Finite Group Databasebeta

← C2×C3×#451 #84863

Subgroups — C2×C3×#451 #84863

Order 384.

740 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 248 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
1C23×C3×Q8C211yesmaximal, characteristic
2C22×C3×Q16C212yesmaximal, complemented
3C2×C3×C22⋊Q8C211yesmaximal, characteristic
4C3×#451C218yesmaximal, direct product factor
5C2×C3×#103C212yesmaximal, complemented
6C2×C3×C22⋊C8C211yesmaximal, characteristic
7C2×#451C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×Q8—28no—
9C22×C3×Q8—22no—
10C22×C3×Q8—21no—
11C22×C3×Q8C2211yescharacteristic
12C22×C3×Q8C2214yes—
13C22×C3×Q8C2212yes—
14C23×C3×C4—21no—
15C23×C3×C4C2211yescharacteristic
16C2×C3×Q16—28no—
17C2×C3×Q16C2218yescomplemented
18C22×C3×C8C2212yes—
19C3×C22⋊Q8—22no—
20C3×C22⋊Q8C2214yes—
21C2×C3×C4⋊C4—21no—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4—21no—
24C3×#103C2218yescomplemented
25C3×C22⋊C8C2214yes—
26C23×Q8C2×C311yescharacteristic
27C22×Q16C2×C312yescomplemented
28C2×C22⋊Q8C2×C311yescharacteristic
29#451C2×C318yesdirect product factor
30C2×#103C2×C312yescomplemented
31C2×C22⋊C8C2×C311yescharacteristic
32C24×C3D411yescharacteristic
33C2×C3×Q8—216no—
34C2×C3×Q8—28no—
35C2×C3×Q8—216no—
36C2×C3×Q8—24no—
37C2×C3×Q8—24no—
38C2×C3×Q8—22no—
39C2×C3×Q8—21no—
40C2×C3×Q8—24no—
41C2×C3×Q8D418yes—
42C2×C3×Q8C2312yes—
43C2×C3×Q8C2314yes—
44C22×C3×C4—28no—
45C22×C3×C4—24no—
46C22×C3×C4—22no—
47C22×C3×C4—24no—
48C22×C3×C4—21no—
49C22×C3×C4—21no—
50C22×C3×C4D412yes—
51C22×C3×C4D411yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C3×Q16—216no—
55C2×C3×C8—24no—
56C2×C3×C8C2314yes—
57C3×C4⋊C4—24no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4C2312yes—
60C3×C22⋊C4—24no—
61C22×Q8—28no—
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8C22×C311yescharacteristic
65C22×Q8C22×C314yes—
66C22×Q8C22×C312yes—
67C23×C4—21no—
68C23×C4C22×C311yescharacteristic
69C2×Q16—28no—
70C2×Q16C22×C318yescomplemented
71C22×C8C22×C312yes—
72C22⋊Q8—22no—
73C22⋊Q8C22×C314yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C22×C311yescharacteristic
76C2×C22⋊C4—21no—
77#103C22×C318yescomplemented
78C22⋊C8C22×C314yes—
79C23×C3—22no—
80C23×C3—22no—
81C23×C3Q1614yescomplemented
82C23×C3C2×D412yes—
83C23×C3C2×D411yescharacteristic
84C3×Q8—216no—
85C3×Q8—28no—
86C3×Q8—24no—
87C3×Q8—24no—
88C3×Q8—42no—
89C3×Q8C2×D418yes—
90C2×C3×C4—216no—
91C2×C3×C4—28no—
92C2×C3×C4—22no—
93C2×C3×C4—24no—
94C2×C3×C4—24no—
95C2×C3×C4—24no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—42no—
99C2×C3×C4—42no—
100C2×C3×C4C2×D412yes—
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4C2×D414yes—
103C2×C3×C4C2411yescharacteristic
104C3×C8—24no—
105C24C3×D411yescharacteristic
106C2×Q8—216no—
107C2×Q8—28no—
108C2×Q8—216no—
109C2×Q8—24no—
110C2×Q8—24no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—24no—
114C2×Q8C3×D418yes—
115C2×Q8C23×C312yes—
116C2×Q8C23×C314yes—
117C22×C4—28no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4C3×D411yescharacteristic
124C22×C4C3×D412yes—
125C22×C4C23×C312yes—
126C22×C4C23×C311yescharacteristic
127Q16—216no—
128C2×C8—24no—
129C2×C8C23×C314yes—
130C4⋊C4—24no—
131C4⋊C4—21no—
132C4⋊C4C23×C312yes—
133C22⋊C4—24no—
134C22×C3—22no—
135C22×C3—22no—
136C22×C3—28no—
137C22×C3#12012yes—
138C22×C3C2×Q1612yes—
139C22×C3C2×Q1614yescomplemented
140C22×C3#13712yes—
141C22×C3C22×D411yescharacteristic
142C3×C4—28no—
143C3×C4—22no—
144C3×C4—42no—
145C3×C4—42no—
146C3×C4#12012yes—
147C3×C4C22×D412yes—
148C23—22no—
149C23—22no—
150C23C3×Q1614yescomplemented
151C23C2×C3×D412yes—
152C23C2×C3×D411yescharacteristic
153Q8—216no—
154Q8—28no—
155Q8—24no—
156Q8—24no—
157Q8—42no—
158Q8C2×C3×D418yes—
159C2×C4—216no—
160C2×C4—28no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4C2×C3×D412yes—
170C2×C4C2×C3×D411yescharacteristic
171C2×C4C2×C3×D414yes—
172C2×C4C24×C311yescharacteristic
173C8—24no—
174C2×C3—24no—
175C2×C3#45114yesdirect product factor
176C2×C3C2×#12011yescharacteristic
177C2×C3C22×Q1611yescharacteristic
178C2×C3C2×#13711yescharacteristic
179C22—22no—
180C22—28no—
181C22—22no—
182C22C3×#12012yes—
183C22C2×C3×Q1612yes—
184C22C2×C3×Q1614yescomplemented
185C22C3×#13712yes—
186C22C22×C3×D411yescharacteristic
187C4—28no—
188C4—22no—
189C4—42no—
190C4—42no—
191C4C3×#12012yes—
192C4C22×C3×D412yes—
193C3C2×#45111yescharacteristic, direct product factor, Sylow
194C2—24no—
195C2C3×#45114yesdirect product factor
196C2C2×C3×#12011yescharacteristic
197C2C22×C3×Q1611yescharacteristic
198C2C2×C3×#13711yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC24×C3D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×Q16C2semidirect
4. C3×#451C2direct (N × H)
5. C2×C3×#103C2semidirect
7. C2×#451C3direct (N × H)
17. C2×C3×Q16C22semidirect
24. C3×#103C22semidirect
27. C22×Q16C2×C3semidirect
29. #451C2×C3direct (N × H)
30. C2×#103C2×C3semidirect
70. C2×Q16C22×C3semidirect
77. #103C22×C3semidirect
81. C23×C3Q16semidirect
139. C22×C3C2×Q16semidirect
150. C23C3×Q16semidirect
175. C2×C3#451direct (N × H)
184. C22C2×C3×Q16semidirect
193. C3C2×#451direct (N × H)
195. C2C3×#451direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×Q82
C22×C3×Q162
C2×C3×C22⋊Q82
C3×#4512
C2×C3×#1032
C2×C3×C22⋊C82
C2×#4513