Finite Group Databasebeta

← #85457

Subgroups — #85457

Order 384.

303 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 34 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#5914C211yesmaximal, characteristic, complemented
2D4×S4C211yesmaximal, characteristic, complemented
3A4×D8C211yesmaximal, characteristic, complemented
4#5415C211yesmaximal, characteristic, complemented
5#5414C211yesmaximal, characteristic, complemented
6#5401C211yesmaximal, characteristic, complemented
7#5400C211yesmaximal, characteristic, complemented
8#3264—31nomaximal, Sylow
9C22×S4—21no—
10D4×A4C2211yescharacteristic
11D4×A4C2211yescharacteristic, complemented
12GL(2,ℤ/4)—21no—
13GL(2,ℤ/4)—21no—
14C2×SL(2,ℤ/4)—21no—
15C4⋊S4C2211yescharacteristic
16C4×S4C2211yescharacteristic
17A4⋊Q8C2211yescharacteristic, complemented
18#911—41nomaximal
19C8×A4C2211yescharacteristic, complemented
20A4⋊C8C2211yescharacteristic, complemented
21C2×AGL(1,ℤ/8)—31no—
22C22×D8D311yescharacteristic, complemented
23#546—31no—
24D42—31no—
25#496—31no—
26#469—31no—
27#465—31no—
28#463—31no—
29#459—31no—
30#450—31no—
31#449—31no—
32#447—31no—
33#447—31no—
34#442—31no—
35#435—31no—
36C22×A4—21no—
37C22×A4—21no—
38C2×S4—21no—
39C2×S4—42no—
40C2×S4D411yescharacteristic
41#237—41no—
42D3×D4—41no—
43C4×A4C2311yescharacteristic
44SL(2,ℤ/4)—21no—
45SL(2,ℤ/4)D411yescharacteristic
46C3×D8—41no—
47#214—41no—
48#213—41no—
49#204—41no—
50ΓL(1,𝔽25)—41no—
51C2×#41—31no—
52C22×D4—31no—
53C22×D4—61no—
54C22×D4C2×D311yescharacteristic
55C22×D4C2×D311yescharacteristic, complemented
56AGL(1,ℤ/8)—61no—
57AGL(1,ℤ/8)—61no—
58AGL(1,ℤ/8)—61no—
59AGL(1,ℤ/8)—61no—
60C2×SD16—31no—
61C2×SD16—31no—
62C2×SD16—31no—
63C2×D8—31no—
64C2×D8—31no—
65C2×D8—31no—
66C2×D8—31no—
67C2×D8—31no—
68C2×D8—31no—
69C2×M16—31no—
70C22×C8C2×D311yescharacteristic, complemented
71#127—31no—
72#124—31no—
73#123—61no—
74C22⋊Q8—31no—
75#121—31no—
76#121—31no—
77#121—31no—
78#121—61no—
79#121—61no—
80#120—61no—
81#120—61no—
82#120—61no—
83C4×D4—31no—
84C4×D4—31no—
85C4×D4—31no—
86C2×C22⋊C4—61no—
87#106—31no—
88C4⋊C8—31no—
89#103—31no—
90#102—31no—
91#102—31no—
92#102—31no—
93#102—31no—
94#102—31no—
95C22⋊C8—31no—
96C22⋊C8—31no—
97#97—31no—
98C22×D3—81no—
99C2×A4—41no—
100C2×A4—41no—
101C2×A4C2×D411yescharacteristic
102S4—21no—
103S4—41no—
104C3×D4—41no—
105C3×D4—41no—
106#67—81no—
107#67—81no—
108C2×Dic3—81no—
109D12—41no—
110C4×D3—41no—
111Dic6—41no—
112C3×C8—41no—
113C3⋊C8—41no—
114C24—21no—
115C24—21no—
116C24—61no—
117#41—61no—
118#41—61no—
119#41—121no—
120C2×Q8—31no—
121C2×D4—31no—
122C2×D4—31no—
123C2×D4—31no—
124C2×D4—31no—
125C2×D4—31no—
126C2×D4—31no—
127C2×D4—61no—
128C2×D4—61no—
129C2×D4—61no—
130C2×D4—61no—
131C2×D4—61no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4—121no—
139C2×D4—122no—
140C2×D4—122no—
141C22×C4—31no—
142C22×C4—61no—
143C22×C4—61no—
144C22×C4—61no—
145C22×C4C22×D311yescharacteristic
146SD16—61no—
