Finite Group Databasebeta

← #80763

Subgroups — #80763

Order 384.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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
1D3×C22⋊Q8C212yesmaximal
2C3×#467C211yesmaximal, characteristic, complemented
3#5186C212yesmaximal, complemented
4#5116C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊C8C211yesmaximal, characteristic
6#5051C212yesmaximal, complemented
7#4884C212yesmaximal, complemented
8#4811C212yesmaximal, complemented
9#4809C212yesmaximal, complemented
10#3189—31nomaximal, Sylow
11C2×D3×Q8C2212yes—
12C22×C4×D3C2211yescharacteristic
13C2×C3×Q16C2211yescharacteristic, complemented
14C22×C3×C8C2211yescharacteristic
15C3×C22⋊Q8C2212yes—
16#946C2212yes—
17C2×#216C2212yescomplemented
18#924C2212yes—
19C22×C3⋊C8C2211yescharacteristic
20C2×Dic12C2211yescharacteristic, complemented
21#897C2212yes—
22#896—22no—
23D3×C4⋊C4—22no—
24D3×C4⋊C4C2212yes—
25D3×C22⋊C4—22no—
26#878—22no—
27C3×#107C2211yescharacteristic, complemented
28C3×#103C2212yescomplemented
29#835C2212yescomplemented
30(C2×D3)⋊C8C2212yes—
31(C2×D3)⋊C8C2212yes—
32#818C2211yescharacteristic, complemented
33#816C2212yescomplemented
34#810C2212yescomplemented
35#807C2212yescomplemented
36C2×C22⋊Q8—32no—
37#484—32no—
38#484—32no—
39#467—31no—
40#467—32no—
41#467D311yescharacteristic, complemented
42#451—32no—
43#451—32no—
44C2×C22⋊C8—31no—
45C23×D3D411yescharacteristic
46C2×C3×Q8C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×Dic3D411yescharacteristic
49D3×Q8—42no—
50D3×Q8—42no—
51C2×C4×D3—22no—
52C2×C4×D3—22no—
53C2×C4×D3—41no—
54C2×C4×D3—41no—
55C2×C4×D3D412yes—
56C2×C4×D3C2312yes—
57C2×Dic6—22no—
58C2×Dic6C2312yes—
59C3×Q16—41no—
60C2×C3×C8—41no—
61C2×C3×C8C2311yescharacteristic
62C2×C3×C8C2311yescharacteristic
63C3×C4⋊C4—22no—
64C3×C4⋊C4C2312yes—
65C3×C22⋊C4—22no—
66#217—22no—
67#216—42no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3C2312yes—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4—22no—
75C2×C3⋊C8—41no—
76C2×C3⋊C8C2312yes—
77Dic12—41no—
78C22×Q8—32no—
79C23×C4—31no—
80C2×Q16—32no—
81C2×Q16—31no—
82C2×Q16C2×D311yescharacteristic, complemented
83C22×C8—31no—
84C22×C8C2×D311yescharacteristic
85C22⋊Q8—32no—
86C22⋊Q8—32no—
87C22⋊Q8—32no—
88C22⋊Q8—62no—
89C22⋊Q8—62no—
90C22⋊Q8C2×D312yes—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—62no—
93C2×C22⋊C4—62no—
94#107—31no—
95#107—32no—
96#107C2×D311yescharacteristic, complemented
97#103—32no—
98#103—32no—
99#103—32no—
100#103C2×D312yescomplemented
101C22⋊C8—32no—
102C22⋊C8—32no—
103C23×C3C2×D411yescharacteristic
104C22×D3—41no—
105C22×D3—41no—
106C22×D3Q1614yescomplemented
107C22×D3C2×D412yes—
108C3×Q8—22no—
109C3×Q8—22no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4C2411yescharacteristic
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—41no—
119C2×Dic3C2×D412yes—
120C4×D3—22no—
121C4×D3—22no—
122C4×D3—42no—
123C4×D3—44no—
124C4×D3—44no—
125Dic6—22no—
126Dic6—22no—
127Dic6—44no—
128C3×C8—21no—
129C3×C8—21no—
130C3⋊C8—22no—
131C24—31no—
132C2×Q8—32no—
133C2×Q8—62no—
134C2×Q8—122no—
135C2×Q8—122no—
136C2×Q8C22×D312yes—
137C22×C4—31no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—62no—
141C22×C4—62no—
142C22×C4—121no—
143C22×C4—121no—
144C22×C4C22×D311yescharacteristic
145Q16—41no—
146Q16—121no—
147Q16—122no—
148C2×C8—32no—
149C2×C8—41no—
150C2×C8—121no—
151C2×C8C22×D311yescharacteristic
152C2×C8C22×D311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—32no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4C22×D312yes—
160C22⋊C4—22no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22×C3—21no—
165C22×C3—21no—
166C22×C3C2×Q1612yes—
167C22×C3C22×D411yescharacteristic
168C2×D3—22no—
169C2×D3—22no—
170C2×D3—44no—
171C2×D3C2×Q1614yes—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—42no—
175C3×C4H(5,𝔽2)11yescharacteristic
176C3×C4H(5,𝔽2)11yescharacteristic
177Dic3—22no—
178Dic3—42no—
179Dic3—42no—
180C23—32no—
181C23—34no—
182C23—121no—
183C23—121no—
184C23D3×D411yescharacteristic
185Q8—22no—
186Q8—22no—
187Q8—62no—
188Q8—62no—
189Q8—124no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—32no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—122no—
200C2×C4—124no—
201C2×C4—124no—
202C2×C4—121no—
203C2×C4D3×D411yescharacteristic
204C2×C4C23×D311yescharacteristic
205C8—21no—
206C8—21no—
207C8—62no—
208C2×C3—22no—
209C2×C3#53411yescharacteristic
210C2×C3C22×Q1611yescharacteristic
211C2×C3#57511yescharacteristic
212D3—24no—
213C22—21no—
214C22—21no—
215C22—34no—
216C22—62no—
217C22—62no—
218C22—124no—
219C22D3×Q1612yes—
220C22C2×D3×D411yescharacteristic
221C4—21no—
222C4—42no—
223C4—42no—
224C4—62no—
225C4—122no—
226C4—122no—
227C4#100411yescharacteristic
228C4#100911yescharacteristic
229C3#318911yescharacteristic, complemented, Sylow
230C2—22no—
231C2—64no—
232C2#560511yescharacteristic
233C2C2×D3×Q1611yescharacteristic
234C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#467C2
socleC22×C3C22×D4
prime radicalC23×D3D4
hypercenter#467D3
hypocenterC3#3189
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. C3×#467C2semidirect
3. #5186C2semidirect
4. #5116C2semidirect
6. #5051C2semidirect
7. #4884C2semidirect
8. #4811C2semidirect
9. #4809C2semidirect
13. C2×C3×Q16C22semidirect
17. C2×#216C22semidirect
20. C2×Dic12C22semidirect
27. C3×#107C22semidirect
28. C3×#103C22semidirect
29. #835C22semidirect
32. #818C22semidirect
33. #816C22semidirect
34. #810C22semidirect
35. #807C22semidirect
41. #467D3semidirect
82. C2×Q16C2×D3semidirect
96. #107C2×D3semidirect
100. #103C2×D3semidirect
106. C22×D3Q16semidirect
229. C3#3189semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊Q82
C3×#4672
#51862
#51162
C2×(C2×D3)⋊C82
#50512
#48842
#48112
#48092
#31893