Finite Group Databasebeta

← #70024

Subgroups — #70024

Order 384.

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

normal subgroups only

This group has 380 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
1#5804C211yesmaximal, characteristic
2#5798C211yesmaximal, characteristic, complemented
3C2×C3×#99C211yesmaximal, characteristic, complemented
4C2×#806C212yesmaximal, complemented
5#4526C212yesmaximal, complemented
6#1913—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C22×#67C2211yescharacteristic
9C2×C3×C22⋊C4C2212yes—
10#953—21no—
11C2×#217—21no—
12C2×#217C2211yescharacteristic
13C2×#217C2212yes—
14#939—22no—
15#937C412yescomplemented
16#935C412yescomplemented
17D4×Dic3—24no—
18#929—21no—
19#926—21no—
20C3×#99—22no—
21#806—24no—
22#534—31no—
23#518—31no—
24C2×#99—32no—
25C2×#99D311yescharacteristic, complemented
26#342—32no—
27C24×C3D411yescharacteristic
28C24×C3C2311yescharacteristic
29C23×D3C2×C411yescharacteristic, complemented
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4C2×C412yes—
33C2×C3×D4D412yes—
34C22×C3×C4—22no—
35C22×C3×C4D411yescharacteristic
36C2×#67—22no—
37C2×#67—21no—
38C2×#67—21no—
39C2×#67—22no—
40C2×#67—22no—
41C22×Dic3—22no—
42C22×Dic3—22no—
43C22×Dic3—21no—
44C22×Dic3C2×C411yescharacteristic, complemented
45C3×C22⋊C4—22no—
46C3×C22⋊C4—22no—
47#217—22no—
48#217—22no—
49#217—22no—
50#217—21no—
51#217C2×C412yescomplemented
52#217D414yescomplemented
53(C2×D3)⋊C4—22no—
54C4⋊Dic3—22no—
55Dic3⋊C4—22no—
56C4×Dic3—22no—
57C22×D4—31no—
58C22×D4C2×D311yescharacteristic
59#123—32no—
60#123—61no—
61#121—62no—
62#120—32no—
63#120—61no—
64C4×D4—64no—
65#117—61no—
66C2×C22⋊C4—32no—
67C2×C22⋊C4—31no—
68C2×C22⋊C4—61no—
69C2×C22⋊C4C2×D312yes—
70#99—22no—
71#99—64no—
72C23×C3—21no—
73C23×C3—21no—
74C23×C3—81no—
75C23×C3—81no—
76C23×C3C22⋊C412yes—
77C23×C3C22×C411yescharacteristic
78C23×C3C2×D411yescharacteristic
79C23×C3C2×D412yes—
80C23×C3C2×D411yescharacteristic
81C23×C3#4112yes—
82C22×D3—21no—
83C22×D3—22no—
84C22×D3—81no—
85C3×D4—24no—
86C3×D4—24no—
87C2×C3×C4—21no—
88C2×C3×C4—21no—
89C2×C3×C4—42no—
90C2×C3×C4—82no—
91C2×C3×C4C22⋊C412yes—
92#67—41no—
93#67—42no—
94#67—41no—
95#67—82no—
96C2×Dic3—21no—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—24no—
100C2×Dic3—82no—
101C2×Dic3—82no—
102C2×Dic3—81no—
103C24—31no—
104C24D1211yescharacteristic
105C24C22×D311yescharacteristic
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—61no—
109C2×D4—62no—
110C2×D4—62no—
111C2×D4—61no—
112C2×D4—62no—
113C2×D4C4×D312yes—
114C2×D4#6712yes—
115C22×C4—22no—
116C22×C4—31no—
117C22×C4—62no—
118C22×C4—62no—
119C22×C4—61no—
120C22×C4D1211yescharacteristic
121C4⋊C4—62no—
122C4⋊C4—62no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—32no—
126C22⋊C4—34no—
127C22⋊C4—62no—
128C22⋊C4—62no—
129C22⋊C4—62no—
130C22⋊C4—61no—
131C22⋊C4—62no—
132C42—62no—
133C22×C3—22no—
134C22×C3—42no—
135C22×C3—41no—
136C22×C3—42no—
137C22×C3—41no—
138C22×C3—41no—
139C22×C3—42no—
140C22×C3—42no—
141C22×C3—81no—
142C22×C3C2×C22⋊C411yescharacteristic
143C22×C3C4×D412yes—
144C22×C3#12011yescharacteristic
145C22×C3#12112yes—
146C22×C3#12311yescharacteristic
147C2×D3—42no—
148C2×D3—41no—
149C2×D3—82no—
150C3×C4—22no—
151C3×C4—82no—
152Dic3—44no—
153Dic3—42no—
154Dic3—41no—
155C23—21no—
156C23—21no—
157C23—61no—
158C23—62no—
159C23—81no—
160C23—81no—
161C23—241no—
162C23(C2×D3)⋊C412yes—
163C23C2×C4×D311yescharacteristic
164C23C2×D1211yescharacteristic
165C23D3×D412yes—
166C23#23712yes—
167C23C2×#6711yescharacteristic
168D4—24no—
169D4—24no—
170D4—121no—
171D4—121no—
172D4—122no—
173D4—242no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—42no—
177C2×C4—62no—
178C2×C4—64no—
179C2×C4—62no—
180C2×C4—61no—
181C2×C4—82no—
182C2×C4—242no—
183C2×C4—242no—
184C2×C4—241no—
185C2×C4(C2×D3)⋊C412yes—
186C2×C3—21no—
187C2×C3—21no—
188C2×C3—22no—
189C2×C3—22no—
190C2×C3—21no—
191C2×C3—42no—
192C2×C3#38611yescharacteristic
193D3—81no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—81no—
203C22—122no—
204C22—121no—
205C22—242no—
206C22#67⋊C412yes—
207C22#88211yescharacteristic
208C22#88611yescharacteristic
209C22C2×(C2×D3)⋊C411yescharacteristic
210C22#93912yes—
211C4—22no—
212C4—82no—
213C4—124no—
214C4—122no—
215C4—121no—
216C3#191311yescharacteristic, complemented, Sylow
217C2—22no—
218C2—21no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—241no—
224C2#495811yescharacteristic
subgroupG/H
centerC2#4958
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC2×C3×#99C2
socleC2×C3#386
prime radical#5804C2
cosocleC24×C3C23
hypercenterC2×#99D3
hypocenterC3#1913
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #5798C2semidirect
3. C2×C3×#99C2semidirect
4. C2×#806C2semidirect
5. #4526C2semidirect
15. #937C4semidirect
16. #935C4semidirect
25. C2×#99D3semidirect
29. C23×D3C2×C4semidirect
44. C22×Dic3C2×C4semidirect
51. #217C2×C4semidirect
52. #217D4semidirect
216. C3#1913semidirect

Maximal subgroups

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

subgroupindex
#58042
#57982
C2×C3×#992
C2×#8062
#45262
#19133