Finite Group Databasebeta

← #70975

Subgroups — #70975

Order 384.

282 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 51 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#5597C211yesmaximal, characteristic
2C3×#448C211yesmaximal, characteristic, complemented
3#5230C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4469C211yesmaximal, characteristic, complemented
8#1604—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C2×#237—21no—
11C2×C3×SD16—21no—
12C3×#121C2211yescharacteristic
13C2×#215—21no—
14#939—21no—
15#939—21no—
16#930C2211yescharacteristic
17C2×Dic3⋊C4—21no—
18C2×Dic3⋊C4C2211yescharacteristic
19#896—21no—
20#884—21no—
21#878—21no—
22C3×#103—21no—
23C3×C22⋊C8C2211yescharacteristic
24#835—21no—
25C3⋊#95—21no—
26C3⋊#95C2211yescharacteristic
27#830C2211yescharacteristic
28#537—31no—
29#448—31no—
30#448D311yescharacteristic, complemented
31#395—31no—
32#327—31no—
33#327—31no—
34#324—31no—
35C2×C3×Q8—21no—
36C2×C3×Q8—21no—
37C2×C3×Q8—42no—
38C2×C3×D4—21no—
39C2×C3×D4D411yescharacteristic
40C22×C3×C4—21no—
41C22×C3×C4C2311yescharacteristic
42C2×#67—21no—
43C2×#67—21no—
44C2×#67—21no—
45C22×Dic3—21no—
46C22×Dic3—21no—
47C22×Dic3D411yescharacteristic
48C22×Dic3D411yescharacteristic
49#237—42no—
50#237—42no—
51C2×D12D411yescharacteristic
52C2×C4×D3—21no—
53C2×Dic6—21no—
54C3×SD16—42no—
55C2×C3×C8—21no—
56C3×C4⋊C4D411yescharacteristic
57C3×C22⋊C4—21no—
58#217—21no—
59#215—42no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—21no—
62C4⋊Dic3D411yescharacteristic
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66Dic3⋊C4—42no—
67C2×C3⋊C8—21no—
68C2×#41—61no—
69C22×Q8C2×D311yescharacteristic
70C2×SD16—21no—
71C2×SD16—61no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74#121—31no—
75#121—61no—
76#121—61no—
77#121—61no—
78#121C2×D311yescharacteristic
79C2×C4⋊C4—31no—
80C2×C4⋊C4—61no—
81#103—21no—
82#103—61no—
83C22⋊C8—31no—
84C22⋊C8C2×D311yescharacteristic
85#95—31no—
86#95—61no—
87C23×C3—21no—
88C23×C3C2×D411yescharacteristic
89C22×D3—21no—
90C3×Q8—22no—
91C3×Q8—22no—
92C3×Q8—41no—
93C3×Q8—41no—
94C3×D4—41no—
95C3×D4—41no—
96C3×D4—41no—
97C2×C3×C4—22no—
98C2×C3×C4—21no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4C2×D411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104#67—42no—
105#67—41no—
106#67—41no—
107#67—81no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—22no—
111C2×Dic3—21no—
112C2×Dic3—21no—
113C2×Dic3—42no—
114C2×Dic3—41no—
115C2×Dic3—41no—
116C2×Dic3—41no—
117C2×Dic3SD1612yescomplemented
118D12—41no—
119C4×D3—42no—
120Dic6—42no—
121C3×C8—41no—
122C3⋊C8—41no—
123#41—122no—
124#41—122no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C2×Q8—61no—
129C2×D4—21no—
130C2×D4—31no—
131C2×D4—61no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4#6711yescharacteristic
135C22×C4—21no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—61no—
139C22×C4—61no—
140C22×C4—61no—
141C22×C4C22×D311yescharacteristic
142SD16—42no—
143SD16—122no—
144C2×C8—21no—
145C2×C8—61no—
146C4⋊C4—31no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C4⋊C4—122no—
151C4⋊C4#6711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—61no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3#12011yescharacteristic
162C22×C3C2×SD1611yescharacteristic
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—42no—
171Dic3—22no—
172Dic3—42no—
173Dic3—41no—
174Dic3—41no—
175C23—21no—
176C23—61no—
177C23D3×D411yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—41no—
181Q8—41no—
182Q8—122no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—121no—
187D4—122no—
188D4—121no—
189D4—121no—
190D4—241no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—32no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—62no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—122no—
203C2×C4—122no—
204C2×C4—121no—
205C2×C4—121no—
206C2×C4—121no—
207C2×C4D3×D411yescharacteristic
208C2×C4C2×#6711yescharacteristic
209C8—41no—
210C8—121no—
211C2×C3—21no—
212C2×C3—21no—
213C2×C3—81no—
214C2×C3#45011yescharacteristic
215C2×C3#45611yescharacteristic
216C2×C3B(4,𝔽2)11yescharacteristic
217D3—81no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—121no—
224C22—121no—
225C22—121no—
226C22D3×SD1611yescharacteristic
227C22#91411yescharacteristic
228C22#93711yescharacteristic
229C4—21no—
230C4—21no—
231C4—41no—
232C4—42no—
233C4—62no—
234C4—122no—
235C4—121no—
236C4—121no—
237C3#160411yescharacteristic, complemented, Sylow
238C2—21no—
239C2—21no—
240C2—81no—
241C2—241no—
242C2#503211yescharacteristic
243C2#516911yescharacteristic
244C2#520511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#448C2
socleC22×C3#120
prime radical#5597C2
hypercenter#448D3
hypocenterC3#1604
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C3×#448C2semidirect
3. #5230C2semidirect
4. #5227C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4469C2semidirect
30. #448D3semidirect
117. C2×Dic3SD16semidirect
237. C3#1604semidirect

Maximal subgroups

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

subgroupindex
#55972
C3×#4482
#52302
#52272
#45372
#44702
#44692
#16043