Finite Group Databasebeta

← #78198

Subgroups — #78198

Order 384.

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

normal subgroups only

This group has 440 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
1C3×#400C211yesmaximal, characteristic, complemented
2#4990C212yesmaximal, complemented
3#4988C211yesmaximal, characteristic
4#4981C211yesmaximal, characteristic, complemented
5#4672C212yesmaximal
6#4670C212yesmaximal, complemented
7#4669C212yesmaximal, complemented
8#4669C212yesmaximal, complemented
9#4649C212yesmaximal, complemented
10#2593—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×(C2×D3)⋊C4C2212yes—
15C2×(C2×D3)⋊C4C2212yes—
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×C4×Dic3C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2212yes—
26C3×#95C2212yes—
27C3×#95C2211yescharacteristic
28C3×#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32#400D311yescharacteristic, complemented
33#398—31no—
34#394—32no—
35#390—31no—
36#388—32no—
37#388—32no—
38#388—32no—
39#388—32no—
40C4×C4⋊C4—32no—
41C23×D3C2312yes—
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C2×D12—22no—
51C2×D12—41no—
52C2×D12—41no—
53C3×C4⋊C4—41no—
54C3×C4⋊C4—41no—
55(C2×D3)⋊C4—24no—
56(C2×D3)⋊C4—24no—
57(C2×D3)⋊C4—24no—
58(C2×D3)⋊C4—24no—
59(C2×D3)⋊C4—42no—
60(C2×D3)⋊C4—42no—
61C4⋊Dic3—41no—
62Dic3⋊C4—24no—
63Dic3⋊C4—24no—
64C4×Dic3—24no—
65C4×Dic3—41no—
66C4×Dic3—42no—
67C22×D4—31no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C42—32no—
80C2×C42—31no—
81C2×C42—32no—
82#95—32no—
83#95—31no—
84#95—32no—
85#95C2×D312yes—
86#95C2×D311yescharacteristic
87#95C2×D312yes—
88C23×C3C2411yescharacteristic
89C22×D3—22no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—22no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4—21no—
108C2×C3×C4#4112yes—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3#4114yes—
117C2×Dic3#4114yes—
118D12—44no—
119C24—32no—
120C2×D4—62no—
121C2×D4—121no—
122C2×D4—121no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—32no—
126C22×C4C22×D311yescharacteristic
127C22×C4C22×D312yes—
128C22×C4C22×D311yescharacteristic
129C22×C4C22×D312yes—
130C22×C4C22×D311yescharacteristic
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—64no—
134C4⋊C4—64no—
135C4⋊C4—121no—
136C22⋊C4—64no—
137C22⋊C4—64no—
138C22⋊C4—64no—
139C22⋊C4—64no—
140C22⋊C4—122no—
141C22⋊C4—122no—
142C42—64no—
143C42—121no—
144C42—122no—
145C22×C3C2×#4111yescharacteristic
146C22×C3C2×#4112yes—
147C22×C3C2×#4112yes—
148C22×C3H(5,𝔽2)11yescharacteristic
149C22×C3H(5,𝔽2)11yescharacteristic
150C2×D3—42no—
151C2×D3—42no—
152C2×D3—42no—
153C2×D3—42no—
154C2×D3—42no—
155C2×D3—42no—
156C2×D3—42no—
157C3×C4—22no—
158C3×C4—42no—
159C3×C4—42no—
160C3×C4—41no—
161C3×C4—41no—
162Dic3—24no—
163Dic3—24no—
164Dic3—42no—
165C23—62no—
166C23—62no—
167C23—62no—
168C23—62no—
169C23—62no—
170C23—62no—
171C23—62no—
172C23C23×D311yescharacteristic
173D4—124no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—34no—
187C2×C4—34no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4#23912yes—
196C2×C3#52912yes—
197C2×C3#53811yescharacteristic
198C2×C3#55312yes—
199C2×C3#55511yescharacteristic
200C2×C3#55911yescharacteristic
201D3—82no—
202C22—122no—
203C22—122no—
204C22—122no—
205C22—122no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22#100411yescharacteristic
210C22C2×#23911yescharacteristic
211C22D3×#4112yes—
212C22D3×#4112yes—
213C22#100911yescharacteristic
214C4—22no—
215C4—41no—
216C4—41no—
217C4—42no—
218C4—42no—
219C4—64no—
220C4—64no—
221C4—122no—
222C3#259311yescharacteristic, complemented, Sylow
223C2—242no—
224C2#564411yescharacteristic
225C2#566411yescharacteristic
226C2#567011yescharacteristic
227C2#569112yes—
228C2#570912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC22×D12C22
hypercenter#400D3
hypocenterC3#2593
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C3×#400C2semidirect
2. #4990C2semidirect
4. #4981C2semidirect
6. #4670C2semidirect
7. #4669C2semidirect
8. #4669C2semidirect
9. #4649C2semidirect
32. #400D3semidirect
222. C3#2593semidirect

Maximal subgroups

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

subgroupindex
C3×#4002
#49902
#49882
#49812
#46722
#46702
#46692
#46492
#25933