Finite Group Databasebeta

← C2×C3×#454 #84880

Subgroups — C2×C3×#454 #84880

Order 384.

728 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 216 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×C3×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×AGL(1,ℤ/8)C212yesmaximal, complemented
3C2×C3×#124C211yesmaximal, characteristic, complemented
4C3×#454C218yesmaximal, direct product factor
5C2×C3×#104C212yesmaximal, complemented
6C2×C3×#101C211yesmaximal, characteristic, complemented
7C2×#454C311yesmaximal, characteristic, direct product factor, Sylow
8C3×2−1+4—24no—
9C3×2−1+4—22no—
10C3×2−1+4C2214yescomplemented
11C2×C3×#41—22no—
12C2×C3×#41—22no—
13C2×C3×#41C2212yes—
14C22×C3×Q8—22no—
15C22×C3×Q8C2211yescharacteristic
16C22×C3×D4C2211yescharacteristic
17C3×AGL(1,ℤ/8)—24no—
18C3×AGL(1,ℤ/8)C2218yescomplemented
19C2×C3×SD16—22no—
20C2×C3×D8—22no—
21C2×C3×M16C2212yes—
22C3×#124—22no—
23C3×#124C2214yescomplemented
24C2×C3×C22⋊C4—22no—
25C2×C3×C42C2211yescharacteristic
26C3×#104C2218yescomplemented
27C3×#101C2214yescomplemented
28C2×2−1+4C2×C311yescharacteristic, complemented
29C2×AGL(1,ℤ/8)C2×C312yescomplemented
30C2×#124C2×C311yescharacteristic, complemented
31#454C2×C318yesdirect product factor
32C2×#104C2×C312yescomplemented
33C2×#101C2×C311yescharacteristic, complemented
34C24×C3—21no—
35C3×#41—28no—
36C3×#41—28no—
37C3×#41—28no—
38C3×#41—28no—
39C3×#41—22no—
40C3×#41—22no—
41C3×#41D414yes—
42C3×#41C2314yes—
43C2×C3×Q8—22no—
44C2×C3×Q8—28no—
45C2×C3×Q8—28no—
46C2×C3×Q8—22no—
47C2×C3×Q8—22no—
48C2×C3×Q8D412yes—
49C2×C3×Q8D412yes—
50C2×C3×Q8C2312yes—
51C2×C3×D4—22no—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4—21no—
56C2×C3×D4D412yes—
57C2×C3×D4C2312yes—
58C22×C3×C4—24no—
59C22×C3×C4—22no—
60C22×C3×C4—21no—
61C22×C3×C4D412yes—
62C22×C3×C4C2311yescharacteristic
63C3×SD16—28no—
64C3×D8—28no—
65C3×M16—22no—
66C3×M16C2314yes—
67C2×C3×C8—22no—
68C3×C22⋊C4—28no—
69C3×C42—21no—
70C3×C42C2312yes—
712−1+4—24no—
722−1+4—22no—
732−1+4C22×C314yescomplemented
74C2×#41—22no—
75C2×#41—22no—
76C2×#41C22×C312yes—
77C22×Q8—22no—
78C22×Q8C22×C311yescharacteristic
79C22×D4C22×C311yescharacteristic
80AGL(1,ℤ/8)—24no—
81AGL(1,ℤ/8)C22×C318yescomplemented
82C2×SD16—22no—
83C2×D8—22no—
84C2×M16C22×C312yes—
85#124—22no—
86#124C22×C314yescomplemented
87C2×C22⋊C4—22no—
88C2×C42C22×C311yescharacteristic
89#104C22×C318yescomplemented
90#101C22×C314yescomplemented
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—44no—
94C23×C3—41no—
95C23×C3—41no—
96C23×C3C2×D411yescharacteristic
97C3×Q8—28no—
98C3×Q8—22no—
99C3×Q8—24no—
100C3×Q8—24no—
101C3×Q8C2×D414yes—
102C3×D4—28no—
103C3×D4—24no—
104C3×D4—24no—
105C3×D4—22no—
106C3×D4—24no—
107C3×D4—42no—
108C3×D4C2×D414yes—
109C2×C3×C4—216no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—24no—
114C2×C3×C4—24no—
115C2×C3×C4—22no—
116C2×C3×C4—42no—
117C2×C3×C4C2×D414yes—
