Finite Group Databasebeta

← #85304

Subgroups — #85304

Order 384.

255 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 40 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
1A4×C4⋊C4C211yesmaximal, characteristic
2C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
4SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
5SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
6SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
7C4×SL(2,ℤ/4)C211yesmaximal, characteristic
8#2473—31nomaximal, Sylow
9C2×C4×A4C2211yescharacteristic
10C2×C4×A4C2211yescharacteristic
11C2×C4×A4C2211yescharacteristic
12C2×SL(2,ℤ/4)C2211yescharacteristic
13C2×SL(2,ℤ/4)C2211yescharacteristic
14C2×SL(2,ℤ/4)C2211yescharacteristic
15C2×SL(2,ℤ/4)C2211yescharacteristic
16#889—41nomaximal
17C2×#125—31no—
18C22×C4⋊C4D311yescharacteristic
19#400—31no—
20#398—31no—
21#398—31no—
22#398—31no—
23#389—31no—
24#387—31no—
25#387—31no—
26#385—31no—
27#385—31no—
28#385—31no—
29#385—31no—
30#380—31no—
31C4×C22⋊C4—31no—
32C22×A4C2311yescharacteristic
33C4×A4—21no—
34C4×A4—21no—
35C4×A4—21no—
36SL(2,ℤ/4)—21no—
37SL(2,ℤ/4)—21no—
38SL(2,ℤ/4)—21no—
39SL(2,ℤ/4)Q812yes—
40C3×C4⋊C4—41no—
41C4⋊Dic3—41no—
42Dic3⋊C4—41no—
43Dic3⋊C4—41no—
44Dic3⋊C4—41no—
45Dic3⋊C4—41no—
46C4×Dic3—41no—
47C23×C4C2×D311yescharacteristic
48C23×C4C2×D311yescharacteristic
49C23×C4C2×D311yescharacteristic
50#125—61no—
51#125—61no—
52#125—61no—
53#125—61no—
54C2×C4⋊C4—31no—
55C2×C4⋊C4—31no—
56C2×C4⋊C4—31no—
57C2×C4⋊C4—31no—
58C2×C4⋊C4—31no—
59C2×C4⋊C4—31no—
60C2×C4⋊C4—31no—
61C2×C4⋊C4—31no—
62C2×C4⋊C4—31no—
63C2×C4⋊C4—31no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—31no—
68C2×C22⋊C4—31no—
69C2×C22⋊C4—31no—
70C2×C22⋊C4—31no—
71C2×C22⋊C4—31no—
72C2×C42—31no—
73C2×C42—31no—
74#95—31no—
75#95—31no—
76#95—31no—
77#95—31no—
78#95—31no—
79#95—31no—
80#95—31no—
81#95—31no—
82#95—31no—
83#95—31no—
84#95—31no—
85#95—31no—
86C2×A4C2×Q811yescharacteristic
87C2×A4#4111yescharacteristic
88C2×A4#4111yescharacteristic
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×Dic3—41no—
93C2×Dic3—41no—
94C2×Dic3—41no—
95C2×Dic3—41no—
96C24C22×D311yescharacteristic
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—31no—
101C22×C4—31no—
102C22×C4—31no—
103C22×C4—31no—
104C22×C4—31no—
105C22×C4—31no—
106C22×C4—31no—
107C22×C4—31no—
108C22×C4—31no—
109C22×C4—31no—
110C22×C4—31no—
111C22×C4—31no—
112C22×C4—31no—
113C22×C4—31no—
114C22×C4—61no—
115C22×C4—61no—
116C22×C4—61no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—61no—
121C4⋊C4—62no—
122C4⋊C4—62no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C4⋊C4—121no—
134C4⋊C4—121no—
135C4⋊C4—121no—
136C4⋊C4S411yescharacteristic
137C22⋊C4—32no—
138C22⋊C4—32no—
139C22⋊C4—61no—
140C22⋊C4—61no—
141C22⋊C4—61no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C42—61no—
146C42—61no—
147C42—121no—
148C22×C3—41no—
149A4#12511yescharacteristic, complemented
150C3×C4—81no—
151C3×C4—81no—
152C3×C4—81no—
153Dic3—42no—
154Dic3—81no—
155Dic3—81no—
156Dic3—81no—
157C23—31no—
158C23—31no—
159C23—31no—
160C23—31no—
161C23#23511yescharacteristic
162C23#23711yescharacteristic
163C23D3×Q811yescharacteristic
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—32no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—31no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4C2×S411yescharacteristic
204C2×C4C2×S411yescharacteristic
205C2×C4C2×S411yescharacteristic
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—41no—
209C22—31no—
210C22—31no—
211C22—31no—
212C22—31no—
213C22—31no—
214C22—31no—
215C22—31no—
216C22—31no—
217C22—31no—
218C22—61no—
219C22#88911yescharacteristic, complemented
220C22C22×S411yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—62no—
225C4—61no—
226C4—61no—
227C4—61no—
228C4—62no—
229C4—121no—
230C4—121no—
231C4—121no—
232C4—121no—
233C4—121no—
234C4—121no—
235C3—41noSylow
236C2—31no—
237C2—31no—
238C2—31no—
239C2—31no—
240C2#591211yescharacteristic
241C2#591411yescharacteristic
242C2Q8×S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, prime radical, cosocleC22×A4C23
FittingC22×C4⋊C4D3
socleC24C22×D3
hypercenterC4⋊C4S4
hypocenterA4#125
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
149. A4#125semidirect
219. C22#889semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊C42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
#24733
#8894