Finite Group Databasebeta

← #77272

Subgroups — #77272

Order 384.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C3×#395C211yesmaximal, characteristic, complemented
2#5231C212yesmaximal, complemented
3#4994C212yesmaximal, complemented
4#4988C211yesmaximal, characteristic
5#4987C212yesmaximal, complemented
6#4986C211yesmaximal, characteristic, complemented
7#4668C212yesmaximal, complemented
8#4665C212yesmaximal
9#4655C212yesmaximal, complemented
10#2784—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C22×C4×D3C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C3×#95C2212yes—
26C3×#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—31no—
33#396—32no—
34#395D311yescharacteristic, complemented
35#393—32no—
36#390—31no—
37#387—32no—
38#386—32no—
39#385—32no—
40#380—32no—
41C23×D3C2312yes—
42C2×C3×Q8—42no—
43C2×C3×Q8—41no—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×D12—41no—
53C2×D12—41no—
54C2×D12Q814yescomplemented
55C2×C4×D3—24no—
56C2×C4×D3—42no—
57C2×C4×D3—42no—
58C3×C4⋊C4—42no—
59C3×C4⋊C4—41no—
60(C2×D3)⋊C4—24no—
61(C2×D3)⋊C4—42no—
62(C2×D3)⋊C4—42no—
63(C2×D3)⋊C4—42no—
64(C2×D3)⋊C4—42no—
65C4⋊Dic3—42no—
66C4⋊Dic3—41no—
67Dic3⋊C4—24no—
68C4×Dic3—41no—
69C22×Q8C2×D311yescharacteristic
70C22×D4—31no—
71C23×C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4C2×D312yes—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C42—31no—
83#95—31no—
84#95—32no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95C2×D311yescharacteristic
89#95C2×D312yes—
90C23×C3C2411yescharacteristic
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—22no—
97C22×D3C2×Q814yes—
98C3×Q8—44no—
99C2×C3×C4—21no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4C2×Q812yes—
111C2×C3×C4#4114yes—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121D12—44no—
122C4×D3—48no—
123C24—32no—
124C2×Q8—41no—
125C2×Q8—42no—
126C2×D4—34no—
127C2×D4—121no—
128C2×D4—121no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—64no—
133C22×C4—122no—
134C22×C4—122no—
135C22×C4C22×D312yes—
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D311yescharacteristic
139C22×C4C22×D312yes—
140C4⋊C4—42no—
141C4⋊C4—41no—
142C4⋊C4—64no—
143C4⋊C4—122no—
144C4⋊C4—121no—
145C22⋊C4—64no—
146C22⋊C4—122no—
147C22⋊C4—122no—
148C22⋊C4—122no—
149C22⋊C4—122no—
150C42—121no—
151C22×C3C22×Q811yescharacteristic
152C22×C3C2×#4112yes—
153C22×C3H(5,𝔽2)12yes—
154C22×C3H(5,𝔽2)11yescharacteristic
155C22×C3H(5,𝔽2)11yescharacteristic
156C2×D3—24no—
157C2×D3—24no—
158C2×D3—24no—
159C2×D3—42no—
160C2×D3—42no—
161C2×D3—42no—
162C2×D3—42no—
163C3×C4—24no—
164C3×C4—22no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—42no—
168Dic3—42no—
169Dic3—42no—
170Dic3—42no—
171C23—34no—
172C23—62no—
173C23—62no—
174C23—62no—
175C23—62no—
176C23—62no—
177C23—62no—
178C23C23×D311yescharacteristic
179Q8—44no—
180D4—124no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—128no—
202C2×C4D3×Q812yes—
203C2×C4#23914yes—
204C2×C3#53812yes—
205C2×C3#54311yescharacteristic
206C2×C3#55412yes—
207C2×C3#55511yescharacteristic
208C2×C3#56011yescharacteristic
209D3—44no—
210C22—64no—
211C22—64no—
212C22—64no—
213C22—122no—
214C22—122no—
215C22—122no—
216C22—122no—
217C22#100411yescharacteristic
218C22#100412yes—
219C22C2×D3×Q811yescharacteristic
220C22C2×#23912yes—
221C22#100911yescharacteristic
222C4—22no—
223C4—24no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—122no—
228C4—122no—
229C4—122no—
230C3#278411yescharacteristic, complemented, Sylow
231C2—124no—
232C2#559511yescharacteristic
233C2#563411yescharacteristic
234C2#564412yes—
235C2#573012yes—
236C2#573411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×D12C22
hypercenter#395D3
hypocenterC3#2784
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C3×#395C2semidirect
2. #5231C2semidirect
3. #4994C2semidirect
5. #4987C2semidirect
6. #4986C2semidirect
7. #4668C2semidirect
9. #4655C2semidirect
34. #395D3semidirect
54. C2×D12Q8semidirect
230. C3#2784semidirect

Maximal subgroups

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

subgroupindex
C3×#3952
#52312
#49942
#49882
#49872
#49862
#46682
#46652
#46552
#27843