Finite Group Databasebeta

← C3×#10593749 #13173220

Subgroups — C3×#10593749 #13173220

Order 1728.

508 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 112 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×#3787C212yesmaximal, complemented
2C3×#64102C211yesmaximal, characteristic
3C3×#63778C212yesmaximal, complemented
4C3×C4⋊#622C211yesmaximal, characteristic, complemented
5C3×C4×#622C211yesmaximal, characteristic, complemented
6#10593749C311yesmaximal, characteristic, direct product factor
7C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
8C2×C3×C4×C3⋊D3C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2212yes—
11C2×C3×#622C2211yescharacteristic
12C3×#3787—24no—
13C33×C4⋊C4C411yescharacteristic
14C3×#3752C412yes—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×#3787C2×C312yescomplemented
17#64102C2×C311yescharacteristic
18C3×#893—62no—
19#63778C2×C312yescomplemented
20C4⋊#622C2×C311yescharacteristic, complemented
21C4×#622C2×C311yescharacteristic, complemented
22C22×C3×C3⋊D3C2×C412yes—
23C2×C33×C4C2×C412yes—
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4⋊(C3⋊D3)C814yescomplemented
28C3×C4×C3⋊D3—22no—
29C3×C4×C3⋊D3—22no—
30C3×#622—22no—
31C3×#622—21no—
32C3×#622D412yescomplemented
33C3×C8×D4—91nomaximal
34C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
35C2×C4×C3⋊D3C22×C312yes—
36C2×C3×D12—62no—
37C2×C3×C4×D3—64no—
38C2×#622C22×C312yes—
39C2×#622C22×C311yescharacteristic
40C2×#622C22×C311yescharacteristic
41#3787—24no—
42C32×C4⋊C4—22no—
43C32×C4⋊C4—42no—
44C32×C4⋊C4C3×C411yescharacteristic
45#3752C3×C412yes—
46C4×C3⋊Dic3C22×C311yescharacteristic
47C3×(C2×D3)⋊C4—64no—
48C3×C4×Dic3—62no—
49C22×C33C22×C411yescharacteristic
50C2×C3×C3⋊D3—22no—
51C2×C3×C3⋊D3—22no—
52C2×C3×C3⋊D3C2×C814yescomplemented
53C33×C4—22no—
54C33×C4C2×C812yescomplemented
55C3×C3⋊Dic3—21no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3#4111yescharacteristic
58C22×C3×C8—92no—
59C3×C4×D4—91no—
60#893—62no—
61C3×C4⋊C8—91no—
62C3×C22⋊C8—92no—
63C3×C4×C8—91no—
64C22×C3⋊D3C2×C3×C412yes—
65C22×C3×D3—64no—
66C2×C32×C4—22no—
67C2×C32×C4—24no—
68C2×C32×C4—42no—
69C2×C32×C4—44no—
70C2×C32×C4C2×C3×C412yes—
71C2×C32×C4C2×C3×C411yescharacteristic
72C2×C3⋊Dic3C2×C3×C411yescharacteristic
73C2×C3⋊Dic3C23×C311yescharacteristic
74C4⋊(C3⋊D3)C3×C814yescomplemented
75C4×C3⋊D3—22no—
76C4×C3⋊D3—22no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C3×D12—68no—
80C3×C4×D3—124no—
81C3×C4×D3—124no—
82#622—22no—
83#622—21no—
84#622C3×D412yescomplemented
85C8×D4—91noSylow
86C2×C33C4×D411yescharacteristic
87C2×C33C22×C811yescharacteristic
88C2×C33#13111yescharacteristic
89C3×C3⋊D3—24no—
90C2×C3×D4—91no—
91C22×C3×C4—92no—
92C2×D12—62no—
93C2×C4×D3—64no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C2×C3×C8—92no—
97C2×C3×C8—184no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4#15711yescharacteristic
101C3×C22⋊C4—92no—
102C3×C42—91no—
103(C2×D3)⋊C4—64no—
104C4×Dic3—62no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32C22×C3×C411yescharacteristic
108C2×C3⋊D3—22no—
109C2×C3⋊D3—22no—
110C2×C3⋊D3C2×C3×C814yescomplemented
111C2×C3×D3—68no—
112C2×C3×D3—124no—
113C2×C3×D3—124no—
114C32×C4—24no—
115C32×C4—22no—
116C32×C4—44no—
117C32×C4—44no—
118C32×C4—84no—
119C32×C4C2×C3×C812yescomplemented
