Finite Group Databasebeta

← C3×#10589335 #13145892

Subgroups — C3×#10589335 #13145892

Order 1728.

668 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 158 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×#3750C211yesmaximal, characteristic
2C33×C22⋊C8C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic
4#10589335C311yesmaximal, characteristic, direct product factor
5C3×#4469—34nomaximal
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C411yescharacteristic
9C2×C33×C8—21no—
10C3×#3750—21no—
11C3×#3750—21no—
12C2×C3×C3⋊(C3⋊C8)—21no—
13C2×#3750C2×C311yescharacteristic
14C2×C3×Dic3⋊C4—34no—
15C32×C22⋊C8—24no—
16C32×C22⋊C8D314yes—
17C32×C22⋊C8C2×C311yescharacteristic
18#63657C2×C311yescharacteristic
19C3×#830—34no—
20C23×C33C2×C411yescharacteristic
21C2×C33×C4—41no—
22C2×C33×C4D411yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—41no—
28C2×C3×C3⋊Dic3C812yescomplemented
29C33×C8—41no—
30C3×C3⋊(C3⋊C8)—41no—
31C3×#324—91no—
32#4469—34no—
33C22×C32×C4—24no—
34C22×C32×C4C2×D314yes—
35C22×C32×C4C22×C311yescharacteristic
36C22×C3⋊Dic3C3×C411yescharacteristic
37C22×C3⋊Dic3C3×C411yescharacteristic
38C22×C3×Dic3—34no—
39C22×C3×Dic3—34no—
40C2×C32×C8—28no—
41C2×C32×C8—24no—
42C2×C32×C8—21no—
43#3750—21no—
44#3750—21no—
45C2×C3⋊(C3⋊C8)—21no—
46C3×Dic3⋊C4—64no—
47C3×Dic3⋊C4—64no—
48C2×C3×C3⋊C8—64no—
49C22×C33—21no—
50C22×C33—21no—
51C22×C33C22⋊C411yescharacteristic
52C22×C33C2×C811yescharacteristic
53C22×C33M1611yescharacteristic
54C33×C4—21no—
55C33×C4—21no—
56C3×C3⋊Dic3—22no—
57C3×C3⋊Dic3—41no—
58C2×C3×C4⋊C4—91no—
59C2×Dic3⋊C4—34no—
60C3×C22⋊C8—24no—
61C3×C22⋊C8—91no—
62C3×C22⋊C8C3×D314yes—
63C3×C22⋊C8C3⋊D311yescharacteristic
64#830—34no—
65C23×C32—24no—
66C23×C32C4×D314yes—
67C23×C32C2×C3×C411yescharacteristic
68C2×C32×C4—24no—
69C2×C32×C4—24no—
70C2×C32×C4—44no—
71C2×C32×C4—41no—
72C2×C32×C4—48no—
73C2×C32×C4D1214yes—
74C2×C32×C4#6714yes—
75C2×C32×C4C3×D411yescharacteristic
76C2×C32×C4C3×D411yescharacteristic
77C2×C3⋊Dic3—21no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3C3×C812yescomplemented
82C2×C3×Dic3—38no—
83C2×C3×Dic3—64no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—124no—
87C32×C8—48no—
88C32×C8—44no—
89C32×C8—41no—
90C3⋊(C3⋊C8)—41no—
91C3×C3⋊C8—124no—
92#324—91noSylow
93C2×C33—21no—
94C2×C33—21no—
95C2×C33C22⋊C811yescharacteristic
96C2×C33#9911yescharacteristic
97C2×C33#10111yescharacteristic
98C22×C3×C4—24no—
99C22×C3×C4—91no—
100C22×C3×C4—91no—
101C22×C3×C4C2×C3×D314yes—
102C22×C3×C4C2×C3⋊D311yescharacteristic
103C22×Dic3—34no—
104C22×Dic3—34no—
105C2×C3×C8—28no—
106C2×C3×C8—24no—
107C2×C3×C8—21no—
108C2×C3×C8—181no—
109C3×C4⋊C4—181no—
110C3×C4⋊C4—181no—
111Dic3⋊C4—64no—
112Dic3⋊C4—64no—
113C2×C3⋊C8—64no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—24no—
117C22×C32—24no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—24no—
121C22×C32—44no—
122C22×C32—44no—
123C22×C32D3×C814yes—
124C22×C32ΓL(1,𝔽25)14yes—
125C22×C32(C2×D3)⋊C414yes—
126C22×C32C3×C22⋊C411yescharacteristic
127C22×C32C2×C3×C811yescharacteristic
128C22×C32C3×M1611yescharacteristic
129C32×C4—28no—
130C32×C4—24no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4—24no—
134C32×C4—44no—
135C3⋊Dic3—22no—
136C3⋊Dic3—41no—
137C3×Dic3—68no—
138C3×Dic3—124no—
139C2×C4⋊C4—91no—
140C22⋊C8—91no—
