Finite Group Databasebeta

← #74291

Subgroups — #74291

Order 384.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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×#876C211yesmaximal, characteristic
2C2×#869C211yesmaximal, characteristic, complemented
3C3×#383C211yesmaximal, characteristic, complemented
4#4677C212yesmaximal
5#4674C212yesmaximal
6#4673C212yesmaximal
7#4661C212yesmaximal, complemented
8#4659C212yesmaximal, complemented
9#4656C212yesmaximal, complemented
10#2821—31nomaximal, Sylow
11C22×Dic6C2212yes—
12C2×C3×C42C2211yescharacteristic
13C2×(C2×D3)⋊C4C2211yescharacteristic
14C2×(C2×D3)⋊C4C2211yescharacteristic
15C2×(C2×D3)⋊C4C2211yescharacteristic
16C2×(C2×D3)⋊C4C2214yes—
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2214yes—
20C2×Dic3⋊C4C2211yescharacteristic
21C2×Dic3⋊C4C2211yescharacteristic
22C2×Dic3⋊C4C2211yescharacteristic
23#876—24no—
24#869—24no—
25C3×#95C2212yes—
26C3×#95C2212yes—
27C3×#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8—31no—
31C2×#126—31no—
32#400—32no—
33#399—32no—
34#398—32no—
35#397—32no—
36#395—32no—
37#393—32no—
38#383D311yescharacteristic, complemented
39C23×D3C2311yescharacteristic
40C22×C3×C4C2314yes—
41C22×C3×C4C2311yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×Dic3C2311yescharacteristic
45C22×Dic3C2311yescharacteristic
46C22×Dic3C2314yes—
47C22×Dic3C2311yescharacteristic
48C2×Dic6—24no—
49C2×Dic6—44no—
50C3×C42D414yes—
51(C2×D3)⋊C4—22no—
52(C2×D3)⋊C4—22no—
53(C2×D3)⋊C4—22no—
54(C2×D3)⋊C4—44no—
55C4⋊Dic3—24no—
56C4⋊Dic3—24no—
57C4⋊Dic3—44no—
58Dic3⋊C4—22no—
59Dic3⋊C4—22no—
60Dic3⋊C4—22no—
61C22×Q8—32no—
62C4⋊Q8—64no—
63#126—64no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—34no—
70C2×C22⋊C4—31no—
71C2×C22⋊C4—31no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4—34no—
74C2×C42C2×D311yescharacteristic
75#95—34no—
76#95—34no—
77#95C2×D312yes—
78#95C2×D312yes—
79#95C2×D312yes—
80C23×C3C2411yescharacteristic
81C22×D3—21no—
82C22×D3—22no—
83C22×D3—22no—
84C22×D3—22no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—24no—
89C2×C3×C4—24no—
90C2×C3×C4—24no—
91C2×C3×C4C2×D412yes—
92C2×C3×C4C2×D412yes—
93C2×C3×C4C2×D412yes—
94C2×Dic3—21no—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—24no—
98C2×Dic3—22no—
99C2×Dic3—24no—
100C2×Dic3—24no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103Dic6—48no—
104C24—31no—
105C2×Q8—64no—
106C2×Q8—124no—
107C22×C4—31no—
108C22×C4—31no—
109C22×C4—34no—
110C22×C4—31no—
111C22×C4C22×D314yes—
112C22×C4C22×D311yescharacteristic
113C22×C4C22×D311yescharacteristic
114C22×C4C22×D311yescharacteristic
115C4⋊C4—62no—
116C4⋊C4—62no—
117C4⋊C4—62no—
118C4⋊C4—64no—
119C4⋊C4—64no—
120C4⋊C4—124no—
121C22⋊C4—62no—
122C22⋊C4—62no—
123C22⋊C4—62no—
124C22⋊C4—124no—
125C42D1214yes—
126C22×C3C22×D411yescharacteristic
127C22×C3H(5,𝔽2)12yes—
128C22×C32−1+412yes—
129C22×C32−1+412yes—
130C2×D3—41no—
131C2×D3—41no—
132C2×D3—41no—
133C2×D3—44no—
134C3×C4—22no—
135C3×C4—22no—
136C3×C4—22no—
137C3×C4—44no—
138Dic3—44no—
139Dic3—41no—
140Dic3—41no—
141Dic3—41no—
142C23—61no—
143C23—62no—
144C23—62no—
145C23—62no—
146C23C23×D311yescharacteristic
147Q8—128no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—64no—
158C2×C4—64no—
159C2×C4—62no—
160C2×C4—64no—
161C2×C4—62no—
162C2×C4—62no—
163C2×C4C2×D1212yes—
164C2×C4C2×D1212yes—
165C2×C4C2×D1212yes—
166C2×C3#53511yescharacteristic
167C2×C3#53611yescharacteristic
168C2×C3#53611yescharacteristic
169C2×C3#56314yes—
170D3—81no—
171C22—124no—
172C22—121no—
173C22—121no—
174C22—121no—
175C22C22×D1211yescharacteristic
176C22#100912yes—
177C22#101012yes—
178C22#101012yes—
179C4—22no—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—124no—
184C4—121no—
185C4—121no—
186C4—121no—
187C3#282111yescharacteristic, complemented, Sylow
188C2—241no—
189C2#552511yescharacteristic
190C2#552611yescharacteristic
191C2#552611yescharacteristic
192C2#571214yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#383C2
prime radicalC23×D3C23
hypercenter#383D3
hypocenterC3#2821
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
2. C2×#869C2semidirect
3. C3×#383C2semidirect
7. #4661C2semidirect
8. #4659C2semidirect
9. #4656C2semidirect
38. #383D3semidirect
187. C3#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#8762
C2×#8692
C3×#3832
#46772
#46742
#46732
#46612
#46592
#46562
#28213