147SD16—61no—
148SD16—61no—
149SD16—61no—
150SD16—121no—
151D8—31no—
152D8—31no—
153D8—31no—
154D8—61no—
155D8—61no—
156D8—61no—
157D8—121no—
158D8S411yescharacteristic, complemented
159M16—61no—
160M16—61no—
161C2×C8—31no—
162C2×C8—31no—
163C2×C8—31no—
164C2×C8—31no—
165C4⋊C4—31no—
166C4⋊C4—31no—
167C4⋊C4—31no—
168C4⋊C4—61no—
169C22⋊C4—31no—
170C22⋊C4—31no—
171C22⋊C4—61no—
172C22⋊C4—61no—
173C22⋊C4—61no—
174C22⋊C4—61no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C42—31no—
180C22×C3—81no—
181C22×C3—81no—
182C2×D3—41no—
183C2×D3—81no—
184C2×D3—162no—
185A4AGL(1,ℤ/8)11yescharacteristic, complemented
186C3×C4—41no—
187Dic3—41no—
188Dic3—81no—
189C23—31no—
190C23—41no—
191C23—41no—
192C23—61no—
193C23—61no—
194C23—61no—
195C23—61no—
196C23—61no—
197C23—121no—
198C23—121no—
199C23—121no—
200C23—121no—
201C23—122no—
202C23—241no—
203C23D3×D411yescharacteristic
204Q8—61no—
205Q8—61no—
206D4—31no—
207D4—31no—
208D4—61no—
209D4—61no—
210D4—61no—
211D4—61no—
212D4—61no—
213D4—61no—
214D4—61no—
215D4—61no—
216D4—121no—
217D4—121no—
218D4—122no—
219D4—121no—
220D4—121no—
221D4—121no—
222D4—122no—
223D4—121no—
224D4—241no—
225D4—241no—
226D4C2×S411yescharacteristic
227D4C2×S411yescharacteristic, complemented
228C2×C4—31no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—31no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—122no—
241C2×C4—241no—
242C8—31no—
243C8—61no—
244C8—61no—
245C8C2×S411yescharacteristic, complemented
246C2×C3—41no—
247C2×C3—161no—
248C2×C3—161no—
249D3—81no—
250D3—161no—
251C22—21no—
252C22—21no—
253C22—31no—
254C22—31no—
255C22—61no—
256C22—61no—
257C22—61no—
258C22—61no—
259C22—61no—
260C22—121no—
261C22—121no—
262C22—121no—
263C22—121no—
264C22—121no—
265C22—121no—
266C22—121no—
267C22—121no—
268C22—121no—
269C22—121no—
270C22—121no—
271C22—242no—
272C22—242no—
273C22#91111yescharacteristic, complemented
274C4—31no—
275C4—61no—
276C4—61no—
277C4—61no—
278C4—121no—
279C4—121no—
280C4—121no—
281C4C22×S411yescharacteristic
282C3—41noSylow
283C2—31no—
284C2—31no—
285C2—41no—
286C2—41no—
287C2—121no—
288C2—121no—
289C2—121no—
290C2—241no—
291C2D4×S411yescharacteristic
subgroupG/H
centerC2D4×S4
commutator, cosocleC4×A4C23
FrattiniC4C22×S4
FittingC22×D8D3
socleC23D3×D4
hypercenterD8S4
hypocenterA4AGL(1,ℤ/8)
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #5914C2semidirect
2. D4×S4C2semidirect
3. A4×D8C2semidirect
4. #5415C2semidirect
5. #5414C2semidirect
6. #5401C2semidirect
7. #5400C2semidirect
11. D4×A4C22semidirect
17. A4⋊Q8C22semidirect
19. C8×A4C22semidirect
20. A4⋊C8C22semidirect
22. C22×D8D3semidirect
55. C22×D4C2×D3semidirect
70. C22×C8C2×D3semidirect
158. D8S4semidirect
185. A4AGL(1,ℤ/8)semidirect
227. D4C2×S4semidirect
245. C8C2×S4semidirect
273. C22#911semidirect

Maximal subgroups

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

subgroupindex
#59142
D4×S42
A4×D82
#54152
#54142
#54012
#54002
#32643
#9114