Finite Group Databasebeta

← #68120

Subgroups — #68120

Order 384.

440 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 71 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×D3×C22⋊C4C211yesmaximal, characteristic
2#4726C212yesmaximal, complemented
3C3×#323C211yesmaximal, characteristic, complemented
4#4525C211yesmaximal, characteristic, complemented
5#4468C212yesmaximal, complemented
6#1462—31nomaximal, Sylow
7C24×D3C411yescharacteristic
8C22×C4×D3C411yescharacteristic
9C22×C4×D3C2211yescharacteristic
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×#217C411yescharacteristic
12C2×(C2×D3)⋊C4C411yescharacteristic
13C2×(C2×D3)⋊C4C2211yescharacteristic
14C2×ΓL(1,𝔽25)—22no—
15D3×C22⋊C4—22no—
16D3×C22⋊C4—22no—
17C3×C22⋊C8C2212yescomplemented
18#830C2212yescomplemented
19(C2×D3)⋊C8—22no—
20C22×C22⋊C4—31no—
21#407—32no—
22#323—31no—
23#323—32no—
24#323D311yescharacteristic, complemented
25C24×C3C2×C411yescharacteristic
26C23×D3—22no—
27C23×D3—21no—
28C23×D3—21no—
29C23×D3—22no—
30C23×D3—42no—
31C23×D3—42no—
32C23×D3C2×C411yescharacteristic
33C23×D3C2×C411yescharacteristic
34C22×C3×C4C2×C411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C22×Dic3C2×C411yescharacteristic
37C22×Dic3C2×C411yescharacteristic
38C2×C4×D3—21no—
39C2×C4×D3—21no—
40C2×C4×D3—42no—
41C2×C4×D3—42no—
42C2×C4×D3D412yes—
43C2×C4×D3D412yes—
44C2×C3×C8—22no—
45C3×C22⋊C4—22no—
46#217—22no—
47(C2×D3)⋊C4—22no—
48(C2×D3)⋊C4—22no—
49C2×C3⋊C8—22no—
50ΓL(1,𝔽25)—44no—
51C25—31no—
52C23×C4—31no—
53C23×C4—31no—
54C2×M16—62no—
55C2×C22⋊C4—31no—
56C2×C22⋊C4—31no—
57C2×C22⋊C4—31no—
58C2×C22⋊C4—62no—
59C2×C22⋊C4—62no—
60C2×C22⋊C4C2×D311yescharacteristic
61C22⋊C8—32no—
62C22⋊C8—62no—
63C22⋊C8C2×D312yescomplemented
64C23×C3—21no—
65C23×C3—21no—
66C23×C3—41no—
67C23×C3—41no—
68C23×C3M1612yescomplemented
69C23×C3C22×C411yescharacteristic
70C22×D3—21no—
71C22×D3—21no—
72C22×D3—22no—
73C22×D3—22no—
74C22×D3—24no—
75C22×D3—24no—
76C22×D3—21no—
77C22×D3—21no—
78C22×D3—48no—
79C22×D3—48no—
80C22×D3—42no—
81C22×D3—42no—
82C22×D3—42no—
83C22×D3—42no—
84C22×D3—41no—
85C22×D3—41no—
86C22×D3C22⋊C411yescharacteristic
87C22×D3C22⋊C411yescharacteristic
88C22×D3M1612yescomplemented
89C2×C3×C4—21no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4C2×D412yes—
93C2×Dic3—21no—
94C2×Dic3—41no—
95C2×Dic3—41no—
96C2×Dic3C22⋊C412yes—
97C4×D3—24no—
98C4×D3—42no—
99C4×D3—42no—
100C4×D3—42no—
101C3×C8—42no—
102C3⋊C8—42no—
103C24—31no—
104C24—31no—
105C24—61no—
106C24—61no—
107C24—62no—
108C24—62no—
109C24—122no—
110C24—122no—
111C24C4×D311yescharacteristic
112C22×C4—31no—
113C22×C4—32no—
114C22×C4—32no—
115C22×C4—31no—
116C22×C4—61no—
117C22×C4—61no—
118C22×C4—122no—
119C22×C4—122no—
