Finite Group Databasebeta

← #79580

Subgroups — #79580

Order 384.

384 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#5559C212yesmaximal, complemented
2C3×C4×D8C211yesmaximal, characteristic, complemented
3#5155C211yesmaximal, characteristic, complemented
4#5151C211yesmaximal, characteristic
5#5016C212yesmaximal, complemented
6#4873C211yesmaximal, characteristic, complemented
7#4872C211yesmaximal, characteristic, complemented
8#4787C212yesmaximal, complemented
9#4762C212yesmaximal, complemented
10#4696C211yesmaximal, characteristic, complemented
11C8×Dic6C211yesmaximal, characteristic, complemented
12#3396—31nomaximal, Sylow
13C2×C3×D8C2211yescharacteristic
14C3×C4×D4C2212yes—
15#940C2212yes—
16#936C2212yes—
17C2×#214C2212yes—
18C2×#213C2212yes—
19#929—22no—
20C4×#67—22no—
21C2×Dic12C2211yescharacteristic
22C2×D24C2211yescharacteristic
23C2×#204C2212yes—
24#893C2212yes—
25#887C2212yes—
26#885—22no—
27#883—22no—
28#875C2212yes—
29C4×Dic6C2211yescharacteristic
30C3×#107C2211yescharacteristic
31C3×#102C2212yes—
32C3×C4×C8C2211yescharacteristic
33Dic3⋊C8C2211yescharacteristic
34Dic3⋊C8C2211yescharacteristic
35C8×Dic3C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37#810C2212yes—
38#809C2212yes—
39C4⋊(C3⋊C8)C2211yescharacteristic
40#559—32no—
41#495—31no—
42#495—31no—
43#493—31no—
44#463—32no—
45#463—32no—
46#460—32no—
47C8×Q8—31no—
48C4×Q16—31no—
49C4×D8—31no—
50C4×D8D311yescharacteristic, complemented
51C2×C3×D4C2312yes—
52C22×C3×C4—22no—
53C2×#67—22no—
54C2×#67—22no—
55C2×D12C2312yes—
56C2×C4×D3—22no—
57C2×Dic6D411yescharacteristic
58C2×Dic6C2312yes—
59C3×D8—22no—
60C2×C3×C8C2311yescharacteristic
61C2×C3×C8C2311yescharacteristic
62C3×C4⋊C4C2312yes—
63C3×C22⋊C4—22no—
64C3×C42C2311yescharacteristic
65#217—22no—
66#217—22no—
67#214—42no—
68#213—42no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72C4⋊Dic3D411yescharacteristic
73Dic3⋊C4—22no—
74Dic3⋊C4D411yescharacteristic
75Dic3⋊C4D411yescharacteristic
76C4×Dic3—22no—
77C4×Dic3C2311yescharacteristic
78C4×Dic3C2311yescharacteristic
79C2×C3⋊C8C2311yescharacteristic
80C2×C3⋊C8C2311yescharacteristic
81Dic12—22no—
82D24—22no—
83#204—42no—
84C2×Q16—31no—
85C2×SD16—32no—
86C2×SD16—32no—
87C2×D8—32no—
88C2×D8—31no—
89C2×D8C2×D311yescharacteristic
90#127—32no—
91#124—32no—
92#124—32no—
93#124—62no—
94#123—62no—
95#123—62no—
96C4×Q8—31no—
97C4×Q8—32no—
98C4×D4—32no—
99C4×D4—62no—
100C4×D4C2×D312yes—
101#107C2×D311yescharacteristic
102C4⋊C8—31no—
103C4⋊C8—31no—
104C4⋊C8—31no—
105#103—32no—
106#102—32no—
107#102C2×D312yes—
108C4×C8—31no—
109C4×C8—31no—
110C4×C8C2×D311yescharacteristic
111C23×C3—22no—
112C22×D3—22no—
113C3×D4—22no—
114C3×D4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—82no—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2411yescharacteristic
120#67—44no—
121#67—82no—
122C2×Dic3—22no—
123C2×Dic3C2×D411yescharacteristic
124C2×Dic3C2×D411yescharacteristic
125C2×Dic3C2×D411yescharacteristic
126C2×Dic3C2×D411yescharacteristic
127D12—22no—
128D12—22no—
129C4×D3—82no—
130Dic6—22no—
131Dic6—22no—
132Dic6—22no—
133C3×C8—21no—
134C3×C8#4112yes—
135C3⋊C8—21no—
136C3⋊C8—21no—
137C2×Q8—31no—
138C2×Q8—32no—
139C2×D4—32no—
140C2×D4—62no—
141C2×D4—62no—
142C2×D4C22×D312yes—
143C22×C4—22no—
144C22×C4—62no—
145Q16—62no—
146SD16—122no—
147SD16—122no—
148D8—22no—
149D8—62no—
150D8—122no—
151C2×C8—31no—
152C2×C8—31no—
153C2×C8C22×D311yescharacteristic
154C2×C8C22×D311yescharacteristic
155C4⋊C4—31no—
156C4⋊C4—31no—
157C4⋊C4—31no—
158C4⋊C4—62no—
159C4⋊C4C22×D312yes—
160C22⋊C4—22no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C42—31no—
167C42—31no—
168C42—62no—
169C42C22×D311yescharacteristic
170C22×C3—42no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3C22×D411yescharacteristic
174C2×D3—42no—
175C2×D3—42no—
176C2×D3—42no—
177C3×C4—21no—
178C3×C4—21no—
179C3×C4—42no—
180C3×C4C2×#4111yescharacteristic
181C3×C4H(5,𝔽2)11yescharacteristic
182Dic3—21no—
183Dic3—21no—
184Dic3—21no—
185Dic3—42no—
186Dic3#13512yes—
187C23—22no—
188C23—62no—
189Q8—62no—
190Q8—62no—
191Q8—62no—
192D4—22no—
193D4—22no—
194D4—62no—
195D4—62no—
196D4—124no—
197D4—242no—
198C2×C4—22no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—62no—
204C2×C4—82no—
205C2×C4—242no—
206C2×C4D3×D411yescharacteristic
207C2×C4D3×D411yescharacteristic
208C2×C4C23×D311yescharacteristic
209C8—21no—
210C8—61no—
211C8—61no—
212C8#23512yes—
213C2×C3—82no—
214C2×C3#55011yescharacteristic
215C2×C3C2×#13511yescharacteristic
216C2×C3#57611yescharacteristic
217D3—82no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—122no—
222C22—122no—
223C22—122no—
224C22C2×D3×D411yescharacteristic
225C4—21no—
226C4—21no—
227C4—32no—
228C4—42no—
229C4—61no—
230C4—61no—
231C4—61no—
232C4—122no—
233C4C2×#23511yescharacteristic
234C4#100911yescharacteristic
235C3#339611yescharacteristic, complemented, Sylow
236C2—82no—
237C2—242no—
238C2#555311yescharacteristic
239C2#575711yescharacteristic
240C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
prime radical#5151C2
hypercenterC4×D8D3
hypocenterC3#3396
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5559C2semidirect
2. C3×C4×D8C2semidirect
3. #5155C2semidirect
5. #5016C2semidirect
6. #4873C2semidirect
7. #4872C2semidirect
8. #4787C2semidirect
9. #4762C2semidirect
10. #4696C2semidirect
11. C8×Dic6C2semidirect
50. C4×D8D3semidirect
235. C3#3396semidirect

Maximal subgroups

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

subgroupindex
#55592
C3×C4×D82
#51552
#51512
#50162
#48732
#48722
#47872
#47622
#46962
C8×Dic62
#33963