Finite Group Databasebeta

← #78032

Subgroups — #78032

Order 384.

412 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 129 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#4980C212yesmaximal, complemented
3#4978C212yesmaximal, complemented
4#4977C212yesmaximal, complemented
5#4953C211yesmaximal, characteristic
6#4953C212yesmaximal
7#4946C211yesmaximal, characteristic
8Dic3×C22⋊C4C212yesmaximal
9#4654C212yesmaximal, complemented
10#2517—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⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4C2212yes—
21C2×C4×Dic3C2212yes—
22C2×C4×Dic3C2212yes—
23C3×#95C2211yescharacteristic
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#399D311yescharacteristic
33#397—31no—
34#397—32no—
35#387—32no—
36#387—32no—
37#387—32no—
38#385—31no—
39#382—32no—
40C4×C22⋊C4—32no—
41C24×C3C2311yescharacteristic
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×Dic3—24no—
47C22×Dic3—41no—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C3×C4⋊C4—24no—
53C3×C4⋊C4—41no—
54C3×C22⋊C4—24no—
55C3×C22⋊C4—41no—
56#217—24no—
57#217—22no—
58C4⋊Dic3—24no—
59C4⋊Dic3—41no—
60Dic3⋊C4—24no—
61C4×Dic3—24no—
62C4×Dic3—42no—
63C23×C4—31no—
64C2×C4⋊C4—32no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4C2×D312yes—
68C2×C4⋊C4C2×D311yescharacteristic
69C2×C22⋊C4—32no—
70C2×C22⋊C4—31no—
71C2×C22⋊C4C2×D312yes—
72C2×C22⋊C4C2×D311yescharacteristic
73C2×C42—32no—
74C2×C42—32no—
75#95—32no—
76#95—32no—
77#95—31no—
78#95—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95C2×D311yescharacteristic
84C23×C3—22no—
85C23×C3—22no—
86C23×C3—21no—
87C23×C3—21no—
88C23×C3#4112yes—
89C23×C3C2411yescharacteristic
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4#4114yes—
100C2×Dic3—22no—
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—44no—
112C2×Dic3#4114yes—
113C24C22×D311yescharacteristic
114C22×C4—32no—
115C22×C4—32no—
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—64no—
119C22×C4—121no—
120C22×C4C22×D312yes—
121C22×C4C22×D311yescharacteristic
122C22×C4C22×D311yescharacteristic
123C22×C4C22×D312yes—
124C4⋊C4—24no—
125C4⋊C4—41no—
126C4⋊C4—64no—
127C4⋊C4—64no—
128C4⋊C4—121no—
129C22⋊C4—24no—
130C22⋊C4—41no—
131C22⋊C4—64no—
132C22⋊C4—62no—
133C42—64no—
134C42—122no—
135C22×C3—24no—
136C22×C3—22no—
137C22×C3—41no—
138C22×C3—42no—
139C22×C3—41no—
140C22×C3C2×#4112yes—
141C22×C3C2×#4111yescharacteristic
142C22×C3C2×#4112yes—
143C22×C3H(5,𝔽2)11yescharacteristic
144C22×C32−1+411yescharacteristic
145C3×C4—24no—
146C3×C4—41no—
147C3×C4—42no—
148C3×C4—41no—
149Dic3—24no—
150Dic3—42no—
151Dic3—42no—
152Dic3—42no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—21no—
157C23#23912yes—
158C23C23×D311yescharacteristic
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—34no—
169C2×C4—62no—
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—124no—
181C2×C4#23714yes—
182C2×C3—42no—
183C2×C3#52912yes—
184C2×C3#53911yescharacteristic
185C2×C3#55111yescharacteristic
186C2×C3#55212yes—
187C2×C3#55911yescharacteristic
188C22—22no—
189C22—24no—
190C22—42no—
191C22—41no—
192C22—41no—
193C22C2×#23712yes—
194C22#100411yescharacteristic
195C22C2×#23911yescharacteristic
196C22D3×#4112yes—
197C22#101011yescharacteristic
198C4—24no—
199C4—41no—
200C4—42no—
201C4—41no—
202C4—64no—
203C4—122no—
204C4—122no—
205C4—122no—
206C3#251711yescharacteristic, complemented, Sylow
207C2—42no—
208C2#562412yes—
209C2#564511yescharacteristic
210C2#565012yes—
211C2#565911yescharacteristic
212C2#571311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×C3C23
hypercenter#399D3
hypocenterC3#2517
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #4980C2semidirect
3. #4978C2semidirect
4. #4977C2semidirect
9. #4654C2semidirect
206. C3#2517semidirect

Maximal subgroups

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

subgroupindex
C3×#3992
#49802
#49782
#49772
#49532
#49462
Dic3×C22⋊C42
#46542
#25173