Finite Group Databasebeta

← #71689

Subgroups — #71689

Order 384.

274 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 81 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×C22⋊Q8C211yesmaximal, characteristic
2#5211C211yesmaximal, characteristic
3#5207C211yesmaximal, characteristic
4#4538C214yesmaximal, complemented
5#1511—31nomaximal, Sylow
6C22×C3×Q8C411yescharacteristic, complemented
7C23×C3×C4C2211yescharacteristic
8C3×C22⋊Q8—21no—
9C3×C22⋊Q8—21no—
10C3×C22⋊Q8C412yescomplemented
11C3×C22⋊Q8C2212yes—
12C2×C3×C4⋊C4—21no—
13C2×C3×C4⋊C4C411yescharacteristic, complemented
14C2×C3×C22⋊C4—21no—
15C2×#217—21no—
16C2×#208—21no—
17C3⋊#95—21no—
18C3⋊#95C2212yes—
19#830—21no—
20#830C2212yes—
21C2×C22⋊Q8D311yescharacteristic
22#407—31no—
23#381—31no—
24#328—34no—
25C24×C3D411yescharacteristic
26C2×C3×Q8—21no—
27C2×C3×Q8—42no—
28C2×C3×Q8C2×C412yes—
29C22×C3×C4—21no—
30C22×C3×C4—21no—
31C22×C3×C4—42no—
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4C2×C411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4C2311yescharacteristic
38C22×Dic3—22no—
39C3×C4⋊C4—22no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4C2×C412yes—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45#217—42no—
46#208—42no—
47C2×C3⋊C8—22no—
48C22×Q8Dic311yescharacteristic, complemented
49C23×C4C2×D311yescharacteristic
50C2×M16—61no—
51C22⋊Q8—21no—
52C22⋊Q8—21no—
53C22⋊Q8Dic312yescomplemented
54C22⋊Q8C2×D312yes—
55C2×C4⋊C4—21no—
56C2×C4⋊C4Dic311yescharacteristic, complemented
57C2×C22⋊C4—21no—
58C2×C22⋊C4—61no—
59C22⋊C8—32no—
60C22⋊C8—61no—
61#95—32no—
62#95—61no—
63C23×C3—22no—
64C23×C3—22no—
65C23×C3—41no—
66C23×C3C22⋊C411yescharacteristic
67C23×C3C22⋊C411yescharacteristic
68C23×C3C2×D411yescharacteristic
69C3×Q8—42no—
70C3×Q8—42no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—22no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4—42no—
80C2×C3×C4—42no—
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C22×C411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85C2×Dic3—42no—
86C2×Dic3—42no—
87C2×Dic3—42no—
88C3⋊C8—42no—
89C24#6711yescharacteristic
90C2×Q8—21no—
91C2×Q8—42no—
92C2×Q8C2×Dic312yes—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—42no—
96C22×C4—62no—
97C22×C4C2×Dic311yescharacteristic
98C22×C4C2×Dic311yescharacteristic
99C22×C4#6711yescharacteristic
100C22×C4#6711yescharacteristic
101C22×C4#6711yescharacteristic
102C22×C4C22×D311yescharacteristic
103M16—122no—
104C2×C8—62no—
105C4⋊C4—22no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4C2×Dic312yes—
109C22⋊C4—22no—
110C22⋊C4—22no—
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—
127C3×C4—42no—
128C3×C4—42no—
129Dic3—82no—
130C23—22no—
131C23—22no—
132C23—41no—
133C23#21711yescharacteristic
134C23#21711yescharacteristic
135C23C2×#6711yescharacteristic
136Q8—42no—
137Q8—42no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—122no—
149C2×C4—122no—
150C2×C4—122no—
151C2×C4#21711yescharacteristic
152C2×C4#21711yescharacteristic
153C2×C4C22×Dic311yescharacteristic
154C2×C4C2×#6711yescharacteristic
155C8—122no—
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#83412yes—
171C22#94212yes—
172C22C2×#21711yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—42no—
178C4—42no—
179C4—242no—
180C3#151111yescharacteristic, complemented, Sylow
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#83411yescharacteristic
185C2#522511yescharacteristic
186C2#523611yescharacteristic
subgroupG/H
centerC22C2×#217
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×C22⋊Q8C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3D4
cosocleC22×C3×C4C23
hypercenterC2×C22⋊Q8D3
hypocenterC3#1511
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. #4538C2semidirect
6. C22×C3×Q8C4semidirect
10. C3×C22⋊Q8C4semidirect
13. C2×C3×C4⋊C4C4semidirect
48. C22×Q8Dic3semidirect
53. C22⋊Q8Dic3semidirect
56. C2×C4⋊C4Dic3semidirect
180. C3#1511semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
#52112
#52072
#45382
#15113