Finite Group Databasebeta

← #11713619

Subgroups — #11713619

Order 768.

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

normal subgroups only

This group has 647 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
2C2×#5416C214yesmaximal
3#85283C211yesmaximal, characteristic
4C4×A4⋊C8C211yesmaximal, characteristic
5#25722—31nomaximal, Sylow
6C2×Q8×A4C2212yes—
7C2×A4⋊Q8C2212yes—
8A4×C4⋊C4—22no—
9C42×A4C2211yescharacteristic
10#5416—28no—
11C4⋊SL(2,ℤ/4)—22no—
12C2×A4⋊C8C2212yes—
13#5092—41nomaximal
14C22×C4⋊Q8D311yescharacteristic
15C2×#494—31no—
16C2×#494—31no—
17C2×#451—34no—
18#2857—31no—
19#2016—34no—
20#1956—31no—
21#1956—31no—
22C4×C22⋊C8—31no—
23Q8×A4—24no—
24C2×C4×A4—22no—
25C2×C4×A4D412yes—
26C2×C4×A4C2311yescharacteristic
27C2×SL(2,ℤ/4)—22no—
28A4⋊Q8—24no—
29C3×C4⋊Q8—41no—
30C2×#216—44no—
31#869—41no—
32A4⋊C8D414yes—
33C4×C3⋊C8—41no—
34C23×Q8C2×D312yes—
35C22×Q16—34no—
36C22×Q16—34no—
37C2×C4⋊Q8—31no—
38C2×C4⋊Q8—31no—
39C2×C4⋊Q8—32no—
40C2×C4⋊Q8—31no—
41C2×C4⋊Q8—62no—
42C2×C22⋊Q8—32no—
43C22×C4⋊C4—22no—
44C22×C42C2×D311yescharacteristic
45#494—64no—
46#494—62no—
47#494—61no—
48#494—61no—
49#451—68no—
50#451—68no—
51C2×#103—34no—
52C2×#103—34no—
53C2×C22⋊C8—32no—
54C2×C4×C8—31no—
55C2×C4×C8—31no—
56#391—32no—
57#389—62no—
58#384—31no—
59#380—62no—
60#336—32no—
61C22×A4C2×D411yescharacteristic
62C2×C3×Q8—42no—
63C2×Dic6—42no—
64C4×A4—42no—
65C4×A4Q1614yescomplemented
66C4×A4C2×D412yes—
67SL(2,ℤ/4)—42no—
68C3×C4⋊C4—82no—
69C3×C42—41no—
70#216—88no—
71C4⋊Dic3—82no—
72C2×C3⋊C8—42no—
73C22×Q8—24no—
74C22×Q8—32no—
75C22×Q8—32no—
76C22×Q8—32no—
77C22×Q8—62no—
78C22×Q8—64no—
79C23×C4—22no—
80C23×C4#6712yes—
81C23×C4C22×D311yescharacteristic
82C2×Q16—68no—
83C2×Q16—68no—
84C2×Q16—68no—
85C2×Q16—64no—
86C2×Q16—64no—
87C2×Q16—124no—
88C2×Q16—124no—
89C22×C8—32no—
90C22×C8—32no—
91C4⋊Q8—34no—
92C4⋊Q8—32no—
93C4⋊Q8—31no—
94C4⋊Q8—62no—
95C4⋊Q8—61no—
96C4⋊Q8—62no—
97C4⋊Q8—61no—
98C4⋊Q8—64no—
99C4⋊Q8—61no—
100C4⋊Q8S411yescharacteristic
101C22⋊Q8—64no—
102C22⋊Q8—64no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—62no—
105C2×C4⋊C4—64no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—62no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—62no—
110C2×C22⋊C4—62no—
111C2×C42—31no—
112C2×C42—31no—
113C2×C42—32no—
114#103—68no—
115#103—68no—
116C22⋊C8—34no—
117C22⋊C8—34no—
118C4×C8—62no—
119C4×C8—61no—
120C4×C8—61no—
121#95—64no—
122C2×A4#12711yescharacteristic
123C2×A4C2×Q1612yes—
124C3×Q8—84no—
125C2×C3×C4—42no—
126C2×C3×C4—41no—
127C2×C3×C4—82no—
128C2×Dic3—82no—
129Dic6—84no—
130C3⋊C8—44no—
131C24C2×#6711yescharacteristic
132C2×Q8—34no—
133C2×Q8—32no—
134C2×Q8—62no—
135C2×Q8—64no—
136C2×Q8—64no—
137C2×Q8—64no—
138C2×Q8—64no—
139C2×Q8—64no—
140C2×Q8—62no—
141C2×Q8—64no—
142C2×Q8—62no—
143C2×Q8—124no—
144C2×Q8—124no—
145C2×Q8C2×S412yes—
146C22×C4—32no—
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—32no—
150C22×C4—31no—
151C22×C4—34no—
152C22×C4—42no—
153C22×C4—62no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4—62no—
157C22×C4—122no—
158C22×C4#21614yescomplemented
159C22×C4D3×D412yes—
160Q16—128no—
161Q16—1216no—
162Q16—128no—
163C2×C8—34no—
164C2×C8—34no—
165C2×C8—62no—
166C2×C8—62no—
167C2×C8—62no—
168C2×C8—62no—
169C4⋊C4—22no—
170C4⋊C4—64no—
171C4⋊C4—64no—
172C4⋊C4—64no—
173C4⋊C4—62no—
174C4⋊C4—124no—
175C4⋊C4—124no—
176C4⋊C4—122no—
177C4⋊C4—122no—
178C22⋊C4—122no—
179C22⋊C4—122no—
180C42—32no—
181C42—31no—
182C42—61no—
183C42C2×S411yescharacteristic
184C22×C3—41no—
185A4#49411yescharacteristic, complemented
186C3×C4—42no—
187C3×C4—44no—
188C3×C4—162no—
189Dic3—162no—
190C23—31no—
191C23—31no—
192C23—32no—
193C23#94011yescharacteristic
194C23C2×#21612yes—
195Q8—24no—
196Q8—68no—
197Q8—64no—
198Q8—124no—
199Q8—124no—
200Q8—124no—
201C2×C4—22no—
202C2×C4—32no—
203C2×C4—31no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—34no—
207C2×C4—34no—
208C2×C4—32no—
209C2×C4—34no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—62no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—124no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4GL(2,ℤ/4)12yes—
223C2×C4C22×S411yescharacteristic
224C8—64no—
225C8—64no—
226C2×C3—41no—
227C2×C3—42no—
228C22—31no—
229C22—32no—
230C22—31no—
231C22—31no—
232C22—32no—
233C22—32no—
234C22—61no—
235C22#509211yescharacteristic, complemented
236C22C2×GL(2,ℤ/4)11yescharacteristic
237C4—32no—
238C4—34no—
239C4—42no—
240C4—122no—
241C4—242no—
242C4—242no—
243C4#541614yescomplemented
244C4D4×S412yes—
245C3—41noSylow
246C2—31no—
247C2—31no—
248C2—32no—
249C2#8549611yescharacteristic
250C2C2×#541612yes—
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#494
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
65. C4×A4Q16semidirect
158. C22×C4#216semidirect
185. A4#494semidirect
235. C22#5092semidirect
243. C4#5416semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
C2×#54162
#852832
C4×A4⋊C82
#257223
#50924