Finite Group Databasebeta

← C3×#2001 #72716

Subgroups — C3×#2001 #72716

Order 384.

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

normal subgroups only

This group has 392 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
1C22×C3×Q16C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊Q8C212yesmaximal
3C2×C3×#103C212yesmaximal, complemented
4C2×C3×C22⋊C8C211yesmaximal, characteristic
5C3×#340C211yesmaximal, characteristic, complemented
6#2001C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q8C2212yes—
8C23×C3×C4C2211yescharacteristic
9C2×C3×Q16—24no—
10C2×C3×Q16—22no—
11C22×C3×C8C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C3×C22⋊Q8—24no—
14C3×C22⋊Q8—24no—
15C2×C3×C4⋊C4—22no—
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C22⋊C4—22no—
18C3×#103—24no—
19C3×C22⋊C8—22no—
20C22×Q16C2×C311yescharacteristic, complemented
21C2×C22⋊Q8C2×C312yes—
22C2×#103C2×C312yescomplemented
23C2×C22⋊C8C2×C311yescharacteristic
24#340C2×C311yescharacteristic, complemented
25C24×C3D411yescharacteristic
26C2×C3×Q8—22no—
27C2×C3×Q8—22no—
28C2×C3×Q8—22no—
29C2×C3×Q8—22no—
30C2×C3×Q8—22no—
31C2×C3×Q8D414yes—
32C22×C3×C4—22no—
33C22×C3×C4—22no—
34C22×C3×C4—21no—
35C22×C3×C4—22no—
36C22×C3×C4—22no—
37C22×C3×C4—21no—
38C22×C3×C4D411yescharacteristic
39C22×C3×C4C2311yescharacteristic
40C3×Q16—28no—
41C2×C3×C8—22no—
42C2×C3×C8—21no—
43C2×C3×C8—22no—
44C2×C3×C8D412yes—
45C3×C4⋊C4—28no—
46C3×C4⋊C4—22no—
47C3×C4⋊C4D414yes—
48C3×C22⋊C4—28no—
49C22×Q8C22×C312yes—
50C23×C4C22×C311yescharacteristic
51C2×Q16—24no—
52C2×Q16—22no—
53C22×C8C22×C311yescharacteristic
54C22×C8C22×C311yescharacteristic
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4C22×C312yes—
59C2×C22⋊C4—22no—
60#103—24no—
61C22⋊C8—22no—
62C23×C3—22no—
63C23×C3—21no—
64C23×C3—21no—
65C23×C3—22no—
66C23×C3Q1612yescomplemented
67C23×C3C2×D411yescharacteristic
68C3×Q8—24no—
69C3×Q8—24no—
70C3×Q8—42no—
71C3×Q8—42no—
72C2×C3×C4—24no—
73C2×C3×C4—24no—
74C2×C3×C4—22no—
75C2×C3×C4—21no—
76C2×C3×C4—42no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4—42no—
80C2×C3×C4—41no—
81C2×C3×C4—42no—
82C2×C3×C4—41no—
83C2×C3×C4C2×D412yes—
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D412yes—
86C2×C3×C4#4111yescharacteristic
87C3×C8—22no—
88C3×C8—41no—
89C24C3×D411yescharacteristic
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8C3×D414yes—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4C3×D411yescharacteristic
103C22×C4C23×C311yescharacteristic
104Q16—28no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8C3×D412yes—
109C4⋊C4—28no—
110C4⋊C4—22no—
111C4⋊C4C3×D414yes—
112C22⋊C4—28no—
113C22×C3—24no—
114C22×C3—22no—
115C22×C3—41no—
116C22×C3—42no—
117C22×C3—41no—
118C22×C3#12012yes—
119C22×C3#12111yescharacteristic
120C22×C3C2×Q1611yescharacteristic
121C22×C3#13511yescharacteristic
122C22×C3#13712yes—
123C3×C4—41no—
124C3×C4—44no—
125C3×C4—44no—
126C3×C4#12011yescharacteristic
127C3×C4#12112yes—
128C3×C4#12711yescharacteristic
129C23—22no—
130C23—21no—
131C23—21no—
132C23—22no—
133C23C3×Q1612yescomplemented
134C23C2×C3×D411yescharacteristic
135Q8—24no—
136Q8—24no—
137Q8—42no—
138Q8—42no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4C2×C3×D412yes—
151C2×C4C2×C3×D411yescharacteristic
152C2×C4C2×C3×D412yes—
153C2×C4C3×#4111yescharacteristic
154C8—22no—
155C8—41no—
156C2×C3—42no—
157C2×C3#39211yescharacteristic
158C2×C3#45112yes—
159C2×C3#45212yes—
160C2×C3#46711yescharacteristic
161C2×C3#47011yescharacteristic
162C22—24no—
163C22—22no—
164C22—41no—
165C22—42no—
166C22—41no—
167C22C3×#12012yes—
168C22C3×#12111yescharacteristic
169C22C2×C3×Q1611yescharacteristic
170C22C3×#13511yescharacteristic
171C22C3×#13712yes—
172C4—41no—
173C4—44no—
174C4—44no—
175C4C3×#12011yescharacteristic
176C4C3×#12112yes—
177C4C3×#12711yescharacteristic
178C3#200111yescharacteristic, direct product factor, Sylow
179C2—42no—
180C2C3×#39211yescharacteristic
181C2C3×#45112yes—
182C2C3×#45212yes—
183C2C3×#46711yescharacteristic
184C2C3×#47011yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC24×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q16C2semidirect
3. C2×C3×#103C2semidirect
5. C3×#340C2semidirect
6. #2001C3direct (N × H)
20. C22×Q16C2×C3semidirect
22. C2×#103C2×C3semidirect
24. #340C2×C3semidirect
66. C23×C3Q16semidirect
133. C23C3×Q16semidirect
178. C3#2001direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q162
C2×C3×C22⋊Q82
C2×C3×#1032
C2×C3×C22⋊C82
C3×#3402
#20013