Finite Group Databasebeta

← C3×#2039 #72754

Subgroups — C3×#2039 #72754

Order 384.

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

normal subgroups only

This group has 306 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×#384C211yesmaximal, characteristic, complemented
5C3×#343C211yesmaximal, characteristic, complemented
6#2039C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41C2212yes—
8C2×C3×M16C2212yes—
9C3×C22⋊Q8—22no—
10C3×#121—22no—
11C3×#121—22no—
12C2×C3×C4⋊C4—22no—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16C3×#104—24no—
17C3×#103—22no—
18C3×#102—22no—
19#537C2×C311yescharacteristic
20C2×#104C2×C312yescomplemented
21#417C2×C312yescomplemented
22#384C2×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×C4⋊C4—44no—
44C3×C22⋊C4—24no—
45C3×C42—21no—
46C3×C42D412yescomplemented
47C2×#41C22×C312yes—
48C2×M16C22×C312yes—
49C22⋊Q8—22no—
50#121—22no—
51#121—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4C22×C311yescharacteristic
54C2×C4⋊C4C22×C311yescharacteristic
55C2×C42C22×C311yescharacteristic
56#104—24no—
57#103—22no—
58#102—22no—
59C23×C3—22no—
60C23×C3C2×D411yescharacteristic
61C3×Q8—22no—
62C3×Q8—22no—
63C3×D4—22no—
64C3×D4—22no—
65C3×D4—42no—
66C3×D4—42no—
67C3×D4—82no—
68C2×C3×C4—22no—
69C2×C3×C4—22no—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—42no—
74C2×C3×C4—42no—
75C2×C3×C4—42no—
76C2×C3×C4—42no—
77C2×C3×C4—41no—
78C2×C3×C4C2×D411yescharacteristic
79C2×C3×C4C2×D412yes—
80C2×C3×C4#4112yes—
81C2×C3×C4#4111yescharacteristic
82C3×C8—42no—
83#41—22no—
84#41—22no—
85#41—22no—
86#41—22no—
87C2×Q8C3×D412yes—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4C3×D412yes—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4C3×D412yes—
95C22×C4C23×C311yescharacteristic
96M16—22no—
97M16—22no—
98C2×C8—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—44no—
103C22⋊C4—24no—
104C42—21no—
105C42C3×D412yescomplemented
106C22×C3—21no—
107C22×C3—21no—
108C22×C3—42no—
109C22×C3—42no—
110C22×C3—42no—
111C22×C3#12011yescharacteristic
112C22×C3#12111yescharacteristic
113C22×C3#12411yescharacteristic
114C3×C4—24no—
115C3×C4—42no—
116C3×C4—42no—
117C3×C4—81no—
118C3×C4#12112yes—
119C3×C4#12312yes—
120C23—22no—
121C23C2×C3×D411yescharacteristic
122Q8—22no—
123Q8—22no—
124D4—22no—
125D4—22no—
126D4—42no—
127D4—42no—
128D4—82no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4C2×C3×D411yescharacteristic
140C2×C4C2×C3×D412yes—
141C2×C4C3×#4112yes—
142C2×C4C3×#4111yescharacteristic
143C8—42no—
144C2×C3—21no—
145C2×C3—21no—
146C2×C3—82no—
147C2×C3#39411yescharacteristic
148C2×C3#45311yescharacteristic
149C2×C3#45611yescharacteristic
150C22—21no—
151C22—21no—
152C22—42no—
153C22—42no—
154C22—42no—
155C22C3×#12011yescharacteristic
156C22C3×#12111yescharacteristic
157C22C3×#12411yescharacteristic
158C4—24no—
159C4—42no—
160C4—42no—
161C4—81no—
162C4C3×#12112yes—
163C4C3×#12312yes—
164C3#203911yescharacteristic, direct product factor, Sylow
165C2—21no—
166C2—21no—
167C2—82no—
168C2C3×#39411yescharacteristic
169C2C3×#45311yescharacteristic
170C2C3×#45611yescharacteristic
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×#384C2semidirect
5. C3×#343C2semidirect
6. #2039C3direct (N × H)
20. C2×#104C2×C3semidirect
21. #417C2×C3semidirect
22. #384C2×C3semidirect
23. #343C2×C3semidirect
46. C3×C42D4semidirect
105. C42C3×D4semidirect
164. C3#2039direct (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×#3842
C3×#3432
#20393