Finite Group Databasebeta

← #71844

Subgroups — #71844

Order 384.

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

normal subgroups only

This group has 288 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×#534C211yesmaximal, characteristic
2#5034C212yesmaximal, complemented
3#4949C211yesmaximal, characteristic, complemented
4#4537C212yesmaximal, complemented
5#4525C211yesmaximal, characteristic, complemented
6#1613—31nomaximal, Sylow
7C22×C3×D4—21no—
8C3×#123—22no—
9C3×#121—21no—
10C3×#121C2212yes—
11C3×#120—22no—
12C2×C3×C22⋊C4C2211yescharacteristic
13C2×#217—21no—
14C2×C4⋊Dic3C2211yescharacteristic, complemented
15C2×Dic3⋊C4—21no—
16#832—22no—
17C3⋊#95C2211yescharacteristic, complemented
18#830C2212yescomplemented
19#808—22no—
20#534D311yescharacteristic
21#481—32no—
22#399—31no—
23#327—32no—
24#323—31no—
25C24×C3—21no—
26C24×C3D411yescharacteristic
27C2×C3×D4—22no—
28C2×C3×D4—22no—
29C2×C3×D4—21no—
30C2×C3×D4—42no—
31C2×C3×D4—42no—
32C2×C3×D4D412yes—
33C22×C3×C4—21no—
34C22×C3×C4D411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C22×Dic3—21no—
37C22×Dic3—21no—
38C3×C4⋊C4—21no—
39C3×C4⋊C4D412yes—
40C3×C22⋊C4—22no—
41C3×C22⋊C4—22no—
42C3×C22⋊C4—22no—
43#217—42no—
44C4⋊Dic3—22no—
45Dic3⋊C4—42no—
46C2×C3⋊C8—22no—
47C22×D4—21no—
48#123—22no—
49#121—21no—
50#121C2×D312yes—
51#120—22no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—61no—
54C2×C22⋊C4—61no—
55C2×C22⋊C4C2×D311yescharacteristic
56#106—62no—
57#102—62no—
58C22⋊C8—32no—
59#95—31no—
60C23×C3—21no—
61C23×C3—21no—
62C23×C3—22no—
63C23×C3—41no—
64C23×C3—41no—
65C23×C3—42no—
66C23×C3—81no—
67C23×C3—81no—
68C23×C3SD1612yes—
69C23×C3C2×D411yescharacteristic
70C3×D4—42no—
71C3×D4—44no—
72C3×D4—42no—
73C3×D4—42no—
74C3×D4—81no—
75C2×C3×C4—21no—
76C2×C3×C4—22no—
77C2×C3×C4—41no—
78C2×C3×C4—42no—
79C2×C3×C4—41no—
80C2×C3×C4C2×D412yes—
81C2×Dic3—22no—
82C2×Dic3—41no—
83C2×Dic3—41no—
84C2×Dic3—41no—
85C2×Dic3—41no—
86C2×Dic3—41no—
87C3⋊C8—42no—
88C24—21no—
89C24#6711yescharacteristic
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—21no—
93C2×D4—42no—
94C2×D4—42no—
95C2×D4#6712yes—
96C22×C4—21no—
97C22×C4—61no—
98C22×C4—61no—
99C22×C4#6711yescharacteristic
100C22×C4C22×D311yescharacteristic
101C2×C8—62no—
102C4⋊C4—21no—
103C4⋊C4—62no—
104C4⋊C4—122no—
105C4⋊C4#6712yes—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—122no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3—22no—
113C22×C3—42no—
114C22×C3—41no—
115C22×C3—41no—
116C22×C3—42no—
117C22×C3—42no—
118C22×C3—42no—
119C22×C3—81no—
120C22×C3—84no—
121C22×C3#12011yescharacteristic
122C22×C3C2×SD1611yescharacteristic
123C22×C3AGL(1,ℤ/8)11yescharacteristic
124C3×C4—22no—
125C3×C4—42no—
126C3×C4—41no—
127Dic3—42no—
128Dic3—81no—
129C23—21no—
130C23—21no—
131C23—22no—
132C23—41no—
133C23—41no—
134C23—42no—
135C23—81no—
136C23—81no—
137C23#21412yes—
138C23C2×#6711yescharacteristic
139D4—42no—
140D4—44no—
141D4—42no—
142D4—42no—
143D4—81no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—62no—
150C2×C4—121no—
151C2×C4—121no—
152C2×C4—121no—
153C2×C4—121no—
154C2×C4—121no—
155C2×C4C2×#6712yes—
156C8—122no—
157C2×C3—21no—
158C2×C3—21no—
159C2×C3—42no—
160C2×C3—82no—
161C2×C3#45011yescharacteristic
162C2×C3#45311yescharacteristic
163C2×C3#45811yescharacteristic
164C22—21no—
165C22—21no—
166C22—22no—
167C22—42no—
168C22—41no—
169C22—41no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—81no—
174C22—84no—
175C22#93211yescharacteristic
176C22C2×#21411yescharacteristic
177C22#95311yescharacteristic
178C4—22no—
179C4—42no—
180C4—41no—
181C4—122no—
182C4—241no—
183C3#161311yescharacteristic, complemented, Sylow
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#521811yescharacteristic
189C2#524111yescharacteristic
190C2#524311yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#534C2
socleC22×C3#120
hypercenter#534D3
hypocenterC3#1613
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
2. #5034C2semidirect
3. #4949C2semidirect
4. #4537C2semidirect
5. #4525C2semidirect
14. C2×C4⋊Dic3C22semidirect
17. C3⋊#95C22semidirect
18. #830C22semidirect
183. C3#1613semidirect

Maximal subgroups

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

subgroupindex
C3×#5342
#50342
#49492
#45372
#45252
#16133