Finite Group Databasebeta

← #71878

Subgroups — #71878

Order 384.

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

normal subgroups only

This group has 310 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×#537C211yesmaximal, characteristic, complemented
2#5036C212yesmaximal, complemented
3#4991C211yesmaximal, characteristic, complemented
4#4537C212yesmaximal, complemented
5#4528C211yesmaximal, characteristic, complemented
6#1598—31nomaximal, Sylow
7C2×C3×#41—21no—
8C22×D12C2211yescharacteristic
9C3×C22⋊Q8—22no—
10C3×#121—22no—
11C3×#121—21no—
12C3×#121C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×#213—22no—
15C2×(C2×D3)⋊C4—21no—
16C2×(C2×D3)⋊C4—21no—
17C3⋊#95C2211yescharacteristic
18#830C2212yes—
19#809—22no—
20#537D311yescharacteristic, complemented
21#447—32no—
22#394—31no—
23#327—32no—
24#324—31no—
25C23×D3—21no—
26C3×#41—42no—
27C3×#41—42no—
28C2×C3×Q8—21no—
29C2×C3×D4—21no—
30C2×C3×D4—22no—
31C2×C3×D4—21no—
32C2×C3×D4D412yes—
33C22×C3×C4—22no—
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4C2311yescharacteristic
37C22×Dic3—21no—
38C2×D12—22no—
39C2×D12—42no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4D412yes—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45#213—44no—
46(C2×D3)⋊C4—42no—
47(C2×D3)⋊C4—42no—
48C2×C3⋊C8—22no—
49C2×#41—21no—
50C22×D4—31no—
51C2×D8—62no—
52C22⋊Q8—22no—
53#121—22no—
54#121—21no—
55#121C2×D312yes—
56C2×C4⋊C4C2×D311yescharacteristic
57C2×C22⋊C4—61no—
58C2×C22⋊C4—61no—
59#102—62no—
60C22⋊C8—32no—
61#95—31no—
62C23×C3—22no—
63C23×C3C2×D411yescharacteristic
64C22×D3—22no—
65C22×D3—41no—
66C22×D3—41no—
67C22×D3—41no—
68C22×D3—41no—
69C22×D3—41no—
70C22×D3—41no—
71C3×Q8—42no—
72C3×D4—42no—
73C3×D4—42no—
74C3×D4—42no—
75C3×D4—42no—
76C3×D4—81no—
77C2×C3×C4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—21no—
80C2×C3×C4—22no—
81C2×C3×C4—41no—
82C2×C3×C4—44no—
83C2×C3×C4—41no—
84C2×C3×C4D812yescomplemented
85C2×C3×C4C2×D412yes—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89D12—24no—
90D12—42no—
91C3⋊C8—42no—
92C24—61no—
93#41—42no—
94#41—42no—
95C2×Q8—21no—
96C2×D4—21no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—62no—
100C2×D4—122no—
101C2×D4#6712yes—
102C22×C4—22no—
103C22×C4—61no—
104C22×C4#6711yescharacteristic
105C22×C4#6711yescharacteristic
106C22×C4C22×D311yescharacteristic
107D8—124no—
108C2×C8—62no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4#6712yes—
112C22⋊C4—22no—
113C22⋊C4—22no—
114C22⋊C4—122no—
115C22⋊C4—122no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—42no—
119C22×C3—42no—
120C22×C3—42no—
121C22×C3#12011yescharacteristic
122C22×C3C2×D811yescharacteristic
123C22×C3AGL(1,ℤ/8)11yescharacteristic
124C2×D3—42no—
125C2×D3—42no—
126C2×D3—42no—
127C2×D3—81no—
128C2×D3—81no—
129C2×D3—81no—
130C2×D3—81no—
131C3×C4—22no—
132C3×C4—22no—
133C3×C4—42no—
134C3×C4—41no—
135Dic3—81no—
136C23—22no—
137C23—62no—
138C23—121no—
139C23—121no—
140C23—121no—
141C23—121no—
142C23—121no—
143C23—121no—
144C23C2×#6711yescharacteristic
145Q8—42no—
146D4—42no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—64no—
151D4—81no—
152D4—122no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—121no—
161C2×C4—121no—
162C2×C4—121no—
163C2×C4#21312yescomplemented
164C2×C4C2×#6712yes—
165C8—122no—
166C2×C3—21no—
167C2×C3—21no—
168C2×C3—82no—
169C2×C3#44711yescharacteristic
170C2×C3#45411yescharacteristic
171C2×C3B(4,𝔽2)11yescharacteristic
172D3—82no—
173C22—21no—
174C22—21no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—122no—
179C22—122no—
180C22—122no—
181C22—241no—
182C22—241no—
183C22—241no—
184C22—241no—
185C22C2×#21311yescharacteristic
186C22#93211yescharacteristic
187C22#95311yescharacteristic
188C4—22no—
189C4—22no—
190C4—42no—
191C4—41no—
192C4—241no—
193C3#159811yescharacteristic, complemented, Sylow
194C2—21no—
195C2—21no—
196C2—82no—
197C2—242no—
198C2#521711yescharacteristic
199C2#524411yescharacteristic
200C2#524511yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#537C2
socleC22×C3#120
hypercenter#537D3
hypocenterC3#1598
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C3×#537C2semidirect
2. #5036C2semidirect
3. #4991C2semidirect
4. #4537C2semidirect
5. #4528C2semidirect
20. #537D3semidirect
84. C2×C3×C4D8semidirect
163. C2×C4#213semidirect
193. C3#1598semidirect

Maximal subgroups

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

subgroupindex
C3×#5372
#50362
#49912
#45372
#45282
#15983