Finite Group Databasebeta

← #11713710

Subgroups — #11713710

Order 768.

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

normal subgroups only

This group has 539 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×#125C211yesmaximal, characteristic, complemented
2#85289C211yesmaximal, characteristic
3#73053C214yesmaximal, complemented
4#73044C211yesmaximal, characteristic, complemented
5#25913—31nomaximal, Sylow
6C2×C4⋊S4—22no—
7C2×C4⋊S4C2212yes—
8A4×C4⋊C4—22no—
9A4×C4⋊C4C2212yes—
10C42×A4C2211yescharacteristic
11C2×A4⋊C8C2212yes—
12#5067—41nomaximal
13C22×#125D311yescharacteristic, complemented
14C2×#490—31no—
15C2×#490—31no—
16#2856—31no—
17#2025—34no—
18#1992—31no—
19#1992—31no—
20#1874—34no—
21#1749—31no—
22C22×S4—22no—
23C2×C4×A4—22no—
24C2×C4×A4D412yes—
25C2×C4×A4C2311yescharacteristic
26C4⋊S4—24no—
27C4⋊S4—44no—
28C3×#125—41no—
29#874—41no—
30A4⋊C8—22no—
31#809—44no—
32#803—41no—
33C2×#127—31no—
34C2×#125—31no—
35C2×#125—31no—
36C2×#125—32no—
37C2×#125—62no—
38C2×#121—32no—
39C2×#121—62no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4C2×D312yes—
42C22×C42C2×D311yescharacteristic
43#490—64no—
44#490—62no—
45#490—61no—
46#490—61no—
47C2×#102—34no—
48C2×#102—34no—
49C2×C22⋊C8—32no—
50C2×#97—31no—
51C2×#97—31no—
52#390—32no—
53#390—62no—
54#384—31no—
55#340—34no—
56#340—34no—
57#336—32no—
58C22×A4C2×D411yescharacteristic
59C2×S4—42no—
60C2×S4—44no—
61C2×D12—42no—
62C2×D12—82no—
63C4×A4—22no—
64C4×A4—42no—
65C4×A4#4112yes—
66C3×C4⋊C4—42no—
67C3×C4⋊C4—82no—
68C3×C42—41no—
69C2×C3⋊C8—42no—
70C22×D4—32no—
71C22×D4—62no—
72C22×D4—62no—
73C23×C4—22no—
74C23×C4#6712yes—
75C23×C4C22×D311yescharacteristic
76C22×C8—32no—
77C22×C8—32no—
78#127—62no—
79#127—61no—
80#127—61no—
81#125—34no—
82#125—32no—
83#125—31no—
84#125—62no—
85#125—61no—
86#125—64no—
87#125S411yescharacteristic, complemented
88#121—64no—
89#121—64no—
90#121—124no—
91#121—124no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—62no—
96C2×C4⋊C4—62no—
97C2×C4⋊C4—64no—
98C2×C4⋊C4—62no—
99C2×C4⋊C4—62no—
100C2×C22⋊C4—62no—
101C2×C22⋊C4—64no—
102C2×C42—31no—
103C2×C42—31no—
104C2×C42—32no—
105#102—68no—
106#102—64no—
107#102—64no—
108C22⋊C8—62no—
109C22⋊C8—62no—
110#97—62no—
111#97—61no—
112#97—61no—
113C22×D3—82no—
114C2×A4#12411yescharacteristic
115C2×A4AGL(1,ℤ/8)12yes—
116S4—82no—
117C2×C3×C4—42no—
118C2×C3×C4—41no—
119C2×C3×C4—82no—
120D12—84no—
121D12—164no—
122C3⋊C8—82no—
123C24—62no—
124C24C2×#6711yescharacteristic
125C2×D4—62no—
126C2×D4—64no—
127C2×D4—64no—
128C2×D4—62no—
129C2×D4—122no—
130C2×D4—124no—
131C2×D4—124no—
132C2×D4—122no—
133C2×D4—124no—
134C2×D4—122no—
135C2×D4—124no—
136C2×D4—122no—
137C22×C4—22no—
138C22×C4—31no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—42no—
144C22×C4—62no—
145C22×C4—62no—
146C22×C4—62no—
147C22×C4—62no—
148C22×C4—122no—
149C22×C4#23912yes—
150C2×C8—62no—
151C2×C8—62no—
152C2×C8—62no—
153C2×C8—62no—
154C2×C8—62no—
155C2×C8—62no—
156C4⋊C4—22no—
157C4⋊C4—34no—
158C4⋊C4—32no—
159C4⋊C4—62no—
160C4⋊C4—64no—
161C4⋊C4—64no—
162C4⋊C4—64no—
163C4⋊C4—62no—
164C4⋊C4—124no—
165C4⋊C4C2×S412yes—
166C22⋊C4—124no—
167C22⋊C4—128no—
168C42—32no—
169C42—31no—
170C42—61no—
171C42C2×S411yescharacteristic
172C22×C3—41no—
173C2×D3—162no—
174C2×D3—164no—
175A4#49011yescharacteristic, complemented
176C3×C4—42no—
177C3×C4—82no—
178C3×C4—162no—
179C23—31no—
180C23—31no—
181C23—32no—
182C23—122no—
183C23—124no—
184C23—122no—
185C23—124no—
186C23—122no—
187C23#94711yescharacteristic
188C23#94912yes—
189D4—124no—
190D4—124no—
191D4—244no—
192D4—244no—
193D4—242no—
194D4—244no—
195D4—242no—
196C2×C4—22no—
197C2×C4—31no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—61no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—122no—
209C2×C4—122no—
210C2×C4—122no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4—124no—
214C2×C4GL(2,ℤ/4)12yes—
215C2×C4C22×S411yescharacteristic
216C8—122no—
217C8—122no—
218C2×C3—41no—
219C2×C3—42no—
220D3—322no—
221C22—31no—
222C22—32no—
223C22—31no—
224C22—31no—
225C22—32no—
226C22—32no—
227C22—61no—
228C22—242no—
229C22—242no—
230C22—244no—
231C22—242no—
232C22—244no—
233C22#506711yescharacteristic, complemented
234C22C2×GL(2,ℤ/4)11yescharacteristic
235C4—22no—
236C4—32no—
237C4—42no—
238C4—62no—
239C4—122no—
240C4—242no—
241C4#591912yes—
242C3—41noSylow
243C2—31no—
244C2—31no—
245C2—32no—
246C2—482no—
247C2#8550811yescharacteristic
248C2#8551312yes—
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#125D3
socleC24C2×#67
prime radical#85289C2
hypercenter#125S4
hypocenterA4#490
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
1. A4×#125C2semidirect
3. #73053C2semidirect
4. #73044C2semidirect
13. C22×#125D3semidirect
87. #125S4semidirect
175. A4#490semidirect
233. C22#5067semidirect

Maximal subgroups

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

subgroupindex
A4×#1252
#852892
#730532
#730442
#259133
#50674