Finite Group Databasebeta

← #85485

Subgroups — #85485

Order 384.

420 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C22×C4×A4C211yesmaximal, characteristic, complemented
2C2×GL(2,ℤ/4)C211yesmaximal, characteristic
3#5432C211yesmaximal, characteristic, complemented
4(C2×S4)⋊C4C212yesmaximal, complemented
5SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6#2849—31nomaximal, Sylow
7C23×A4C2211yescharacteristic
8C22×S4C2211yescharacteristic
9C2×C4×A4—22no—
10C2×C4×A4C2212yes—
11GL(2,ℤ/4)—22no—
12C2×SL(2,ℤ/4)C2211yescharacteristic
13C2×SL(2,ℤ/4)C2212yes—
14#929—41nomaximal
15C24×C4D311yescharacteristic, complemented
16C2×#123—31no—
17C2×#123—31no—
18C2×#120—31no—
19#386—32no—
20#386—32no—
21#385—32no—
22#385—32no—
23#381—31no—
24#381—31no—
25#379—31no—
26C22×A4—22no—
27C22×A4D412yes—
28C22×A4C2311yescharacteristic
29C2×S4—21no—
30C2×S4—22no—
31C22×C3×C4—41no—
32C2×#67—41no—
33C4×A4—22no—
34SL(2,ℤ/4)—21no—
35SL(2,ℤ/4)—22no—
36#217—41no—
37(C2×D3)⋊C4—42no—
38Dic3⋊C4—42no—
39C25C2×D311yescharacteristic
40C22×D4—31no—
41C22×D4—31no—
42C22×D4—31no—
43C23×C4—22no—
44C23×C4—31no—
45C23×C4—31no—
46C23×C4—32no—
47C23×C4—62no—
48C23×C4C2×D312yes—
49#123—64no—
50#123—61no—
51#123—62no—
52#123—61no—
53#120—62no—
54#120—62no—
55C2×C4⋊C4—32no—
56C2×C4⋊C4—32no—
57C2×C22⋊C4—31no—
58C2×C22⋊C4—31no—
59C2×C22⋊C4—31no—
60C2×C22⋊C4—31no—
61C2×C22⋊C4—32no—
62C2×C22⋊C4—32no—
63C2×C22⋊C4—32no—
64C2×C22⋊C4—31no—
65C2×C22⋊C4—32no—
66#95—32no—
67#95—32no—
68#95—32no—
69#95—32no—
70C23×C3—41no—
71C22×D3—41no—
72C2×A4—22no—
73C2×A4C2×D411yescharacteristic
74C2×A4#4112yes—
75S4—41no—
76C2×C3×C4—42no—
77C2×C3×C4—82no—
78#67—82no—
79C2×Dic3—41no—
80C2×Dic3—42no—
81C24—22no—
82C24—31no—
83C24—31no—
84C24—32no—
85C24—31no—
86C24—62no—
87C24#6712yes—
88C24C22×D311yescharacteristic
89C2×D4—62no—
90C2×D4—61no—
91C2×D4—62no—
92C2×D4—62no—
93C2×D4—61no—
94C2×D4—62no—
95C2×D4—64no—
96C2×D4—62no—
97C2×D4—61no—
98C2×D4—61no—
99C22×C4—22no—
100C22×C4—32no—
101C22×C4—34no—
102C22×C4—31no—
103C22×C4—32no—
104C22×C4—32no—
105C22×C4—31no—
106C22×C4—31no—
107C22×C4—32no—
108C22×C4—32no—
109C22×C4—31no—
110C22×C4—62no—
111C22×C4—62no—
112C22×C4—61no—
113C22×C4—62no—
114C22×C4—64no—
115C22×C4—64no—
116C22×C4—62no—
117C22×C4S411yescharacteristic, complemented
118C4⋊C4—64no—
119C4⋊C4—62no—
120C4⋊C4—62no—
121C22⋊C4—64no—
122C22⋊C4—62no—
123C22⋊C4—62no—
124C22⋊C4—64no—
125C22⋊C4—62no—
126C22⋊C4—61no—
127C22⋊C4—62no—
128C22⋊C4—62no—
129C22⋊C4—61no—
130C22⋊C4—62no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—62no—
134C22×C3—42no—
135C22×C3—41no—
136C22×C3—82no—
137C2×D3—81no—
138C2×D3—82no—
139A4#12311yescharacteristic, complemented
140C3×C4—82no—
141Dic3—81no—
142Dic3—82no—
143C23—22no—
144C23—31no—
145C23—32no—
146C23—31no—
147C23—32no—
148C23—34no—
149C23—32no—
150C23—32no—
151C23—31no—
152C23—62no—
153C23—62no—
154C23—62no—
155C23—61no—
156C23—64no—
157C23—62no—
158C23—64no—
159C23—61no—
160C23—61no—
161C23—62no—
162C23—62no—
163C23—61no—
164C23#23512yes—
165C23C2×#6711yescharacteristic
166C23C2×S411yescharacteristic
167D4—122no—
168D4—121no—
169D4—124no—
170D4—122no—
171D4—121no—
172D4—122no—
173C2×C4—22no—
174C2×C4—34no—
175C2×C4—32no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—64no—
179C2×C4—64no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—61no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4C2×S412yes—
196C2×C3—42no—
197C2×C3—41no—
198C2×C3—82no—
199D3—161no—
200C22—22no—
201C22—32no—
202C22—31no—
203C22—32no—
204C22—32no—
205C22—31no—
206C22—31no—
207C22—34no—
208C22—32no—
209C22—62no—
210C22—62no—
211C22—64no—
212C22—61no—
213C22—62no—
214C22—64no—
215C22—62no—
216C22—121no—
217C22—121no—
218C22—122no—
219C22—122no—
220C22—121no—
221C22#92911yescharacteristic, complemented
222C22GL(2,ℤ/4)12yes—
223C22C22×S411yescharacteristic
224C4—22no—
225C4—62no—
226C4—122no—
227C4—121no—
228C4—121no—
229C4—122no—
230C4—121no—
231C3—41noSylow
232C2—22no—
233C2—31no—
234C2—31no—
235C2—32no—
236C2—62no—
237C2—241no—
238C2#591212yes—
239C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC24×C4D3
socleC24C22×D3
prime radicalC2×GL(2,ℤ/4)C2
hypercenterC22×C4S4
hypocenterA4#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×A4C2semidirect
3. #5432C2semidirect
4. (C2×S4)⋊C4C2semidirect
5. SL(2,ℤ/4)⋊C4C2semidirect
15. C24×C4D3semidirect
117. C22×C4S4semidirect
139. A4#123semidirect
221. C22#929semidirect

Maximal subgroups

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

subgroupindex
C22×C4×A42
C2×GL(2,ℤ/4)2
#54322
(C2×S4)⋊C42
SL(2,ℤ/4)⋊C42
#28493
#9294