Finite Group Databasebeta

← #74027

Subgroups — #74027

Order 384.

624 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 173 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×C4×#67C211yesmaximal, characteristic
2C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#886C212yesmaximal
4C3×#380C211yesmaximal, characteristic, complemented
5#4987C212yesmaximal, complemented
6#4984C212yesmaximal, complemented
7#4943C212yesmaximal, complemented
8#4674C212yesmaximal, complemented
9#4661C212yesmaximal, complemented
10#2486—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×#67C2211yescharacteristic
13C23×Dic3C2212yes—
14C22×C4×D3C2211yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2211yescharacteristic
18C4×#67—24no—
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2214yes—
21C2×C4⋊Dic3—24no—
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2214yes—
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×C4×Dic3C2211yescharacteristic
27#886—28no—
28C3×#95C2212yes—
29C3⋊#95C2214yes—
30C3⋊#95C2214yes—
31C2×#123—32no—
32C2×C4×D4—31no—
33C22×C4⋊C4—31no—
34#398—32no—
35#397—32no—
36#389—32no—
37#381—32no—
38#380—32no—
39#380D311yescharacteristic, complemented
40C24×C3C2311yescharacteristic
41C23×D3C2311yescharacteristic
42C22×C3×C4—24no—
43C22×C3×C4D414yes—
44C22×C3×C4C2314yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C2×#67—22no—
48C2×#67—22no—
49C2×#67D414yes—
50C22×Dic3—24no—
51C22×Dic3—24no—
52C22×Dic3—24no—
53C22×Dic3C2311yescharacteristic
54C22×Dic3C2311yescharacteristic
55C22×Dic3C2314yes—
56C22×Dic3C2311yescharacteristic
57C2×C4×D3—22no—
58C2×C4×D3—24no—
59C3×C4⋊C4—24no—
60C3×C22⋊C4—24no—
61#217—22no—
62(C2×D3)⋊C4—28no—
63(C2×D3)⋊C4—22no—
64C4⋊Dic3—24no—
65C4⋊Dic3—24no—
66C4⋊Dic3—28no—
67Dic3⋊C4—22no—
68C4×Dic3—22no—
69C22×D4—31no—
70C23×C4—32no—
71C23×C4—31no—
72C23×C4C2×D311yescharacteristic
73#123—68no—
74C4×D4—64no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—34no—
79C2×C4⋊C4—64no—
80C2×C4⋊C4C2×D312yes—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—31no—
83C2×C22⋊C4—34no—
84C2×C22⋊C4C2×D312yes—
85C2×C42—31no—
86#95—34no—
87#95—34no—
88#95C2×D312yes—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3C2×D412yes—
92C23×C3C2×D412yes—
93C23×C3#4112yes—
94C23×C3C2411yescharacteristic
95C22×D3—21no—
96C22×D3—22no—
97C22×D3—21no—
98C22×D3—22no—
99C22×D3C2×D412yes—
100C2×C3×C4—22no—
101C2×C3×C4—28no—
102C2×C3×C4—24no—
103C2×C3×C4—24no—
104C2×C3×C4—24no—
105C2×C3×C4C2×D412yes—
106C2×C3×C4C2×D412yes—
107C2×C3×C4#4114yes—
108#67—28no—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—21no—
113C2×Dic3—24no—
114C2×Dic3—24no—
115C2×Dic3—22no—
116C2×Dic3—24no—
117C2×Dic3—22no—
118C2×Dic3—48no—
119C2×Dic3C2×D412yes—
120C4×D3—44no—
121C24—31no—
122C24C22×D311yescharacteristic
123C2×D4—34no—
124C2×D4—62no—
125C2×D4—62no—
126C22×C4—24no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—34no—
131C22×C4—64no—
132C22×C4—64no—
133C22×C4—64no—
134C22×C4—62no—
135C22×C4—64no—
136C22×C4D1214yes—
137C22×C4C22×D314yes—
138C22×C4C22×D311yescharacteristic
139C22×C4C22×D311yescharacteristic
140C4⋊C4—24no—
141C4⋊C4—68no—
142C4⋊C4—64no—
143C4⋊C4—64no—
144C4⋊C4—62no—
145C22⋊C4—24no—
146C22⋊C4—68no—
147C22⋊C4—62no—
148C22⋊C4—62no—
149C42—62no—
150C22×C3—22no—
151C22×C3—22no—
152C22×C3—24no—
153C22×C3—24no—
154C22×C3#12114yes—
155C22×C3C22×D411yescharacteristic
156C22×C3C22×D411yescharacteristic
157C22×C3C2×#4112yes—
158C22×C3C2×#4111yescharacteristic
159C22×C32−1+412yes—
160C2×D3—22no—
161C2×D3—24no—
162C2×D3—41no—
163C2×D3—41no—
164C2×D3—42no—
165C3×C4—22no—
166C3×C4—44no—
167C3×C4#12314yescomplemented
168Dic3—22no—
169Dic3—44no—
170Dic3—41no—
171Dic3—41no—
172C23—22no—
173C23—22no—
174C23—32no—
175C23—61no—
176C23—62no—
177C23—61no—
178C23—62no—
179C23C2×D1212yes—
180C23D3×D412yes—
181C23#23912yes—
182C23C23×D311yescharacteristic
183D4—68no—
184C2×C4—22no—
185C2×C4—28no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—32no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—64no—
195C2×C4—64no—
196C2×C4—62no—
197C2×C4—64no—
198C2×C4—62no—
199C2×C4—128no—
200C2×C4—124no—
201C2×C4C2×D1212yes—
202C2×C4C2×D1212yes—
203C2×C4#23714yes—
204C2×C3—24no—
205C2×C3C2×#12111yescharacteristic
206C2×C3C2×#12311yescharacteristic
207C2×C3#53611yescharacteristic
208C2×C3#54712yes—
209C2×C3#55212yes—
210D3—42no—
211C22—24no—
212C22—22no—
213C22—22no—
214C22—24no—
215C22—62no—
216C22—64no—
217C22—122no—
218C22—121no—
219C22—121no—
220C22#89514yes—
221C22C22×D1211yescharacteristic
222C22C2×D3×D411yescharacteristic
223C22C2×#23712yes—
224C22C2×#23911yescharacteristic
225C22#101012yes—
226C4—22no—
227C4—44no—
228C4—62no—
229C4—121no—
230C4—124no—
231C4—121no—
232C4#88614yescomplemented
233C3#248611yescharacteristic, complemented, Sylow
234C2—24no—
235C2—122no—
236C2C2×#88611yescharacteristic
237C2C2×#89511yescharacteristic
238C2#552611yescharacteristic
239C2#561112yes—
240C2#564212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
prime radicalC22×#67C22
hypercenter#380D3
hypocenterC3#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic3C2semidirect
4. C3×#380C2semidirect
5. #4987C2semidirect
6. #4984C2semidirect
7. #4943C2semidirect
8. #4674C2semidirect
9. #4661C2semidirect
39. #380D3semidirect
167. C3×C4#123semidirect
232. C4#886semidirect
233. C3#2486semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#672
C22×C4⋊Dic32
C2×#8862
C3×#3802
#49872
#49842
#49432
#46742
#46612
#24863