Finite Group Databasebeta

← C3×#10593768 #13173239

Subgroups — C3×#10593768 #13173239

Order 1728.

424 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 100 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×#64286C212yesmaximal
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
3C3×#63769C212yesmaximal, complemented
4C3×#63768C212yesmaximal, complemented
5#10593768C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×C22⋊#157—22no—
8C3×C4⋊#157C2214yes—
9C3×C4×#157—22no—
10C2×C33×C8C411yescharacteristic, complemented
11C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
12C3×C8×C3⋊D3C411yescharacteristic, complemented
13C3×C8×C3⋊D3C411yescharacteristic, complemented
14C3×C8×C3⋊D3C2211yescharacteristic
15C3×C8×C3⋊D3C2211yescharacteristic
16#64286C2×C312yes—
17C2×C8×C3⋊D3C2×C311yescharacteristic
18C2×C3×D3×C8—62no—
19#63769C2×C312yescomplemented
20#63768C2×C312yescomplemented
21C22×C3×C3⋊D3D411yescharacteristic
22C2×C3×#157—24no—
23C2×C33×C4C2×C411yescharacteristic
24C2×C3×C3⋊Dic3Q811yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3Q811yescharacteristic
28C3×C4×C3⋊D3C2311yescharacteristic
29C33×C8C2×C411yescharacteristic
30C33×C8C2×C411yescharacteristic
31C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33C3×#427—91nomaximal
34C2×C4×C3⋊D3C22×C311yescharacteristic
35C2×C3×C4×D3—62no—
36C22⋊#157—22no—
37C4⋊#157C22×C314yes—
38C4×#157—22no—
39C2×C32×C8—22no—
40C2×C32×C8—42no—
41C2×C32×C8C3×C411yescharacteristic, complemented
42C2×C3⋊(C3⋊C8)C3×C411yescharacteristic, complemented
43C8×C3⋊D3C3×C411yescharacteristic, complemented
44C8×C3⋊D3C3×C411yescharacteristic, complemented
45C8×C3⋊D3C22×C311yescharacteristic
46C8×C3⋊D3C22×C311yescharacteristic
47C2×C3×C3⋊C8—62no—
48C3×D3×C8—62no—
49C3×D3×C8—62no—
50C3×D3×C8—62no—
51C3×D3×C8—62no—
52C22×C33C4⋊C411yescharacteristic
53C2×C3×C3⋊D3—22no—
54C2×C3×C3⋊D3C4⋊C411yescharacteristic
55C2×C3×C3⋊D3C2×D411yescharacteristic
56C3×#157—44no—
57C33×C4C4⋊C411yescharacteristic
58C33×C4C22×C411yescharacteristic
59C3×C3⋊Dic3C4⋊C411yescharacteristic
60C3×C3⋊Dic3C2×Q811yescharacteristic
61C22×C3×C8—91no—
62C3×#117—92no—
63C2×D3×C8—62no—
64C3×#107—92no—
65C3×#106—92no—
66C22×C3⋊D3C3×D411yescharacteristic
67C22×C3×D3—62no—
68C2×#157—24no—
69C2×C32×C4—22no—
70C2×C32×C4—42no—
71C2×C32×C4C2×C3×C411yescharacteristic
72C2×C3⋊Dic3C3×Q811yescharacteristic
73C4×C3⋊D3C2×C3×C411yescharacteristic
74C4×C3⋊D3C3×D411yescharacteristic
75C4×C3⋊D3C3×Q811yescharacteristic
76C4×C3⋊D3C23×C311yescharacteristic
77C2×C3×Dic3—62no—
78C3×C4×D3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C32×C8—22no—
83C32×C8—22no—
84C32×C8—42no—
85C32×C8—42no—
86C32×C8C2×C3×C411yescharacteristic
87C32×C8C2×C3×C411yescharacteristic
88C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
89C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
90C3×C3⋊C8—62no—
91C3×C3⋊C8—62no—
92#427—91noSylow
93C2×C33—21no—
94C2×C33C2×C4⋊C411yescharacteristic
95C3×C3⋊D3—21no—
96C3×C3⋊D3#13512yes—
97C22×C3×C4—91no—
98C2×C4×D3—62no—
99C2×C3×C8—22no—
100C2×C3×C8—42no—
101C2×C3×C8—91no—
102C2×C3×C8—91no—
103C2×C3×C8—91no—
104C2×C3×C8—91no—
105C2×C3×C8—91no—
106C2×C3×C8#15711yescharacteristic, complemented
107C3×C4⋊C4—94no—
108C3×C22⋊C4—182no—
109C3×C42—182no—
110C2×C3⋊C8—62no—
111D3×C8—62no—
112D3×C8—62no—
113D3×C8—62no—
114D3×C8—62no—
115C22×C32—22no—
116C22×C32—42no—
