Finite Group Databasebeta

← C3×#2191 #72906

Subgroups — C3×#2191 #72906

Order 384.

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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×#575C211yesmaximal, characteristic
2C3×#510C212yesmaximal, complemented
3C3×#470C211yesmaximal, characteristic, complemented
4C3×#360C212yesmaximal, complemented
5C3×#349C211yesmaximal, characteristic, complemented
6#2191C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41—21no—
8C3×#137—22no—
9C3×AGL(1,ℤ/8)—22no—
10C3×#135—21no—
11C3×#135C2212yes—
12C2×C3×Q16C2211yescharacteristic, complemented
13C2×C3×M16C2211yescharacteristic
14C3×C22⋊Q8—21no—
15C3×Q32—22no—
16C3×SD32—22no—
17C3×M32C2212yescomplemented
18C3×#106C2211yescharacteristic, complemented
19C3×#103—21no—
20#575C2×C311yescharacteristic
21#510C2×C312yescomplemented
22#470C2×C311yescharacteristic, complemented
23#360C2×C312yescomplemented
24#349C2×C311yescharacteristic, complemented
25C3×#41—22no—
26C3×#41—22no—
27C3×#41—41no—
28C3×#41—41no—
29C2×C3×Q8—21no—
30C2×C3×Q8—21no—
31C2×C3×D4—21no—
32C2×C3×D4—41no—
33C22×C3×C4—41no—
34C22×C3×C4D411yescharacteristic
35C3×Q16—21no—
36C3×Q16—22no—
37C3×Q16D412yes—
38C3×SD16—22no—
39C3×SD16—22no—
40C3×D8—21no—
41C3×D8D412yes—
42C3×M16—22no—
43C2×C3×C8D411yescharacteristic
44C2×C3×C8C2311yescharacteristic
45C3×C4⋊C4—21no—
46C3×C4⋊C4—41no—
47C3×C22⋊C4—41no—
48C3×C16—22no—
49C2×#41—21no—
50#137—22no—
51AGL(1,ℤ/8)—22no—
52#135—21no—
53#135C22×C312yes—
54C2×Q16C22×C311yescharacteristic, complemented
55C2×M16C22×C311yescharacteristic
56C22⋊Q8—21no—
57Q32—22no—
58SD32—22no—
59M32C22×C312yescomplemented
60#106C22×C311yescharacteristic, complemented
61#103—21no—
62C23×C3—41no—
63C23×C3D811yescharacteristic
64C3×Q8—22no—
65C3×Q8—41no—
66C3×Q8—41no—
67C3×Q8—41no—
68C3×D4—22no—
69C3×D4—41no—
70C3×D4—42no—
71C3×D4—42no—
72C2×C3×C4—21no—
73C2×C3×C4—21no—
74C2×C3×C4—42no—
75C2×C3×C4—41no—
76C2×C3×C4—42no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4D811yescharacteristic
80C2×C3×C4C2×D411yescharacteristic
81C3×C8—21no—
82C3×C8C2×D412yes—
83#41—22no—
84#41—22no—
85#41—41no—
86#41—41no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×D4—21no—
90C2×D4—41no—
91C22×C4—41no—
92C22×C4C3×D411yescharacteristic
93Q16—21no—
94Q16—22no—
95Q16C3×D412yes—
96SD16—22no—
97SD16—22no—
98D8—21no—
99D8C3×D412yes—
100M16—22no—
101C2×C8C3×D411yescharacteristic
102C2×C8C23×C311yescharacteristic
103C4⋊C4—21no—
104C4⋊C4—41no—
105C22⋊C4—41no—
106C16—22no—
107C22×C3—21no—
108C22×C3—41no—
109C22×C3—42no—
110C22×C3—82no—
111C22×C3C2×D811yescharacteristic
112C3×C4—21no—
113C3×C4—42no—
114C3×C4—81no—
115C3×C4—81no—
116C3×C4#12011yescharacteristic
117C3×C4AGL(1,ℤ/8)11yescharacteristic
118C23—41no—
119C23C3×D811yescharacteristic
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—41no—
124D4—22no—
125D4—41no—
126D4—42no—
127D4—42no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4C3×D811yescharacteristic
136C2×C4C2×C3×D411yescharacteristic
137C8—21no—
138C8C2×C3×D412yes—
139C2×C3—21no—
140C2×C3—41no—
141C2×C3—82no—
142C2×C3#44711yescharacteristic
143C22—21no—
144C22—41no—
145C22—42no—
146C22—82no—
147C22C2×C3×D811yescharacteristic
148C4—21no—
149C4—42no—
150C4—81no—
151C4—81no—
152C4C3×#12011yescharacteristic
153C4C3×AGL(1,ℤ/8)11yescharacteristic
154C3#219111yescharacteristic, direct product factor, Sylow
155C2—21no—
156C2—41no—
157C2—82no—
158C2C3×#44711yescharacteristic
subgroupG/H
center, socleC2×C3#447
commutator, Frattini, cosocleC2×C8C23×C3
prime radicalC3×#575C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#510C2semidirect
3. C3×#470C2semidirect
4. C3×#360C2semidirect
5. C3×#349C2semidirect
6. #2191C3direct (N × H)
12. C2×C3×Q16C22semidirect
17. C3×M32C22semidirect
18. C3×#106C22semidirect
21. #510C2×C3semidirect
22. #470C2×C3semidirect
23. #360C2×C3semidirect
24. #349C2×C3semidirect
54. C2×Q16C22×C3semidirect
59. M32C22×C3semidirect
60. #106C22×C3semidirect
154. C3#2191direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5752
C3×#5102
C3×#4702
C3×#3602
C3×#3492
#21913