Finite Group Databasebeta

← #71688

Subgroups — #71688

Order 384.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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
2C4×#217C211yesmaximal, characteristic
3#5207C211yesmaximal, characteristic
4#4538C214yesmaximal, complemented
5#1508—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×C4×Dic3—21no—
17C2×#208—21no—
18C3⋊#95C2212yes—
19#830—21no—
20#830C2212yes—
21C2×C22⋊Q8D311yescharacteristic
22#407—31no—
23C4×C22⋊C4—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—24no—
46C4×Dic3—42no—
47#208—42no—
48C2×C3⋊C8—22no—
49C22×Q8Dic311yescharacteristic, complemented
50C23×C4C2×D311yescharacteristic
51C2×M16—61no—
52C22⋊Q8—21no—
53C22⋊Q8—21no—
54C22⋊Q8Dic312yescomplemented
55C22⋊Q8C2×D312yes—
56C2×C4⋊C4—21no—
57C2×C4⋊C4Dic311yescharacteristic, complemented
58C2×C22⋊C4—21no—
59C2×C22⋊C4—61no—
60C2×C42—61no—
61C22⋊C8—32no—
62C22⋊C8—61no—
63#95—32no—
64C23×C3—22no—
65C23×C3—22no—
66C23×C3—41no—
67C23×C3C22⋊C411yescharacteristic
68C23×C3C22⋊C411yescharacteristic
69C23×C3C2×D411yescharacteristic
70C3×Q8—42no—
71C3×Q8—42no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—22no—
75C2×C3×C4—22no—
76C2×C3×C4—22no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4—42no—
81C2×C3×C4—42no—
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C22⋊C411yescharacteristic
84C2×C3×C4C22×C411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86C2×Dic3—24no—
87C2×Dic3—42no—
88C2×Dic3—42no—
89C3⋊C8—42no—
90C24#6711yescharacteristic
91C2×Q8—21no—
92C2×Q8—42no—
93C2×Q8C2×Dic312yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—62no—
98C22×C4C2×Dic311yescharacteristic
99C22×C4C2×Dic311yescharacteristic
100C22×C4#6711yescharacteristic
101C22×C4#6711yescharacteristic
102C22×C4#6711yescharacteristic
103C22×C4C22×D311yescharacteristic
104M16—122no—
105C2×C8—62no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4C2×Dic312yes—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—64no—
113C42—122no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—21no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3—44no—
122C22×C3#10412yes—
123C22×C3C2×C22⋊C411yescharacteristic
124C22×C3#13712yes—
125C3×C4—21no—
126C3×C4—21no—
127C3×C4—21no—
128C3×C4—42no—
129C3×C4—42no—
130C3×C4#9912yes—
131Dic3—44no—
132C23—22no—
133C23—22no—
134C23—41no—
135C23#21711yescharacteristic
136C23#21711yescharacteristic
137C23C2×#6711yescharacteristic
138Q8—42no—
139Q8—42no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—64no—
151C2×C4—122no—
152C2×C4—122no—
153C2×C4#21711yescharacteristic
154C2×C4#21711yescharacteristic
155C2×C4C22×Dic311yescharacteristic
156C2×C4C2×#6711yescharacteristic
157C8—122no—
158C2×C3—22no—
159C2×C3—22no—
160C2×C3—41no—
161C2×C3C2×#9911yescharacteristic
162C2×C3#41911yescharacteristic
163C2×C3C2×#10411yescharacteristic
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—44no—
172C22#83712yes—
173C22#94212yes—
174C22C2×#21711yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—42no—
180C4—124no—
181C4#83412yes—
182C3#150811yescharacteristic, complemented, Sylow
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×#83411yescharacteristic
187C2#522511yescharacteristic
188C2C2×#83711yescharacteristic
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#1508
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
49. C22×Q8Dic3semidirect
54. C22⋊Q8Dic3semidirect
57. C2×C4⋊C4Dic3semidirect
182. C3#1508semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C4×#2172
#52072
#45382
#15083