141C22⋊C8C3×C3⋊D311yescharacteristic
142C33#32411yescharacteristic, complemented, Sylow
143C23×C3—24no—
144C23×C3C3×C4×D314yes—
145C23×C3C4×C3⋊D311yescharacteristic
146C2×C3×C4—24no—
147C2×C3×C4—24no—
148C2×C3×C4—44no—
149C2×C3×C4—41no—
150C2×C3×C4—48no—
151C2×C3×C4—92no—
152C2×C3×C4—181no—
153C2×C3×C4—181no—
154C2×C3×C4—181no—
155C2×C3×C4—361no—
156C2×C3×C4C3×D1214yes—
157C2×C3×C4C3×#6714yes—
158C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
159C2×C3×C4#63811yescharacteristic
160C2×Dic3—38no—
161C2×Dic3—64no—
162C2×Dic3—64no—
163C2×Dic3—64no—
164C2×Dic3—124no—
165C3×C8—48no—
166C3×C8—44no—
167C3×C8—41no—
168C3×C8—361no—
169C3⋊C8—124no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—24no—
173C2×C32—24no—
174C2×C32—24no—
175C2×C32—24no—
176C2×C32—24no—
177C2×C32—44no—
178C2×C32—44no—
179C2×C32#80614yes—
180C2×C32(C2×D3)⋊C814yes—
181C2×C32#82414yes—
182C2×C32C3×C22⋊C811yescharacteristic
183C2×C32C3×#9911yescharacteristic
184C2×C32C3×#10111yescharacteristic
185C22×C4—91no—
186C22×C4—91no—
187C22×C4C2×C3×C3⋊D311yescharacteristic
188C2×C8—21no—
189C2×C8—181no—
190C4⋊C4—181no—
191C4⋊C4—181no—
192C22×C3—24no—
193C22×C3—24no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—24no—
197C22×C3—24no—
198C22×C3—24no—
199C22×C3—44no—
200C22×C3—44no—
201C22×C3C3×D3×C814yes—
202C22×C3C3×ΓL(1,𝔽25)14yes—
203C22×C3C3×(C2×D3)⋊C414yes—
204C22×C3C8×C3⋊D311yescharacteristic
205C22×C3#374311yescharacteristic
206C22×C3#375211yescharacteristic
207C3×C4—28no—
208C3×C4—24no—
209C3×C4—24no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—44no—
213C3×C4—182no—
214C3×C4—361no—
215Dic3—68no—
216Dic3—124no—
217C32—24no—
218C32#446914yescomplemented
219C32C3×#32411yescharacteristic, complemented
220C23C3×C4×C3⋊D311yescharacteristic
221C2×C4—41no—
222C2×C4—92no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C4—181no—
226C2×C4—361no—
227C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
228C2×C4C3×#63811yescharacteristic
229C8—41no—
230C8—361no—
231C2×C3—24no—
232C2×C3—24no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—24no—
236C2×C3—24no—
237C2×C3—24no—
238C2×C3—44no—
239C2×C3—44no—
240C2×C3C3×#80614yes—
241C2×C3C3×(C2×D3)⋊C814yes—
242C2×C3C3×#82414yes—
243C2×C3#6363311yescharacteristic
244C2×C3#6364711yescharacteristic
245C2×C3#6365111yescharacteristic
246C22—21no—
247C22—21no—
248C22C3×C8×C3⋊D311yescharacteristic
249C22C3×#374311yescharacteristic
250C22C3×#375211yescharacteristic
251C4—21no—
252C4—21no—
253C4—182no—
254C4—361no—
255C3—24no—
256C3C3×#446914yescomplemented
257C3#1058933511yescharacteristic, direct product factor
258C2—21no—
259C2—21no—
260C2C3×#6363311yescharacteristic
261C2C3×#6364711yescharacteristic
262C2C3×#6365111yescharacteristic
subgroupG/H
centerC22×C3#3752
commutatorC22×C32C2×C3×C8
FrattiniC22×C4C2×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
cosocleC22×C32×C4C22×C3
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#324
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #10589335C3direct (N × H)
28. C2×C3×C3⋊Dic3C8semidirect
81. C2×C3⋊Dic3C3×C8semidirect
142. C33#324semidirect
218. C32#4469semidirect
219. C32C3×#324semidirect
256. C3C3×#4469semidirect
257. C3#10589335direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C33×C22⋊C82
C3×#636572
#105893353
C3×#44693