Finite Group Databasebeta

← #11713617

Subgroups — #11713617

Order 768.

737 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 69 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, complemented
2C2×#5417C214yesmaximal, complemented
3#85289C211yesmaximal, characteristic
4C4×A4⋊C8C211yesmaximal, characteristic, complemented
5#25718—31nomaximal, Sylow
6C2×Q8×A4C2212yes—
7C2×C4⋊S4—22no—
8C2×C4⋊S4C2212yes—
9A4×C4⋊C4—22no—
10C42×A4C2211yescharacteristic
11#5417—28no—
12C2×A4⋊C8C2212yes—
13#5085—41nomaximal
14C22×C4⋊Q8D311yescharacteristic, complemented
15C2×#492—31no—
16C2×#492—31no—
17C2×#448—34no—
18#2856—31no—
19#2013—34no—
20#1956—31no—
21#1956—31no—
22C4×C22⋊C8—31no—
23C22×S4—22no—
24Q8×A4—24no—
25C2×C4×A4—22no—
26C2×C4×A4D412yes—
27C2×C4×A4C2311yescharacteristic
28C4⋊S4—24no—
29C4⋊S4—44no—
30C3×C4⋊Q8—41no—
31C2×#215—44no—
32#874—41no—
33A4⋊C8D414yescomplemented
34C4×C3⋊C8—41no—
35C23×Q8C2×D312yes—
36C22×SD16—34no—
37C22×SD16—34no—
38C2×C4⋊Q8—31no—
39C2×C4⋊Q8—31no—
40C2×C4⋊Q8—32no—
41C2×C4⋊Q8—62no—
42C2×#127—31no—
43C2×#121—32no—
44C2×#121—62no—
45C22×C4⋊C4—22no—
46C22×C42C2×D311yescharacteristic
47#492—64no—
48#492—62no—
49#492—61no—
50#492—61no—
51#448—68no—
52#448—68no—
53C2×#103—34no—
54C2×#103—34no—
55C2×C22⋊C8—32no—
56C2×C4×C8—31no—
57C2×C4×C8—31no—
58#390—32no—
59#390—62no—
60#384—31no—
61#336—32no—
62C22×A4C2×D411yescharacteristic
63C2×S4—42no—
64C2×S4—44no—
65C2×C3×Q8—42no—
66C2×D12—42no—
67C2×D12—82no—
68C4×A4—42no—
69C4×A4SD1614yescomplemented
70C4×A4C2×D412yes—
71C3×C4⋊C4—82no—
72C3×C42—41no—
73#215—88no—
74C2×C3⋊C8—42no—
75C22×Q8—24no—
76C22×Q8—32no—
77C22×Q8—32no—
78C22×Q8—62no—
79C22×Q8—64no—
80C22×D4—32no—
81C22×D4—62no—
82C22×D4—62no—
83C23×C4—22no—
84C23×C4#6712yes—
85C23×C4C22×D311yescharacteristic
86C2×SD16—68no—
87C2×SD16—68no—
88C2×SD16—68no—
89C2×SD16—64no—
90C2×SD16—64no—
91C2×SD16—124no—
92C2×SD16—124no—
93C22×C8—32no—
94C22×C8—32no—
95C4⋊Q8—34no—
96C4⋊Q8—32no—
97C4⋊Q8—31no—
98C4⋊Q8—62no—
99C4⋊Q8—61no—
100C4⋊Q8—64no—
101C4⋊Q8S411yescharacteristic, complemented
102#127—62no—
103#127—61no—
104#127—61no—
105#121—64no—
106#121—64no—
107#121—124no—
108#121—124no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—64no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4—62no—
114C2×C22⋊C4—62no—
115C2×C22⋊C4—64no—
116C2×C42—31no—
117C2×C42—31no—
118C2×C42—32no—
119#103—68no—
120#103—68no—
121C22⋊C8—34no—
122C22⋊C8—34no—
123C4×C8—62no—
124C4×C8—61no—
125C4×C8—61no—
126C22×D3—82no—
127C2×A4#12711yescharacteristic
128C2×A4C2×SD1612yes—
129S4—82no—
130C3×Q8—84no—
131C2×C3×C4—42no—
132C2×C3×C4—41no—
133C2×C3×C4—82no—
134D12—84no—
135D12—164no—
136C3⋊C8—44no—
137C24—62no—
138C24C2×#6711yescharacteristic
139C2×Q8—34no—
140C2×Q8—32no—
141C2×Q8—62no—
