Finite Group Databasebeta

← #85294

Subgroups — #85294

Order 384.

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

normal subgroups only

This group has 291 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×C22⋊C4C211yesmaximal, characteristic
2#5432C211yesmaximal, characteristic
3#5432C211yesmaximal, characteristic
4C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
6SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8#2480—31nomaximal, Sylow
9C23×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#879—41nomaximal
17C2×#126—31no—
18C22×C22⋊C4D311yescharacteristic
19#397—31no—
20#397—31no—
21#397—31no—
22#396—31no—
23#388—31no—
24#388—31no—
25#388—31no—
26#385—31no—
27#385—31no—
28#380—31no—
29#379—31no—
30#379—31no—
31C4×C22⋊C4—31no—
32C22×A4—21no—
33C22×A4—21no—
34C22×A4—21no—
35C22×A4C2311yescharacteristic
36C4×A4—21no—
37C4×A4—21no—
38SL(2,ℤ/4)—21no—
39SL(2,ℤ/4)—21no—
40SL(2,ℤ/4)—21no—
41SL(2,ℤ/4)—21no—
42C3×C22⋊C4—41no—
43#217—41no—
44#217—41no—
45C4⋊Dic3—41no—
46Dic3⋊C4—41no—
47Dic3⋊C4—41no—
48C4×Dic3—41no—
49C25C2×D311yescharacteristic
50C23×C4C2×D311yescharacteristic
51C23×C4C2×D311yescharacteristic
52#126—61no—
53#126—61no—
54#126—61no—
55#126—61no—
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×C22⋊C4—31no—
63C2×C22⋊C4—31no—
64C2×C22⋊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×C22⋊C4—31no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—31no—
78C2×C42—31no—
79C2×C42—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—
88C23×C3—41no—
89C2×A4—41no—
90C2×A4#4111yescharacteristic
91C2×A4#4111yescharacteristic
92C2×A4#4111yescharacteristic
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×Dic3—41no—
96C2×Dic3—41no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C24—21no—
100C24—21no—
101C24—21no—
102C24—31no—
103C24—31no—
104C24—61no—
105C24—61no—
106C24—61no—
107C24C22×D311yescharacteristic
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—61no—
123C22×C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—121no—
131C4⋊C4—121no—
132C4⋊C4—121no—
133C22⋊C4—31no—
134C22⋊C4—31no—
135C22⋊C4—31no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C22⋊C4—61no—
139C22⋊C4—61no—
140C22⋊C4—61no—
141C22⋊C4—62no—
142C22⋊C4—62no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—121no—
152C22⋊C4—121no—
153C22⋊C4—121no—
154C22⋊C4—121no—
155C22⋊C4S411yescharacteristic
156C42—61no—
157C42—61no—
158C42—121no—
159C22×C3—41no—
160C22×C3—81no—
161C22×C3—81no—
162C22×C3—81no—
163A4#12611yescharacteristic, complemented
164C3×C4—81no—
165C3×C4—81no—
166Dic3—81no—
167Dic3—81no—
168Dic3—81no—
169Dic3—81no—
170C23—31no—
171C23—31no—
172C23—31no—
173C23—31no—
174C23—32no—
175C23—31no—
176C23—41no—
177C23—61no—
178C23—61no—
179C23—61no—
180C23—61no—
181C23—61no—
182C23—61no—
183C23—61no—
184C23—61no—
185C23—61no—
186C23—61no—
187C23—61no—
188C23—121no—
189C23—121no—
190C23—121no—
191C23—122no—
192C23#23511yescharacteristic
193C23#23711yescharacteristic
194C23#23711yescharacteristic
195C23C2×S411yescharacteristic
196C2×C4—31no—
197C2×C4—31no—
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×C4—61no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4C2×S411yescharacteristic
231C2×C4C2×S411yescharacteristic
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—161no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—31no—
240C22—31no—
241C22—31no—
242C22—31no—
243C22—31no—
244C22—31no—
245C22—31no—
246C22—31no—
247C22—31no—
248C22—61no—
249C22—62no—
250C22—62no—
251C22—61no—
252C22—61no—
253C22—61no—
254C22—121no—
255C22—121no—
256C22—121no—
257C22—121no—
258C22—121no—
259C22—121no—
260C22#87911yescharacteristic, complemented
261C22C22×S411yescharacteristic
262C4—21no—
263C4—21no—
264C4—61no—
265C4—61no—
266C4—121no—
267C4—121no—
268C4—121no—
269C4—121no—
270C4—121no—
271C4—121no—
272C4—121no—
273C4—121no—
274C3—41noSylow
275C2—31no—
276C2—31no—
277C2—31no—
278C2—31no—
279C2—41no—
280C2—121no—
281C2#591211yescharacteristic
282C2#591411yescharacteristic
283C2#591411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC22×C22⋊C4D3
socleC24C22×D3
prime radicalC23×A4C22
hypercenterC22⋊C4S4
hypocenterA4#126
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C4⋊SL(2,ℤ/4)C2semidirect
5. SL(2,ℤ/4)⋊C4C2semidirect
6. SL(2,ℤ/4)⋊C4C2semidirect
7. C4×SL(2,ℤ/4)C2semidirect
163. A4#126semidirect
260. C22#879semidirect

Maximal subgroups

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

subgroupindex
A4×C22⋊C42
#54322
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
#24803
#8794