Finite Group Databasebeta

← #77277

Subgroups — #77277

Order 384.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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#4992C212yesmaximal, complemented
5#4988C211yesmaximal, characteristic
6#4981C211yesmaximal, characteristic, complemented
7#4675C212yesmaximal, complemented
8#4672C212yesmaximal
9#4652C212yesmaximal, complemented
10#2830—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2212yes—
15C2×(C2×D3)⋊C4C2212yes—
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C3×#95C2212yes—
26C3×#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#398—31no—
33#397—32no—
34#397—32no—
35#396—32no—
36#395D311yescharacteristic, complemented
37#394—32no—
38#393—32no—
39#390—31no—
40#387—32no—
41C23×D3C2312yes—
42C2×C3×Q8—41no—
43C2×C3×Q8—42no—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×D12—22no—
53C2×D12—41no—
54C2×D12—41no—
55C3×C4⋊C4—41no—
56C3×C4⋊C4—42no—
57(C2×D3)⋊C4—42no—
58(C2×D3)⋊C4—42no—
59(C2×D3)⋊C4—42no—
60(C2×D3)⋊C4—42no—
61(C2×D3)⋊C4—42no—
62(C2×D3)⋊C4—42no—
63C4⋊Dic3—41no—
64Dic3⋊C4—24no—
65Dic3⋊C4—42no—
66C4×Dic3—41no—
67C22×Q8C2×D311yescharacteristic
68C22×D4—31no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C4⋊C4C2×D312yes—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C42—31no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95—31no—
86#95C2×D312yes—
87#95C2×D311yescharacteristic
88C23×C3C2411yescharacteristic
89C22×D3—22no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—22no—
96C3×Q8—44no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4#4112yes—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119D12—44no—
120C24—32no—
121C2×Q8—41no—
122C2×Q8—42no—
123C2×D4—62no—
124C2×D4—121no—
125C2×D4—121no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—32no—
129C22×C4C22×D312yes—
130C22×C4C22×D311yescharacteristic
131C22×C4C22×D312yes—
132C22×C4C22×D311yescharacteristic
133C22×C4C22×D311yescharacteristic
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—64no—
137C4⋊C4—122no—
138C4⋊C4—121no—
139C22⋊C4—122no—
140C22⋊C4—122no—
141C22⋊C4—122no—
142C22⋊C4—122no—
143C22⋊C4—122no—
144C22⋊C4—122no—
145C42—121no—
146C22×C3C2×#4111yescharacteristic
147C22×C3H(5,𝔽2)11yescharacteristic
148C22×C3H(5,𝔽2)12yes—
149C22×C3H(5,𝔽2)11yescharacteristic
150C22×C32−1+412yes—
151C2×D3—42no—
152C2×D3—42no—
153C2×D3—42no—
154C2×D3—42no—
155C2×D3—42no—
156C2×D3—42no—
157C2×D3—42no—
158C3×C4—22no—
159C3×C4—41no—
160C3×C4—42no—
161C3×C4—41no—
162C3×C4—42no—
163Dic3—42no—
164Dic3—42no—
165Dic3—42no—
166C23—62no—
167C23—62no—
168C23—62no—
169C23—62no—
170C23—62no—
171C23—62no—
172C23—62no—
173C23C23×D311yescharacteristic
174Q8—44no—
175D4—124no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4#23912yes—
198C2×C3#53811yescharacteristic
199C2×C3#54212yes—
200C2×C3#56011yescharacteristic
201C2×C3#56212yes—
202C2×C3#56311yescharacteristic
203D3—82no—
204C22—122no—
205C22—122no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22—122no—
210C22—122no—
211C22#100411yescharacteristic
212C22#100412yes—
213C22C2×#23911yescharacteristic
214C22#100712yes—
215C22#100911yescharacteristic
216C4—22no—
217C4—42no—
218C4—42no—
219C4—41no—
220C4—41no—
221C4—122no—
222C4—122no—
223C4—122no—
224C3#283011yescharacteristic, complemented, Sylow
225C2—242no—
226C2#559511yescharacteristic
227C2#564411yescharacteristic
228C2#564712yes—
229C2#573312yes—
230C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×D12C22
hypercenter#395D3
hypocenterC3#2830
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
4. #4992C2semidirect
6. #4981C2semidirect
7. #4675C2semidirect
9. #4652C2semidirect
36. #395D3semidirect
224. C3#2830semidirect

Maximal subgroups

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

subgroupindex
C3×#3952
#52312
#49942
#49922
#49882
#49812
#46752
#46722
#46522
#28303