Finite Group Databasebeta

← #79559

Subgroups — #79559

Order 384.

476 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 129 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×#909C212yesmaximal, complemented
2#5552C212yesmaximal, complemented
3C3×#436C211yesmaximal, characteristic, complemented
4#5134C212yesmaximal, complemented
5#4845C212yesmaximal, complemented
6#4739C212yesmaximal, complemented
7#4734C212yesmaximal, complemented
8C4×Dic12C211yesmaximal, characteristic, complemented
9#4696C211yesmaximal, characteristic, complemented
10#3277—31nomaximal, Sylow
11C2×#235C2212yes—
12C22×Dic6C2212yes—
13C2×C3×M16C2212yes—
14C3×C4×D4C2211yescharacteristic
15#939—22no—
16#930C2212yes—
17#924C2212yes—
18#909—28no—
19C2×Dic12C2212yes—
20C2×Dic12C2211yescharacteristic
21C2×Dic12C2211yescharacteristic
22C2×D24C2211yescharacteristic
23C2×#204C2212yes—
24C2×#204C2212yes—
25#896—22no—
26#885—22no—
27#67⋊C4—22no—
28#875C2212yes—
29C4×Dic6C2212yes—
30C3×C4⋊C8C2211yescharacteristic
31C3×C22⋊C8C2212yes—
32C3×C4×C8C2211yescharacteristic
33#821C2212yes—
34#818C2211yescharacteristic
35#816C2212yes—
36#816C2212yes—
37C2×#137—32no—
38#548—32no—
39#495—31no—
40#468—32no—
41#464—32no—
42#451—32no—
43#449—32no—
44C4×Q16—31no—
45#436D311yescharacteristic, complemented
46C2×C3×D4D411yescharacteristic
47C22×C3×C4C2312yes—
48C2×#67—22no—
49C2×#67—22no—
50C22×Dic3—22no—
51#235—24no—
52#235—42no—
53C2×D12C2312yes—
54C2×C4×D3—22no—
55C2×Dic6—24no—
56C2×Dic6—22no—
57C2×Dic6—42no—
58C2×Dic6C2312yes—
59C2×Dic6C2312yes—
60C3×M16—24no—
61C2×C3×C8C2311yescharacteristic
62C2×C3×C8C2312yes—
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4D411yescharacteristic
65C3×C22⋊C4D412yes—
66C3×C42C2311yescharacteristic
67#217—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71C4⋊Dic3C2312yes—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75Dic12—24no—
76Dic12—41no—
77Dic12D414yescomplemented
78D24—41no—
79#204—24no—
80#204—24no—
81C2×#41—32no—
82C22×Q8—32no—
83#137—68no—
84C2×Q16—31no—
85C2×Q16—32no—
86C2×Q16—31no—
87C2×SD16—32no—
88C2×SD16—32no—
89C2×D8—31no—
90C2×M16C2×D312yes—
91#124—32no—
92#124—62no—
93C22⋊Q8—32no—
94C22⋊Q8—62no—
95#121—32no—
96#121—62no—
97C4×Q8—32no—
98C4×D4—62no—
99C4×D4C2×D311yescharacteristic
100#107—31no—
101C4⋊C8C2×D311yescharacteristic
102#103—32no—
103#103—32no—
104#102—32no—
105C22⋊C8C2×D312yes—
106C4×C8C2×D311yescharacteristic
107C23×C3C2×D412yes—
108C22×D3—22no—
109C3×D4—41no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2411yescharacteristic
116#67—44no—
117#67—44no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—44no—
122D12—22no—
123D12—22no—
124C4×D3—44no—
125Dic6—22no—
126Dic6—22no—
127Dic6—22no—
128Dic6—44no—
129Dic6C2×D414yes—
130C3×C8—22no—
131C3×C8—21no—
132C3×C8C2×D412yes—
133#41—64no—
134#41—122no—
135C2×Q8—32no—
136C2×Q8—32no—
137C2×Q8—64no—
138C2×Q8—62no—
139C2×Q8—122no—
140C2×D4—32no—
141C2×D4—62no—
142C2×D4—62no—
143C2×D4D1211yescharacteristic
144C22×C4—62no—
145C22×C4—62no—
146C22×C4C22×D312yes—
147Q16—34no—
148Q16—64no—
149Q16—121no—
150SD16—64no—
151SD16—64no—
152D8—121no—
153M16—24no—
154C2×C8C22×D311yescharacteristic
155C2×C8C22×D312yes—
156C2×C8C22×D311yescharacteristic
157C4⋊C4—32no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4D1211yescharacteristic
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4D1212yes—
166C42—62no—
167C42C22×D311yescharacteristic
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—22no—
171C22×C3C22×D411yescharacteristic
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C3×C4—22no—
176C3×C4—21no—
177C3×C4—21no—
178C3×C4C22×D411yescharacteristic
179C3×C4H(5,𝔽2)11yescharacteristic
180Dic3—24no—
181Dic3—42no—
182Dic3—42no—
183C23—62no—
184C23C2×D1212yes—
185Q8—34no—
186Q8—62no—
187Q8—62no—
188Q8—62no—
189Q8—124no—
190D4—41no—
191D4—62no—
192D4—62no—
193D4—124no—
194D4—124no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—124no—
201C2×C4—124no—
202C2×C4C2×D1211yescharacteristic
203C2×C4C2×D1211yescharacteristic
204C2×C4C2×D1212yes—
205C2×C4C23×D311yescharacteristic
206C8—22no—
207C8—21no—
208C8D3×D412yes—
209C2×C3—42no—
210C2×C3D4211yescharacteristic
211C2×C3#57511yescharacteristic
212C2×C3#57811yescharacteristic
213D3—82no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—122no—
218C22—122no—
219C22—122no—
220C22C22×D1211yescharacteristic
221C4—21no—
222C4—21no—
223C4—22no—
224C4—64no—
225C4—122no—
226C4—122no—
227C4C2×D3×D411yescharacteristic
228C4#100911yescharacteristic
229C3#327711yescharacteristic, complemented, Sylow
230C2—42no—
231C2—242no—
232C2D4×D1211yescharacteristic
233C2#574811yescharacteristic
234C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#436C2
socleC22×C3C22×D4
hypercenter#436D3
hypocenterC3#3277
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#909C2semidirect
2. #5552C2semidirect
3. C3×#436C2semidirect
4. #5134C2semidirect
5. #4845C2semidirect
6. #4739C2semidirect
7. #4734C2semidirect
8. C4×Dic12C2semidirect
9. #4696C2semidirect
45. #436D3semidirect
77. Dic12D4semidirect
229. C3#3277semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
#55522
C3×#4362
#51342
#48452
#47392
#47342
C4×Dic122
#46962
#32773