Finite Group Databasebeta

← #69981

Subgroups — #69981

Order 384.

274 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 73 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×#924C211yesmaximal, characteristic
2C3×#407C211yesmaximal, characteristic
3#5211C211yesmaximal, characteristic
4#4467C214yesmaximal, complemented
5#1511—31nomaximal, Sylow
6C23×C3×C4C2211yescharacteristic
7C22×Dic6C411yescharacteristic, complemented
8C2×C3×M16—21no—
9C2×#217—21no—
10C2×#217—21no—
11C2×C4⋊Dic3C411yescharacteristic, complemented
12#924—21no—
13#924—21no—
14#924C412yescomplemented
15#924C2212yes—
16C2×Dic3⋊C4—21no—
17C3×C22⋊C8—21no—
18C3×C22⋊C8C2212yes—
19C3⋊#95—21no—
20C3⋊#95C2212yes—
21C2×C22⋊Q8—31no—
22#407D311yescharacteristic
23#381—31no—
24#328—34no—
25C24×C3D411yescharacteristic
26C22×C3×C4—42no—
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2×C411yescharacteristic
29C22×C3×C4D411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4D411yescharacteristic
32C22×C3×C4C2311yescharacteristic
33C22×Dic3—21no—
34C22×Dic3—21no—
35C22×Dic3—22no—
36C2×Dic6—21no—
37C2×Dic6—42no—
38C2×Dic6C2×C412yes—
39C3×M16—42no—
40C2×C3×C8—22no—
41#217—22no—
42#217—22no—
43#217—42no—
44C4⋊Dic3—21no—
45C4⋊Dic3C2×C412yes—
46Dic3⋊C4—22no—
47Dic3⋊C4—22no—
48C22×Q8—31no—
49C23×C4C2×D311yescharacteristic
50C2×M16—21no—
51C22⋊Q8—32no—
52C22⋊Q8—32no—
53C22⋊Q8—61no—
54C22⋊Q8—61no—
55C2×C4⋊C4—31no—
56C2×C4⋊C4—61no—
57C2×C22⋊C4—61no—
58C2×C22⋊C4—61no—
59C22⋊C8—21no—
60C22⋊C8C2×D312yes—
61#95—32no—
62#95—61no—
63C23×C3—22no—
64C23×C3—22no—
65C23×C3—41no—
66C23×C3C22⋊C411yescharacteristic
67C23×C3C22⋊C411yescharacteristic
68C23×C3C2×D411yescharacteristic
69C2×C3×C4—22no—
70C2×C3×C4—22no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4—41no—
74C2×C3×C4—41no—
75C2×C3×C4C22⋊C411yescharacteristic
76C2×C3×C4C22⋊C411yescharacteristic
77C2×C3×C4C22×C411yescharacteristic
78C2×C3×C4C2×D411yescharacteristic
79C2×Dic3—22no—
80C2×Dic3—22no—
81C2×Dic3—42no—
82C2×Dic3—42no—
83C2×Dic3—42no—
84C2×Dic3—42no—
85C2×Dic3—42no—
86Dic6—42no—
87Dic6—42no—
88C3×C8—42no—
89C24D1211yescharacteristic
90C2×Q8—32no—
91C2×Q8—61no—
92C2×Q8—122no—
93C22×C4—42no—
94C22×C4—61no—
95C22×C4—61no—
96C22×C4—62no—
97C22×C4C4×D311yescharacteristic
98C22×C4C4×D311yescharacteristic
99C22×C4D1211yescharacteristic
100C22×C4#6711yescharacteristic
101C22×C4#6711yescharacteristic
102C22×C4C22×D311yescharacteristic
103M16—42no—
104C2×C8—22no—
105C4⋊C4—32no—
106C4⋊C4—61no—
107C4⋊C4—62no—
108C4⋊C4—62no—
109C22⋊C4—62no—
110C22⋊C4—62no—
111C22⋊C4—122no—
112C22×C3—21no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—21no—
119C22×C3—44no—
120C22×C3#9912yes—
121C22×C3C2×C22⋊C411yescharacteristic
122C22×C3#13712yes—
123C3×C4—21no—
124C3×C4—21no—
125C3×C4—21no—
126C3×C4—21no—
127Dic3—42no—
128Dic3—42no—
129Dic3—82no—
130C23—22no—
131C23—22no—
132C23—41no—
133C23(C2×D3)⋊C411yescharacteristic
134C23(C2×D3)⋊C411yescharacteristic
135C23C2×D1211yescharacteristic
136Q8—122no—
137Q8—122no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—62no—
145C2×C4—62no—
146C2×C4—122no—
147C2×C4—122no—
148C2×C4—122no—
149C2×C4—122no—
150C2×C4—122no—
151C2×C4(C2×D3)⋊C411yescharacteristic
152C2×C4(C2×D3)⋊C411yescharacteristic
153C2×C4C2×C4×D311yescharacteristic
154C2×C4C2×#6711yescharacteristic
155C8—42no—
156C2×C3—22no—
157C2×C3—22no—
158C2×C3—41no—
159C2×C3C2×#9911yescharacteristic
160C2×C3#41911yescharacteristic
161C2×C3#42111yescharacteristic
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—44no—
170C22#80612yes—
171C22#90912yes—
172C22C2×(C2×D3)⋊C411yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—122no—
178C4—122no—
179C4—242no—
180C3#151111yescharacteristic, complemented, Sylow
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#80611yescharacteristic
185C2#512011yescharacteristic
186C2#513911yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#407C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3D4
cosocleC22×C3×C4C23
hypercenter#407D3
hypocenterC3#1511
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
4. #4467C2semidirect
7. C22×Dic6C4semidirect
11. C2×C4⋊Dic3C4semidirect
14. #924C4semidirect
180. C3#1511semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C3×#4072
#52112
#44672
#15113