Finite Group Databasebeta

← C3×#2027 #72742

Subgroups — C3×#2027 #72742

Order 384.

282 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C3×#544C211yesmaximal, characteristic, complemented
2C3×#418C212yesmaximal
3C3×#390C211yesmaximal, characteristic
4C3×#343C211yesmaximal, characteristic, complemented
5C3×#339C212yesmaximal, complemented
6#2027C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4C2211yescharacteristic
8C2×C3×M16C2212yes—
9C3×C4⋊Q8—22no—
10C3×#125—22no—
11C3×C22⋊Q8—22no—
12C3×#117C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4—22no—
15C2×C3×C42C2211yescharacteristic
16C3×#102—22no—
17C3×#102—22no—
18#544C2×C311yescharacteristic, complemented
19#418C2×C312yes—
20#390C2×C311yescharacteristic
21#343C2×C311yescharacteristic, complemented
22#339C2×C312yescomplemented
23C24×C3—21no—
24C2×C3×Q8—22no—
25C2×C3×D4—22no—
26C2×C3×D4—22no—
27C2×C3×D4—21no—
28C2×C3×D4Q812yescomplemented
29C22×C3×C4—21no—
30C22×C3×C4D412yes—
31C22×C3×C4C2311yescharacteristic
32C3×M16—22no—
33C3×M16—22no—
34C2×C3×C8—22no—
35C3×C4⋊C4—22no—
36C3×C4⋊C4—24no—
37C3×C4⋊C4—22no—
38C3×C4⋊C4D412yes—
39C3×C4⋊C4D412yes—
40C3×C22⋊C4—22no—
41C3×C22⋊C4—44no—
42C3×C42—22no—
43C3×C42—22no—
44C22×D4C22×C311yescharacteristic
45C2×M16C22×C312yes—
46C4⋊Q8—22no—
47#125—22no—
48C22⋊Q8—22no—
49#117C22×C312yes—
50C2×C4⋊C4C22×C311yescharacteristic
51C2×C22⋊C4—22no—
52C2×C42C22×C311yescharacteristic
53#102—22no—
54#102—22no—
55C23×C3—22no—
56C23×C3—42no—
57C23×C3—42no—
58C23×C3—41no—
59C23×C3—41no—
60C23×C3C2×D411yescharacteristic
61C3×Q8—82no—
62C3×D4—24no—
63C3×D4—42no—
64C2×C3×C4—22no—
65C2×C3×C4—22no—
66C2×C3×C4—22no—
67C2×C3×C4—22no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—42no—
71C2×C3×C4C2×D412yes—
72C2×C3×C4C2×Q811yescharacteristic
73C2×C3×C4#4112yes—
74C2×C3×C4#4111yescharacteristic
75C3×C8—42no—
76C24—21no—
77C2×Q8—22no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—21no—
81C2×D4C3×Q812yescomplemented
82C22×C4—21no—
83C22×C4C3×D412yes—
84C22×C4C23×C311yescharacteristic
85M16—22no—
86M16—22no—
87C2×C8—22no—
88C4⋊C4—22no—
89C4⋊C4—24no—
90C4⋊C4—22no—
91C4⋊C4C3×D412yes—
92C4⋊C4C3×D412yes—
93C22⋊C4—22no—
94C22⋊C4—44no—
95C42—22no—
96C42—22no—
97C22×C3—21no—
98C22×C3—21no—
99C22×C3—44no—
100C22×C3—42no—
101C22×C3—82no—
102C22×C3—81no—
103C22×C3—81no—
104C22×C3#12011yescharacteristic
105C22×C3C22⋊Q811yescharacteristic
106C22×C3#12411yescharacteristic
107C3×C4—42no—
108C3×C4—42no—
109C3×C4—42no—
110C3×C4—42no—
111C3×C4C22⋊Q812yes—
112C3×C4#12412yes—
113C23—22no—
114C23—42no—
115C23—42no—
116C23—41no—
117C23—41no—
118C23C2×C3×D411yescharacteristic
119Q8—82no—
120D4—24no—
121D4—42no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4C2×C3×D412yes—
130C2×C4C2×C3×Q811yescharacteristic
131C2×C4C3×#4112yes—
132C2×C4C3×#4111yescharacteristic
133C8—42no—
134C2×C3—21no—
135C2×C3—21no—
136C2×C3—82no—
137C2×C3#39311yescharacteristic
138C2×C3#45311yescharacteristic
139C2×C3#45411yescharacteristic
140C22—21no—
141C22—21no—
142C22—44no—
143C22—42no—
144C22—82no—
145C22—81no—
146C22—81no—
147C22C3×#12011yescharacteristic
148C22C3×C22⋊Q811yescharacteristic
149C22C3×#12411yescharacteristic
150C4—42no—
151C4—42no—
152C4—42no—
153C4—42no—
154C4C3×C22⋊Q812yes—
155C4C3×#12412yes—
156C3#202711yescharacteristic, direct product factor, Sylow
157C2—21no—
158C2—21no—
159C2—82no—
160C2C3×#39311yescharacteristic
161C2C3×#45311yescharacteristic
162C2C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C22×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. C3×#544C2semidirect
4. C3×#343C2semidirect
5. C3×#339C2semidirect
6. #2027C3direct (N × H)
18. #544C2×C3semidirect
21. #343C2×C3semidirect
22. #339C2×C3semidirect
28. C2×C3×D4Q8semidirect
81. C2×D4C3×Q8semidirect
156. C3#2027direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5442
C3×#4182
C3×#3902
C3×#3432
C3×#3392
#20273