Finite Group Databasebeta

← C3×#1786 #72501

Subgroups — C3×#1786 #72501

Order 384.

848 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 168 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×D4C211yesmaximal, characteristic
2C2×C3×#102C214yesmaximal, complemented
3C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5#1786C311yesmaximal, characteristic, direct product factor, Sylow
6C25×C3—21no—
7C22×C3×D4—28no—
8C22×C3×D4—21no—
9C22×C3×D4—22no—
10C22×C3×D4C414yescomplemented
11C22×C3×D4C2212yes—
12C23×C3×C4C2211yescharacteristic
13C22×C3×C8C2212yescomplemented
14C2×C3×C4⋊C4C2212yescomplemented
15C2×C3×C22⋊C4—21no—
16C3×#102—28no—
17C3×C22⋊C8—22no—
18C3×#95—21no—
19C23×D4C2×C311yescharacteristic
20C2×#102C2×C314yescomplemented
21C2×C22⋊C8C2×C311yescharacteristic, complemented
22#380C2×C311yescharacteristic, complemented
23C24×C3—28no—
24C24×C3—24no—
25C24×C3—22no—
26C24×C3—44no—
27C24×C3—42no—
28C24×C3—42no—
29C24×C3D411yescharacteristic
30C2×C3×D4—28no—
31C2×C3×D4—216no—
32C2×C3×D4—216no—
33C2×C3×D4—24no—
34C2×C3×D4—24no—
35C2×C3×D4—22no—
36C2×C3×D4C2×C414yes—
37C2×C3×D4D418yes—
38C22×C3×C4—24no—
39C22×C3×C4—22no—
40C22×C3×C4C2×C412yes—
41C22×C3×C4D411yescharacteristic
42C22×C3×C4D412yes—
43C22×C3×C4C2311yescharacteristic
44C2×C3×C8—22no—
45C2×C3×C8—24no—
46C3×C4⋊C4—24no—
47C3×C22⋊C4—42no—
48C25—21no—
49C22×D4—28no—
50C22×D4—21no—
51C22×D4—22no—
52C22×D4C3×C414yescomplemented
53C22×D4C22×C312yes—
54C23×C4C22×C311yescharacteristic
55C22×C8C22×C312yescomplemented
56C2×C4⋊C4C22×C312yescomplemented
57C2×C22⋊C4—21no—
58#102—28no—
59C22⋊C8—22no—
60#95—21no—
61C23×C3—22no—
62C23×C3—22no—
63C23×C3—216no—
64C23×C3—24no—
65C23×C3—28no—
66C23×C3—48no—
67C23×C3—416no—
68C23×C3—416no—
69C23×C3—48no—
70C23×C3—42no—
71C23×C3—42no—
72C23×C3—44no—
73C23×C3C22⋊C412yes—
74C23×C3D812yes—
75C23×C3SD1612yes—
76C23×C3C2×D411yescharacteristic
77C3×D4—216no—
78C3×D4—24no—
79C3×D4—28no—
80C3×D4C22⋊C418yescomplemented
81C2×C3×C4—22no—
82C2×C3×C4—24no—
83C2×C3×C4—24no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4—42no—
87C2×C3×C4C22⋊C412yes—
88C2×C3×C4C22×C411yescharacteristic
89C2×C3×C4C2×D414yes—
90C2×C3×C4C2×D411yescharacteristic
91C3×C8—42no—
92C24—28no—
93C24—24no—
94C24—22no—
95C24—44no—
96C24—42no—
97C24—42no—
98C24C3×D411yescharacteristic
99C2×D4—28no—
100C2×D4—216no—
101C2×D4—216no—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4C2×C3×C414yes—
106C2×D4C3×D418yes—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4C2×C3×C412yes—
110C22×C4C3×D411yescharacteristic
111C22×C4C3×D412yes—
112C22×C4C23×C311yescharacteristic
113C2×C8—22no—
114C2×C8—24no—
115C4⋊C4—24no—
116C22⋊C4—42no—
117C22×C3—22no—
118C22×C3—22no—
119C22×C3—24no—
120C22×C3—24no—
121C22×C3—28no—
122C22×C3—432no—
123C22×C3—44no—
124C22×C3—44no—
125C22×C3—48no—
126C22×C3—48no—
127C22×C3#10214yescomplemented
128C22×C3C2×C22⋊C411yescharacteristic
129C22×C3#12012yes—
130C22×C3C2×D811yescharacteristic
131C22×C3C2×SD1611yescharacteristic
132C22×C3AGL(1,ℤ/8)12yes—
133C3×C4—22no—
134C3×C4—82no—
135C3×C4C2×C22⋊C412yes—
136C3×C4#12012yes—
137C23—22no—
138C23—22no—
139C23—216no—
140C23—24no—
141C23—28no—
142C23—48no—
143C23—416no—
144C23—416no—
145C23—48no—
146C23—42no—
147C23—42no—
148C23—44no—
149C23C3×C22⋊C412yes—
150C23C3×D812yes—
151C23C3×SD1612yes—
152C23C2×C3×D411yescharacteristic
153D4—216no—
154D4—24no—
155D4—28no—
156D4C3×C22⋊C418yescomplemented
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4C3×C22⋊C412yes—
164C2×C4C22×C3×C411yescharacteristic
165C2×C4C2×C3×D414yes—
166C2×C4C2×C3×D411yescharacteristic
167C8—42no—
168C2×C3—24no—
169C2×C3—48no—
170C2×C3#37911yescharacteristic
171C2×C3C2×#10211yescharacteristic
172C2×C3#41811yescharacteristic
173C2×C3#44712yes—
174C2×C3#45012yes—
175C22—22no—
176C22—22no—
177C22—24no—
178C22—24no—
179C22—28no—
180C22—432no—
181C22—44no—
182C22—44no—
183C22—48no—
184C22—48no—
185C22C3×#10214yescomplemented
186C22C2×C3×C22⋊C411yescharacteristic
187C22C3×#12012yes—
188C22C2×C3×D811yescharacteristic
189C22C2×C3×SD1611yescharacteristic
190C22C3×AGL(1,ℤ/8)12yes—
191C4—22no—
192C4—82no—
193C4C2×C3×C22⋊C412yes—
194C4C3×#12012yes—
195C3#178611yescharacteristic, direct product factor, Sylow
196C2—24no—
197C2—48no—
198C2C3×#37911yescharacteristic
199C2C2×C3×#10211yescharacteristic
200C2C3×#41811yescharacteristic
201C2C3×#44712yes—
202C2C3×#45012yes—
subgroupG/H
center, socleC23×C3C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC23×C3×D4C2
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. C2×C3×#102C2semidirect
3. C2×C3×C22⋊C8C2semidirect
4. C3×#380C2semidirect
5. #1786C3direct (N × H)
10. C22×C3×D4C4semidirect
13. C22×C3×C8C22semidirect
14. C2×C3×C4⋊C4C22semidirect
20. C2×#102C2×C3semidirect
21. C2×C22⋊C8C2×C3semidirect
22. #380C2×C3semidirect
52. C22×D4C3×C4semidirect
55. C22×C8C22×C3semidirect
56. C2×C4⋊C4C22×C3semidirect
80. C3×D4C22⋊C4semidirect
127. C22×C3#102semidirect
156. D4C3×C22⋊C4semidirect
185. C22C3×#102semidirect
195. C3#1786direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D42
C2×C3×#1022
C2×C3×C22⋊C82
C3×#3802
#17863