Finite Group Databasebeta

← C3×#2024 #72739

Subgroups — C3×#2024 #72739

Order 384.

290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C2×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#103C212yesmaximal
3C3×#391C211yesmaximal, characteristic
4C3×#340C212yesmaximal
5C3×#336C211yesmaximal, characteristic
6#2024C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q8—21no—
8C22×C3×Q8C2211yescharacteristic
9C22×C3×C8C2212yes—
10C3×C4⋊Q8—24no—
11C2×C3×C4⋊C4—21no—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C42C2211yescharacteristic
15C3×#103—24no—
16C3×#95—21no—
17C3×#95—21no—
18C2×C4⋊Q8C2×C311yescharacteristic
19C2×#103C2×C312yes—
20#391C2×C311yescharacteristic
21#340C2×C312yes—
22#336C2×C311yescharacteristic
23C2×C3×Q8—24no—
24C2×C3×Q8—21no—
25C2×C3×Q8—22no—
26C2×C3×Q8—22no—
27C2×C3×Q8—42no—
28C2×C3×Q8—42no—
29C2×C3×Q8Q812yescomplemented
30C22×C3×C4—21no—
31C22×C3×C4—22no—
32C22×C3×C4—21no—
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C2×C3×C8—22no—
37C2×C3×C8—22no—
38C2×C3×C8—22no—
39C3×C4⋊C4—24no—
40C3×C4⋊C4—22no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4D414yes—
43C3×C42—22no—
44C22×Q8—21no—
45C22×Q8C22×C311yescharacteristic
46C22×C8C22×C312yes—
47C4⋊Q8—24no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4C22×C311yescharacteristic
50C2×C4⋊C4C22×C312yes—
51C2×C42C22×C311yescharacteristic
52#103—24no—
53#95—21no—
54#95—21no—
55C23×C3C2×D411yescharacteristic
56C3×Q8—24no—
57C3×Q8—48no—
58C3×Q8—42no—
59C2×C3×C4—21no—
60C2×C3×C4—21no—
61C2×C3×C4—21no—
62C2×C3×C4—21no—
63C2×C3×C4—24no—
64C2×C3×C4—22no—
65C2×C3×C4—41no—
66C2×C3×C4—41no—
67C2×C3×C4—42no—
68C2×C3×C4—42no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4SD1612yes—
73C2×C3×C4Q1612yes—
74C2×C3×C4C2×D412yes—
75C2×C3×C4C2×Q811yescharacteristic
76C2×C3×C4#4112yes—
77C2×C3×C4#4111yescharacteristic
78C3×C8—42no—
79C2×Q8—24no—
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—42no—
84C2×Q8—42no—
85C2×Q8C3×Q812yescomplemented
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4C3×D411yescharacteristic
90C22×C4C3×D411yescharacteristic
91C22×C4C23×C311yescharacteristic
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4C3×D414yes—
99C42—22no—
100C22×C3#12011yescharacteristic
101C22×C3C22⋊Q811yescharacteristic
102C22×C3#12411yescharacteristic
103C22×C3C2×SD1611yescharacteristic
104C22×C3C2×Q1611yescharacteristic
105C22×C3AGL(1,ℤ/8)11yescharacteristic
106C22×C3#13711yescharacteristic
107C3×C4—22no—
108C3×C4—22no—
109C3×C4—42no—
110C3×C4—44no—
111C3×C4—81no—
112C3×C4C22⋊Q812yes—
113C3×C4#12412yes—
114C23C2×C3×D411yescharacteristic
115Q8—24no—
116Q8—48no—
117Q8—42no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—24no—
123C2×C4—22no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4C3×SD1612yes—
132C2×C4C3×Q1612yes—
133C2×C4C2×C3×D412yes—
134C2×C4C2×C3×Q811yescharacteristic
135C2×C4C3×#4112yes—
136C2×C4C3×#4111yescharacteristic
137C8—42no—
138C2×C3#39311yescharacteristic
139C2×C3#45011yescharacteristic
140C2×C3#45111yescharacteristic
141C2×C3#47511yescharacteristic
142C2×C3#47711yescharacteristic
143C2×C3#48711yescharacteristic
144C2×C3#48911yescharacteristic
145C22C3×#12011yescharacteristic
146C22C3×C22⋊Q811yescharacteristic
147C22C3×#12411yescharacteristic
148C22C2×C3×SD1611yescharacteristic
149C22C2×C3×Q1611yescharacteristic
150C22C3×AGL(1,ℤ/8)11yescharacteristic
151C22C3×#13711yescharacteristic
152C4—22no—
153C4—22no—
154C4—42no—
155C4—44no—
156C4—81no—
157C4C3×C22⋊Q812yes—
158C4C3×#12412yes—
159C3#202411yescharacteristic, direct product factor, Sylow
160C2C3×#39311yescharacteristic
161C2C3×#45011yescharacteristic
162C2C3×#45111yescharacteristic
163C2C3×#47511yescharacteristic
164C2C3×#47711yescharacteristic
165C2C3×#48711yescharacteristic
166C2C3×#48911yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
6. #2024C3direct (N × H)
29. C2×C3×Q8Q8semidirect
85. C2×Q8C3×Q8semidirect
159. C3#2024direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
C2×C3×#1032
C3×#3912
C3×#3402
C3×#3362
#20243