Finite Group Databasebeta

← #69702

Subgroups — #69702

Order 384.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#5515C211yesmaximal, characteristic, complemented
2C3×#394C211yesmaximal, characteristic, complemented
3#5218C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic
5#4528C211yesmaximal, characteristic, complemented
6#4452C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1597—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11C2×C3×C22⋊C4—21no—
12C2×C3×C22⋊C4—21no—
13C2×#214—21no—
14C2×#213—21no—
15#930C2211yescharacteristic
16#929—21no—
17C4×#67—21no—
18#924C2211yescharacteristic
19#898—21no—
20#896—21no—
21#894—21no—
22#889—21no—
23C3×#95C2211yescharacteristic
24#832—21no—
25#832—21no—
26#830C2211yescharacteristic
27#830C2211yescharacteristic
28#539—31no—
29#450—31no—
30#447—31no—
31#394D311yescharacteristic, complemented
32#328—31no—
33#327—31no—
34#324—31no—
35C24×C3—21no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4—42no—
39C22×C3×C4—21no—
40C22×C3×C4D411yescharacteristic
41C22×C3×C4D411yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C2×#67—21no—
44C2×D12D411yescharacteristic
45C2×C4×D3—21no—
46C2×Dic6D411yescharacteristic
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C22⋊C4—42no—
50C3×C22⋊C4—42no—
51#217—21no—
52#214—42no—
53#213—42no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58C4⋊Dic3D411yescharacteristic
59C4⋊Dic3D411yescharacteristic
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64C4×Dic3—21no—
65C2×C3⋊C8—21no—
66C2×C3⋊C8—21no—
67C22×D4C2×D311yescharacteristic
68C2×SD16—61no—
69C2×D8—61no—
70#126—61no—
71#125—61no—
72#123—61no—
73#123—61no—
74C22⋊Q8—31no—
75C22⋊Q8—61no—
76#121—31no—
77C4×D4—61no—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81#102—61no—
82#102—61no—
83C22⋊C8—31no—
84C22⋊C8—31no—
85#95C2×D311yescharacteristic
86C23×C3—22no—
87C23×C3—41no—
88C23×C3—41no—
89C23×C3—41no—
90C23×C3—41no—
91C23×C3—41no—
92C23×C3—41no—
93C23×C3C2×D411yescharacteristic
94C22×D3—21no—
95C3×D4—22no—
96C3×D4—22no—
97C3×D4—41no—
98C3×D4—41no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4—41no—
107C2×C3×C4—41no—
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110#67—41no—
111#67—41no—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115D12—41no—
116C4×D3—81no—
117Dic6—41no—
118C3⋊C8—41no—
119C3⋊C8—41no—
120C24—21no—
121C2×Q8—31no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—31no—
125C2×D4—42no—
126C2×D4—61no—
127C22×C4—21no—
128C22×C4—61no—
129C22×C4#6711yescharacteristic
130C22×C4#6711yescharacteristic
131C22×C4C22×D311yescharacteristic
132SD16—122no—
133D8—122no—
134C2×C8—61no—
135C2×C8—61no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—31no—
139C4⋊C4—31no—
140C4⋊C4—61no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C42—61no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—42no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—81no—
158C22×C3—81no—
159C22×C3—81no—
160C22×C3—81no—
161C22×C3#12011yescharacteristic
162C22×C3#13511yescharacteristic
163C22×C3AGL(1,ℤ/8)11yescharacteristic
164C2×D3—41no—
165C2×D3—41no—
166C2×D3—41no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—81no—
172Dic3—41no—
173Dic3—41no—
174Dic3—41no—
175C23—22no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—61no—
183C23C2×#6711yescharacteristic
184Q8—121no—
185D4—22no—
186D4—22no—
187D4—41no—
188D4—41no—
189D4—121no—
190D4—121no—
191D4—121no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—241no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C8—121no—
208C8—121no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—82no—
212C2×C3#44911yescharacteristic
213C2×C3#45411yescharacteristic
214C2×C3B(4,𝔽2)11yescharacteristic
215D3—81no—
216C22—21no—
217C22—21no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—121no—
226C22—121no—
227C22—121no—
228C22#93211yescharacteristic
229C22#93711yescharacteristic
230C22#95011yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—81no—
236C4—121no—
237C4—121no—
238C4—121no—
239C3#159711yescharacteristic, complemented, Sylow
240C2—21no—
241C2—21no—
242C2—82no—
243C2—241no—
244C2#503211yescharacteristic
245C2#503711yescharacteristic
246C2#506211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#394C2
socleC22×C3#120
prime radical#5217C2
hypercenter#394D3
hypocenterC3#1597
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #5515C2semidirect
2. C3×#394C2semidirect
3. #5218C2semidirect
5. #4528C2semidirect
6. #4452C2semidirect
7. #4451C2semidirect
31. #394D3semidirect
239. C3#1597semidirect

Maximal subgroups

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

subgroupindex
#55152
C3×#3942
#52182
#52172
#45282
#44522
#44512
#15973