Finite Group Databasebeta

← #79350

Subgroups — #79350

Order 384.

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

normal subgroups only

This group has 434 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
1#5477C211yesmaximal, characteristic, complemented
2#5477C211yesmaximal, characteristic, complemented
3C3×#432C211yesmaximal, characteristic, complemented
4#4843C212yesmaximal, complemented
5#4841C212yesmaximal, complemented
6#4734C214yesmaximal, complemented
7#4714C212yesmaximal, complemented
8#4712C212yesmaximal, complemented
9#3214—31nomaximal, Sylow
10C2×#235—21no—
11C2×#235C2212yes—
12C2×C3×C42C2211yescharacteristic
13#930—21no—
14#930—21no—
15#930—21no—
16#930C2212yes—
17C2×D24C2214yes—
18C2×#204C2214yes—
19#875—21no—
20#875C2212yes—
21#874C2211yescharacteristic
22#874C2211yescharacteristic
23C4×D12—21no—
24C4×D12—21no—
25C4×D12—21no—
26C4×D12C2212yes—
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29C3×C4⋊C8C2212yes—
30C3×C22⋊C8C2212yes—
31C3×#97C2212yes—
32#821C2214yes—
33#816C2214yes—
34#532—31no—
35#532—31no—
36#496—32no—
37#489—32no—
38#461—32no—
39#459—32no—
40#449—34no—
41#432D311yescharacteristic, complemented
42C22×C3×C4D411yescharacteristic
43C22×C3×C4D411yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C2×#67—22no—
46C2×#67—22no—
47#235—44no—
48#235—44no—
49C2×D12—21no—
50C2×D12—21no—
51C2×D12—21no—
52C2×D12—21no—
53C2×D12C2312yes—
54C2×D12C2312yes—
55C2×C4×D3—22no—
56C2×C4×D3—22no—
57C2×Dic6—21no—
58C2×Dic6C2312yes—
59C2×C3×C8C2314yes—
60C3×C42D411yescharacteristic
61C3×C42D411yescharacteristic
62C3×C42C2311yescharacteristic
63C3×C42C2311yescharacteristic
64(C2×D3)⋊C4—22no—
65(C2×D3)⋊C4—22no—
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—22no—
68C4⋊Dic3—21no—
69C4⋊Dic3—21no—
70C4⋊Dic3—21no—
71C4⋊Dic3C2312yes—
72D24—44no—
73#204—44no—
74C2×#41—32no—
75C2×#41—61no—
76C2×SD16—34no—
77C2×D8—34no—
78C4⋊Q8—31no—
79C4⋊Q8—31no—
80#127—31no—
81#127—31no—
82#124—32no—
83#124—61no—
84#121—32no—
85#121—61no—
86#121—61no—
87#121—61no—
88C4×D4—32no—
89C4×D4—61no—
90C4×D4—61no—
91C4×D4—61no—
92C2×C42C2×D311yescharacteristic
93C4⋊C8C2×D312yes—
94#103—34no—
95#102—34no—
96C22⋊C8C2×D312yes—
97#97C2×D312yes—
98C23×C3C2×D411yescharacteristic
99C22×D3—22no—
100C22×D3—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—41no—
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2411yescharacteristic
110#67—44no—
111#67—44no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114D12—22no—
115D12—22no—
116D12—24no—
117D12—42no—
118D12—42no—
119D12—42no—
120D12—42no—
121C4×D3—44no—
122C4×D3—44no—
123Dic6—24no—
124Dic6—42no—
125C3×C8—24no—
126#41—124no—
127#41—124no—
128C2×Q8—32no—
129C2×Q8—61no—
130C2×D4—32no—
131C2×D4—32no—
132C2×D4—62no—
133C2×D4—62no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C22×C4—62no—
139C22×C4—62no—
140C22×C4D1211yescharacteristic
141C22×C4D1211yescharacteristic
142C22×C4C22×D311yescharacteristic
143SD16—124no—
144D8—124no—
145C2×C8C22×D314yes—
146C4⋊C4—32no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C22⋊C4—62no—
151C22⋊C4—62no—
152C22⋊C4—62no—
153C22⋊C4—62no—
154C42D1211yescharacteristic
155C42D1211yescharacteristic
156C42C22×D311yescharacteristic
157C42C22×D311yescharacteristic
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3C22×D411yescharacteristic
162C2×D3—42no—
163C2×D3—42no—
164C2×D3—42no—
165C2×D3—42no—
166C2×D3—42no—
167C2×D3—42no—
168C3×C4—21no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—21no—
172C3×C4AGL(1,ℤ/8)12yes—
173C3×C4H(5,𝔽2)12yes—
174Dic3—42no—
175Dic3—42no—
176C23—62no—
177C23—62no—
178C23C2×D1211yescharacteristic
179Q8—64no—
180Q8—122no—
181D4—62no—
182D4—64no—
183D4—62no—
184D4—122no—
185D4—122no—
186D4—122no—
187D4—124no—
188D4—124no—
189D4—122no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—124no—
196C2×C4—124no—
197C2×C4C2×D1211yescharacteristic
198C2×C4C2×D1211yescharacteristic
199C2×C4C2×D1211yescharacteristic
200C2×C4C2×D1211yescharacteristic
201C2×C4C2×D1211yescharacteristic
202C2×C4C23×D311yescharacteristic
203C8—24no—
204C2×C3—41no—
205C2×C3#53511yescharacteristic
206C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C3#57511yescharacteristic
208D3—82no—
209D3—82no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—122no—
214C22—122no—
215C22—122no—
216C22—122no—
217C22—122no—
218C22—122no—
219C22C22×D1211yescharacteristic
220C4—21no—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—122no—
225C4—122no—
226C4#90812yes—
227C4#100912yes—
228C3#321411yescharacteristic, complemented, Sylow
229C2—41no—
230C2—242no—
231C2—242no—
232C2#552511yescharacteristic
233C2C2×#90811yescharacteristic
234C2#574811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#432C2
socleC22×C3C22×D4
hypercenter#432D3
hypocenterC3#3214
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #5477C2semidirect
2. #5477C2semidirect
3. C3×#432C2semidirect
4. #4843C2semidirect
5. #4841C2semidirect
6. #4734C2semidirect
7. #4714C2semidirect
8. #4712C2semidirect
41. #432D3semidirect
228. C3#3214semidirect

Maximal subgroups

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

subgroupindex
#54772
C3×#4322
#48432
#48412
#47342
#47142
#47122
#32143