Finite Group Databasebeta

← #11713688

Subgroups — #11713688

Order 768.

483 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 53 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⋊Q8C211yesmaximal, characteristic
2#85284C211yesmaximal, characteristic
3#73147C214yesmaximal
4#73044C211yesmaximal, characteristic
5#25861—31nomaximal, Sylow
6C2×Q8×A4C2212yes—
7A4×C4⋊C4—22no—
8C42×A4C2211yescharacteristic
9C4⋊SL(2,ℤ/4)C2212yes—
10SL(2,ℤ/4)⋊C4—22no—
11C2×A4⋊C8C2212yes—
12#5082—41nomaximal
13C22×C4⋊Q8D311yescharacteristic
14C2×#491—31no—
15C2×#491—31no—
16#2851—31no—
17#2043—34no—
18#1960—31no—
19#1960—31no—
20#1787—34no—
21#1749—31no—
22Q8×A4—24no—
23C2×C4×A4—22no—
24C2×C4×A4D412yes—
25C2×C4×A4C2311yescharacteristic
26C2×SL(2,ℤ/4)—22no—
27C3×C4⋊Q8—41no—
28#870—41no—
29A4⋊C8—22no—
30#835—44no—
31#803—41no—
32C23×Q8C2×D312yes—
33C2×C4⋊Q8—31no—
34C2×C4⋊Q8—31no—
35C2×C4⋊Q8—32no—
36C2×C4⋊Q8—62no—
37C2×#125—31no—
38C22×C4⋊C4—22no—
39C22×C42C2×D311yescharacteristic
40#491—64no—
41#491—62no—
42#491—61no—
43#491—61no—
44C2×#103—34no—
45C2×#103—34no—
46C2×#103—34no—
47C2×#103—34no—
48C2×C22⋊C8—32no—
49C2×#97—31no—
50C2×#97—31no—
51#389—32no—
52#387—62no—
53#385—62no—
54#382—31no—
55#380—32no—
56#336—32no—
57C22×A4C2×D411yescharacteristic
58C2×C3×Q8—42no—
59C4×A4—22no—
60C4×A4—42no—
61C4×A4#4112yes—
62SL(2,ℤ/4)—42no—
63C3×C4⋊C4—82no—
64C3×C42—41no—
65C4⋊Dic3—42no—
66Dic3⋊C4—82no—
67C2×C3⋊C8—42no—
68C22×Q8—24no—
69C22×Q8—32no—
70C22×Q8—32no—
71C22×Q8—62no—
72C22×Q8—64no—
73C23×C4—22no—
74C23×C4#6712yes—
75C23×C4C22×D311yescharacteristic
76C22×C8—32no—
77C22×C8—32no—
78C4⋊Q8—34no—
79C4⋊Q8—32no—
80C4⋊Q8—31no—
81C4⋊Q8—62no—
82C4⋊Q8—61no—
83C4⋊Q8—64no—
84C4⋊Q8S411yescharacteristic
85#125—62no—
86#125—61no—
87#125—61no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—62no—
91C2×C4⋊C4—64no—
92C2×C4⋊C4—62no—
93C2×C4⋊C4—62no—
94C2×C4⋊C4—62no—
95C2×C22⋊C4—62no—
96C2×C42—31no—
97C2×C42—31no—
98C2×C42—32no—
99#103—68no—
100#103—64no—
101#103—64no—
102#103—124no—
103#103—124no—
104C22⋊C8—62no—
105C22⋊C8—62no—
106#97—62no—
107#97—61no—
108#97—61no—
109#95—62no—
110#95—62no—
111#95—62no—
112C2×A4#12411yescharacteristic
113C2×A4#13712yes—
114C3×Q8—84no—
115C2×C3×C4—42no—
116C2×C3×C4—41no—
117C2×C3×C4—82no—
118C2×Dic3—82no—
119C3⋊C8—82no—
120C24C2×#6711yescharacteristic
121C2×Q8—34no—
122C2×Q8—32no—
123C2×Q8—62no—
124C2×Q8—64no—
125C2×Q8—64no—
126C2×Q8—64no—
127C2×Q8—64no—
128C2×Q8—124no—
129C2×Q8—124no—
130C2×Q8C2×S412yes—
131C22×C4—22no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—32no—
137C22×C4—42no—
138C22×C4—62no—
139C22×C4—62no—
140C22×C4—62no—
141C22×C4—62no—
142C22×C4—62no—
143C22×C4—122no—
144C22×C4#23712yes—
145C2×C8—62no—
146C2×C8—62no—
147C2×C8—62no—
148C2×C8—62no—
149C2×C8—62no—
150C2×C8—62no—
151C4⋊C4—22no—
152C4⋊C4—64no—
153C4⋊C4—64no—
154C4⋊C4—62no—
155C4⋊C4—64no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—124no—
159C4⋊C4—122no—
160C4⋊C4—122no—
161C22⋊C4—122no—
162C22⋊C4—122no—
163C42—32no—
164C42—31no—
165C42—61no—
166C42C2×S411yescharacteristic
167C22×C3—41no—
168A4#49111yescharacteristic, complemented
169C3×C4—42no—
170C3×C4—82no—
171C3×C4—162no—
172Dic3—162no—
173C23—31no—
174C23—32no—
175C23—31no—
176C23#93611yescharacteristic
177C23#94212yes—
178Q8—24no—
179Q8—68no—
180Q8—64no—
181Q8—124no—
182C2×C4—22no—
183C2×C4—31no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—32no—
187C2×C4—34no—
188C2×C4—32no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—61no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4—122no—
201C2×C4—122no—
202C2×C4—122no—
203C2×C4—122no—
204C2×C4GL(2,ℤ/4)12yes—
205C2×C4C22×S411yescharacteristic
206C8—122no—
207C8—122no—
208C2×C3—41no—
209C2×C3—42no—
210C22—31no—
211C22—32no—
212C22—31no—
213C22—31no—
214C22—32no—
215C22—32no—
216C22—61no—
217C22#508211yescharacteristic, complemented
218C22C2×GL(2,ℤ/4)11yescharacteristic
219C4—22no—
220C4—32no—
221C4—42no—
222C4—62no—
223C4—122no—
224C4—242no—
225C4—242no—
226C4#591412yes—
227C3—41noSylow
228C2—32no—
229C2—31no—
230C2—31no—
231C2#8549211yescharacteristic
232C2#8549912yes—
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4⋊Q8D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenterC4⋊Q8S4
hypocenterA4#491
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
168. A4#491semidirect
217. C22#5082semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#852842
#731472
#730442
#258613
#50824