Finite Group Databasebeta

← #71892

Subgroups — #71892

Order 384.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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×#543C211yesmaximal, characteristic, complemented
2#5046C212yesmaximal, complemented
3#4960C211yesmaximal, characteristic
4#4538C212yesmaximal, complemented
5#4525C211yesmaximal, characteristic, complemented
6#1596—31nomaximal, Sylow
7C22×#67—21no—
8C22×D12C2211yescharacteristic
9C3×C22⋊Q8—22no—
10C3×C22⋊Q8—21no—
11C3×C22⋊Q8—22no—
12C3×C22⋊Q8C2212yes—
13C2×C3×C22⋊C4—21no—
14C2×C3×C22⋊C4C2211yescharacteristic
15C2×#215—22no—
16C2×(C2×D3)⋊C4—21no—
17C3⋊#95C2211yescharacteristic
18#830C2212yes—
19#809—22no—
20#543D311yescharacteristic, complemented
21#450—32no—
22#392—31no—
23#328—32no—
24#323—31no—
25C24×C3D411yescharacteristic
26C23×D3—21no—
27C2×C3×Q8—21no—
28C2×C3×Q8D412yescomplemented
29C22×C3×C4—21no—
30C22×C3×C4—21no—
31C22×C3×C4D411yescharacteristic
32C22×C3×C4C2311yescharacteristic
33C2×#67—42no—
34C2×#67—42no—
35C2×#67—42no—
36C22×Dic3—21no—
37C2×D12—22no—
38C2×D12—42no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4—22no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4D412yescomplemented
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45C3×C22⋊C4—22no—
46#215—44no—
47(C2×D3)⋊C4—42no—
48C2×C3⋊C8—22no—
49C22×D4—31no—
50C22×D4—61no—
51C2×SD16—62no—
52C22⋊Q8—22no—
53C22⋊Q8—21no—
54C22⋊Q8—22no—
55C22⋊Q8C2×D312yes—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—61no—
58C2×C22⋊C4C2×D311yescharacteristic
59#102—62no—
60C22⋊C8—32no—
61#95—31no—
62C23×C3—21no—
63C23×C3—21no—
64C23×C3—41no—
65C23×C3—41no—
66C23×C3SD1612yescomplemented
67C23×C3C2×D411yescharacteristic
68C22×D3—22no—
69C22×D3—41no—
70C22×D3—41no—
71C22×D3—41no—
72C22×D3—41no—
73C22×D3—41no—
74C22×D3—41no—
75C3×Q8—42no—
76C3×Q8—81no—
77C2×C3×C4—22no—
78C2×C3×C4—21no—
79C2×C3×C4—22no—
80C2×C3×C4—41no—
81C2×C3×C4—41no—
82C2×C3×C4—81no—
83C2×C3×C4—81no—
84C2×C3×C4C2×D412yes—
85#67—48no—
86C2×Dic3—22no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89D12—24no—
90D12—42no—
91C3⋊C8—42no—
92C24—61no—
93C24#6711yescharacteristic
94C2×Q8—21no—
95C2×Q8#6712yescomplemented
96C2×D4—62no—
97C2×D4—122no—
98C2×D4—122no—
99C2×D4—122no—
100C2×D4—122no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—61no—
104C22×C4#6711yescharacteristic
105C22×C4C22×D311yescharacteristic
106SD16—124no—
107C2×C8—62no—
108C4⋊C4—21no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4#6712yescomplemented
112C22⋊C4—22no—
113C22⋊C4—22no—
114C22⋊C4—22no—
115C22⋊C4—122no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—22no—
119C22×C3—42no—
120C22×C3—41no—
121C22×C3—41no—
122C22×C3—81no—
123C22×C3#12011yescharacteristic
124C22×C3C2×SD1611yescharacteristic
125C22×C3#13711yescharacteristic
126C2×D3—42no—
127C2×D3—42no—
128C2×D3—42no—
129C2×D3—81no—
130C2×D3—81no—
131C2×D3—81no—
132C2×D3—81no—
133C3×C4—22no—
134C3×C4—41no—
135C3×C4—42no—
136C3×C4—42no—
137Dic3—42no—
138C23—21no—
139C23—21no—
140C23—41no—
141C23—41no—
142C23—62no—
143C23—121no—
144C23—121no—
145C23—121no—
146C23—121no—
147C23—121no—
148C23—121no—
149C23#21512yescomplemented
150C23C2×#6711yescharacteristic
151Q8—42no—
152Q8—81no—
153D4—64no—
154D4—122no—
155D4—128no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—62no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4C2×#6712yes—
167C8—122no—
168C2×C3—21no—
169C2×C3—21no—
170C2×C3—42no—
171C2×C3#44811yescharacteristic
172C2×C3#45311yescharacteristic
173C2×C3B(4,𝔽2)11yescharacteristic
174D3—82no—
175C22—21no—
176C22—21no—
177C22—22no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—81no—
182C22—122no—
183C22—122no—
184C22—122no—
185C22—241no—
186C22—241no—
187C22—241no—
188C22—241no—
189C22C2×#21511yescharacteristic
190C22#94211yescharacteristic
191C22#95311yescharacteristic
192C4—22no—
193C4—41no—
194C4—42no—
195C4—42no—
196C4—122no—
197C3#159611yescharacteristic, complemented, Sylow
198C2—21no—
199C2—21no—
200C2—42no—
201C2—242no—
202C2#522711yescharacteristic
203C2#524111yescharacteristic
204C2#524411yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#543C2
socleC22×C3#120
prime radical#4960C2
hypercenter#543D3
hypocenterC3#1596
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. C3×#543C2semidirect
2. #5046C2semidirect
4. #4538C2semidirect
5. #4525C2semidirect
20. #543D3semidirect
28. C2×C3×Q8D4semidirect
42. C3×C4⋊C4D4semidirect
66. C23×C3SD16semidirect
95. C2×Q8#67semidirect
111. C4⋊C4#67semidirect
149. C23#215semidirect
197. C3#1596semidirect

Maximal subgroups

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

subgroupindex
C3×#5432
#50462
#49602
#45382
#45252
#15963