Finite Group Databasebeta

← C3×#2198 #72913

Subgroups — C3×#2198 #72913

Order 384.

246 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 56 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×#559C211yesmaximal, characteristic, complemented
2C3×#458C212yesmaximal, complemented
3C3×#453C211yesmaximal, characteristic
4C3×#353C211yesmaximal, characteristic, complemented
5C3×#352C212yesmaximal, complemented
6#2198C311yesmaximal, characteristic, direct product factor, Sylow
7C3×H(5,𝔽2)C2211yescharacteristic
8C3×AGL(1,ℤ/8)—21no—
9C3×#127C2211yescharacteristic
10C3×#124—22no—
11C3×#123—21no—
12C3×#123—22no—
13C3×#123C2212yes—
14C3×C4×Q8—21no—
15C3×C4×D4—22no—
16C3×#104—21no—
17C3×#100C2211yescharacteristic
18C3×#99—22no—
19C3×#99C2212yes—
20#559C2×C311yescharacteristic, complemented
21#458C2×C312yescomplemented
22#453C2×C311yescharacteristic
23#353C2×C311yescharacteristic, complemented
24#352C2×C312yescomplemented
25C3×#41—21no—
26C3×#41—41no—
27C2×C3×Q8—21no—
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4—41no—
31C2×C3×D4D411yescharacteristic
32C2×C3×D4C2311yescharacteristic
33C22×C3×C4—21no—
34C22×C3×C4D412yes—
35C3×SD16—41no—
36C3×D8—41no—
37C3×M16—21no—
38C3×C4⋊C4—21no—
39C3×C4⋊C4—22no—
40C3×C4⋊C4—42no—
41C3×C22⋊C4—22no—
42C3×C22⋊C4—21no—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45C3×C22⋊C4—42no—
46C3×C22⋊C4D412yes—
47C3×C42—22no—
48C3×C42D411yescharacteristic
49H(5,𝔽2)C22×C311yescharacteristic
50AGL(1,ℤ/8)—21no—
51#127C22×C311yescharacteristic
52#124—22no—
53#123—21no—
54#123—22no—
55#123C22×C312yes—
56C4×Q8—21no—
57C4×D4—22no—
58#104—21no—
59#100C22×C311yescharacteristic
60#99—22no—
61#99C22×C312yes—
62C23×C3—22no—
63C23×C3—21no—
64C23×C3C2×D412yes—
65C3×Q8—21no—
66C3×Q8—42no—
67C3×D4—21no—
68C3×D4—41no—
69C3×D4—41no—
70C3×D4—44no—
71C3×D4—42no—
72C3×D4—41no—
73C2×C3×C4—21no—
74C2×C3×C4—22no—
75C2×C3×C4—22no—
76C2×C3×C4—22no—
77C2×C3×C4—41no—
78C2×C3×C4—42no—
79C2×C3×C4—42no—
80C2×C3×C4—81no—
81C2×C3×C4C2×D411yescharacteristic
82C3×C8—41no—
83#41—21no—
84#41—41no—
85C2×Q8—21no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—41no—
89C2×D4C3×D411yescharacteristic
90C2×D4C23×C311yescharacteristic
91C22×C4—21no—
92C22×C4C3×D412yes—
93SD16—41no—
94D8—41no—
95M16—21no—
96C4⋊C4—21no—
97C4⋊C4—22no—
98C4⋊C4—42no—
99C22⋊C4—22no—
100C22⋊C4—21no—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C22⋊C4—42no—
104C22⋊C4C3×D412yes—
105C42—22no—
106C42C3×D411yescharacteristic
107C22×C3—22no—
108C22×C3—42no—
109C22×C3—42no—
110C22×C3—41no—
111C22×C3—41no—
112C22×C3—82no—
113C22×C3#12011yescharacteristic
114C3×C4—22no—
115C3×C4—21no—
116C3×C4—42no—
117C3×C4—41no—
118C3×C4—42no—
119C3×C4—82no—
120C23—22no—
121C23—21no—
122C23C2×C3×D412yes—
123Q8—21no—
124Q8—42no—
125D4—21no—
126D4—41no—
127D4—41no—
128D4—44no—
129D4—42no—
130D4—41no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—81no—
139C2×C4C2×C3×D411yescharacteristic
140C8—41no—
141C2×C3—21no—
142C2×C3—42no—
143C2×C3—81no—
144C2×C3—81no—
145C2×C3B(4,𝔽2)11yescharacteristic
146C22—22no—
147C22—42no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—82no—
152C22C3×#12011yescharacteristic
153C4—22no—
154C4—21no—
155C4—42no—
156C4—41no—
157C4—42no—
158C4—82no—
159C3#219811yescharacteristic, direct product factor, Sylow
160C2—21no—
161C2—42no—
162C2—81no—
163C2—81no—
164C2C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C3
prime radicalC3×#453C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#559C2semidirect
2. C3×#458C2semidirect
4. C3×#353C2semidirect
5. C3×#352C2semidirect
6. #2198C3direct (N × H)
20. #559C2×C3semidirect
21. #458C2×C3semidirect
23. #353C2×C3semidirect
24. #352C2×C3semidirect
159. C3#2198direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5592
C3×#4582
C3×#4532
C3×#3532
C3×#3522
#21983