Finite Group Databasebeta

← #85494

Subgroups — #85494

Order 384.

501 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C2×D4×A4C211yesmaximal, characteristic, complemented
2C2×GL(2,ℤ/4)C212yesmaximal, complemented
3C2×C4×S4C211yesmaximal, characteristic, complemented
4#5432C212yesmaximal, complemented
5C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6#2408—31nomaximal, Sylow
7C23×A4C2212yes—
8C22×S4C2211yescharacteristic
9D4×A4—22no—
10C2×C4×A4C2211yescharacteristic
11GL(2,ℤ/4)—24no—
12C2×SL(2,ℤ/4)C2211yescharacteristic
13C2×SL(2,ℤ/4)C2212yes—
14C4×S4—22no—
15#938—41nomaximal
16C23×D4D311yescharacteristic, complemented
17C2×#121—31no—
18C2×#121—31no—
19C2×#120—32no—
20C2×C4×D4—31no—
21#394—32no—
22#392—32no—
23#390—31no—
24#390—31no—
25#380—31no—
26#379—32no—
27C22×A4—22no—
28C22×A4—22no—
29C22×A4—22no—
30C22×A4C2311yescharacteristic
31C2×S4—21no—
32C2×S4—21no—
33C2×S4D412yescomplemented
34C2×C3×D4—41no—
35C2×#67—42no—
36C2×C4×D3—41no—
37C4×A4D412yescomplemented
38SL(2,ℤ/4)—21no—
39SL(2,ℤ/4)—22no—
40#217—42no—
41C4⋊Dic3—41no—
42C25C2×D312yes—
43C22×D4—22no—
44C22×D4—31no—
45C22×D4—31no—
46C22×D4—31no—
47C22×D4—32no—
48C22×D4—32no—
49C22×D4—32no—
50C22×D4—62no—
51C23×C4—31no—
52C23×C4C2×D311yescharacteristic
53#121—64no—
54#121—61no—
55#121—62no—
56#121—61no—
57#120—64no—
58#120—64no—
59C4×D4—62no—
60C4×D4—62no—
61C2×C4⋊C4—31no—
62C2×C4⋊C4—31no—
63C2×C4⋊C4—31no—
64C2×C22⋊C4—31no—
65C2×C22⋊C4—32no—
66C2×C22⋊C4—32no—
67C2×C22⋊C4—32no—
68C2×C22⋊C4—32no—
69C2×C22⋊C4—31no—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—32no—
72C2×C22⋊C4—32no—
73C2×C42—31no—
74#95—32no—
75C23×C3—42no—
76C22×D3—41no—
77C2×A4—42no—
78C2×A4C2×D411yescharacteristic
79C2×A4C2×D411yescharacteristic
80C2×A4#4111yescharacteristic
81S4—22no—
82C3×D4—82no—
83C2×C3×C4—41no—
84#67—84no—
85C2×Dic3—42no—
86C2×Dic3—41no—
87C4×D3—82no—
88C24—22no—
89C24—22no—
90C24—22no—
91C24—32no—
92C24—31no—
93C24—32no—
94C24—62no—
95C24—62no—
96C24—62no—
97C24C22×D311yescharacteristic
98C2×D4—32no—
99C2×D4—34no—
100C2×D4—32no—
101C2×D4—32no—
102C2×D4—31no—
103C2×D4—62no—
104C2×D4—62no—
105C2×D4—61no—
106C2×D4—62no—
107C2×D4—64no—
108C2×D4—61no—
109C2×D4—61no—
110C2×D4—64no—
111C2×D4—61no—
112C2×D4—61no—
113C2×D4—64no—
114C2×D4—64no—
115C2×D4—64no—
116C2×D4—62no—
117C2×D4—62no—
118C2×D4—122no—
119C2×D4—122no—
120C2×D4S411yescharacteristic, complemented
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—32no—
124C22×C4—31no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—62no—
130C22×C4—61no—
131C22×C4—62no—
132C22×C4—61no—
133C22×C4#6712yescomplemented
134C4⋊C4—62no—
