Finite Group Databasebeta

← #68107

Subgroups — #68107

Order 384.

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

normal subgroups only

This group has 320 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#5494C211yesmaximal, characteristic, complemented
2#4960C211yesmaximal, characteristic, complemented
3#4732C211yesmaximal, characteristic, complemented
4#4731C211yesmaximal, characteristic, complemented
5C3×#323C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4467C211yesmaximal, characteristic, complemented
8#1600—31nomaximal, Sylow
9C22×#67—21no—
10C22×D12C2211yescharacteristic
11C2×C3×C22⋊C4C2211yescharacteristic
12#953—21no—
13#930C2211yescharacteristic
14C4×#67—21no—
15C2×(C2×D3)⋊C4—21no—
16#924C2211yescharacteristic
17C2×D24—21no—
18C2×#204—21no—
19#885—21no—
20#884—21no—
21#883—21no—
22#879—21no—
23C3×C22⋊C8C2211yescharacteristic
24C3×C22⋊C8C2211yescharacteristic
25C3⋊#95C2211yescharacteristic
26#821—21no—
27#821—21no—
28#538—31no—
29#450—31no—
30#447—31no—
31#392—31no—
32#328—31no—
33#327—31no—
34#323D311yescharacteristic, complemented
35C24×C3D411yescharacteristic
36C23×D3—21no—
37C22×C3×C4D411yescharacteristic
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C2×#67—21no—
41C2×#67—21no—
42C2×#67—42no—
43C2×#67—42no—
44C2×#67—42no—
45C22×Dic3—21no—
46C2×D12—21no—
47C2×D12—21no—
48C2×D12—42no—
49C2×D12D411yescharacteristic, complemented
50C2×C4×D3—21no—
51C2×Dic6D411yescharacteristic, complemented
52C2×C3×C8—21no—
53C2×C3×C8—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—21no—
56#217—21no—
57#217—21no—
58#217—21no—
59(C2×D3)⋊C4—21no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—42no—
62C4⋊Dic3D411yescharacteristic, complemented
63C4⋊Dic3D411yescharacteristic, complemented
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66C4×Dic3—21no—
67D24—42no—
68#204—42no—
69C22×D4—31no—
70C22×D4—61no—
71C2×SD16—61no—
72C2×D8—61no—
73#126—61no—
74#124—61no—
75#123—61no—
76C22⋊Q8—31no—
77#121—31no—
78#121—61no—
79#120—61no—
80C4×D4—61no—
81C2×C22⋊C4—61no—
82C2×C22⋊C4C2×D311yescharacteristic
83#102—61no—
84#102—61no—
85C22⋊C8C2×D311yescharacteristic
86C22⋊C8C2×D311yescharacteristic
87#95—31no—
88C23×C3—21no—
89C23×C3—21no—
90C23×C3—21no—
91C23×C3—41no—
92C23×C3—41no—
93C23×C3C2×D411yescharacteristic
94C22×D3—22no—
95C22×D3—21no—
96C22×D3—41no—
97C22×D3—41no—
98C22×D3—41no—
99C22×D3—41no—
100C22×D3—41no—
101C22×D3—41no—
102C2×C3×C4—21no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107#67—48no—
108#67—41no—
109#67—41no—
110#67—81no—
111#67—81no—
112C2×Dic3—22no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—41no—
117C2×Dic3—41no—
118D12—22no—
119D12—22no—
120D12—41no—
121D12—41no—
122D12—41no—
123C4×D3—81no—
124Dic6—41no—
125C3×C8—41no—
126C3×C8—41no—
127C24—61no—
128C24D1211yescharacteristic
129C2×Q8—31no—
130C2×D4—31no—
131C2×D4—61no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—122no—
137C2×D4—122no—
138C2×D4—122no—
139C2×D4—122no—
140C22×C4—61no—
141C22×C4—61no—
142C22×C4D1211yescharacteristic
143C22×C4C22×D311yescharacteristic
144SD16—122no—
145D8—122no—
146C2×C8—21no—
147C2×C8—21no—
148C4⋊C4—31no—
149C4⋊C4—31no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—122no—
160C42—61no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—42no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3—81no—
168C22×C3#12011yescharacteristic
169C22×C3#13511yescharacteristic
170C22×C3AGL(1,ℤ/8)11yescharacteristic
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—41no—
174C2×D3—41no—
175C2×D3—42no—
176C2×D3—41no—
177C2×D3—81no—
178C2×D3—81no—
179C2×D3—81no—
180C2×D3—81no—
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—41no—
184Dic3—42no—
185Dic3—41no—
186Dic3—41no—
187Dic3—41no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—41no—
192C23—41no—
193C23—62no—
194C23—61no—
195C23—121no—
196C23—121no—
197C23—121no—
198C23—121no—
199C23—121no—
200C23—121no—
201C23C2×D1211yescharacteristic
202Q8—121no—
203D4—62no—
204D4—62no—
205D4—121no—
206D4—121no—
207D4—121no—
208D4—128no—
209D4—121no—
210D4—121no—
211D4—241no—
212D4—241no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—62no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—121no—
221C2×C4—121no—
222C2×C4—241no—
223C2×C4D3×D411yescharacteristic
224C2×C4D3×D411yescharacteristic
225C8—41no—
226C8—41no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—81no—
230C2×C3#44911yescharacteristic
231C2×C3#45311yescharacteristic
232C2×C3B(4,𝔽2)11yescharacteristic
233D3—81no—
234D3—82no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22—122no—
243C22—122no—
244C22—121no—
245C22—122no—
246C22—121no—
247C22—121no—
248C22—241no—
249C22—241no—
250C22—241no—
251C22—241no—
252C22#88211yescharacteristic
253C22#90411yescharacteristic
254C22#90811yescharacteristic
255C4—21no—
256C4—21no—
257C4—41no—
258C4—122no—
259C4—121no—
260C4—121no—
261C4—121no—
262C3#160011yescharacteristic, complemented, Sylow
263C2—21no—
264C2—21no—
265C2—81no—
266C2—242no—
267C2—241no—
268C2#473411yescharacteristic
269C2#474111yescharacteristic
270C2#474711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3#120
hypercenter#323D3
hypocenterC3#1600
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #5494C2semidirect
2. #4960C2semidirect
3. #4732C2semidirect
4. #4731C2semidirect
5. C3×#323C2semidirect
6. #4470C2semidirect
7. #4467C2semidirect
34. #323D3semidirect
49. C2×D12D4semidirect
51. C2×Dic6D4semidirect
62. C4⋊Dic3D4semidirect
63. C4⋊Dic3D4semidirect
262. C3#1600semidirect

Maximal subgroups

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

subgroupindex
#54942
#49602
#47322
#47312
C3×#3232
#44702
#44672
#16003