142C2×Q8—64no—
143C2×Q8—64no—
144C2×Q8—64no—
145C2×Q8—64no—
146C2×Q8—124no—
147C2×Q8—124no—
148C2×Q8C2×S412yes—
149C2×D4—64no—
150C2×D4—62no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—124no—
154C2×D4—122no—
155C2×D4—124no—
156C2×D4—122no—
157C2×D4—124no—
158C2×D4—122no—
159C2×D4—124no—
160C2×D4—122no—
161C22×C4—32no—
162C22×C4—31no—
163C22×C4—32no—
164C22×C4—32no—
165C22×C4—31no—
166C22×C4—34no—
167C22×C4—42no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4—122no—
172C22×C4#21514yescomplemented
173C22×C4D3×D412yes—
174SD16—128no—
175SD16—1216no—
176SD16—128no—
177C2×C8—34no—
178C2×C8—34no—
179C2×C8—62no—
180C2×C8—62no—
181C2×C8—62no—
182C2×C8—62no—
183C4⋊C4—22no—
184C4⋊C4—64no—
185C4⋊C4—64no—
186C4⋊C4—64no—
187C4⋊C4—62no—
188C4⋊C4—124no—
189C22⋊C4—124no—
190C22⋊C4—128no—
191C42—32no—
192C42—31no—
193C42—61no—
194C42C2×S411yescharacteristic
195C22×C3—41no—
196C2×D3—162no—
197C2×D3—164no—
198A4#49211yescharacteristic, complemented
199C3×C4—42no—
200C3×C4—44no—
201C3×C4—162no—
202C23—31no—
203C23—31no—
204C23—32no—
205C23—122no—
206C23—124no—
207C23—122no—
208C23—124no—
209C23—122no—
210C23#94011yescharacteristic
211C23C2×#21512yes—
212Q8—24no—
213Q8—68no—
214Q8—64no—
215Q8—124no—
216D4—124no—
217D4—124no—
218D4—244no—
219D4—242no—
220D4—244no—
221D4—242no—
222D4—244no—
223C2×C4—22no—
224C2×C4—32no—
225C2×C4—31no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—34no—
229C2×C4—34no—
230C2×C4—34no—
231C2×C4—32no—
232C2×C4—62no—
233C2×C4—61no—
234C2×C4—62no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—122no—
238C2×C4—122no—
239C2×C4—122no—
240C2×C4—124no—
241C2×C4GL(2,ℤ/4)12yes—
242C2×C4C22×S411yescharacteristic
243C8—64no—
244C8—64no—
245C2×C3—41no—
246C2×C3—42no—
247D3—322no—
248C22—31no—
249C22—32no—
250C22—31no—
251C22—31no—
252C22—32no—
253C22—32no—
254C22—61no—
255C22—242no—
256C22—242no—
257C22—244no—
258C22—242no—
259C22—244no—
260C22#508511yescharacteristic, complemented
261C22C2×GL(2,ℤ/4)11yescharacteristic
262C4—32no—
263C4—34no—
264C4—42no—
265C4—122no—
266C4—242no—
267C4#541714yescomplemented
268C4D4×S412yes—
269C3—41noSylow
270C2—31no—
271C2—31no—
272C2—32no—
273C2—482no—
274C2#8549611yescharacteristic
275C2C2×#541712yes—
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4⋊Q8D3
socleC24C2×#67
prime radical#85289C2
hypercenterC4⋊Q8S4
hypocenterA4#492
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
1. A4×C4⋊Q8C2semidirect
2. C2×#5417C2semidirect
4. C4×A4⋊C8C2semidirect
14. C22×C4⋊Q8D3semidirect
33. A4⋊C8D4semidirect
69. C4×A4SD16semidirect
101. C4⋊Q8S4semidirect
172. C22×C4#215semidirect
198. A4#492semidirect
260. C22#5085semidirect
267. C4#5417semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
C2×#54172
#852892
C4×A4⋊C82
#257183
#50854