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2411yescharacteristic
122C3×C8—24no—
123C24—21no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41C3×D414yes—
131#41C23×C314yes—
132C2×Q8—22no—
133C2×Q8—28no—
134C2×Q8—28no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C3×D412yes—
138C2×Q8C3×D412yes—
139C2×Q8C23×C312yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4C3×D412yes—
146C2×D4C23×C312yes—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4C3×D412yes—
151C22×C4C23×C311yescharacteristic
152SD16—28no—
153D8—28no—
154M16—22no—
155M16C23×C314yes—
156C2×C8—22no—
157C22⋊C4—28no—
158C42—21no—
159C42C23×C312yes—
160C22×C3—22no—
161C22×C3—24no—
162C22×C3—44no—
163C22×C3—44no—
164C22×C3—42no—
165C22×C3—82no—
166C22×C3—82no—
167C22×C3#12011yescharacteristic
168C22×C3#12011yescharacteristic
169C22×C3C22×D411yescharacteristic
170C3×C4—24no—
171C3×C4—24no—
172C3×C4—42no—
173C3×C4#12012yes—
174C3×C4C22×D412yes—
175C23—22no—
176C23—22no—
177C23—44no—
178C23—41no—
179C23—41no—
180C23C2×C3×D411yescharacteristic
181Q8—28no—
182Q8—22no—
183Q8—24no—
184Q8—24no—
185Q8C2×C3×D414yes—
186D4—28no—
187D4—24no—
188D4—24no—
189D4—22no—
190D4—24no—
191D4—42no—
192D4C2×C3×D414yes—
193C2×C4—216no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4C2×C3×D414yes—
202C2×C4C2×C3×D411yescharacteristic
203C2×C4C2×C3×D412yes—
204C2×C4C2×C3×D412yes—
205C2×C4C24×C311yescharacteristic
206C8—24no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3—44no—
210C2×C3—82no—
211C2×C3#45412yesdirect product factor
212C2×C3C2×#12011yescharacteristic
213C22—22no—
214C22—24no—
215C22—44no—
216C22—44no—
217C22—42no—
218C22—82no—
219C22—82no—
220C22C3×#12011yescharacteristic
221C22C3×#12011yescharacteristic
222C22C22×C3×D411yescharacteristic
223C4—24no—
224C4—24no—
225C4—42no—
226C4C3×#12012yes—
227C4C22×C3×D412yes—
228C3C2×#45411yescharacteristic, direct product factor, Sylow
229C2—21no—
230C2—21no—
231C2—44no—
232C2—82no—
233C2C3×#45412yesdirect product factor
234C2C2×C3×#12011yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC2×C4C24×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4C2semidirect
2. C2×C3×AGL(1,ℤ/8)C2semidirect
3. C2×C3×#124C2semidirect
4. C3×#454C2direct (N × H)
5. C2×C3×#104C2semidirect
6. C2×C3×#101C2semidirect
7. C2×#454C3direct (N × H)
10. C3×2−1+4C22semidirect
18. C3×AGL(1,ℤ/8)C22semidirect
23. C3×#124C22semidirect
26. C3×#104C22semidirect
27. C3×#101C22semidirect
28. C2×2−1+4C2×C3semidirect
29. C2×AGL(1,ℤ/8)C2×C3semidirect
30. C2×#124C2×C3semidirect
31. #454C2×C3direct (N × H)
32. C2×#104C2×C3semidirect
33. C2×#101C2×C3semidirect
73. 2−1+4C22×C3semidirect
81. AGL(1,ℤ/8)C22×C3semidirect
86. #124C22×C3semidirect
89. #104C22×C3semidirect
90. #101C22×C3semidirect
211. C2×C3#454direct (N × H)
228. C3C2×#454direct (N × H)
233. C2C3×#454direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+42
C2×C3×AGL(1,ℤ/8)2
C2×C3×#1242
C3×#4542
C2×C3×#1042
C2×C3×#1012
C2×#4543