135C4⋊C4—61no—
136C4⋊C4—62no—
137C4⋊C4—61no—
138C22⋊C4—62no—
139C22⋊C4—64no—
140C22⋊C4—64no—
141C22⋊C4—61no—
142C22⋊C4—62no—
143C22⋊C4—64no—
144C22⋊C4—62no—
145C22⋊C4—62no—
146C22⋊C4—61no—
147C22⋊C4—62no—
148C22⋊C4—122no—
149C22⋊C4—122no—
150C42—62no—
151C22×C3—41no—
152C22×C3—82no—
153C22×C3—82no—
154C22×C3—82no—
155C2×D3—42no—
156C2×D3—81no—
157C2×D3—81no—
158A4#12111yescharacteristic, complemented
159C3×C4—42no—
160Dic3—81no—
161Dic3—82no—
162C23—31no—
163C23—31no—
164C23—31no—
165C23—31no—
166C23—34no—
167C23—32no—
168C23—32no—
169C23—42no—
170C23—62no—
171C23—62no—
172C23—62no—
173C23—62no—
174C23—62no—
175C23—62no—
176C23—62no—
177C23—62no—
178C23—62no—
179C23—62no—
180C23—61no—
181C23—61no—
182C23—61no—
183C23—61no—
184C23—61no—
185C23—62no—
186C23—61no—
187C23—122no—
188C23—122no—
189C23—122no—
190C23—124no—
191C23D3×D411yescharacteristic
192C23#23711yescharacteristic
193C23C2×#6711yescharacteristic
194C23C2×S412yes—
195D4—22no—
196D4—64no—
197D4—64no—
198D4—62no—
199D4—62no—
200D4—62no—
201D4—62no—
202D4—62no—
203D4—124no—
204D4—128no—
205D4—124no—
206C2×C4—31no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4C2×S411yescharacteristic
229C2×C3—41no—
230C2×C3—41no—
231C2×C3—41no—
232C2×C3—162no—
233D3—82no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—31no—
238C22—31no—
239C22—31no—
240C22—31no—
241C22—31no—
242C22—31no—
243C22—31no—
244C22—31no—
245C22—31no—
246C22—61no—
247C22—62no—
248C22—62no—
249C22—64no—
250C22—64no—
251C22—62no—
252C22—62no—
253C22—62no—
254C22—62no—
255C22—122no—
256C22—122no—
257C22—122no—
258C22—122no—
259C22—122no—
260C22—122no—
261C22—121no—
262C22—121no—
263C22—121no—
264C22—121no—
265C22#93811yescharacteristic, complemented
266C22C22×S411yescharacteristic
267C4—32no—
268C4—62no—
269C4—121no—
270C4—122no—
271C4—122no—
272C4—121no—
273C4GL(2,ℤ/4)12yescomplemented
274C3—41noSylow
275C2—31no—
276C2—31no—
277C2—31no—
278C2—31no—
279C2—42no—
280C2—122no—
281C2—122no—
282C2D4×S411yescharacteristic
283C2#591411yescharacteristic
284C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC23×D4D3
socleC24C22×D3
hypercenterC2×D4S4
hypocenterA4#121
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×A4C2semidirect
2. C2×GL(2,ℤ/4)C2semidirect
3. C2×C4×S4C2semidirect
4. #5432C2semidirect
5. C4⋊SL(2,ℤ/4)C2semidirect
16. C23×D4D3semidirect
33. C2×S4D4semidirect
37. C4×A4D4semidirect
120. C2×D4S4semidirect
133. C22×C4#67semidirect
158. A4#121semidirect
265. C22#938semidirect
273. C4GL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×D4×A42
C2×GL(2,ℤ/4)2
C2×C4×S42
#54322
C4⋊SL(2,ℤ/4)2
#24083
#9384