Finite Group Databasebeta

← C2×C3×#354 #72840

Subgroups — C2×C3×#354 #72840

Order 384.

284 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 88 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×#124C211yesmaximal, characteristic
2C2×C3×#99C212yesmaximal
3C3×#354C214yesmaximal, direct product factor
4C2×#354C311yesmaximal, characteristic, direct product factor, Sylow
5C22×C3×Q8C411yescharacteristic, complemented
6C22×C3×D4C2211yescharacteristic
7C3×#124—22no—
8C3×#124C412yescomplemented
9C3×#124C2212yes—
10C2×C3×C22⋊C4—22no—
11C2×C3×C22⋊C4—22no—
12C2×C3×C42C411yescharacteristic, complemented
13C3×#99—22no—
14C3×#99C2214yes—
15C2×#124C2×C311yescharacteristic
16C2×#99C2×C312yes—
17#354C2×C314yesdirect product factor
18C24×C3D412yes—
19C2×C3×Q8—21no—
20C2×C3×Q8—41no—
21C2×C3×Q8—41no—
22C2×C3×Q8C2×C412yescomplemented
23C2×C3×D4—41no—
24C2×C3×D4—41no—
25C2×C3×D4C2×C411yescharacteristic
26C2×C3×D4D412yes—
27C2×C3×D4C2311yescharacteristic
28C22×C3×C4—21no—
29C22×C3×C4—21no—
30C22×C3×C4—42no—
31C22×C3×C4C2×C411yescharacteristic
32C3×C22⋊C4—24no—
33C3×C22⋊C4—24no—
34C3×C22⋊C4—24no—
35C3×C22⋊C4—42no—
36C3×C42—21no—
37C3×C42C2×C412yescomplemented
38C22×Q8C3×C411yescharacteristic, complemented
39C22×D4C22×C311yescharacteristic
40#124—22no—
41#124C3×C412yescomplemented
42#124C22×C312yes—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C42C3×C411yescharacteristic, complemented
46#99—22no—
47#99C22×C314yes—
48C23×C3—22no—
49C23×C3—22no—
50C23×C3—44no—
51C23×C3C22⋊C412yes—
52C23×C3C22⋊C411yescharacteristic
53C23×C3C2×D412yes—
54C3×Q8—42no—
55C3×Q8—42no—
56C3×D4—41no—
57C3×D4—42no—
58C3×D4—41no—
59C2×C3×C4—21no—
60C2×C3×C4—21no—
61C2×C3×C4—22no—
62C2×C3×C4—22no—
63C2×C3×C4—42no—
64C2×C3×C4—41no—
65C2×C3×C4—41no—
66C2×C3×C4—44no—
67C2×C3×C4—84no—
68C2×C3×C4C22⋊C411yescharacteristic
69C2×C3×C4C22×C411yescharacteristic
70C24C3×D412yes—
71C2×Q8—21no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8C2×C3×C412yescomplemented
75C2×D4—41no—
76C2×D4—41no—
77C2×D4C2×C3×C411yescharacteristic
78C2×D4C3×D412yes—
79C2×D4C23×C311yescharacteristic
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—42no—
83C22×C4C2×C3×C411yescharacteristic
84C22⋊C4—24no—
85C22⋊C4—24no—
86C22⋊C4—24no—
87C22⋊C4—42no—
88C42—21no—
89C42C2×C3×C412yescomplemented
90C22×C3—22no—
91C22×C3—22no—
92C22×C3—22no—
93C22×C3—42no—
94C22×C3—42no—
95C22×C3—44no—
96C22×C3—82no—
97C22×C3#9911yescharacteristic
98C22×C3#9911yescharacteristic
99C22×C3C2×C22⋊C411yescharacteristic
100C3×C4—21no—
101C3×C4—21no—
102C3×C4—42no—
103C3×C4—42no—
104C3×C4—84no—
105C23—22no—
106C23—22no—
107C23—44no—
108C23C3×C22⋊C411yescharacteristic
109C23C3×C22⋊C412yes—
110C23C2×C3×D412yes—
111Q8—42no—
112Q8—42no—
113D4—41no—
114D4—42no—
115D4—41no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—44no—
124C2×C4—84no—
125C2×C4C3×C22⋊C411yescharacteristic
126C2×C4C22×C3×C411yescharacteristic
127C2×C3—21no—
128C2×C3—21no—
129C2×C3—42no—
130C2×C3—42no—
131C2×C3#35412yesdirect product factor
132C2×C3C2×#9911yescharacteristic
133C22—22no—
134C22—22no—
135C22—22no—
136C22—42no—
137C22—42no—
138C22—44no—
139C22—82no—
140C22C3×#9911yescharacteristic
141C22C3×#9911yescharacteristic
142C22C2×C3×C22⋊C411yescharacteristic
143C4—21no—
144C4—21no—
145C4—42no—
146C4—42no—
147C4—84no—
148C3C2×#35411yescharacteristic, direct product factor, Sylow
149C2—21no—
150C2—21no—
151C2—42no—
152C2—42no—
153C2C3×#35412yesdirect product factor
154C2C2×C3×#9911yescharacteristic
subgroupG/H
center, socleC22×C3#99
commutatorC2×C4C22×C3×C4
Frattini, cosocleC2×D4C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#354C2direct (N × H)
4. C2×#354C3direct (N × H)
5. C22×C3×Q8C4semidirect
8. C3×#124C4semidirect
12. C2×C3×C42C4semidirect
17. #354C2×C3direct (N × H)
22. C2×C3×Q8C2×C4semidirect
37. C3×C42C2×C4semidirect
38. C22×Q8C3×C4semidirect
41. #124C3×C4semidirect
45. C2×C42C3×C4semidirect
74. C2×Q8C2×C3×C4semidirect
89. C42C2×C3×C4semidirect
131. C2×C3#354direct (N × H)
148. C3C2×#354direct (N × H)
153. C2C3×#354direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×C3×#992
C3×#3542
C2×#3543