Finite Group Databasebeta

← C3×#2040 #72755

Subgroups — C3×#2040 #72755

Order 384.

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

normal subgroups only

This group has 290 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×#537C211yesmaximal, characteristic
2C2×C3×#104C212yesmaximal, complemented
3C3×#417C212yesmaximal, complemented
4C3×#382C211yesmaximal, characteristic, complemented
5C3×#343C211yesmaximal, characteristic, complemented
6#2040C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41C2212yes—
8C2×C3×M16C2212yes—
9C3×C22⋊Q8—22no—
10C3×#121—22no—
11C3×#121—22no—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15C3×#104—24no—
16C3×#103—22no—
17C3×#102—22no—
18C3×#95—22no—
19#537C2×C311yescharacteristic
20C2×#104C2×C312yescomplemented
21#417C2×C312yescomplemented
22#382C2×C311yescharacteristic, complemented
23#343C2×C311yescharacteristic, complemented
24C3×#41—22no—
25C3×#41—22no—
26C3×#41—22no—
27C3×#41—22no—
28C2×C3×Q8D412yes—
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4D412yes—
32C22×C3×C4—21no—
33C22×C3×C4—22no—
34C22×C3×C4—21no—
35C22×C3×C4D412yes—
36C22×C3×C4C2311yescharacteristic
37C3×M16—22no—
38C3×M16—22no—
39C2×C3×C8—22no—
40C3×C4⋊C4—22no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4—22no—
43C3×C22⋊C4—24no—
44C3×C42—21no—
45C3×C42D412yescomplemented
46C2×#41C22×C312yes—
47C2×M16C22×C312yes—
48C22⋊Q8—22no—
49#121—22no—
50#121—22no—
51C2×C4⋊C4C22×C311yescharacteristic
52C2×C4⋊C4C22×C311yescharacteristic
53C2×C42C22×C311yescharacteristic
54#104—24no—
55#103—22no—
56#102—22no—
57#95—22no—
58C23×C3—22no—
59C23×C3C2×D411yescharacteristic
60C3×Q8—22no—
61C3×Q8—22no—
62C3×D4—22no—
63C3×D4—22no—
64C3×D4—42no—
65C3×D4—42no—
66C3×D4—82no—
67C2×C3×C4—22no—
68C2×C3×C4—22no—
69C2×C3×C4—22no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4—42no—
73C2×C3×C4—42no—
74C2×C3×C4—42no—
75C2×C3×C4—42no—
76C2×C3×C4—41no—
77C2×C3×C4C2×D411yescharacteristic
78C2×C3×C4C2×D412yes—
79C2×C3×C4#4112yes—
80C2×C3×C4#4111yescharacteristic
81C3×C8—42no—
82#41—22no—
83#41—22no—
84#41—22no—
85#41—22no—
86C2×Q8C3×D412yes—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4C3×D412yes—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4C3×D412yes—
94C22×C4C23×C311yescharacteristic
95M16—22no—
96M16—22no—
97C2×C8—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C22⋊C4—24no—
102C42—21no—
103C42C3×D412yescomplemented
104C22×C3—21no—
105C22×C3—21no—
106C22×C3—42no—
107C22×C3—42no—
108C22×C3—42no—
109C22×C3#12011yescharacteristic
110C22×C3#12111yescharacteristic
111C22×C3#12411yescharacteristic
112C3×C4—42no—
113C3×C4—42no—
114C3×C4—42no—
115C3×C4—81no—
116C3×C4#12112yes—
117C3×C4#12312yes—
118C23—22no—
119C23C2×C3×D411yescharacteristic
120Q8—22no—
121Q8—22no—
122D4—22no—
123D4—22no—
124D4—42no—
125D4—42no—
126D4—82no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4C2×C3×D411yescharacteristic
138C2×C4C2×C3×D412yes—
139C2×C4C3×#4112yes—
140C2×C4C3×#4111yescharacteristic
141C8—42no—
142C2×C3—21no—
143C2×C3—21no—
144C2×C3—82no—
145C2×C3#39411yescharacteristic
146C2×C3#45411yescharacteristic
147C2×C3#45511yescharacteristic
148C22—21no—
149C22—21no—
150C22—42no—
151C22—42no—
152C22—42no—
153C22C3×#12011yescharacteristic
154C22C3×#12111yescharacteristic
155C22C3×#12411yescharacteristic
156C4—42no—
157C4—42no—
158C4—42no—
159C4—81no—
160C4C3×#12112yes—
161C4C3×#12312yes—
162C3#204011yescharacteristic, direct product factor, Sylow
163C2—21no—
164C2—21no—
165C2—82no—
166C2C3×#39411yescharacteristic
167C2C3×#45411yescharacteristic
168C2C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC3×#537C2
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
2. C2×C3×#104C2semidirect
3. C3×#417C2semidirect
4. C3×#382C2semidirect
5. C3×#343C2semidirect
6. #2040C3direct (N × H)
20. C2×#104C2×C3semidirect
21. #417C2×C3semidirect
22. #382C2×C3semidirect
23. #343C2×C3semidirect
45. C3×C42D4semidirect
103. C42C3×D4semidirect
162. C3#2040direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5372
C2×C3×#1042
C3×#4172
C3×#3822
C3×#3432
#20403