Finite Group Databasebeta

← C3×#10593775 #13173245

Subgroups — C3×#10593775 #13173245

Order 1728.

436 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C3×#64286C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4C3×#63767C212yesmaximal, complemented
5C3×C8×#157C212yesmaximal, complemented
6#10593775C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C3×C22⋊#157C412yes—
9C2×C3×#622C412yescomplemented
10C3×C4⋊#157C412yes—
11C3×C4×#157C2212yes—
12C3×#3787C412yescomplemented
13C3×#3787C2211yescharacteristic
14C3×#3787C2211yescharacteristic
15C33×M16C411yescharacteristic, complemented
16C3×#3748C411yescharacteristic, complemented
17C3×#3743C412yescomplemented
18C3×C8×C3⋊D3C2212yes—
19#64286C2×C311yescharacteristic
20C2×#3787C2×C311yescharacteristic
21M16×C3⋊D3C2×C311yescharacteristic
22C3×D3×M16—62no—
23#63767C2×C312yescomplemented
24C8×#157C2×C312yescomplemented
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C2×C3×#157C2×C414yes—
27C2×C33×C4C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2311yescharacteristic
33C3×#622C2×C412yes—
34C3×#622C2×C412yes—
35C33×C8C2×C412yes—
36C3×C3⋊(C3⋊C8)C2×C412yes—
37C3×#405—91nomaximal
38C2×C4×C3⋊D3C22×C311yescharacteristic
39C2×C3×C4×D3—62no—
40C22⋊#157C3×C412yes—
41C2×#622C3×C412yescomplemented
42C4⋊#157C3×C412yes—
43C4×#157C22×C312yes—
44#3787C3×C412yescomplemented
45#3787C22×C311yescharacteristic
46#3787C22×C311yescharacteristic
47C32×M16—22no—
48C32×M16—42no—
49C32×M16C3×C411yescharacteristic, complemented
50#3748C3×C411yescharacteristic, complemented
51#3743C3×C412yescomplemented
52C8×C3⋊D3C22×C312yes—
53C3×#208—62no—
54C3×ΓL(1,𝔽25)—64no—
55C3×D3×C8—64no—
56C22×C33C4211yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3C4211yescharacteristic
59C2×C3×C3⋊D3C22×C411yescharacteristic
60C3×#157—24no—
61C33×C4C4211yescharacteristic
62C33×C4C22×C411yescharacteristic
63C3×C3⋊Dic3C4211yescharacteristic
64C3×C3⋊Dic3C22×C411yescharacteristic
65C2×C3×M16—91no—
66C22×C3×C8—91no—
67C3×#117—91no—
68D3×M16—62no—
69C3×#97—92no—
70C3×C4×C8—92no—
71C22×C3⋊D3C2×C3×C411yescharacteristic
72C22×C3×D3—62no—
73C2×#157C2×C3×C414yes—
74C2×C32×C4—22no—
75C2×C32×C4—42no—
76C2×C32×C4C2×C3×C411yescharacteristic
77C2×C3⋊Dic3C2×C3×C411yescharacteristic
78C4×C3⋊D3C2×C3×C411yescharacteristic
79C4×C3⋊D3C2×C3×C411yescharacteristic
80C4×C3⋊D3C2×C3×C411yescharacteristic
81C4×C3⋊D3C23×C311yescharacteristic
82C2×C3×Dic3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85C3×C4×D3—62no—
86C3×C4×D3—62no—
87#622C2×C3×C412yes—
88#622C2×C3×C412yes—
89C32×C8—24no—
90C32×C8—44no—
91C32×C8C2×C3×C412yes—
92C3⋊(C3⋊C8)C2×C3×C412yes—
93C3×C3⋊C8—64no—
94#405—91noSylow
95C2×C33—21no—
96C2×C33C2×C4211yescharacteristic
97C3×C3⋊D3—21no—
98C3×C3⋊D3#13112yes—
99C22×C3×C4—91no—
100C2×C4×D3—62no—
101C3×M16—22no—
102C3×M16—42no—
103C3×M16—91no—
104C3×M16—92no—
105C3×M16#15711yescharacteristic, complemented
106C2×C3×C8—91no—
107C2×C3×C8—91no—
108C2×C3×C8—92no—
109C2×C3×C8—92no—
110C2×C3×C8—92no—
111C3×C4⋊C4—92no—
112C3×C22⋊C4—92no—
113C3×C42—92no—
114#208—62no—
115ΓL(1,𝔽25)—64no—
116D3×C8—64no—
117C22×C32—22no—
118C22×C32—42no—
119C22×C32C3×C4211yescharacteristic
120C2×C3⋊D3—22no—
121C2×C3⋊D3C3×C4211yescharacteristic
122C2×C3⋊D3C22×C3×C411yescharacteristic
