Finite Group Databasebeta

← #69664

Subgroups — #69664

Order 384.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#932C212yesmaximal, complemented
2#5510C211yesmaximal, characteristic, complemented
3C3×#390C211yesmaximal, characteristic, complemented
4C2×#805C212yesmaximal, complemented
5#4530C211yesmaximal, characteristic, complemented
6#2006—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C2×#235C2212yes—
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×C3×C22⋊C4—22no—
11C2×C3×C42C2211yescharacteristic
12C2×#214—22no—
13#932—24no—
14#932—24no—
15C2×#213—22no—
16#930—22no—
17C2×#208C2212yes—
18#897—22no—
19#895—22no—
20#805—24no—
21C2×AGL(1,ℤ/8)—32no—
22#537—31no—
23C2×#104—32no—
24#390D311yescharacteristic, complemented
25#343—31no—
26C24×C3—21no—
27C2×C3×D4—22no—
28C2×C3×D4—22no—
29C2×C3×D4—21no—
30C2×C3×D4D412yescomplemented
31C22×C3×C4—21no—
32C22×C3×C4D412yes—
33C22×C3×C4C2311yescharacteristic
34C2×#67—22no—
35#235—22no—
36#235—22no—
37#235—22no—
38#235—22no—
39C2×D12—22no—
40C2×D12D412yes—
41C2×C4×D3—22no—
42C2×Dic6D412yes—
43C3×C4⋊C4—22no—
44C3×C22⋊C4—44no—
45C3×C42—22no—
46#214—28no—
47#213—28no—
48(C2×D3)⋊C4—24no—
49C4⋊Dic3—22no—
50#208—22no—
51#208D414yescomplemented
52C2×C3⋊C8—22no—
53C2×#41—32no—
54C22×D4C2×D311yescharacteristic
55AGL(1,ℤ/8)—64no—
56AGL(1,ℤ/8)—64no—
57C2×SD16—62no—
58C2×D8—62no—
59C2×M16—32no—
60C22⋊Q8—62no—
61#121—62no—
62#121—62no—
63C2×C4⋊C4C2×D311yescharacteristic
64C2×C22⋊C4—22no—
65C2×C42C2×D311yescharacteristic
66#104—64no—
67C23×C3—22no—
68C23×C3—42no—
69C23×C3—42no—
70C23×C3—41no—
71C23×C3—41no—
72C23×C3C2×D411yescharacteristic
73C22×D3—22no—
74C3×D4—24no—
75C3×D4—42no—
76C2×C3×C4—22no—
77C2×C3×C4—22no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4—42no—
81C2×C3×C4C2×D411yescharacteristic
82C2×C3×C4C2×D412yes—
83C2×C3×C4C2×D412yes—
84C2×C3×C4#4111yescharacteristic
85#67—42no—
86#67—42no—
87C2×Dic3—22no—
88D12—22no—
89D12—22no—
90D12—82no—
91C4×D3—42no—
92C4×D3—42no—
93Dic6—22no—
94Dic6—22no—
95C3⋊C8—24no—
96C24—21no—
97#41—62no—
98#41—62no—
99#41—62no—
100#41—62no—
101C2×Q8—32no—
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—32no—
106C2×D4—62no—
107C2×D4—62no—
108C2×D4D1212yescomplemented
109C22×C4—21no—
110C22×C4—62no—
111C22×C4#6712yes—
112C22×C4C22×D311yescharacteristic
113SD16—68no—
114D8—68no—
115M16—34no—
116M16—62no—
117C2×C8—62no—
118C4⋊C4—22no—
119C4⋊C4—62no—
120C22⋊C4—44no—
121C22⋊C4—64no—
122C42—22no—
123C22×C3—21no—
124C22×C3—21no—
125C22×C3—44no—
126C22×C3—42no—
127C22×C3—82no—
128C22×C3—81no—
129C22×C3—81no—
130C22×C3#12011yescharacteristic
131C22×C3#12111yescharacteristic
132C22×C3#12711yescharacteristic
133C2×D3—42no—
134C2×D3—42no—
135C2×D3—42no—
136C3×C4—42no—
137C3×C4—42no—
138C3×C4#12012yes—
139C3×C4#12112yes—
140Dic3—42no—
141C23—22no—
142C23—42no—
143C23—42no—
144C23—41no—
145C23—41no—
146C23—62no—
147C23C2×#6711yescharacteristic
148Q8—62no—
149Q8—62no—
150D4—24no—
151D4—42no—
152D4—62no—
153D4—62no—
154D4—122no—
155D4—122no—
156D4—242no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—62no—
163C2×C4—122no—
164C2×C4—122no—
165C2×C4C2×D1211yescharacteristic
166C2×C4#23511yescharacteristic
167C2×C4D3×D412yes—
168C2×C4D3×D412yes—
169C8—64no—
170C2×C3—21no—
171C2×C3—21no—
172C2×C3—82no—
173C2×C3#39211yescharacteristic
174C2×C3#45311yescharacteristic
175C2×C3#45411yescharacteristic
176D3—82no—
177C22—21no—
178C22—21no—
179C22—44no—
180C22—42no—
181C22—82no—
182C22—81no—
183C22—81no—
184C22—122no—
185C22—122no—
186C22—122no—
187C22#93011yescharacteristic
188C22#93711yescharacteristic
189C22#94011yescharacteristic
190C4—42no—
191C4—42no—
192C4—122no—
193C4#88212yes—
194C4#88412yes—
195C3#200611yescharacteristic, complemented, Sylow
196C2—21no—
197C2—21no—
198C2—82no—
199C2—242no—
200C2#496011yescharacteristic
201C2#506211yescharacteristic
202C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#390C2
socleC22×C3#120
hypercenter#390D3
hypocenterC3#2006
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#932C2semidirect
2. #5510C2semidirect
3. C3×#390C2semidirect
4. C2×#805C2semidirect
5. #4530C2semidirect
24. #390D3semidirect
30. C2×C3×D4D4semidirect
51. #208D4semidirect
108. C2×D4D12semidirect
195. C3#2006semidirect

Maximal subgroups

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

subgroupindex
C2×#9322
#55102
C3×#3902
C2×#8052
#45302
#20063