Finite Group Databasebeta

← #68130

Subgroups — #68130

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
2#4991C211yesmaximal, characteristic, complemented
3#4732C211yesmaximal, characteristic
4#4731C211yesmaximal, characteristic, complemented
5C3×#324C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4467C211yesmaximal, characteristic, complemented
8#1597—31nomaximal, Sylow
9C22×D12C2211yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11#930C2211yescharacteristic
12#929—21no—
13C4×#67—21no—
14C2×(C2×D3)⋊C4—21no—
15C2×(C2×D3)⋊C4—21no—
16#924C2211yescharacteristic
17C2×D24—21no—
18C2×#204—21no—
19#898—21no—
20#896—21no—
21#894—21no—
22#889—21no—
23C3×C22⋊C8C2211yescharacteristic
24C3×C22⋊C8C2211yescharacteristic
25C3⋊#95C2211yescharacteristic
26#821—21no—
27#821—21no—
28#539—31no—
29#450—31no—
30#447—31no—
31#394—31no—
32#328—31no—
33#327—31no—
34#324D311yescharacteristic, complemented
35C23×D3—21no—
36C22×C3×C4D411yescharacteristic
37C22×C3×C4D411yescharacteristic
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C22×Dic3—21no—
41C2×D12—21no—
42C2×D12—21no—
43C2×D12—42no—
44C2×D12D411yescharacteristic
45C2×C4×D3—21no—
46C2×Dic6D411yescharacteristic
47C2×C3×C8—21no—
48C2×C3×C8—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51#217—21no—
52(C2×D3)⋊C4—21no—
53(C2×D3)⋊C4—21no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—42no—
57(C2×D3)⋊C4—42no—
58C4⋊Dic3D411yescharacteristic
59C4⋊Dic3D411yescharacteristic
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64C4×Dic3—21no—
65D24—42no—
66#204—42no—
67C22×D4—31no—
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—61no—
80C2×C22⋊C4—61no—
81#102—61no—
82#102—61no—
83C22⋊C8C2×D311yescharacteristic
84C22⋊C8C2×D311yescharacteristic
85#95—31no—
86C23×C3C2×D411yescharacteristic
87C22×D3—22no—
88C22×D3—21no—
89C22×D3—41no—
90C22×D3—41no—
91C22×D3—41no—
92C22×D3—41no—
93C22×D3—41no—
94C22×D3—41no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4C2×D411yescharacteristic
103#67—41no—
104#67—41no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111D12—22no—
112D12—22no—
113D12—41no—
114D12—41no—
115D12—41no—
116C4×D3—81no—
117Dic6—41no—
118C3×C8—41no—
119C3×C8—41no—
120C24—61no—
121C2×Q8—31no—
122C2×D4—31no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4—61no—
126C2×D4—122no—
127C22×C4—61no—
128C22×C4—61no—
129C22×C4D1211yescharacteristic
130C22×C4D1211yescharacteristic
131C22×C4C22×D311yescharacteristic
132SD16—122no—
133D8—122no—
134C2×C8—21no—
135C2×C8—21no—
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—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—122no—
150C22⋊C4—122no—
151C42—61no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3#12011yescharacteristic
155C22×C3#13511yescharacteristic
156C22×C3AGL(1,ℤ/8)11yescharacteristic
157C2×D3—42no—
158C2×D3—42no—
159C2×D3—41no—
160C2×D3—41no—
161C2×D3—41no—
162C2×D3—42no—
163C2×D3—81no—
164C2×D3—81no—
165C2×D3—81no—
166C2×D3—81no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—41no—
170C3×C4—41no—
171Dic3—41no—
172Dic3—41no—
173Dic3—41no—
174Dic3—81no—
175C23—62no—
176C23—61no—
177C23—121no—
178C23—121no—
179C23—121no—
180C23—121no—
181C23—121no—
182C23—121no—
183C23C2×D1211yescharacteristic
184Q8—121no—
185D4—62no—
186D4—62no—
187D4—121no—
188D4—121no—
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—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—121no—
202C2×C4—121no—
203C2×C4—121no—
204C2×C4—241no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C8—41no—
208C8—41no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3#44911yescharacteristic
212C2×C3#45411yescharacteristic
213C2×C3B(4,𝔽2)11yescharacteristic
214D3—81no—
215D3—82no—
216C22—21no—
217C22—21no—
218C22—122no—
219C22—122no—
220C22—121no—
221C22—121no—
222C22—122no—
223C22—121no—
224C22—241no—
225C22—241no—
226C22—241no—
227C22—241no—
228C22#88211yescharacteristic
229C22#90411yescharacteristic
230C22#90811yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—121no—
236C4—121no—
237C4—121no—
238C4—241no—
239C3#159711yescharacteristic, complemented, Sylow
240C2—21no—
241C2—21no—
242C2—242no—
243C2—241no—
244C2#473411yescharacteristic
245C2#474111yescharacteristic
246C2#475311yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radical#4732C2
hypercenter#324D3
hypocenterC3#1597
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #5515C2semidirect
2. #4991C2semidirect
4. #4731C2semidirect
5. C3×#324C2semidirect
6. #4470C2semidirect
7. #4467C2semidirect
34. #324D3semidirect
239. C3#1597semidirect

Maximal subgroups

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

subgroupindex
#55152
#49912
#47322
#47312
C3×#3242
#44702
#44672
#15973