Finite Group Databasebeta

← C3×#2013 #72728

Subgroups — C3×#2013 #72728

Order 384.

406 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 116 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×SD16C212yesmaximal
2C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×#103C212yesmaximal, complemented
4C3×#390C211yesmaximal, characteristic
5C3×#336C211yesmaximal, characteristic, complemented
6#2013C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q8C2212yes—
8C22×C3×D4C2211yescharacteristic
9C2×C3×SD16—24no—
10C2×C3×SD16—24no—
11C2×C3×SD16—24no—
12C22×C3×C8C2212yes—
13C3×C4⋊Q8—24no—
14C2×C3×C4⋊C4—22no—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4—22no—
17C2×C3×C42C2211yescharacteristic
18C3×#103—24no—
19C22×SD16C2×C312yes—
20C2×C4⋊Q8C2×C311yescharacteristic, complemented
21C2×#103C2×C312yescomplemented
22#390C2×C311yescharacteristic
23#336C2×C311yescharacteristic, complemented
24C24×C3—21no—
25C2×C3×Q8—22no—
26C2×C3×Q8—22no—
27C2×C3×Q8—22no—
28C2×C3×Q8—22no—
29C2×C3×Q8—22no—
30C2×C3×Q8D414yes—
31C2×C3×D4—22no—
32C2×C3×D4—22no—
33C2×C3×D4—21no—
34C2×C3×D4D412yes—
35C22×C3×C4—22no—
36C22×C3×C4—21no—
37C22×C3×C4D412yes—
38C22×C3×C4C2311yescharacteristic
39C3×SD16—216no—
40C2×C3×C8—22no—
41C2×C3×C8—22no—
42C2×C3×C8D414yes—
43C3×C4⋊C4—28no—
44C3×C4⋊C4—22no—
45C3×C22⋊C4—44no—
46C3×C42—22no—
47C22×Q8C22×C312yes—
48C22×D4C22×C311yescharacteristic
49C2×SD16—24no—
50C2×SD16—24no—
51C2×SD16—24no—
52C22×C8C22×C312yes—
53C4⋊Q8—24no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4C22×C311yescharacteristic
56C2×C22⋊C4—22no—
57C2×C42C22×C311yescharacteristic
58#103—24no—
59C23×C3—22no—
60C23×C3—41no—
61C23×C3—41no—
62C23×C3—42no—
63C23×C3—42no—
64C23×C3C2×D411yescharacteristic
65C3×Q8—24no—
66C3×Q8—24no—
67C3×Q8—42no—
68C3×Q8—42no—
69C3×D4—24no—
70C3×D4—42no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—24no—
74C2×C3×C4—42no—
75C2×C3×C4—42no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×C3×C4SD1614yescomplemented
79C2×C3×C4C2×D411yescharacteristic
80C2×C3×C4C2×D412yes—
81C2×C3×C4C2×D412yes—
82C2×C3×C4#4111yescharacteristic
83C3×C8—24no—
84C24—21no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8C3×D414yes—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—21no—
94C2×D4C3×D412yes—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4C3×D412yes—
98C22×C4C23×C311yescharacteristic
99SD16—216no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8C3×D414yes—
103C4⋊C4—28no—
104C4⋊C4—22no—
105C22⋊C4—44no—
106C42—22no—
107C22×C3—42no—
108C22×C3—44no—
109C22×C3—82no—
110C22×C3—81no—
111C22×C3—81no—
112C22×C3#12011yescharacteristic
113C22×C3#12111yescharacteristic
114C22×C3#12711yescharacteristic
115C22×C3C2×SD1612yes—
116C22×C3#13712yes—
117C3×C4—24no—
118C3×C4—44no—
119C3×C4—81no—
120C3×C4#12012yes—
121C3×C4#12112yes—
122C23—22no—
123C23—41no—
124C23—41no—
125C23—42no—
126C23—42no—
127C23C2×C3×D411yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—42no—
131Q8—42no—
132D4—24no—
133D4—42no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4C3×SD1614yescomplemented
142C2×C4C2×C3×D411yescharacteristic
143C2×C4C2×C3×D412yes—
144C2×C4C2×C3×D412yes—
145C2×C4C3×#4111yescharacteristic
146C8—24no—
147C2×C3—82no—
148C2×C3#39211yescharacteristic
149C2×C3#44812yes—
150C2×C3#46112yes—
151C2×C3#49211yescharacteristic
152C2×C3#49711yescharacteristic
153C22—42no—
154C22—44no—
155C22—82no—
156C22—81no—
157C22—81no—
158C22C3×#12011yescharacteristic
159C22C3×#12111yescharacteristic
160C22C3×#12711yescharacteristic
161C22C2×C3×SD1612yes—
162C22C3×#13712yes—
163C4—24no—
164C4—44no—
165C4—81no—
166C4C3×#12012yes—
167C4C3×#12112yes—
168C3#201311yescharacteristic, direct product factor, Sylow
169C2—82no—
170C2C3×#39211yescharacteristic
171C2C3×#44812yes—
172C2C3×#46112yes—
173C2C3×#49211yescharacteristic
174C2C3×#49711yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊Q8C2semidirect
3. C2×C3×#103C2semidirect
5. C3×#336C2semidirect
6. #2013C3direct (N × H)
20. C2×C4⋊Q8C2×C3semidirect
21. C2×#103C2×C3semidirect
23. #336C2×C3semidirect
78. C2×C3×C4SD16semidirect
141. C2×C4C3×SD16semidirect
168. C3#2013direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×SD162
C2×C3×C4⋊Q82
C2×C3×#1032
C3×#3902
C3×#3362
#20133