Finite Group Databasebeta

← C3×#2204 #72919

Subgroups — C3×#2204 #72919

Order 384.

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

normal subgroups only

This group has 236 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×#555C211yesmaximal, characteristic, complemented
2C3×#456C211yesmaximal, characteristic, complemented
3C3×#454C212yesmaximal, complemented
4C3×#356C211yesmaximal, characteristic, complemented
5C3×#355C212yesmaximal, complemented
6#2204C311yesmaximal, characteristic, direct product factor, Sylow
7C3×2−1+4C2211yescharacteristic
8C3×#137—21no—
9C3×AGL(1,ℤ/8)—22no—
10C3×C4⋊Q8C2211yescharacteristic
11C3×#124—21no—
12C3×#124C2212yes—
13C3×#121—22no—
14C3×C4×D4—21no—
15C3×#117—22no—
16C3×#104—22no—
17C3×#104—21no—
18C3×#101C2212yes—
19C3×#101C2211yescharacteristic
20#555C2×C311yescharacteristic, complemented
21#456C2×C311yescharacteristic, complemented
22#454C2×C312yescomplemented
23#356C2×C311yescharacteristic, complemented
24#355C2×C312yescomplemented
25C3×#41—22no—
26C3×#41—21no—
27C3×#41—41no—
28C2×C3×Q8—41no—
29C2×C3×Q8D411yescharacteristic
30C2×C3×Q8C2311yescharacteristic
31C2×C3×D4—21no—
32C2×C3×D4—21no—
33C2×C3×D4D412yes—
34C22×C3×C4—22no—
35C3×Q16—41no—
36C3×SD16—41no—
37C3×SD16—42no—
38C3×D8—42no—
39C3×M16—22no—
40C3×M16—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4—22no—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45C3×C22⋊C4—22no—
46C3×C42—21no—
47C3×C42D412yes—
48C3×C42D411yescharacteristic
492−1+4C22×C311yescharacteristic
50#137—21no—
51AGL(1,ℤ/8)—22no—
52C4⋊Q8C22×C311yescharacteristic
53#124—21no—
54#124C22×C312yes—
55#121—22no—
56C4×D4—21no—
57#117—22no—
58#104—22no—
59#104—21no—
60#101C22×C311yescharacteristic
61#101C22×C312yes—
62C23×C3—22no—
63C3×Q8—22no—
64C3×Q8—21no—
65C3×Q8—41no—
66C3×Q8—41no—
67C3×D4—22no—
68C3×D4—21no—
69C3×D4—42no—
70C3×D4—42no—
71C3×D4—41no—
72C3×D4—81no—
73C2×C3×C4—21no—
74C2×C3×C4—22no—
75C2×C3×C4—21no—
76C2×C3×C4—44no—
77C2×C3×C4—42no—
78C2×C3×C4—41no—
79C2×C3×C4C2×D411yescharacteristic
80C2×C3×C4C2×D412yes—
81C3×C8—42no—
82C3×C8—41no—
83#41—22no—
84#41—21no—
85#41—41no—
86C2×Q8—41no—
87C2×Q8C3×D411yescharacteristic
88C2×Q8C23×C311yescharacteristic
89C2×D4—21no—
90C2×D4—21no—
91C2×D4C3×D412yes—
92C22×C4—22no—
93Q16—41no—
94SD16—41no—
95SD16—42no—
96D8—42no—
97M16—22no—
98M16—21no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C42—21no—
105C42C3×D412yes—
106C42C3×D411yescharacteristic
107C22×C3—42no—
108C22×C3—42no—
109C22×C3—41no—
110C22×C3—42no—
111C22×C3#12011yescharacteristic
112C3×C4—22no—
113C3×C4—22no—
114C3×C4—21no—
115C3×C4—42no—
116C3×C4—41no—
117C3×C4—41no—
118C23—22no—
119Q8—22no—
120Q8—21no—
121Q8—41no—
122Q8—41no—
123D4—22no—
124D4—21no—
125D4—42no—
126D4—42no—
127D4—41no—
128D4—81no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—44no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4C2×C3×D411yescharacteristic
136C2×C4C2×C3×D412yes—
137C8—42no—
138C8—41no—
139C2×C3—21no—
140C2×C3—81no—
141C2×C3—82no—
142C2×C3B(4,𝔽2)11yescharacteristic
143C22—42no—
144C22—42no—
145C22—41no—
146C22—42no—
147C22C3×#12011yescharacteristic
148C4—22no—
149C4—22no—
150C4—21no—
151C4—42no—
152C4—41no—
153C4—41no—
154C3#220411yescharacteristic, direct product factor, Sylow
155C2—21no—
156C2—82no—
157C2—81no—
158C2C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C3
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#555C2semidirect
2. C3×#456C2semidirect
3. C3×#454C2semidirect
4. C3×#356C2semidirect
5. C3×#355C2semidirect
6. #2204C3direct (N × H)
20. #555C2×C3semidirect
21. #456C2×C3semidirect
22. #454C2×C3semidirect
23. #356C2×C3semidirect
24. #355C2×C3semidirect
154. C3#2204direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5552
C3×#4562
C3×#4542
C3×#3562
C3×#3552
#22043