Finite Group Databasebeta

← #79304

Subgroups — #79304

Order 384.

438 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 125 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#5477C211yesmaximal, characteristic, complemented
2#5477C211yesmaximal, characteristic, complemented
3C3×#431C211yesmaximal, characteristic, complemented
4#4844C214yesmaximal, complemented
5#4734C214yesmaximal, complemented
6#4695C211yesmaximal, characteristic, complemented
7#4694C211yesmaximal, characteristic, complemented
8#4693C211yesmaximal, characteristic, complemented
9#4681C211yesmaximal, characteristic, complemented
10#3207—31nomaximal, Sylow
11C2×#235—21no—
12C2×#235C2212yes—
13C2×C3×C42C2211yescharacteristic
14#930—21no—
15#930—21no—
16#930—21no—
17#930C2212yes—
18C2×D24C2214yes—
19C2×#204C2214yes—
20#875—21no—
21#875C2212yes—
22#874C2211yescharacteristic
23#874C2211yescharacteristic
24C4×D12—21no—
25C4×D12—21no—
26C4×D12—21no—
27C4×D12C2212yes—
28#869C2211yescharacteristic
29#869C2211yescharacteristic
30C3×C4⋊C8C2212yes—
31C3×C22⋊C8C2212yes—
32C3×C4×C8C2211yescharacteristic
33C3×C4×C8C2211yescharacteristic
34#821C2214yes—
35#816C2214yes—
36#532—31no—
37#532—31no—
38#493—31no—
39#492—31no—
40#487—31no—
41#486—31no—
42#463—34no—
43#449—34no—
44#431D311yescharacteristic, complemented
45C22×C3×C4D411yescharacteristic
46C22×C3×C4D411yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49C2×#67—22no—
50#235—44no—
51#235—44no—
52C2×D12—21no—
53C2×D12—21no—
54C2×D12—21no—
55C2×D12—21no—
56C2×D12C2312yes—
57C2×D12C2312yes—
58C2×C4×D3—22no—
59C2×C4×D3—22no—
60C2×Dic6—21no—
61C2×Dic6C2312yes—
62C2×C3×C8C2312yes—
63C2×C3×C8C2312yes—
64C3×C42D411yescharacteristic
65C3×C42D411yescharacteristic
66C3×C42C2311yescharacteristic
67C3×C42C2311yescharacteristic
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72C4⋊Dic3—21no—
73C4⋊Dic3—21no—
74C4⋊Dic3—21no—
75C4⋊Dic3C2312yes—
76D24—44no—
77#204—44no—
78C2×#41—32no—
79C2×#41—61no—
80C2×SD16—34no—
81C2×D8—34no—
82C4⋊Q8—31no—
83C4⋊Q8—31no—
84#127—31no—
85#127—31no—
86#124—32no—
87#124—61no—
88#121—32no—
89#121—61no—
90#121—61no—
91#121—61no—
92C4×D4—32no—
93C4×D4—61no—
94C4×D4—61no—
95C4×D4—61no—
96C2×C42C2×D311yescharacteristic
97C4⋊C8C2×D312yes—
98#103—34no—
99#102—34no—
100C22⋊C8C2×D312yes—
101C4×C8C2×D311yescharacteristic
102C4×C8C2×D311yescharacteristic
103C23×C3C2×D411yescharacteristic
104C22×D3—22no—
105C22×D3—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2411yescharacteristic
115#67—44no—
116#67—44no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119D12—24no—
120D12—24no—
121D12—42no—
122D12—42no—
123D12—42no—
124D12—42no—
125C4×D3—44no—
126C4×D3—44no—
127Dic6—24no—
128Dic6—42no—
129C3×C8—22no—
130C3×C8—22no—
131#41—124no—
132#41—124no—
133C2×Q8—32no—
134C2×Q8—61no—
135C2×D4—32no—
136C2×D4—32no—
137C2×D4—62no—
138C2×D4—62no—
139C2×D4—61no—
140C2×D4—61no—
141C2×D4—61no—
142C2×D4—61no—
143C22×C4—62no—
144C22×C4—62no—
145C22×C4D1211yescharacteristic
146C22×C4D1211yescharacteristic
147C22×C4C22×D311yescharacteristic
148SD16—124no—
149D8—124no—
150C2×C8C22×D312yes—
151C2×C8C22×D312yes—
152C4⋊C4—32no—
153C4⋊C4—61no—
154C4⋊C4—61no—
155C4⋊C4—61no—
156C22⋊C4—62no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C42D1211yescharacteristic
161C42D1211yescharacteristic
162C42C22×D311yescharacteristic
163C42C22×D311yescharacteristic
164C22×C3—21no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3C22×D411yescharacteristic
168C2×D3—42no—
169C2×D3—42no—
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—21no—
175C3×C4—21no—
176C3×C4—21no—
177C3×C4#13512yes—
178C3×C4AGL(1,ℤ/8)12yes—
179C3×C4H(5,𝔽2)12yes—
180Dic3—42no—
181Dic3—42no—
182C23—62no—
183C23—62no—
184C23C2×D1211yescharacteristic
185Q8—64no—
186Q8—122no—
187D4—64no—
188D4—64no—
189D4—124no—
190D4—124no—
191D4—122no—
192D4—122no—
193D4—122no—
194D4—122no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—124no—
201C2×C4—124no—
202C2×C4C2×D1211yescharacteristic
203C2×C4C2×D1211yescharacteristic
204C2×C4C2×D1211yescharacteristic
205C2×C4C2×D1211yescharacteristic
206C2×C4C2×D1211yescharacteristic
207C2×C4C23×D311yescharacteristic
208C8—22no—
209C8—22no—
210C2×C3—41no—
211C2×C3#53511yescharacteristic
212C2×C3C2×#13511yescharacteristic
213C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
214D3—82no—
215D3—82no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—122no—
220C22—122no—
221C22—122no—
222C22—122no—
223C22—122no—
224C22—122no—
225C22C22×D1211yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—122no—
230C4—122no—
231C4#90412yes—
232C4#90812yes—
233C4#100912yes—
234C3#320711yescharacteristic, complemented, Sylow
235C2—41no—
236C2—242no—
237C2—242no—
238C2#552511yescharacteristic
239C2C2×#90411yescharacteristic
240C2C2×#90811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#431C2
socleC22×C3C22×D4
hypercenter#431D3
hypocenterC3#3207
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #5477C2semidirect
2. #5477C2semidirect
3. C3×#431C2semidirect
4. #4844C2semidirect
5. #4734C2semidirect
6. #4695C2semidirect
7. #4694C2semidirect
8. #4693C2semidirect
9. #4681C2semidirect
44. #431D3semidirect
234. C3#3207semidirect

Maximal subgroups

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

subgroupindex
#54772
C3×#4312
#48442
#47342
#46952
#46942
#46932
#46812
#32073