123C2×C3×D3—62no—
124C2×C3×D3—62no—
125C2×C3×D3—124no—
126#157—24no—
127C32×C4—22no—
128C32×C4—22no—
129C32×C4—42no—
130C32×C4—42no—
131C32×C4C3×C4211yescharacteristic
132C32×C4C22×C3×C411yescharacteristic
133C3⋊Dic3C3×C4211yescharacteristic
134C3⋊Dic3C22×C3×C411yescharacteristic
135C3×Dic3—62no—
136C3×Dic3—62no—
137C2×M16—91no—
138C22×C8—91no—
139#117—91no—
140#97—92no—
141C4×C8—92no—
142C33#40511yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C22×D3—62no—
145C2×C3×C4—22no—
146C2×C3×C4—42no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—94no—
153C2×C3×C4C2×#15711yescharacteristic
154C2×Dic3—62no—
155C4×D3—62no—
156C4×D3—62no—
157C4×D3—62no—
158C4×D3—62no—
159C3×C8—24no—
160C3×C8—44no—
161C3×C8—92no—
162C3×C8—92no—
163C3×C8—92no—
164C3×C8C2×#15712yes—
165C3⋊C8—64no—
166C2×C32—22no—
167C2×C32—21no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—82no—
171C2×C32C2×C3×C4211yescharacteristic
172C3⋊D3—21no—
173C3⋊D3C3×#13112yes—
174C3×D3—64no—
175C3×D3—122no—
176C22×C4—91no—
177M16—91no—
178M16—92no—
179M16C3×#15711yescharacteristic, complemented
180C2×C8—91no—
181C2×C8—91no—
182C2×C8—92no—
183C2×C8—92no—
184C2×C8—92no—
185C4⋊C4—92no—
186C22⋊C4—92no—
187C42—92no—
188C22×C3—22no—
189C22×C3—42no—
190C22×C3—91no—
191C22×C3—91no—
192C22×C3—182no—
193C22×C3C4×#15711yescharacteristic
194C2×D3—62no—
195C2×D3—62no—
196C2×D3—124no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4—91no—
202C3×C4—91no—
203C3×C4—184no—
204C3×C4C4×#15711yescharacteristic
205C3×C4C22×#15711yescharacteristic
206Dic3—62no—
207Dic3—62no—
208C32—22no—
209C32—42no—
210C32C3×#40511yescharacteristic, complemented
211C23—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—94no—
218C2×C4C2×C3×#15711yescharacteristic
219C8—92no—
220C8—92no—
221C8—92no—
222C8C2×C3×#15712yes—
223C2×C3—21no—
224C2×C3—22no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—82no—
228C2×C3—92no—
229C2×C3—181no—
230C2×C3C2×C4×#15711yescharacteristic
231D3—64no—
232D3—122no—
233C22—91no—
234C22—91no—
235C22—182no—
236C22C3×C4×#15711yescharacteristic
237C4—91no—
238C4—91no—
239C4—184no—
240C4C3×C4×#15711yescharacteristic
241C4C22×C3×#15711yescharacteristic
242C3—22no—
243C3—42no—
244C3#1059377511yescharacteristic, direct product factor
245C2—21no—
246C2—92no—
247C2—181no—
248C2C2×C3×C4×#15711yescharacteristic
subgroupG/H
centerC3×C4C22×#157
commutatorC2×C32C2×C3×C42
FrattiniC4C22×C3×#157
FittingC33×M16C4
socleC2×C33C2×C42
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypercenterC3×M16#157
hypocenterC32C3×#405
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
4. C3×#63767C2semidirect
5. C3×C8×#157C2semidirect
6. #10593775C3direct (N × H)
9. C2×C3×#622C4semidirect
12. C3×#3787C4semidirect
15. C33×M16C4semidirect
16. C3×#3748C4semidirect
17. C3×#3743C4semidirect
23. #63767C2×C3semidirect
24. C8×#157C2×C3semidirect
41. C2×#622C3×C4semidirect
44. #3787C3×C4semidirect
49. C32×M16C3×C4semidirect
50. #3748C3×C4semidirect
51. #3743C3×C4semidirect
105. C3×M16#157semidirect
142. C33#405semidirect
179. M16C3×#157semidirect
210. C32C3×#405semidirect
244. C3#10593775direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642862
C2×C3×#37872
C3×M16×C3⋊D32
C3×#637672
C3×C8×#1572
#105937753
C3×#4059