Finite Group Databasebeta

← #82139

Subgroups — #82139

Order 384.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2#5791C211yesmaximal, characteristic
3#5207C211yesmaximal, characteristic, complemented
4#4971C212yesmaximal, complemented
5#4970C212yesmaximal, complemented
6#4968C212yesmaximal, complemented
7#4967C212yesmaximal, complemented
8#4966C212yesmaximal, complemented
9#4964C212yesmaximal, complemented
10#3092—31nomaximal, Sylow
11C23×C3×C4—21no—
12C23×C3×C4C2211yescharacteristic
13C2×#235C2211yescharacteristic
14C2×C3×C4⋊C4—21no—
15C2×C3×C4⋊C4—21no—
16C2×C3×C4⋊C4—21no—
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C4⋊C4C2212yes—
19C2×C3×C4⋊C4C2212yes—
20#953—21no—
21#930C2211yescharacteristic
22#930C2211yescharacteristic
23#929—21no—
24C4×#67—21no—
25C4×#67—21no—
26#926C2211yescharacteristic
27#924C2211yescharacteristic
28#924C2211yescharacteristic
29C2×#208C2212yes—
30#830C2212yes—
31#830C2212yes—
32#810C2214yescomplemented
33#809C2214yescomplemented
34#808C2214yescomplemented
35#807C2214yescomplemented
36#525—31no—
37C22×C4⋊C4D311yescharacteristic, complemented
38#484—32no—
39#483—32no—
40#481—32no—
41#480—32no—
42#428—32no—
43#417—32no—
44#407—31no—
45C24×C3D411yescharacteristic
46C22×C3×C4—24no—
47C22×C3×C4—22no—
48C22×C3×C4—22no—
49C22×C3×C4—22no—
50C22×C3×C4—44no—
51C22×C3×C4D411yescharacteristic
52C22×C3×C4D411yescharacteristic
53C22×C3×C4D411yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C2×#67—21no—
58C2×#67—21no—
59C2×#67—21no—
60#235—24no—
61C2×D12C2311yescharacteristic
62C2×C4×D3—21no—
63C2×Dic6C2311yescharacteristic
64C3×C4⋊C4—22no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4C2314yes—
68#217—21no—
69#217—21no—
70#217—21no—
71(C2×D3)⋊C4—21no—
72(C2×D3)⋊C4—21no—
73C4⋊Dic3C2311yescharacteristic
74C4⋊Dic3C2311yescharacteristic
75Dic3⋊C4—21no—
76Dic3⋊C4—21no—
77C4×Dic3—21no—
78#208—24no—
79C2×C3⋊C8C2314yes—
80C2×#41—31no—
81C23×C4—21no—
82C23×C4C2×D311yescharacteristic
83C2×M16—32no—
84#123—61no—
85C22⋊Q8—31no—
86C22⋊Q8—31no—
87#121—31no—
88#121—31no—
89#120—61no—
90C4×D4—61no—
91C4×D4—61no—
92#117—31no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×D312yes—
97C2×C4⋊C4C2×D312yes—
98C2×C4⋊C4C2×D312yes—
99#107—34no—
100#106—34no—
101#103—34no—
102#102—34no—
103C22⋊C8—32no—
104C22⋊C8—32no—
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3C2×D411yescharacteristic
109C23×C3C2×D411yescharacteristic
110C23×C3C2×D411yescharacteristic
111C22×D3—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—24no—
118C2×C3×C4—44no—
119C2×C3×C4—44no—
120C2×C3×C4—44no—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4#4114yes—
125C2×C3×C4C2411yescharacteristic
126#67—42no—
127#67—42no—
128#67—42no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132D12—22no—
133C4×D3—42no—
134Dic6—22no—
135C3⋊C8—24no—
136C24#6711yescharacteristic
137#41—64no—
138C2×Q8—31no—
139C2×D4—31no—
140C2×D4—61no—
141C2×D4—61no—
142C2×D4—61no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—44no—
148C22×C4—61no—
149C22×C4#6711yescharacteristic
150C22×C4#6711yescharacteristic
151C22×C4#6711yescharacteristic
152C22×C4C22×D311yescharacteristic
153C22×C4C22×D311yescharacteristic
154C22×C4C22×D311yescharacteristic
155M16—64no—
156C2×C8—34no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—31no—
161C4⋊C4—31no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4C22×D314yes—
165C22⋊C4—61no—
166C22⋊C4—61no—
167C22⋊C4—61no—
168C22⋊C4—61no—
169C22⋊C4—61no—
170C42—61no—
171C22×C3—28no—
172C22×C3—21no—
173C22×C3—21no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—21no—
177C22×C3—21no—
178C22×C3#12312yes—
179C22×C3AGL(1,ℤ/8)12yes—
180C22×C3#13712yes—
181C22×C3C22×D411yescharacteristic
182C2×D3—41no—
183C2×D3—41no—
184C2×D3—41no—
185C3×C4—21no—
186C3×C4—21no—
187C3×C4—44no—
188C3×C4#12312yes—
189C3×C4C2×#4112yes—
190Dic3—41no—
191Dic3—41no—
192Dic3—41no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—61no—
197C23C2×#6711yescharacteristic
198C23C2×#6711yescharacteristic
199C23C2×#6711yescharacteristic
200Q8—62no—
201D4—62no—
202D4—122no—
203D4—122no—
204D4—122no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—122no—
218C2×C4#23514yes—
219C2×C4C2×#6711yescharacteristic
220C2×C4C2×#6711yescharacteristic
221C2×C4C2×#6711yescharacteristic
222C2×C4C23×D311yescharacteristic
223C8—64no—
224C2×C3—22no—
225C2×C3—22no—
226C2×C3—22no—
227C2×C3C2×#12311yescharacteristic
228C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C3C2×#13711yescharacteristic
230D3—81no—
231C22—28no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—121no—
239C22—121no—
240C22—121no—
241C22#92912yes—
242C22#93212yes—
243C22#94212yes—
244C22C22×#6711yescharacteristic
245C4—21no—
246C4—21no—
247C4—44no—
248C4—121no—
249C4—121no—
250C4—121no—
251C4#92912yes—
252C4C2×#23512yes—
253C3#309211yescharacteristic, complemented, Sylow
254C2—22no—
255C2—22no—
256C2—22no—
257C2—241no—
258C2C2×#92911yescharacteristic
259C2C2×#93211yescharacteristic
260C2C2×#94211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC22×C3×C4⋊C4C2
socleC22×C3C22×D4
prime radical#5791C2
hypercenterC22×C4⋊C4D3
hypocenterC3#3092
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊C4C2semidirect
3. #5207C2semidirect
4. #4971C2semidirect
5. #4970C2semidirect
6. #4968C2semidirect
7. #4967C2semidirect
8. #4966C2semidirect
9. #4964C2semidirect
32. #810C22semidirect
33. #809C22semidirect
34. #808C22semidirect
35. #807C22semidirect
37. C22×C4⋊C4D3semidirect
253. C3#3092semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
#57912
#52072
#49712
#49702
#49682
#49672
#49662
#49642
#30923