Finite Group Databasebeta

← #77194

Subgroups — #77194

Order 384.

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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
2#5229C212yesmaximal
3#4984C212yesmaximal
4#4981C211yesmaximal, characteristic
5#4978C212yesmaximal
6#4973C211yesmaximal, characteristic
7#4656C212yesmaximal
8#4655C212yesmaximal
9#4649C212yesmaximal
10#2586—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic3C2212yes—
16C2×C4⋊Dic3C2212yes—
17C2×C4⋊Dic3C2211yescharacteristic
18C2×Dic3⋊C4C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×C4×Dic3C2212yes—
21C2×C4×Dic3C2211yescharacteristic
22C2×C4×Dic3C2212yes—
23C3×#95C2212yes—
24C3×#95C2211yescharacteristic
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#398—31no—
33#395—32no—
34#395D311yescharacteristic
35#391—32no—
36#391—31no—
37#389—32no—
38#387—32no—
39#387—32no—
40C4×C4⋊C4—32no—
41C2×C3×Q8—24no—
42C2×C3×Q8—41no—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×Dic6—22no—
53C2×Dic6—41no—
54C2×Dic6—41no—
55C3×C4⋊C4—24no—
56C3×C4⋊C4—41no—
57C4⋊Dic3—42no—
58C4⋊Dic3—41no—
59C4⋊Dic3—42no—
60Dic3⋊C4—24no—
61Dic3⋊C4Q818yescomplemented
62C4×Dic3—24no—
63C4×Dic3—41no—
64C4×Dic3—42no—
65C22×Q8—31no—
66C22×Q8C2×D311yescharacteristic
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4C2×D312yes—
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C42—32no—
75C2×C42—31no—
76C2×C42—32no—
77#95—32no—
78#95—32no—
79#95—31no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95C2×D312yes—
85#95C2×D311yescharacteristic
86C23×C3C2411yescharacteristic
87C3×Q8—44no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—21no—
92C2×C3×C4—22no—
93C2×C3×C4—21no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4C2×Q814yes—
99C2×C3×C4#4114yes—
100C2×C3×C4#4112yes—
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×Dic3C2×Q814yes—
112C2×Dic3C2×Q814yes—
113Dic6—44no—
114C2×Q8—24no—
115C2×Q8—41no—
116C2×Q8—62no—
117C2×Q8—121no—
118C2×Q8—121no—
119C22×C4—32no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C22×C4C22×D312yes—
126C22×C4C22×D312yes—
127C22×C4C22×D311yescharacteristic
128C4⋊C4—24no—
129C4⋊C4—38no—
130C4⋊C4—41no—
131C4⋊C4—64no—
132C4⋊C4—122no—
133C4⋊C4—122no—
134C4⋊C4—121no—
135C42—64no—
136C42—122no—
137C42—121no—
138C22×C3C22×Q812yes—
139C22×C3C2×#4112yes—
140C22×C3C2×#4111yescharacteristic
141C22×C3H(5,𝔽2)11yescharacteristic
142C22×C3H(5,𝔽2)11yescharacteristic
143C3×C4—22no—
144C3×C4—24no—
145C3×C4—24no—
146C3×C4—41no—
147C3×C4—41no—
148Dic3—24no—
149Dic3—24no—
150Dic3—42no—
151Dic3—42no—
152C23C23×D311yescharacteristic
153Q8—44no—
154Q8—124no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—34no—
166C2×C4—34no—
167C2×C4—62no—
168C2×C4—62no—
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#23714yes—
178C2×C4D3×Q814yes—
179C2×C4#23912yes—
180C2×C3#53312yes—
181C2×C3#53811yescharacteristic
182C2×C3#55111yescharacteristic
183C2×C3#55412yes—
184C2×C3Q8211yescharacteristic
185C22C2×#23712yes—
186C22#100411yescharacteristic
187C22C2×D3×Q812yes—
188C22C2×#23911yescharacteristic
189C22#100911yescharacteristic
190C4—24no—
191C4—22no—
192C4—24no—
193C4—41no—
194C4—41no—
195C4—64no—
196C4—64no—
197C4—122no—
198C4—122no—
199C3#258611yescharacteristic, complemented, Sylow
200C2#558711yescharacteristic
201C2#563512yes—
202C2#564411yescharacteristic
203C2#572211yescharacteristic
204C2#572812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#395C2
hypercenter#395D3
hypocenterC3#2586
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
61. Dic3⋊C4Q8semidirect
199. C3#2586semidirect

Maximal subgroups

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

subgroupindex
C3×#3952
#52292
#49842
#49812
#49782
#49732
#46562
#46552
#46492
#25863