120C22×C4C4×D311yescharacteristic
121C22×C4C22×D311yescharacteristic
122M16—124no—
123C2×C8—22no—
124C2×C8—62no—
125C22⋊C4—22no—
126C22⋊C4—62no—
127C22⋊C4—62no—
128C22⋊C4—62no—
129C22×C3—21no—
130C22×C3—21no—
131C22×C3—22no—
132C22×C3—42no—
133C22×C3—42no—
134C22×C3—41no—
135C22×C3—41no—
136C22×C3C2×C22⋊C411yescharacteristic
137C22×C3C2×M1611yescharacteristic
138C22×C3C2×M1611yescharacteristic
139C2×D3—24no—
140C2×D3—24no—
141C2×D3—21no—
142C2×D3—21no—
143C2×D3—21no—
144C2×D3—21no—
145C2×D3—22no—
146C2×D3—48no—
147C2×D3—48no—
148C2×D3—42no—
149C2×D3—41no—
150C2×D3—41no—
151C2×D3—42no—
152C2×D3#9912yes—
153C2×D3#10012yes—
154C3×C4—22no—
155C3×C4—41no—
156Dic3—22no—
157Dic3—41no—
158C23—21no—
159C23—21no—
160C23—31no—
161C23—31no—
162C23—32no—
163C23—41no—
164C23—41no—
165C23—62no—
166C23—62no—
167C23—64no—
168C23—64no—
169C23—61no—
170C23—61no—
171C23—61no—
172C23—61no—
173C23—121no—
174C23—121no—
175C23—122no—
176C23—122no—
177C23—128no—
178C23—128no—
179C23—122no—
180C23—122no—
181C23ΓL(1,𝔽25)12yescomplemented
182C23C2×C4×D311yescharacteristic
183C2×C4—21no—
184C2×C4—32no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—64no—
188C2×C4—61no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—121no—
193C2×C4—121no—
194C2×C4D3×D412yes—
195C8—42no—
196C8—122no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—42no—
200C2×C3#40711yescharacteristic
201C2×C3C2×#9911yescharacteristic
202C2×C3C2×#10011yescharacteristic
203D3—22no—
204D3—22no—
205D3—42no—
206C22—21no—
207C22—21no—
208C22—22no—
209C22—32no—
210C22—32no—
211C22—42no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22—64no—
216C22—64no—
217C22—61no—
218C22—61no—
219C22—61no—
220C22—61no—
221C22—62no—
222C22—122no—
223C22—122no—
224C22—128no—
225C22—128no—
226C22—121no—
227C22—121no—
228C22D3×C22⋊C411yescharacteristic
229C22C2×ΓL(1,𝔽25)11yescharacteristic
230C22D3×M1611yescharacteristic
231C4—22no—
232C4—41no—
233C4—62no—
234C4—121no—
235C3#146211yescharacteristic, complemented, Sylow
236C2—21no—
237C2—21no—
238C2—42no—
239C2—62no—
240C2—62no—
241C2—122no—
242C2#472611yescharacteristic
243C2D3×#9911yescharacteristic
244C2D3×#10011yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3C2×C22⋊C4
prime radicalC24×D3C4
cosocleC22×C3×C4C23
hypercenter#323D3
hypocenterC3#1462
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #4726C2semidirect
3. C3×#323C2semidirect
4. #4525C2semidirect
5. #4468C2semidirect
17. C3×C22⋊C8C22semidirect
18. #830C22semidirect
24. #323D3semidirect
63. C22⋊C8C2×D3semidirect
68. C23×C3M16semidirect
88. C22×D3M16semidirect
181. C23ΓL(1,𝔽25)semidirect
235. C3#1462semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊C42
#47262
C3×#3232
#45252
#44682
#14623