Finite Group Databasebeta

← #85313

Subgroups — #85313

Order 384.

272 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 38 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, complemented
2(C2×S4)⋊C4C211yesmaximal, characteristic
3(C2×S4)⋊C4C211yesmaximal, characteristic
4(C2×S4)⋊C4C211yesmaximal, characteristic
5C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8#2481—31nomaximal, Sylow
9C22×S4C2211yescharacteristic
10C2×C4×A4C2211yescharacteristic
11C2×C4×A4C2211yescharacteristic
12C2×C4×A4C2211yescharacteristic
13C2×SL(2,ℤ/4)C2211yescharacteristic
14C2×SL(2,ℤ/4)C2211yescharacteristic
15C2×SL(2,ℤ/4)C2211yescharacteristic
16#898—41nomaximal
17C2×#126—31no—
18C22×C4⋊C4D311yescharacteristic, complemented
19#400—31no—
20#400—31no—
21#397—31no—
22#396—31no—
23#388—31no—
24#387—31no—
25#387—31no—
26#386—31no—
27#386—31no—
28#386—31no—
29#385—31no—
30#380—31no—
31C4×C22⋊C4—31no—
32C22×A4C2311yescharacteristic
33C2×S4—21no—
34C2×S4—21no—
35C2×S4—21no—
36C4×A4—21no—
37C4×A4—21no—
38C4×A4—21no—
39SL(2,ℤ/4)—21no—
40SL(2,ℤ/4)—21no—
41SL(2,ℤ/4)—21no—
42C3×C4⋊C4—41no—
43(C2×D3)⋊C4—41no—
44(C2×D3)⋊C4—41no—
45(C2×D3)⋊C4—41no—
46C4⋊Dic3—41no—
47Dic3⋊C4—41no—
48C4×Dic3—41no—
49C22×D4—31no—
50C23×C4C2×D311yescharacteristic
51C23×C4C2×D311yescharacteristic
52C23×C4C2×D311yescharacteristic
53#126—61no—
54#126—61no—
55#126—61no—
56#126—61no—
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×C22⋊C4—31no—
66C2×C22⋊C4—31no—
67C2×C22⋊C4—31no—
68C2×C22⋊C4—31no—
69C2×C22⋊C4—31no—
70C2×C22⋊C4—31no—
71C2×C22⋊C4—31no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4—31no—
74C2×C42—31no—
75C2×C42—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—
86#95—31no—
87#95—31no—
88C22×D3—41no—
89C2×A4#4111yescharacteristic
90C2×A4#4111yescharacteristic
91C2×A4#4111yescharacteristic
92S4—41no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×Dic3—41no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C24—31no—
100C24C22×D311yescharacteristic
101C2×D4—61no—
102C2×D4—61no—
103C2×D4—61no—
104C2×D4—61no—
105C2×D4—61no—
106C2×D4—61no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—31no—
111C22×C4—31no—
112C22×C4—31no—
113C22×C4—31no—
114C22×C4—31no—
115C22×C4—31no—
116C22×C4—31no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—61no—
124C22×C4—61no—
125C22×C4—61no—
126C4⋊C4—31no—
127C4⋊C4—31no—
128C4⋊C4—31no—
129C4⋊C4—61no—
130C4⋊C4—62no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C4⋊C4—121no—
136C4⋊C4S411yescharacteristic, complemented
137C22⋊C4—62no—
138C22⋊C4—61no—
139C22⋊C4—61no—
140C22⋊C4—61no—
141C22⋊C4—61no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—121no—
151C22⋊C4—121no—
152C42—61no—
153C42—61no—
154C42—121no—
155C22×C3—41no—
156C2×D3—81no—
157C2×D3—81no—
158C2×D3—81no—
159A4#12611yescharacteristic, complemented
160C3×C4—81no—
161C3×C4—81no—
162C3×C4—81no—
163Dic3—81no—
164Dic3—81no—
165Dic3—81no—
166C23—31no—
167C23—31no—
168C23—31no—
169C23—31no—
170C23—61no—
171C23—61no—
172C23—61no—
173C23—61no—
174C23—61no—
175C23—61no—
176C23—61no—
177C23#23511yescharacteristic
178C23#23711yescharacteristic
179C23#23911yescharacteristic
180D4—121no—
181D4—121no—
182D4—121no—
183D4—121no—
184C2×C4—32no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
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×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—61no—
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×C4C2×S411yescharacteristic
221C2×C4C2×S411yescharacteristic
222C2×C4C2×S411yescharacteristic
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—41no—
226D3—161no—
227C22—31no—
228C22—31no—
229C22—31no—
230C22—31no—
231C22—31no—
232C22—31no—
233C22—31no—
234C22—31no—
235C22—31no—
236C22—61no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22—121no—
241C22—121no—
242C22—121no—
243C22—121no—
244C22#89811yescharacteristic, complemented
245C22C22×S411yescharacteristic
246C4—21no—
247C4—21no—
248C4—21no—
249C4—61no—
250C4—61no—
251C4—61no—
252C4—121no—
253C4—121no—
254C4—121no—
255C4—121no—
256C4—121no—
257C4—121no—
258C4—121no—
259C3—41noSylow
260C2—31no—
261C2—31no—
262C2—31no—
263C2—31no—
264C2—241no—
265C2#591211yescharacteristic
266C2#591411yescharacteristic
267C2#591911yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC22×C4⋊C4D3
socleC24C22×D3
prime radicalC22×S4C22
hypercenterC4⋊C4S4
hypocenterA4#126
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. A4×C4⋊C4C2semidirect
5. C4⋊SL(2,ℤ/4)C2semidirect
6. SL(2,ℤ/4)⋊C4C2semidirect
7. C4×SL(2,ℤ/4)C2semidirect
18. C22×C4⋊C4D3semidirect
136. C4⋊C4S4semidirect
159. A4#126semidirect
244. C22#898semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊C42
(C2×S4)⋊C42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
#24813
#8984