic
181C2×Dic3—34no—
182C2×Dic3—68no—
183C4×D3—64no—
184C4×D3—64no—
185C3×C8—28no—
186C3×C8—181no—
187C3×C8C3×#6718yes—
188C3×C8#63812yes—
189C3⋊C8—64no—
190C2×C32—24no—
191C2×C32—24no—
192C2×C32—24no—
193C2×C32D3×Q1614yes—
194C2×C32#91914yes—
195C2×C32#93814yes—
196C2×C32C3×#12111yescharacteristic
197C2×C32C2×C3×Q1611yescharacteristic
198C2×C32C3×#13511yescharacteristic
199C3⋊D3—22no—
200C3×D3—68no—
201C2×Q8C2×C3×C3⋊D312yes—
202C22×C4—91no—
203Q16—22no—
204C2×C8—91no—
205C2×C8—182no—
206C2×C8C2×C3×C3⋊D311yescharacteristic
207C4⋊C4—92no—
208C4⋊C4—182no—
209C22⋊C4—182no—
210C22×C3—24no—
211C22×C3—92no—
212C22×C3—181no—
213C22×C3—181no—
214C22×C3C3×D3×D414yes—
215C22×C3D4×C3⋊D311yescharacteristic
216C2×D3—38no—
217C2×D3—64no—
218C2×D3—64no—
219C3×C4—24no—
220C3×C4—24no—
221C3×C4—416no—
222C3×C4—48no—
223C3×C4—42no—
224C3×C4—181no—
225C3×C4—362no—
226C3×C4C3×#23714yes—
227C3×C4C2×C3×#6714yes—
228C3×C4#383011yescharacteristic
229C3×C4C2×#63811yescharacteristic
230Dic3—64no—
231Dic3—128no—
232C32—24no—
233C32#518814yescomplemented
234C32C3×#46711yescharacteristic, complemented
235C23—91no—
236Q8—22no—
237Q8—22no—
238C2×C4—22no—
239C2×C4—91no—
240C2×C4—181no—
241C2×C4—181no—
242C2×C4—182no—
243C2×C4C22×C3×C3⋊D311yescharacteristic
244C8—181no—
245C8C3×#63812yes—
246C2×C3—24no—
247C2×C3—24no—
248C2×C3—24no—
249C2×C3—182no—
250C2×C3C3×D3×Q1614yes—
251C2×C3C3×#91914yes—
252C2×C3C3×#93814yes—
253C2×C3Q16×C3⋊D311yescharacteristic
254C2×C3#6412811yescharacteristic
255C2×C3#6414711yescharacteristic
256D3—68no—
257C22—92no—
258C22—181no—
259C22—181no—
260C22C3×D4×C3⋊D311yescharacteristic
261C4—42no—
262C4—181no—
263C4—362no—
264C4C3×#383011yescharacteristic
265C4C2×C3×#63811yescharacteristic
266C3—24no—
267C3C3×#518814yescomplemented
268C3#1059288711yescharacteristic, direct product factor
269C2—182no—
270C2C3×Q16×C3⋊D311yescharacteristic
271C2C3×#6412811yescharacteristic
272C2C3×#6414711yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×Q16C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×Q16C3⋊D3
hypocenterC32C3×#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C33×Q16C2semidirect
4. C3×#63662C2semidirect
5. C3×#63645C2semidirect
6. #10592887C3direct (N × H)
18. C2×C32×Q16D3semidirect
19. C2×C32×Q16C2×C3semidirect
24. #63662C2×C3semidirect
25. #63645C2×C3semidirect
65. C2×C3×C3⋊D3Q16semidirect
72. C2×C3×Q16C3×D3semidirect
73. C2×C3×Q16C3⋊D3semidirect
140. C2×C3⋊D3C3×Q16semidirect
157. C2×Q16C3×C3⋊D3semidirect
162. C33#467semidirect
233. C32#5188semidirect
234. C32C3×#467semidirect
267. C3C3×#5188semidirect
268. C3#10592887direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×Q162
C3×#641552
C2×C3×C8×C3⋊D32
C3×#636622
C3×#636452
#105928873
C3×#51883