Finite Group Databasebeta

← #77872

Subgroups — #77872

Order 384.

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

normal subgroups only

This group has 456 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
1C2×#888C212yesmaximal
2C3×#398C211yesmaximal, characteristic
3#4981C211yesmaximal, characteristic
4#4979C212yesmaximal
5#4976C212yesmaximal
6#4973C211yesmaximal, characteristic
7#4656C212yesmaximal
8#4655C212yesmaximal
9#4652C212yesmaximal
10#2734—31nomaximal, Sylow
11C22×Dic6C2212yes—
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic3C2212yes—
17C2×C4⋊Dic3C2211yescharacteristic
18C2×Dic3⋊C4C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×C4×Dic3C2212yes—
23C2×C4×Dic3C2211yescharacteristic
24#888—28no—
25C3×#95C2212yes—
26C3×#95C2211yescharacteristic
27C3⋊#95C2211yescharacteristic
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×C4⋊Q8—32no—
34#398—31no—
35#398D311yescharacteristic
36#395—32no—
37#395—32no—
38#391—31no—
39#389—32no—
40#387—32no—
41#387—32no—
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C2×Dic6—24no—
52C2×Dic6—41no—
53C2×Dic6—41no—
54C2×Dic6—42no—
55C2×Dic6—42no—
56C2×Dic6D414yes—
57C3×C4⋊C4—24no—
58C3×C4⋊C4—41no—
59C3×C4⋊C4—41no—
60C4⋊Dic3—24no—
61C4⋊Dic3—41no—
62Dic3⋊C4—24no—
63Dic3⋊C4—24no—
64Dic3⋊C4—42no—
65Dic3⋊C4Q818yescomplemented
66C4×Dic3—24no—
67C4×Dic3—41no—
68C22×Q8—31no—
69C22×Q8—32no—
70C4⋊Q8—68no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4C2×D312yes—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C4⋊C4C2×D311yescharacteristic
80C2×C42—32no—
81C2×C42—31no—
82#95—31no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95C2×D311yescharacteristic
89#95C2×D312yes—
90C23×C3C2411yescharacteristic
91C2×C3×C4—22no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4C2×D412yes—
103C2×C3×C4C2×Q814yes—
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×Dic3C2×D414yes—
114C2×Dic3C2×Q814yes—
115C2×Dic3C2×Q814yes—
116Dic6—44no—
117Dic6—48no—
118C2×Q8—34no—
119C2×Q8—64no—
120C2×Q8—121no—
121C2×Q8—121no—
122C2×Q8—122no—
123C2×Q8—122no—
124C22×C4—32no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4C22×D311yescharacteristic
129C22×C4C22×D311yescharacteristic
130C22×C4C22×D312yes—
131C22×C4C22×D311yescharacteristic
132C22×C4C22×D312yes—
133C4⋊C4—24no—
134C4⋊C4—38no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—64no—
138C4⋊C4—64no—
139C4⋊C4—64no—
140C4⋊C4—122no—
141C4⋊C4—121no—
142C42—64no—
143C42—121no—
144C22×C3C22×D411yescharacteristic
145C22×C3C22×Q812yes—
146C22×C3H(5,𝔽2)11yescharacteristic
147C22×C3H(5,𝔽2)11yescharacteristic
148C22×C32−1+412yes—
149C3×C4—22no—
150C3×C4—24no—
151C3×C4—42no—
152C3×C4—41no—
153C3×C4—41no—
154Dic3—24no—
155Dic3—24no—
156Dic3—24no—
157Dic3—42no—
158C23C23×D311yescharacteristic
159Q8—124no—
160Q8—128no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—34no—
173C2×C4—34no—
174C2×C4—34no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4D3×D412yes—
185C2×C4D3×Q814yes—
186C2×C3#53411yescharacteristic
187C2×C3#54412yes—
188C2×C3D4×Q812yes—
189C2×C3Q8211yescharacteristic
190C2×C3#56311yescharacteristic
191C22C2×D3×D411yescharacteristic
192C22#100411yescharacteristic
193C22C2×D3×Q812yes—
194C22#100911yescharacteristic
195C22#101012yes—
196C4—24no—
197C4—22no—
198C4—41no—
199C4—41no—
200C4—42no—
201C4—64no—
202C4—64no—
203C4—64no—
204C4—122no—
205C3#273411yescharacteristic, complemented, Sylow
206C2#560511yescharacteristic
207C2#563212yes—
208C2#565111yescharacteristic
209C2#568912yes—
210C2#572211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#398C2
hypercenter#398D3
hypocenterC3#2734
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
65. Dic3⋊C4Q8semidirect
205. C3#2734semidirect

Maximal subgroups

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

subgroupindex
C2×#8882
C3×#3982
#49812
#49792
#49762
#49732
#46562
#46552
#46522
#27343