Finite Group Databasebeta

← #69535

Subgroups — #69535

Order 384.

378 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 89 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×#929C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3C3×#381C211yesmaximal, characteristic, complemented
4C2×#806C212yesmaximal
5#4527C212yesmaximal, complemented
6#1899—31nomaximal, Sylow
7C23×C3×C4C2211yescharacteristic
8C22×#67C2211yescharacteristic
9C2×C3×C22⋊C4C2212yes—
10C2×#217C2211yescharacteristic
11C2×#217C2212yes—
12#929—21no—
13#929—21no—
14#929—22no—
15C2×(C2×D3)⋊C4C412yescomplemented
16#926—24no—
17C2×C4⋊Dic3—21no—
18C2×Dic3⋊C4C412yescomplemented
19C2×C4×Dic3—21no—
20C3×#95—22no—
21#806—24no—
22C2×#123—31no—
23C2×#117—31no—
24C2×#99—32no—
25#381D311yescharacteristic, complemented
26#342—32no—
27C24×C3C2311yescharacteristic
28C23×D3C2×C411yescharacteristic
29C22×C3×C4—22no—
30C22×C3×C4—22no—
31C22×C3×C4—22no—
32C22×C3×C4C2×C412yes—
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D412yes—
36C2×#67—22no—
37C2×#67—21no—
38C2×#67—21no—
39C2×#67—22no—
40C22×Dic3—22no—
41C22×Dic3C2×C411yescharacteristic
42C22×Dic3C2×C412yes—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45#217—22no—
46#217—21no—
47#217—21no—
48#217D414yes—
49(C2×D3)⋊C4—22no—
50(C2×D3)⋊C4—22no—
51C4⋊Dic3—24no—
52Dic3⋊C4—22no—
53Dic3⋊C4—22no—
54C4×Dic3—24no—
55C22×D4—31no—
56C23×C4C2×D311yescharacteristic
57#123—61no—
58#123—61no—
59#123—62no—
60#117—64no—
61C2×C4⋊C4—32no—
62C2×C4⋊C4—61no—
63C2×C22⋊C4—32no—
64C2×C22⋊C4—31no—
65C2×C22⋊C4—32no—
66C2×C22⋊C4C2×D312yes—
67C2×C42—61no—
68#99—64no—
69#95—22no—
70C23×C3—42no—
71C23×C3C22×C411yescharacteristic
72C23×C3C2×D411yescharacteristic
73C23×C3C2×D412yes—
74C23×C3C2×D411yescharacteristic
75C23×C3#4112yes—
76C22×D3—21no—
77C22×D3—22no—
78C22×D3—42no—
79C2×C3×C4—24no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—24no—
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4—44no—
86C2×C3×C4—42no—
87C2×C3×C4C22⋊C412yes—
88C2×C3×C4C22⋊C412yes—
89#67—41no—
90#67—42no—
91#67—41no—
92C2×Dic3—21no—
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—24no—
96C2×Dic3—44no—
97C2×Dic3—42no—
98C24—31no—
99C24C22×D311yescharacteristic
100C2×D4—61no—
101C2×D4—62no—
102C2×D4—62no—
103C2×D4—61no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—31no—
108C22×C4—32no—
109C22×C4—62no—
110C22×C4C4×D312yes—
111C22×C4D1211yescharacteristic
112C22×C4D1211yescharacteristic
113C22×C4#6712yes—
114C4⋊C4—64no—
115C4⋊C4—62no—
116C4⋊C4—62no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—34no—
120C22⋊C4—61no—
121C22⋊C4—62no—
122C22⋊C4—61no—
123C22⋊C4—62no—
124C22⋊C4—62no—
125C42—64no—
126C22×C3—22no—
127C22×C3—22no—
128C22×C3—22no—
129C22×C3—42no—
130C22×C3—42no—
131C22×C3C2×C22⋊C411yescharacteristic
132C22×C3C4×D412yes—
133C22×C3#12011yescharacteristic
134C22×C3#12112yes—
135C22×C3#12311yescharacteristic
136C2×D3—41no—
137C2×D3—42no—
138C2×D3—82no—
139C3×C4—22no—
140C3×C4—22no—
141C3×C4—82no—
142Dic3—44no—
143Dic3—41no—
144Dic3—42no—
145C23—42no—
146C23—61no—
147C23—62no—
148C23—122no—
149C23C2×C4×D311yescharacteristic
150C23C2×D1211yescharacteristic
151C23D3×D412yes—
152C23#23712yes—
153C23C2×#6711yescharacteristic
154D4—121no—
155D4—121no—
156D4—122no—
157C2×C4—24no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—62no—
166C2×C4—64no—
167C2×C4—62no—
168C2×C4—61no—
169C2×C4—124no—
170C2×C4—122no—
171C2×C4(C2×D3)⋊C412yes—
172C2×C4(C2×D3)⋊C412yes—
173C2×C3—21no—
174C2×C3—22no—
175C2×C3—22no—
176C2×C3—21no—
177C2×C3#38611yescharacteristic
178C2×C3#41012yes—
179D3—81no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—42no—
185C22—121no—
186C22—122no—
187C22—242no—
188C22#67⋊C412yes—
189C22#88211yescharacteristic
190C22#88611yescharacteristic
191C22C2×(C2×D3)⋊C411yescharacteristic
192C22#93912yes—
193C4—22no—
194C4—22no—
195C4—82no—
196C4—124no—
197C4—121no—
198C4—122no—
199C3#189911yescharacteristic, complemented, Sylow
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—241no—
205C2#495811yescharacteristic
206C2#500612yes—
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#381C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#67C22
cosocleC24×C3C23
hypercenter#381D3
hypocenterC3#1899
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. C3×#381C2semidirect
5. #4527C2semidirect
15. C2×(C2×D3)⋊C4C4semidirect
18. C2×Dic3⋊C4C4semidirect
25. #381D3semidirect
199. C3#1899semidirect

Maximal subgroups

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

subgroupindex
C2×#9292
C2×#9262
C3×#3812
C2×#8062
#45272
#18993