Finite Group Databasebeta

← #11714070

Subgroups — #11714070

Order 768.

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

normal subgroups only

This group has 431 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
1#85284C211yesmaximal, characteristic
2#85283C211yesmaximal, characteristic
3A4×#97C211yesmaximal, characteristic
4#73109C214yesmaximal
5#33053—31nomaximal, Sylow
6C2×A4⋊Q8C2212yes—
7C2×C8×A4C2212yes—
8C42×A4C2211yescharacteristic
9C4⋊SL(2,ℤ/4)—22no—
10C4⋊SL(2,ℤ/4)C2212yes—
11SL(2,ℤ/4)⋊C4—22no—
12#4703—41nomaximal
13C2×#491—31no—
14C22×#97D311yescharacteristic
15#2857—31no—
16#2851—31no—
17#1992—31no—
18#1960—31no—
19#1935—34no—
20#1892—34no—
21#1836—31no—
22C2×C4×A4D412yes—
23C2×C4×A4C2311yescharacteristic
24C2×SL(2,ℤ/4)—22no—
25C2×SL(2,ℤ/4)—22no—
26A4⋊Q8—24no—
27#870—41no—
28#869—41no—
29C8×A4—22no—
30C3×#97—41no—
31#816—44no—
32C23×C8C2×D312yes—
33C2×C4⋊Q8—31no—
34C2×#125—31no—
35C2×C22⋊Q8—32no—
36C22×C42C2×D311yescharacteristic
37#491—62no—
38#491—61no—
39#491—61no—
40C2×#103—34no—
41C2×#103—34no—
42C2×#97—31no—
43C2×#97—32no—
44C2×#97—31no—
45#391—32no—
46#389—32no—
47#389—62no—
48#387—62no—
49#385—62no—
50#384—31no—
51#382—31no—
52#380—32no—
53#380—62no—
54#340—34no—
55#340—34no—
56C22×A4C2×D411yescharacteristic
57C2×Dic6—42no—
58C4×A4—22no—
59C4×A4#4112yes—
60SL(2,ℤ/4)—42no—
61SL(2,ℤ/4)—42no—
62C2×C3×C8—42no—
63C3×C42—41no—
64C4⋊Dic3—42no—
65C4⋊Dic3—82no—
66Dic3⋊C4—82no—
67C22×Q8—32no—
68C23×C4D1212yes—
69C23×C4C22×D311yescharacteristic
70C22×C8—22no—
71C22×C8—32no—
72C22×C8—32no—
73C22×C8—62no—
74C4⋊Q8—62no—
75C4⋊Q8—61no—
76C4⋊Q8—61no—
77#125—62no—
78#125—61no—
79#125—61no—
80C22⋊Q8—64no—
81C22⋊Q8—64no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—62no—
86C2×C4⋊C4—62no—
87C2×C4⋊C4—62no—
88C2×C4⋊C4—62no—
89C2×C4⋊C4—62no—
90C2×C22⋊C4—62no—
91C2×C22⋊C4—62no—
92C2×C42—31no—
93C2×C42—32no—
94C2×C42—31no—
95#103—68no—
96#103—64no—
97#103—64no—
98#97—32no—
99#97—31no—
100#97—61no—
101#97S411yescharacteristic
102#95—62no—
103#95—64no—
104#95—62no—
105#95—62no—
106C2×A4#12411yescharacteristic
107C2×A4#13712yes—
108C2×C3×C4—42no—
109C2×C3×C4—41no—
110C2×Dic3—82no—
111C2×Dic3—82no—
112Dic6—84no—
113C3×C8—82no—
114C24C2×D1211yescharacteristic
115C2×Q8—64no—
116C2×Q8—64no—
117C2×Q8—62no—
118C2×Q8—62no—
119C22×C4—22no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—62no—
126C22×C4—62no—
127C22×C4—62no—
128C22×C4—62no—
129C22×C4—62no—
130C22×C4#23512yes—
131C2×C8—34no—
132C2×C8—32no—
133C2×C8—62no—
134C2×C8—62no—
135C2×C8—62no—
136C2×C8C2×S412yes—
137C4⋊C4—64no—
138C4⋊C4—64no—
139C4⋊C4—62no—
140C4⋊C4—62no—
141C4⋊C4—124no—
142C4⋊C4—122no—
143C4⋊C4—122no—
144C4⋊C4—124no—
145C4⋊C4—124no—
146C4⋊C4—122no—
147C4⋊C4—122no—
148C22⋊C4—122no—
149C22⋊C4—122no—
150C22⋊C4—122no—
151C22⋊C4—122no—
152C42—32no—
153C42—31no—
154C42—61no—
155C42C2×S411yescharacteristic
156C22×C3—41no—
157A4#49111yescharacteristic, complemented
158C3×C4—42no—
159C3×C4—82no—
160Dic3—162no—
161Dic3—162no—
162C23—31no—
163C23—31no—
164C23—32no—
165C23#87511yescharacteristic
166C23#90912yes—
167Q8—124no—
168Q8—124no—
169C2×C4—31no—
170C2×C4—32no—
171C2×C4—32no—
172C2×C4—32no—
173C2×C4—32no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—61no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—122no—
180C2×C4—122no—
181C2×C4—122no—
182C2×C4—122no—
183C2×C4—122no—
184C2×C4—122no—
185C2×C4—124no—
186C2×C4—122no—
187C2×C4—122no—
188C2×C4—122no—
189C2×C4—122no—
190C2×C4C4⋊S412yes—
191C2×C4C22×S411yescharacteristic
192C8—22no—
193C8—62no—
194C2×C3—41no—
195C2×C3—42no—
196C22—31no—
197C22—32no—
198C22—31no—
199C22—31no—
200C22—32no—
201C22—32no—
202C22—61no—
203C22#470311yescharacteristic, complemented
204C22C2×C4⋊S411yescharacteristic
205C4—22no—
206C4—32no—
207C4—62no—
208C4—242no—
209C4—242no—
210C4—242no—
211C4—242no—
212C4#591212yes—
213C3—41noSylow
214C2—31no—
215C2—32no—
216C2—31no—
217C2#8529011yescharacteristic
218C2#8544412yes—
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#97D3
socleC24C2×D12
prime radicalC22×A4C2×D4
hypercenter#97S4
hypocenterA4#491
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
157. A4#491semidirect
203. C22#4703semidirect

Maximal subgroups

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

subgroupindex
#852842
#852832
A4×#972
#731092
#330533
#47034