117C22×C32C3×C4⋊C411yescharacteristic
118C2×C3⋊D3—22no—
119C2×C3⋊D3C3×C4⋊C411yescharacteristic
120C2×C3⋊D3C2×C3×D411yescharacteristic
121C2×C3×D3—62no—
122C2×C3×D3—62no—
123C2×C3×D3—68no—
124#157—44no—
125C32×C4—22no—
126C32×C4—22no—
127C32×C4—42no—
128C32×C4—42no—
129C32×C4C3×C4⋊C411yescharacteristic
130C32×C4C22×C3×C411yescharacteristic
131C3⋊Dic3C3×C4⋊C411yescharacteristic
132C3⋊Dic3C2×C3×Q811yescharacteristic
133C3×Dic3—62no—
134C3×Dic3—62no—
135C22×C8—91no—
136#117—92no—
137#107—92no—
138#106—92no—
139C33#42711yescharacteristic, complemented, Sylow
140C23×C3—91no—
141C22×D3—62no—
142C2×C3×C4—22no—
143C2×C3×C4—42no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—184no—
150C2×C3×C4C2×#15711yescharacteristic
151C2×Dic3—62no—
152C4×D3—62no—
153C4×D3—62no—
154C4×D3—62no—
155C4×D3—62no—
156C3×C8—22no—
157C3×C8—22no—
158C3×C8—42no—
159C3×C8—42no—
160C3×C8—91no—
161C3×C8—91no—
162C3×C8C2×#15711yescharacteristic
163C3×C8C2×#15711yescharacteristic
164C3⋊C8—62no—
165C3⋊C8—62no—
166C2×C32—21no—
167C2×C32—24no—
168C2×C32—22no—
169C2×C32—44no—
170C2×C32—42no—
171C2×C32C2×C3×C4⋊C411yescharacteristic
172C3⋊D3—21no—
173C3⋊D3C3×#13512yes—
174C3×D3—64no—
175C3×D3—64no—
176C22×C4—91no—
177C2×C8—91no—
178C2×C8—91no—
179C2×C8—91no—
180C2×C8—91no—
181C2×C8—91no—
182C2×C8C3×#15711yescharacteristic, complemented
183C4⋊C4—94no—
184C22⋊C4—182no—
185C42—182no—
186C22×C3—22no—
187C22×C3—42no—
188C22×C3—91no—
189C22×C3—91no—
190C22×C3—182no—
191C22×C3C4⋊#15711yescharacteristic
192C2×D3—62no—
193C2×D3—62no—
194C2×D3—68no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—42no—
198C3×C4—42no—
199C3×C4—91no—
200C3×C4—91no—
201C3×C4—364no—
202C3×C4C4⋊#15711yescharacteristic
203C3×C4C22×#15711yescharacteristic
204Dic3—62no—
205Dic3—62no—
206C32—22no—
207C32—42no—
208C32C3×#42711yescharacteristic, complemented
209C23—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—184no—
216C2×C4C2×C3×#15711yescharacteristic
217C8—91no—
218C8—91no—
219C8C2×C3×#15711yescharacteristic
220C8C2×C3×#15711yescharacteristic
221C2×C3—24no—
222C2×C3—21no—
223C2×C3—22no—
224C2×C3—44no—
225C2×C3—42no—
226C2×C3—92no—
227C2×C3—181no—
228C2×C3C2×C4⋊#15711yescharacteristic
229D3—64no—
230D3—64no—
231C22—91no—
232C22—91no—
233C22—182no—
234C22C3×C4⋊#15711yescharacteristic
235C4—91no—
236C4—91no—
237C4—364no—
238C4C3×C4⋊#15711yescharacteristic
239C4C22×C3×#15711yescharacteristic
240C3—22no—
241C3—42no—
242C3#1059376811yescharacteristic, direct product factor
243C2—21no—
244C2—92no—
245C2—181no—
246C2C2×C3×C4⋊#15711yescharacteristic
subgroupG/H
centerC3×C4C4⋊#157
commutatorC32×C4C22×C3×C4
FrattiniC4C22×C3×#157
FittingC2×C33×C8C4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C8#157
hypocenterC32C3×#427
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63769C2semidirect
4. C3×#63768C2semidirect
5. #10593768C3direct (N × H)
10. C2×C33×C8C4semidirect
11. C2×C3×C3⋊(C3⋊C8)C4semidirect
12. C3×C8×C3⋊D3C4semidirect
13. C3×C8×C3⋊D3C4semidirect
19. #63769C2×C3semidirect
20. #63768C2×C3semidirect
41. C2×C32×C8C3×C4semidirect
42. C2×C3⋊(C3⋊C8)C3×C4semidirect
43. C8×C3⋊D3C3×C4semidirect
44. C8×C3⋊D3C3×C4semidirect
106. C2×C3×C8#157semidirect
139. C33#427semidirect
182. C2×C8C3×#157semidirect
208. C32C3×#427semidirect
242. C3#10593768direct (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×C8×C3⋊D32
C3×#637692
C3×#637682
#105937683
C3×#4279