Finite Group Databasebeta

← #71846

Subgroups — #71846

Order 384.

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

normal subgroups only

This group has 308 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#5040C212yesmaximal, complemented
3#4947C211yesmaximal, characteristic, complemented
4#4537C212yesmaximal, complemented
5#4525C211yesmaximal, characteristic, complemented
6#1601—31nomaximal, Sylow
7C22×C3×D4—21no—
8C22×Dic6C2211yescharacteristic, complemented
9C3×#123—22no—
10C3×#121—21no—
11C3×#121C2212yes—
12C3×#120—22no—
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×#217—21no—
15C2×#214—22no—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic, complemented
18#830C2212yescomplemented
19#810—22no—
20#534D311yescharacteristic
21#448—32no—
22#393—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—
38C2×Dic6—22no—
39C2×Dic6—42no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4D412yes—
42C3×C22⋊C4—22no—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45#217—42no—
46#214—44no—
47C2×C3⋊C8—22no—
48C22×Q8—31no—
49C22×D4—21no—
50C2×SD16—62no—
51#123—22no—
52#121—21no—
53#121C2×D312yes—
54#120—22no—
55C2×C22⋊C4—61no—
56C2×C22⋊C4C2×D311yescharacteristic
57#103—62no—
58C22⋊C8—32no—
59#95—31no—
60#95—61no—
61C23×C3—21no—
62C23×C3—21no—
63C23×C3—22no—
64C23×C3—41no—
65C23×C3—41no—
66C23×C3—42no—
67C23×C3—81no—
68C23×C3—81no—
69C23×C3SD1612yescomplemented
70C23×C3C2×D411yescharacteristic
71C3×D4—42no—
72C3×D4—44no—
73C3×D4—42no—
74C3×D4—42no—
75C3×D4—81no—
76C2×C3×C4—21no—
77C2×C3×C4—22no—
78C2×C3×C4—41no—
79C2×C3×C4—42no—
80C2×C3×C4—41no—
81C2×C3×C4C2×D412yes—
82C2×Dic3—22no—
83C2×Dic3—41no—
84C2×Dic3—41no—
85C2×Dic3—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88Dic6—24no—
89Dic6—42no—
90C3⋊C8—42no—
91C24—21no—
92C24#6711yescharacteristic
93C2×Q8—62no—
94C2×Q8—122no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—21no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4#6712yes—
101C22×C4—21no—
102C22×C4—61no—
103C22×C4—61no—
104C22×C4#6711yescharacteristic
105C22×C4C22×D311yescharacteristic
106SD16—124no—
107C2×C8—62no—
108C4⋊C4—21no—
109C4⋊C4#6712yes—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—22no—
113C22⋊C4—122no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—22no—
117C22×C3—42no—
118C22×C3—41no—
119C22×C3—41no—
120C22×C3—42no—
121C22×C3—42no—
122C22×C3—42no—
123C22×C3—81no—
124C22×C3—84no—
125C22×C3#12011yescharacteristic
126C22×C3C2×SD1611yescharacteristic
127C22×C3AGL(1,ℤ/8)11yescharacteristic
128C3×C4—22no—
129C3×C4—42no—
130C3×C4—41no—
131Dic3—42no—
132Dic3—81no—
133C23—21no—
134C23—21no—
135C23—22no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—81no—
140C23—81no—
141C23#21412yescomplemented
142C23C2×#6711yescharacteristic
143Q8—64no—
144Q8—122no—
145D4—42no—
146D4—44no—
147D4—42no—
148D4—42no—
149D4—81no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—62no—
156C2×C4—121no—
157C2×C4—121no—
158C2×C4—121no—
159C2×C4—121no—
160C2×C4—121no—
161C2×C4C2×#6712yes—
162C8—122no—
163C2×C3—21no—
164C2×C3—21no—
165C2×C3—42no—
166C2×C3—82no—
167C2×C3#45011yescharacteristic
168C2×C3#45511yescharacteristic
169C2×C3B(4,𝔽2)11yescharacteristic
170C22—21no—
171C22—21no—
172C22—22no—
173C22—42no—
174C22—41no—
175C22—41no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—84no—
181C22#93211yescharacteristic
182C22C2×#21411yescharacteristic
183C22#95311yescharacteristic
184C4—22no—
185C4—42no—
186C4—41no—
187C4—122no—
188C4—241no—
189C3#160111yescharacteristic, complemented, Sylow
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#521811yescharacteristic
195C2#524211yescharacteristic
196C2#524411yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#534C2
socleC22×C3#120
hypercenter#534D3
hypocenterC3#1601
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. #5040C2semidirect
3. #4947C2semidirect
4. #4537C2semidirect
5. #4525C2semidirect
8. C22×Dic6C22semidirect
17. C3⋊#95C22semidirect
18. #830C22semidirect
69. C23×C3SD16semidirect
141. C23#214semidirect
189. C3#1601semidirect

Maximal subgroups

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

subgroupindex
C3×#5342
#50402
#49472
#45372
#45252
#16013