Finite Group Databasebeta

← #71843

Subgroups — #71843

Order 384.

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

normal subgroups only

This group has 280 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#5033C212yesmaximal, complemented
3#4953C211yesmaximal, characteristic, complemented
4#4537C212yesmaximal, complemented
5#4525C211yesmaximal, characteristic, complemented
6#1607—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
15#832—22no—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic, complemented
18#830C2212yescomplemented
19#807—22no—
20#534D311yescharacteristic
21#480—32no—
22#397—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—
45C2×C3⋊C8—22no—
46C22×D4—21no—
47#123—22no—
48#121—21no—
49#121C2×D312yes—
50#120—22no—
51C2×C4⋊C4—31no—
52C2×C22⋊C4—61no—
53C2×C22⋊C4C2×D311yescharacteristic
54#107—62no—
55#102—62no—
56C22⋊C8—32no—
57#95—31no—
58#95—61no—
59C23×C3—21no—
60C23×C3—21no—
61C23×C3—22no—
62C23×C3—41no—
63C23×C3—41no—
64C23×C3—42no—
65C23×C3—81no—
66C23×C3—81no—
67C23×C3D812yes—
68C23×C3C2×D411yescharacteristic
69C3×D4—42no—
70C3×D4—44no—
71C3×D4—42no—
72C3×D4—42no—
73C3×D4—81no—
74C2×C3×C4—21no—
75C2×C3×C4—22no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×C3×C4—41no—
79C2×C3×C4C2×D412yes—
80C2×Dic3—41no—
81C2×Dic3—41no—
82C2×Dic3—41no—
83C2×Dic3—41no—
84C2×Dic3—41no—
85C2×Dic3—41no—
86C3⋊C8—42no—
87C24—21no—
88C24#6711yescharacteristic
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—21no—
92C2×D4—42no—
93C2×D4—42no—
94C2×D4#6712yes—
95C22×C4—21no—
96C22×C4—61no—
97C22×C4—61no—
98C22×C4#6711yescharacteristic
99C22×C4C22×D311yescharacteristic
100C2×C8—62no—
101C4⋊C4—21no—
102C4⋊C4—62no—
103C4⋊C4#6712yes—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—122no—
108C22×C3—21no—
109C22×C3—21no—
110C22×C3—22no—
111C22×C3—42no—
112C22×C3—41no—
113C22×C3—41no—
114C22×C3—42no—
115C22×C3—42no—
116C22×C3—42no—
117C22×C3—81no—
118C22×C3—84no—
119C22×C3#12011yescharacteristic
120C22×C3C2×D811yescharacteristic
121C22×C3AGL(1,ℤ/8)11yescharacteristic
122C3×C4—22no—
123C3×C4—42no—
124C3×C4—41no—
125Dic3—81no—
126Dic3—81no—
127C23—21no—
128C23—21no—
129C23—22no—
130C23—41no—
131C23—41no—
132C23—42no—
133C23—81no—
134C23—81no—
135C23#21312yes—
136C23C2×#6711yescharacteristic
137D4—42no—
138D4—44no—
139D4—42no—
140D4—42no—
141D4—81no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—121no—
148C2×C4—121no—
149C2×C4—121no—
150C2×C4—121no—
151C2×C4—121no—
152C2×C4—121no—
153C2×C4C2×#6712yes—
154C8—122no—
155C2×C3—21no—
156C2×C3—21no—
157C2×C3—42no—
158C2×C3—82no—
159C2×C3#44711yescharacteristic
160C2×C3#45511yescharacteristic
161C2×C3#45811yescharacteristic
162C22—21no—
163C22—21no—
164C22—22no—
165C22—42no—
166C22—41no—
167C22—41no—
168C22—42no—
169C22—42no—
170C22—42no—
171C22—81no—
172C22—84no—
173C22C2×#21311yescharacteristic
174C22#93211yescharacteristic
175C22#95311yescharacteristic
176C4—22no—
177C4—42no—
178C4—41no—
179C4—241no—
180C4—241no—
181C3#160711yescharacteristic, complemented, Sylow
182C2—21no—
183C2—21no—
184C2—42no—
185C2—82no—
186C2#521711yescharacteristic
187C2#524211yescharacteristic
188C2#524311yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#534C2
socleC22×C3#120
hypercenter#534D3
hypocenterC3#1607
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. #5033C2semidirect
3. #4953C2semidirect
4. #4537C2semidirect
5. #4525C2semidirect
14. C2×C4⋊Dic3C22semidirect
17. C3⋊#95C22semidirect
18. #830C22semidirect
181. C3#1607semidirect

Maximal subgroups

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

subgroupindex
C3×#5342
#50332
#49532
#45372
#45252
#16073