Finite Group Databasebeta

← #11713666

Subgroups — #11713666

Order 768.

429 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 57 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×#125C211yesmaximal, characteristic
2#85284C211yesmaximal, characteristic
3#73051C212yesmaximal
4#73050C212yesmaximal
5C4×A4⋊C8C211yesmaximal, characteristic
6#25815—31nomaximal, Sylow
7A4×C4⋊C4—22no—
8A4×C4⋊C4C2212yes—
9C42×A4C2211yescharacteristic
10C4⋊SL(2,ℤ/4)C2212yes—
11SL(2,ℤ/4)⋊C4—22no—
12C2×A4⋊C8C2211yescharacteristic
13C2×A4⋊C8C2211yescharacteristic
14#5065—41nomaximal
15C22×#125D311yescharacteristic
16C2×#499—31no—
17C2×#499—31no—
18#2851—31no—
19#2059—32no—
20#2059—32no—
21#1991—31no—
22#1991—31no—
23#1853—32no—
24#1852—32no—
25C4×C22⋊C8—31no—
26C2×C4×A4—22no—
27C2×C4×A4D412yes—
28C2×C4×A4C2311yescharacteristic
29C2×SL(2,ℤ/4)—22no—
30C3×#125—41no—
31#870—41no—
32A4⋊C8Q812yes—
33A4⋊C8Q812yes—
34#808—42no—
35#807—42no—
36C4×C3⋊C8—41no—
37C2×#125—31no—
38C2×#125—31no—
39C2×#125—32no—
40C2×#125—31no—
41C2×#125—62no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2×D312yes—
44C22×C42C2×D311yescharacteristic
45#499—64no—
46#499—62no—
47#499—61no—
48#499—61no—
49C2×#107—32no—
50C2×#107—32no—
51C2×#106—32no—
52C2×#106—32no—
53C2×C22⋊C8—31no—
54C2×C22⋊C8—31no—
55C2×C4×C8—31no—
56C2×C4×C8—31no—
57#389—32no—
58#387—62no—
59#385—62no—
60#382—31no—
61#380—32no—
62#340—32no—
63#340—32no—
64#340—32no—
65#340—32no—
66#336—31no—
67#336—31no—
68C22×A4C2×D411yescharacteristic
69C4×A4—22no—
70C4×A4—42no—
71C4×A4C2×Q811yescharacteristic
72C4×A4C2×Q811yescharacteristic
73SL(2,ℤ/4)—42no—
74C3×C4⋊C4—42no—
75C3×C4⋊C4—82no—
76C3×C42—41no—
77C4⋊Dic3—42no—
78Dic3⋊C4—82no—
79C2×C3⋊C8—41no—
80C2×C3⋊C8—41no—
81C23×C4—22no—
82C23×C4#6712yes—
83C23×C4C22×D311yescharacteristic
84C22×C8—31no—
85C22×C8—31no—
86C22×C8—31no—
87C22×C8—31no—
88#125—34no—
89#125—32no—
90#125—31no—
91#125—62no—
92#125—61no—
93#125—62no—
94#125—61no—
95#125—64no—
96#125—61no—
97#125S411yescharacteristic
98C2×C4⋊C4—32no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—62no—
103C2×C4⋊C4—62no—
104C2×C4⋊C4—64no—
105C2×C4⋊C4—62no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—62no—
108C2×C22⋊C4—62no—
109C2×C42—31no—
110C2×C42—31no—
111C2×C42—32no—
112#107—64no—
113#107—62no—
114#107—62no—
115#106—64no—
116#106—62no—
117#106—62no—
118C22⋊C8—32no—
119C22⋊C8—32no—
120C22⋊C8—32no—
121C22⋊C8—32no—
122C4×C8—62no—
123C4×C8—61no—
124C4×C8—61no—
125#95—62no—
126#95—62no—
127#95—62no—
128C2×A4C4⋊Q811yescharacteristic
129C2×A4#13512yes—
130C2×C3×C4—42no—
131C2×C3×C4—41no—
132C2×C3×C4—82no—
133C2×Dic3—82no—
134C3⋊C8—42no—
135C3⋊C8—42no—
136C24C2×#6711yescharacteristic
137C22×C4—22no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—42no—
145C22×C4—62no—
146C22×C4—62no—
147C22×C4—62no—
148C22×C4—62no—
149C22×C4—62no—
150C22×C4—122no—
151C22×C4D3×Q811yescharacteristic
152C22×C4D3×Q811yescharacteristic
153C2×C8—32no—
154C2×C8—32no—
155C2×C8—32no—
156C2×C8—32no—
157C2×C8—61no—
158C2×C8—61no—
159C2×C8—61no—
160C2×C8—61no—
161C2×C8—61no—
162C2×C8—61no—
163C2×C8—61no—
164C2×C8—61no—
165C4⋊C4—22no—
166C4⋊C4—34no—
167C4⋊C4—32no—
168C4⋊C4—62no—
169C4⋊C4—64no—
170C4⋊C4—64no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—62no—
175C4⋊C4—62no—
176C4⋊C4—122no—
177C4⋊C4—124no—
178C4⋊C4—122no—
179C4⋊C4C2×S412yes—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C42—32no—
183C42—31no—
184C42—61no—
185C42C2×S411yescharacteristic
186C22×C3—41no—
187A4#49911yescharacteristic, complemented
188C3×C4—41no—
189C3×C4—41no—
190C3×C4—82no—
191C3×C4—162no—
192Dic3—162no—
193C23—31no—
194C23—32no—
195C23—31no—
196C23#94411yescharacteristic
197C23#95012yes—
198C2×C4—22no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—32no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4GL(2,ℤ/4)12yes—
224C2×C4C22×S411yescharacteristic
225C8—62no—
226C8—62no—
227C8—62no—
228C8—62no—
229C2×C3—42no—
230C2×C3—41no—
231C22—31no—
232C22—32no—
233C22—32no—
234C22—32no—
235C22—31no—
236C22—31no—
237C22—61no—
238C22#506511yescharacteristic, complemented
239C22C2×GL(2,ℤ/4)11yescharacteristic
240C4—22no—
241C4—31no—
242C4—31no—
243C4—42no—
244C4—62no—
245C4—122no—
246C4—242no—
247C4—242no—
248C4Q8×S411yescharacteristic
249C4Q8×S411yescharacteristic
250C3—41noSylow
251C2—32no—
252C2—31no—
253C2—31no—
254C2#8550211yescharacteristic
255C2#8551412yes—
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#125D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenter#125S4
hypocenterA4#499
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
187. A4#499semidirect
238. C22#5065semidirect

Maximal subgroups

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

subgroupindex
A4×#1252
#852842
#730512
#730502
C4×A4⋊C82
#258153
#50654