120C3⋊Dic3—21no—
121C3⋊Dic3C2×C3×D411yescharacteristic
122C3⋊Dic3C3×#4111yescharacteristic
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—122no—
126C22×C8—92no—
127C4×D4—91no—
128C4⋊C8—91no—
129C22⋊C8—92no—
130C4×C8—91no—
131C33C8×D411yescharacteristic, complemented, Sylow
132C23×C3—92no—
133C22×D3—64no—
134C3×D4—94no—
135C2×C3×C4—22no—
136C2×C3×C4—24no—
137C2×C3×C4—42no—
138C2×C3×C4—44no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—182no—
142C2×C3×C4—182no—
143C2×C3×C4C2×#15712yes—
144C2×C3×C4C2×#15711yescharacteristic
145C2×Dic3—62no—
146C2×Dic3—62no—
147D12—68no—
148C4×D3—124no—
149C4×D3—124no—
150C3×C8—92no—
151C3×C8—181no—
152C3×C8—182no—
153C2×C32—22no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—42no—
157C2×C32—42no—
158C2×C32—42no—
159C2×C32C3×C4×D411yescharacteristic
160C2×C32C22×C3×C811yescharacteristic
161C2×C32C3×#13111yescharacteristic
162C3⋊D3—24no—
163C3×D3—128no—
164C2×D4—91no—
165C22×C4—92no—
166C2×C8—91no—
167C2×C8—91no—
168C2×C8—92no—
169C2×C8—184no—
170C4⋊C4C3×#15711yescharacteristic
171C22⋊C4—92no—
172C42—91no—
173C22×C3—22no—
174C22×C3—42no—
175C22×C3—94no—
176C22×C3—182no—
177C22×C3—182no—
178C22×C3C22×#15711yescharacteristic
179C2×D3—68no—
180C2×D3—124no—
181C2×D3—124no—
182C3×C4—22no—
183C3×C4—24no—
184C3×C4—44no—
185C3×C4—44no—
186C3×C4—84no—
187C3×C4—91no—
188C3×C4—91no—
189C3×C4—181no—
190C3×C4#378712yescomplemented
191Dic3—62no—
192Dic3—62no—
193Dic3—122no—
194C32—22no—
195C32—42no—
196C32C3×C8×D411yescharacteristic, complemented
197C23—92no—
198D4—94no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—182no—
202C2×C4—182no—
203C2×C4C2×C3×#15712yes—
204C2×C4C2×C3×#15711yescharacteristic
205C8—92no—
206C8—182no—
207C8—181no—
208C2×C3—22no—
209C2×C3—22no—
210C2×C3—22no—
211C2×C3—42no—
212C2×C3—42no—
213C2×C3—42no—
214C2×C3—184no—
215C2×C3C2×#378711yescharacteristic
216C2×C3D4×#15711yescharacteristic
217C2×C3#6428911yescharacteristic
218D3—128no—
219C22—94no—
220C22—182no—
221C22—182no—
222C22C22×C3×#15711yescharacteristic
223C4—22no—
224C4—91no—
225C4—91no—
226C4—181no—
227C4C3×#378712yescomplemented
228C3—22no—
229C3—42no—
230C3#1059374911yescharacteristic, direct product factor
231C2—184no—
232C2C2×C3×#378711yescharacteristic
233C2C3×D4×#15711yescharacteristic
234C2C3×#6428911yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC2×C32C22×C3×C8
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socleC22×C33C22×C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×C8×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3787C2semidirect
3. C3×#63778C2semidirect
4. C3×C4⋊#622C2semidirect
5. C3×C4×#622C2semidirect
6. #10593749C3direct (N × H)
16. C2×#3787C2×C3semidirect
19. #63778C2×C3semidirect
20. C4⋊#622C2×C3semidirect
21. C4×#622C2×C3semidirect
27. C3×C4⋊(C3⋊D3)C8semidirect
32. C3×#622D4semidirect
52. C2×C3×C3⋊D3C2×C8semidirect
54. C33×C4C2×C8semidirect
74. C4⋊(C3⋊D3)C3×C8semidirect
84. #622C3×D4semidirect
110. C2×C3⋊D3C2×C3×C8semidirect
119. C32×C4C2×C3×C8semidirect
131. C33C8×D4semidirect
190. C3×C4#3787semidirect
196. C32C3×C8×D4semidirect
227. C4C3×#3787semidirect
230. C3#10593749direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C3×#641022
C3×#637782
C3×C4⋊#6222
C3×C4×#6222
#105937493
C3×C8×D49