Finite Group Databasebeta

← #69699

Subgroups — #69699

Order 384.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#5493C211yesmaximal, characteristic
2C3×#393C211yesmaximal, characteristic, complemented
3#5227C212yesmaximal, complemented
4#4525C211yesmaximal, characteristic, complemented
5#4451C212yesmaximal, complemented
6#1601—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic, complemented
8C22×#67—21no—
9C2×C3×C22⋊C4—21no—
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×#215—22no—
12#930—21no—
13#930C2212yes—
14#886—22no—
15#882—22no—
16C3×#95—21no—
17C3×#95C2211yescharacteristic, complemented
18#835—22no—
19#830C2212yescomplemented
20#534—31no—
21#448—32no—
22#393D311yescharacteristic, complemented
23#327—32no—
24#323—31no—
25C24×C3D411yescharacteristic
26C23×D3—21no—
27C2×C3×Q8—22no—
28C2×C3×Q8—42no—
29C22×C3×C4—21no—
30C22×C3×C4—21no—
31C22×C3×C4D411yescharacteristic
32C22×C3×C4C2311yescharacteristic
33C2×#67—22no—
34C2×#67—22no—
35C2×#67—42no—
36C2×#67—42no—
37C22×Dic3—21no—
38C2×D12—21no—
39C2×D12D412yes—
40C3×C22⋊C4—22no—
41C3×C22⋊C4—42no—
42#215—44no—
43(C2×D3)⋊C4—22no—
44(C2×D3)⋊C4—22no—
45C4⋊Dic3—21no—
46C4⋊Dic3D412yes—
47C2×C3⋊C8—22no—
48C22×Q8C2×D311yescharacteristic, complemented
49C22×D4—61no—
50C2×SD16—62no—
51#123—62no—
52#121—32no—
53#121—61no—
54#120—62no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4C2×D311yescharacteristic
57#103—62no—
58C22⋊C8—32no—
59#95—21no—
60#95C2×D311yescharacteristic, complemented
61C23×C3—21no—
62C23×C3—21no—
63C23×C3—41no—
64C23×C3—41no—
65C23×C3SD1612yescomplemented
66C23×C3C2×D411yescharacteristic
67C22×D3—22no—
68C22×D3—42no—
69C22×D3—81no—
70C22×D3—81no—
71C3×Q8—24no—
72C3×Q8—42no—
73C2×C3×C4—22no—
74C2×C3×C4—21no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4—41no—
81C2×C3×C4—41no—
82C2×C3×C4C2×D412yes—
83#67—44no—
84#67—42no—
85#67—42no—
86C2×Dic3—22no—
87C2×Dic3—42no—
88D12—42no—
89D12—81no—
90C3⋊C8—42no—
91C24—61no—
92C24#6711yescharacteristic
93C2×Q8—22no—
94C2×Q8—42no—
95C2×D4—32no—
96C2×D4—61no—
97C2×D4—62no—
98C2×D4—62no—
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—32no—
109C4⋊C4—61no—
110C22⋊C4—22no—
111C22⋊C4—42no—
112C22⋊C4—62no—
113C22⋊C4—62no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—22no—
117C22×C3—42no—
118C22×C3—41no—
119C22×C3—41no—
120C22×C3—81no—
121C22×C3#12011yescharacteristic
122C22×C3C2×SD1611yescharacteristic
123C22×C3AGL(1,ℤ/8)11yescharacteristic
124C2×D3—42no—
125C2×D3—42no—
126C2×D3—42no—
127C2×D3—84no—
128C3×C4—22no—
129C3×C4—41no—
130C3×C4—42no—
131C3×C4—81no—
132Dic3—42no—
133C23—21no—
134C23—21no—
135C23—41no—
136C23—41no—
137C23—62no—
138C23—122no—
139C23—241no—
140C23—241no—
141C23#21512yescomplemented
142C23C2×#6711yescharacteristic
143Q8—24no—
144Q8—42no—
145D4—122no—
146D4—124no—
147D4—122no—
148D4—122no—
149D4—241no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—62no—
160C2×C4—122no—
161C2×C4D3×D412yes—
162C8—122no—
163C2×C3—21no—
164C2×C3—21no—
165C2×C3—42no—
166C2×C3#45011yescharacteristic
167C2×C3#45511yescharacteristic
168C2×C3B(4,𝔽2)11yescharacteristic
169D3—82no—
170C22—21no—
171C22—21no—
172C22—22no—
173C22—42no—
174C22—41no—
175C22—41no—
176C22—81no—
177C22—122no—
178C22—122no—
179C22—122no—
180C22—244no—
181C22#93711yescharacteristic
182C22C2×#21511yescharacteristic
183C22#94911yescharacteristic
184C4—22no—
185C4—42no—
186C4—41no—
187C4—81no—
188C4—122no—
189C3#160111yescharacteristic, complemented, Sylow
190C2—21no—
191C2—21no—
192C2—42no—
193C2—242no—
194C2#503211yescharacteristic
195C2#504611yescharacteristic
196C2#506111yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#393C2
socleC22×C3#120
prime radical#5493C2
hypercenter#393D3
hypocenterC3#1601
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. C3×#393C2semidirect
3. #5227C2semidirect
4. #4525C2semidirect
5. #4451C2semidirect
7. C22×C3×Q8C22semidirect
17. C3×#95C22semidirect
19. #830C22semidirect
22. #393D3semidirect
48. C22×Q8C2×D3semidirect
60. #95C2×D3semidirect
65. C23×C3SD16semidirect
141. C23#215semidirect
189. C3#1601semidirect

Maximal subgroups

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

subgroupindex
#54932
C3×#3932
#52272
#45252
#44512
#16013