Finite Group Databasebeta

← #78081

Subgroups — #78081

Order 384.

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

normal subgroups only

This group has 392 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×#399C211yesmaximal, characteristic
2#4986C212yesmaximal, complemented
3#4985C212yesmaximal, complemented
4#4984C212yesmaximal, complemented
5#4953C211yesmaximal, characteristic
6#4950C212yesmaximal
7#4945C212yesmaximal
8#4943C211yesmaximal, characteristic
9#4655C212yesmaximal, complemented
10#2771—31nomaximal, Sylow
11C23×Dic3C2211yescharacteristic
12C2×C3×C4⋊C4C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×C4×Dic3C2212yes—
23C3×#95C2211yescharacteristic
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#400—32no—
34#399D311yescharacteristic
35#397—31no—
36#389—32no—
37#388—32no—
38#387—32no—
39#385—32no—
40#381—31no—
41C24×C3C2311yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×Dic3—22no—
47C22×Dic3—41no—
48C22×Dic3—41no—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C3×C4⋊C4—24no—
54C3×C4⋊C4—41no—
55C3×C22⋊C4—24no—
56C3×C22⋊C4—41no—
57#217—24no—
58#217—41no—
59C4⋊Dic3—42no—
60C4⋊Dic3—41no—
61C4⋊Dic3—42no—
62Dic3⋊C4—24no—
63C4×Dic3—42no—
64C23×C4—31no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4C2×D312yes—
70C2×C4⋊C4C2×D311yescharacteristic
71C2×C22⋊C4—32no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4C2×D312yes—
74C2×C22⋊C4C2×D311yescharacteristic
75C2×C42—32no—
76#95—32no—
77#95—32no—
78#95—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—31no—
84#95C2×D311yescharacteristic
85C23×C3—22no—
86C23×C3—21no—
87C23×C3—21no—
88C23×C3—22no—
89C23×C3#4112yes—
90C23×C3C2411yescharacteristic
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—21no—
97C2×C3×C4—22no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4#4114yes—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—44no—
114C24C22×D311yescharacteristic
115C22×C4—32no—
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—62no—
120C22×C4—121no—
121C22×C4—121no—
122C22×C4C22×D311yescharacteristic
123C22×C4C22×D312yes—
124C22×C4C22×D311yescharacteristic
125C22×C4C22×D312yes—
126C4⋊C4—24no—
127C4⋊C4—41no—
128C4⋊C4—64no—
129C4⋊C4—122no—
130C4⋊C4—121no—
131C4⋊C4—122no—
132C22⋊C4—24no—
133C22⋊C4—41no—
134C22⋊C4—64no—
135C22⋊C4—121no—
136C42—122no—
137C22×C3—22no—
138C22×C3—24no—
139C22×C3—42no—
140C22×C3—41no—
141C22×C3—41no—
142C22×C3C2×#4112yes—
143C22×C3C2×#4111yescharacteristic
144C22×C3H(5,𝔽2)11yescharacteristic
145C22×C3H(5,𝔽2)11yescharacteristic
146C22×C32−1+412yes—
147C3×C4—24no—
148C3×C4—42no—
149C3×C4—41no—
150C3×C4—41no—
151Dic3—42no—
152Dic3—42no—
153Dic3—42no—
154Dic3—42no—
155C23—22no—
156C23—21no—
157C23—21no—
158C23—22no—
159C23#23912yes—
160C23C23×D311yescharacteristic
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—124no—
183C2×C4#23714yes—
184C2×C3—42no—
185C2×C3#53811yescharacteristic
186C2×C3#53912yes—
187C2×C3#55212yes—
188C2×C3#55911yescharacteristic
189C2×C3#56311yescharacteristic
190C22—24no—
191C22—22no—
192C22—41no—
193C22—42no—
194C22—41no—
195C22C2×#23712yes—
196C22#100411yescharacteristic
197C22C2×#23911yescharacteristic
198C22#100911yescharacteristic
199C22#101012yes—
200C4—24no—
201C4—41no—
202C4—42no—
203C4—41no—
204C4—122no—
205C4—122no—
206C4—122no—
207C4—122no—
208C3#277111yescharacteristic, complemented, Sylow
209C2—42no—
210C2#564212yes—
211C2#564411yescharacteristic
212C2#565111yescharacteristic
213C2#566512yes—
214C2#571311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×C3C23
hypercenter#399D3
hypocenterC3#2771
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #4986C2semidirect
3. #4985C2semidirect
4. #4984C2semidirect
9. #4655C2semidirect
208. C3#2771semidirect

Maximal subgroups

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

subgroupindex
C3×#3992
#49862
#49852
#49842
#49532
#49502
#49452
#49432
#46552
#27713