Finite Group Databasebeta

← #69529

Subgroups — #69529

Order 384.

324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 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×#380C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic
4#4527C214yesmaximal
5#1996—31nomaximal, Sylow
6C23×C3×C4C2211yescharacteristic
7C22×Dic6C411yescharacteristic, complemented
8C2×C3×C4⋊C4—21no—
9C2×C3×C22⋊C4C2212yes—
10C2×#217C2212yes—
11C2×#217C2212yes—
12C2×C4⋊Dic3C411yescharacteristic, complemented
13#924—21no—
14#924—21no—
15#924—21no—
16#924—21no—
17C2×Dic3⋊C4C412yescomplemented
18C2×C4×Dic3—21no—
19C3×#95—21no—
20C3⋊#95—21no—
21C2×C22⋊Q8—31no—
22#380D311yescharacteristic
23C4×C22⋊C4—31no—
24#342—34no—
25C24×C3C2311yescharacteristic
26C22×C3×C4—22no—
27C22×C3×C4—42no—
28C22×C3×C4C2×C411yescharacteristic
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4D411yescharacteristic
32C22×C3×C4D411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×Dic3—22no—
35C22×Dic3C2×C412yes—
36C22×Dic3C2×C412yes—
37C2×Dic6—21no—
38C2×Dic6—21no—
39C2×Dic6—42no—
40C3×C4⋊C4—42no—
41C3×C22⋊C4—24no—
42#217—22no—
43#217—22no—
44#217—22no—
45#217Q814yescomplemented
46C4⋊Dic3—21no—
47C4⋊Dic3—21no—
48Dic3⋊C4—22no—
49Dic3⋊C4—22no—
50C4×Dic3—42no—
51C22×Q8—31no—
52C23×C4C2×D311yescharacteristic
53C22⋊Q8—61no—
54C22⋊Q8—61no—
55C22⋊Q8—61no—
56C22⋊Q8—61no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—32no—
59C2×C4⋊C4—31no—
60C2×C22⋊C4—32no—
61C2×C22⋊C4—32no—
62C2×C22⋊C4C2×D312yes—
63C2×C42—61no—
64#95—21no—
65#95—61no—
66C23×C3—41no—
67C23×C3—41no—
68C23×C3C22×C411yescharacteristic
69C23×C3C2×D411yescharacteristic
70C23×C3C2×D411yescharacteristic
71C23×C3C2×Q812yes—
72C23×C3#4112yes—
73C2×C3×C4—22no—
74C2×C3×C4—22no—
75C2×C3×C4—22no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4—42no—
80C2×C3×C4—42no—
81C2×C3×C4—42no—
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C22⋊C411yescharacteristic
84C2×C3×C4C22⋊C411yescharacteristic
85C2×C3×C4C22⋊C411yescharacteristic
86C2×Dic3—22no—
87C2×Dic3—22no—
88C2×Dic3—24no—
89C2×Dic3—42no—
90C2×Dic3—42no—
91C2×Dic3—42no—
92C2×Dic3—42no—
93Dic6—42no—
94Dic6—42no—
95C24C22×D311yescharacteristic
96C2×Q8—61no—
97C2×Q8—61no—
98C2×Q8—122no—
99C22×C4—22no—
100C22×C4—32no—
101C22×C4—32no—
102C22×C4—42no—
103C22×C4—62no—
104C22×C4C4×D311yescharacteristic
105C22×C4C4×D311yescharacteristic
106C22×C4D1211yescharacteristic
107C22×C4D1211yescharacteristic
108C22×C4#6711yescharacteristic
109C22×C4#6711yescharacteristic
110C4⋊C4—42no—
111C4⋊C4—62no—
112C4⋊C4—62no—
113C4⋊C4—61no—
114C4⋊C4—61no—
115C22⋊C4—24no—
116C22⋊C4—34no—
117C22⋊C4—62no—
118C22⋊C4—62no—
119C22⋊C4—62no—
120C42—122no—
121C22×C3—21no—
122C22×C3—21no—
123C22×C3—21no—
124C22×C3—21no—
125C22×C3—21no—
126C22×C3—21no—
127C22×C3—42no—
128C22×C3—42no—
129C22×C3C2×C22⋊C411yescharacteristic
130C22×C3C4×Q812yes—
131C22×C3C22⋊Q812yes—
132C22×C3#12411yescharacteristic
133C22×C3C4⋊Q811yescharacteristic
134C3×C4—21no—
135C3×C4—21no—
136C3×C4—21no—
137C3×C4—82no—
138C3×C4#9912yes—
139Dic3—44no—
140Dic3—42no—
141Dic3—42no—
142C23—41no—
143C23—41no—
144C23C2×C4×D311yescharacteristic
145C23C2×D1211yescharacteristic
146C23#23712yes—
147C23D3×Q812yes—
148C23C2×#6711yescharacteristic
149Q8—122no—
150Q8—122no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—64no—
161C2×C4—62no—
162C2×C4—62no—
163C2×C4—122no—
164C2×C4—122no—
165C2×C4—122no—
166C2×C4—122no—
167C2×C4(C2×D3)⋊C411yescharacteristic
168C2×C4(C2×D3)⋊C411yescharacteristic
169C2×C4(C2×D3)⋊C411yescharacteristic
170C2×C4(C2×D3)⋊C411yescharacteristic
171C2×C3—22no—
172C2×C3—21no—
173C2×C3—22no—
174C2×C3—21no—
175C2×C3#39111yescharacteristic
176C2×C3C2×#9911yescharacteristic
177C2×C3#41011yescharacteristic
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—42no—
185C22—42no—
186C22#88712yes—
187C22#89712yes—
188C22C2×(C2×D3)⋊C411yescharacteristic
189C22#93611yescharacteristic
190C22#94411yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—82no—
195C4—124no—
196C4—122no—
197C4—122no—
198C4#80612yes—
199C3#199611yescharacteristic, complemented, Sylow
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2C2×#80611yescharacteristic
205C2#497311yescharacteristic
206C2#500611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#380C2
socleC22×C3C2×C22⋊C4
prime radical, cosocleC24×C3C23
hypercenter#380D3
hypocenterC3#1996
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
7. C22×Dic6C4semidirect
12. C2×C4⋊Dic3C4semidirect
17. C2×Dic3⋊C4C4semidirect
45. #217Q8semidirect
199. C3#1996semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C3×#3802
C4×#2172
#45272
#19963