Finite Group Databasebeta

← #11713709

Subgroups — #11713709

Order 768.

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

normal subgroups only

This group has 449 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
2#85283C211yesmaximal, characteristic
3#73052C214yesmaximal
4#73044C211yesmaximal, characteristic
5#25912—31nomaximal, Sylow
6C2×A4⋊Q8C2212yes—
7A4×C4⋊C4—22no—
8A4×C4⋊C4C2212yes—
9C42×A4C2211yescharacteristic
10C4⋊SL(2,ℤ/4)—22no—
11C2×A4⋊C8C2212yes—
12#5069—41nomaximal
13C22×#125D311yescharacteristic
14C2×#491—31no—
15C2×#491—31no—
16#2857—31no—
17#2026—34no—
18#1992—31no—
19#1992—31no—
20#1872—34no—
21#1749—31no—
22C2×C4×A4—22no—
23C2×C4×A4D412yes—
24C2×C4×A4C2311yescharacteristic
25C2×SL(2,ℤ/4)—22no—
26A4⋊Q8—24no—
27C3×#125—41no—
28#869—41no—
29A4⋊C8—22no—
30#810—44no—
31#803—41no—
32C2×C4⋊Q8—31no—
33C2×#125—31no—
34C2×#125—31no—
35C2×#125—32no—
36C2×#125—62no—
37C2×C22⋊Q8—32no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×D312yes—
40C22×C42C2×D311yescharacteristic
41#491—64no—
42#491—62no—
43#491—61no—
44#491—61no—
45C2×#103—34no—
46C2×#103—34no—
47C2×C22⋊C8—32no—
48C2×#97—31no—
49C2×#97—31no—
50#391—32no—
51#389—62no—
52#384—31no—
53#380—62no—
54#340—34no—
55#340—34no—
56#336—32no—
57C22×A4C2×D411yescharacteristic
58C2×Dic6—42no—
59C4×A4—22no—
60C4×A4—42no—
61C4×A4#4112yes—
62SL(2,ℤ/4)—42no—
63C3×C4⋊C4—42no—
64C3×C4⋊C4—82no—
65C3×C42—41no—
66C4⋊Dic3—82no—
67C2×C3⋊C8—42no—
68C22×Q8—32no—
69C23×C4—22no—
70C23×C4#6712yes—
71C23×C4C22×D311yescharacteristic
72C22×C8—32no—
73C22×C8—32no—
74C4⋊Q8—62no—
75C4⋊Q8—61no—
76C4⋊Q8—61no—
77#125—34no—
78#125—32no—
79#125—31no—
80#125—62no—
81#125—61no—
82#125—64no—
83#125S411yescharacteristic
84C22⋊Q8—64no—
85C22⋊Q8—64no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—62no—
90C2×C4⋊C4—62no—
91C2×C4⋊C4—64no—
92C2×C4⋊C4—62no—
93C2×C4⋊C4—62no—
94C2×C4⋊C4—62no—
95C2×C4⋊C4—62no—
96C2×C22⋊C4—62no—
97C2×C42—31no—
98C2×C42—31no—
99C2×C42—32no—
100#103—68no—
101#103—64no—
102#103—64no—
103C22⋊C8—62no—
104C22⋊C8—62no—
105#97—62no—
106#97—61no—
107#97—61no—
108#95—64no—
109C2×A4#12411yescharacteristic
110C2×A4#13712yes—
111C2×C3×C4—42no—
112C2×C3×C4—41no—
113C2×C3×C4—82no—
114C2×Dic3—82no—
115Dic6—84no—
116C3⋊C8—82no—
117C24C2×#6711yescharacteristic
118C2×Q8—62no—
119C2×Q8—64no—
120C2×Q8—64no—
121C2×Q8—62no—
122C22×C4—22no—
123C22×C4—31no—
124C22×C4—32no—
125C22×C4—31no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—42no—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4—62no—
132C22×C4—62no—
133C22×C4—62no—
134C22×C4—122no—
135C22×C4#23912yes—
136C2×C8—62no—
137C2×C8—62no—
138C2×C8—62no—
139C2×C8—62no—
140C2×C8—62no—
141C2×C8—62no—
142C4⋊C4—22no—
143C4⋊C4—34no—
144C4⋊C4—32no—
145C4⋊C4—62no—
146C4⋊C4—64no—
147C4⋊C4—64no—
148C4⋊C4—64no—
149C4⋊C4—62no—
150C4⋊C4—124no—
151C4⋊C4—122no—
152C4⋊C4—124no—
153C4⋊C4—122no—
154C4⋊C4C2×S412yes—
155C22⋊C4—122no—
156C22⋊C4—122no—
157C42—32no—
158C42—31no—
159C42—61no—
160C42C2×S411yescharacteristic
161C22×C3—41no—
162A4#49111yescharacteristic, complemented
163C3×C4—42no—
164C3×C4—82no—
165C3×C4—162no—
166Dic3—162no—
167C23—31no—
168C23—31no—
169C23—32no—
170C23#94711yescharacteristic
171C23#95112yes—
172Q8—124no—
173Q8—124no—
174C2×C4—22no—
175C2×C4—31no—
176C2×C4—32no—
177C2×C4—32no—
178C2×C4—32no—
179C2×C4—32no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—61no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—122no—
187C2×C4—122no—
188C2×C4—122no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—124no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4GL(2,ℤ/4)12yes—
196C2×C4C22×S411yescharacteristic
197C8—122no—
198C8—122no—
199C2×C3—41no—
200C2×C3—42no—
201C22—31no—
202C22—32no—
203C22—31no—
204C22—31no—
205C22—32no—
206C22—32no—
207C22—61no—
208C22#506911yescharacteristic, complemented
209C22C2×GL(2,ℤ/4)11yescharacteristic
210C4—22no—
211C4—32no—
212C4—42no—
213C4—62no—
214C4—122no—
215C4—242no—
216C4—242no—
217C4#591912yes—
218C3—41noSylow
219C2—31no—
220C2—31no—
221C2—32no—
222C2#8550811yescharacteristic
223C2#8551212yes—
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#491
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
162. A4#491semidirect
208. C22#5069semidirect

Maximal subgroups

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

subgroupindex
A4×#1252
#852832
#730522
#730442
#259123
#50694