Finite Group Databasebeta

← #85464

Subgroups — #85464

Order 384.

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

normal subgroups only

This group has 259 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#5919C211yesmaximal, characteristic
2Q8×S4C211yesmaximal, characteristic, complemented
3A4×Q16C211yesmaximal, characteristic, complemented
4#5417C211yesmaximal, characteristic, complemented
5#5416C211yesmaximal, characteristic, complemented
6#5401C211yesmaximal, characteristic, complemented
7#5400C211yesmaximal, characteristic, complemented
8#3270—31nomaximal, Sylow
9Q8×A4C2211yescharacteristic
10Q8×A4C2211yescharacteristic
11C4⋊S4—21no—
12C4⋊S4C2211yescharacteristic
13C4×S4—21no—
14C4×S4—21no—
15C4×S4C2211yescharacteristic
16A4⋊Q8—21no—
17A4⋊Q8C2211yescharacteristic
18#918—41nomaximal
19C8×A4C2211yescharacteristic
20A4⋊C8C2211yescharacteristic
21C2×#137—31no—
22C22×Q16D311yescharacteristic, complemented
23D4×Q8—31no—
24#548—31no—
25#497—31no—
26#470—31no—
27#465—31no—
28#464—31no—
29#462—31no—
30#452—31no—
31#451—31no—
32#451—31no—
33#448—31no—
34#441—31no—
35#435—31no—
36C2×S4—21no—
37C2×S4D411yescharacteristic
38#239—41no—
39D3×Q8—41no—
40C4×A4—21no—
41C4×A4—21no—
42C4×A4C2311yescharacteristic
43SL(2,ℤ/4)—21no—
44SL(2,ℤ/4)D411yescharacteristic
45C3×Q16—41no—
46#216—41no—
47#215—41no—
48#204—41no—
49ΓL(1,𝔽25)—41no—
50C2×#41—31no—
51C22×Q8—31no—
52C22×Q8C2×D311yescharacteristic
53C22×Q8C2×D311yescharacteristic
54#137—61no—
55#137—61no—
56#137—61no—
57#137—61no—
58C2×Q16—31no—
59C2×Q16—31no—
60C2×Q16—31no—
61C2×Q16—31no—
62C2×Q16—31no—
63C2×Q16—31no—
64C2×SD16—31no—
65C2×SD16—31no—
66C2×SD16—31no—
67C2×M16—31no—
68C22×C8C2×D311yescharacteristic
69C4⋊Q8—31no—
70C4⋊Q8—61no—
71#124—31no—
72#124—61no—
73C22⋊Q8—31no—
74C22⋊Q8—31no—
75C22⋊Q8—31no—
76C22⋊Q8—61no—
77C22⋊Q8—61no—
78C22⋊Q8—61no—
79#121—31no—
80#121—61no—
81C4×Q8—31no—
82C4×Q8—31no—
83C4×D4—31no—
84C4×D4—61no—
85C4×D4—61no—
86#106—31no—
87C4⋊C8—31no—
88#103—31no—
89#103—31no—
90#103—31no—
91#103—31no—
92#103—31no—
93#102—31no—
94C22⋊C8—31no—
95C22⋊C8—31no—
96#97—31no—
97C2×A4C2×D411yescharacteristic
98S4—21no—
99S4—41no—
100C3×Q8—41no—
101C3×Q8—41no—
102D12—41no—
103D12—81no—
104C4×D3—41no—
105C4×D3—81no—
106C4×D3—81no—
107Dic6—41no—
108Dic6—81no—
109C3×C8—41no—
110C3⋊C8—41no—
111#41—61no—
112#41—61no—
113#41—121no—
114C2×Q8—31no—
115C2×Q8—31no—
116C2×Q8—31no—
117C2×Q8—31no—
118C2×Q8—31no—
119C2×Q8—61no—
120C2×Q8—61no—
121C2×Q8—61no—
122C2×Q8—61no—
123C2×Q8—61no—
124C2×Q8—121no—
125C2×D4—31no—
126C2×D4—31no—
127C2×D4—61no—
128C2×D4—61no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—31no—
132C22×C4—61no—
133C22×C4—61no—
134C22×C4C22×D311yescharacteristic
135Q16—31no—
136Q16—31no—
137Q16—31no—
138Q16—61no—
139Q16—61no—
140Q16—61no—
141Q16—121no—
142Q16S411yescharacteristic, complemented
143SD16—61no—
144SD16—61no—
145SD16—61no—
146SD16—61no—
147SD16—121no—
148M16—61no—
149M16—61no—
150C2×C8—31no—
151C2×C8—31no—
152C2×C8—31no—
153C2×C8—31no—
154C4⋊C4—31no—
155C4⋊C4—31no—
156C4⋊C4—31no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4—61no—
161C4⋊C4—61no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C22⋊C4—31no—
165C22⋊C4—31no—
166C22⋊C4—61no—
167C22⋊C4—61no—
168C22⋊C4—61no—
169C22⋊C4—61no—
170C22⋊C4—61no—
171C22⋊C4—61no—
172C42—31no—
173C42—61no—
174C42—61no—
175C2×D3—41no—
176C2×D3—81no—
177A4#13711yescharacteristic, complemented
178C3×C4—41no—
179C3×C4—81no—
180C3×C4—81no—
181Dic3—41no—
182Dic3—81no—
183C23—31no—
184C23—61no—
185C23D3×D411yescharacteristic
186Q8—31no—
187Q8—31no—
188Q8—61no—
189Q8—61no—
190Q8—61no—
191Q8—61no—
192Q8—61no—
193Q8—61no—
194Q8—121no—
195Q8—121no—
196Q8C2×S411yescharacteristic
197Q8C2×S411yescharacteristic
198D4—61no—
199D4—61no—
200D4—61no—
201D4—61no—
202D4—121no—
203D4—121no—
204D4—121no—
205D4—121no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—121no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4—121no—
224C8—31no—
225C8—61no—
226C8—61no—
227C8C2×S411yescharacteristic
228C2×C3—41no—
229D3—81no—
230D3—161no—
231C22—31no—
232C22—31no—
233C22—61no—
234C22—61no—
235C22—61no—
236C22—121no—
237C22—121no—
238C22—121no—
239C22#91811yescharacteristic, complemented
240C4—21no—
241C4—21no—
242C4—31no—
243C4—61no—
244C4—61no—
245C4—61no—
246C4—61no—
247C4—61no—
248C4—121no—
249C4—121no—
250C4—121no—
251C4C22×S411yescharacteristic
252C3—41noSylow
253C2—31no—
254C2—31no—
255C2—121no—
256C2—241no—
257C2D4×S411yescharacteristic
subgroupG/H
centerC2D4×S4
commutator, cosocleC4×A4C23
FrattiniC4C22×S4
FittingC22×Q16D3
socleC23D3×D4
prime radical#5919C2
hypercenterQ16S4
hypocenterA4#137
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
2. Q8×S4C2semidirect
3. A4×Q16C2semidirect
4. #5417C2semidirect
5. #5416C2semidirect
6. #5401C2semidirect
7. #5400C2semidirect
22. C22×Q16D3semidirect
142. Q16S4semidirect
177. A4#137semidirect
239. C22#918semidirect

Maximal subgroups

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

subgroupindex
#59192
Q8×S42
A4×Q162
#54172
#54162
#54012
#54002
#32703
#9184