Finite Group Databasebeta

← #11712648

Subgroups — #11712648

Order 768.

557 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×A4×M16C211yesmaximal, characteristic
2C2×#5409C212yesmaximal
3#73105C214yesmaximal, complemented
4#20667—31nomaximal, Sylow
5C22×C4×A4C2211yescharacteristic
6A4×M16C414yes—
7C2×C8×A4C2212yes—
8#5409—22no—
9#5409C2214yes—
10C2×A4⋊C8—22no—
11#5125—41nomaximal
12C23×M16D311yescharacteristic
13C2×#430—31no—
14C2×#430—31no—
15C2×#407—32no—
16#1811—34no—
17#1810—34no—
18#1738—32no—
19C23×A4Q811yescharacteristic
20C2×C4×A4C2×C412yes—
21C2×C4×A4D412yes—
22C2×C4×A4Q811yescharacteristic
23C2×C4×A4C2311yescharacteristic
24C2×C3×M16—41no—
25C2×#208—42no—
26C8×A4C2×C414yes—
27A4⋊C8—24no—
28#819—44no—
29C24×C4C2×D311yescharacteristic
30C22×M16—31no—
31C22×M16—32no—
32C22×M16—31no—
33C22×M16—34no—
34C22×M16—32no—
35C22×M16—32no—
36C22×M16Dic314yes—
37C23×C8C2×D312yes—
38#430—61no—
39#430—64no—
40#430—62no—
41#430—61no—
42C2×#108—34no—
43C2×#108—34no—
44#407—34no—
45#407—34no—
46#407—62no—
47#407—62no—
48C2×C22⋊C8—62no—
49C2×C22⋊C8—62no—
50#341—38no—
51C22×A4—41no—
52C22×A4C4⋊C412yes—
53C22×A4C2×Q811yescharacteristic
54C22×C3×C4—41no—
55C4×A4C4⋊C412yes—
56C4×A4C22×C411yescharacteristic
57C4×A4C2×D411yescharacteristic
58C3×M16—44no—
59C2×C3×C8—42no—
60#208—44no—
61#208—82no—
62C2×C3⋊C8—82no—
63C25Dic611yescharacteristic
64C23×C4—31no—
65C23×C4—32no—
66C23×C4—31no—
67C23×C4—31no—
68C23×C4—31no—
69C23×C4—32no—
70C23×C4Dic611yescharacteristic
71C23×C4D1212yes—
72C23×C4C2×Dic312yes—
73C23×C4C22×D311yescharacteristic
74C2×M16—31no—
75C2×M16—34no—
76C2×M16—34no—
77C2×M16—38no—
78C2×M16—34no—
79C2×M16—32no—
80C2×M16—34no—
81C2×M16—34no—
82C2×M16—61no—
83C2×M16—64no—
84C2×M16—62no—
85C2×M16—62no—
86C2×M16—62no—
87C2×M16—62no—
88C2×M16—62no—
89C2×M16—62no—
90C2×M16—62no—
91C2×M16—62no—
92C2×M16—62no—
93C2×M16—62no—
94C2×M16—62no—
95C2×M16S411yescharacteristic
96C22×C8—32no—
97C22×C8—34no—
98C22×C8—32no—
99C22×C8—62no—
100C22×C8—62no—
101C22×C8C2×Dic314yes—
102#108—64no—
103#108—68no—
104#108—64no—
105C22⋊C8—64no—
106C22⋊C8—64no—
107C22⋊C8—64no—
108C22⋊C8—64no—
109C23×C3—41no—
110C2×A4—22no—
111C2×A4—21no—
112C2×A4C2×C4⋊C411yescharacteristic
113C2×C3×C4—41no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—41no—
117C3×C8—44no—
118C3⋊C8—84no—
119C24—31no—
120C24—32no—
121C24—31no—
122C24—41no—
123C24—121no—
124C24C4⋊Dic312yes—
125C24C2×Dic611yescharacteristic
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—32no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—32no—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—31no—
146C22×C4—61no—
147C22×C4—61no—
148C22×C4—62no—
149C22×C4—61no—
150C22×C4—62no—
151C22×C4C4⋊Dic312yes—
152C22×C4C2×D1211yescharacteristic
153C22×C4C22×Dic311yescharacteristic
154C22×C4C2×S411yescharacteristic
155M16—34no—
156M16—38no—
157M16—64no—
158M16—64no—
159M16—64no—
160M16—64no—
161M16—64no—
162M16—122no—
163M16—122no—
164M16—122no—
165M16—122no—
166M16SL(2,ℤ/4)14yes—
167C2×C8—32no—
168C2×C8—34no—
169C2×C8—34no—
170C2×C8—34no—
171C2×C8—62no—
172C2×C8—64no—
173C2×C8—64no—
174C2×C8—122no—
175C2×C8—122no—
176C2×C8—122no—
177C2×C8—122no—
178C2×C8C2×S412yes—
179C22×C3—41no—
180C22×C3—42no—
181C22×C3—161no—
182A4#43011yescharacteristic, complemented
183C3×C4—41no—
184C3×C4—42no—
185C3×C4—41no—
186C23—22no—
187C23—21no—
188C23—31no—
189C23—32no—
190C23—31no—
191C23—32no—
192C23—31no—
193C23—32no—
194C23—61no—
195C23—61no—
196C23—62no—
197C23—64no—
198C23—121no—
199C23—121no—
200C23—122no—
201C23—121no—
202C23—121no—
203C23C2×C4⋊Dic311yescharacteristic
204C23A4⋊Q811yescharacteristic
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—31no—
210C2×C4—32no—
211C2×C4—31no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C4—62no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4A4⋊Q811yescharacteristic
227C2×C4C4⋊S412yes—
228C2×C4C2×SL(2,ℤ/4)12yes—
229C2×C4C22×S411yescharacteristic
230C8—34no—
231C8—124no—
232C8—124no—
233C8C2×SL(2,ℤ/4)14yes—
234C2×C3—41no—
235C2×C3—81no—
236C2×C3—82no—
237C22—31no—
238C22—31no—
239C22—32no—
240C22—31no—
241C22—32no—
242C22—41no—
243C22—61no—
244C22—62no—
245C22—62no—
246C22—61no—
247C22—61no—
248C22—62no—
249C22—62no—
250C22—122no—
251C22—121no—
252C22—122no—
253C22#512511yescharacteristic, complemented
254C22C4⋊SL(2,ℤ/4)12yes—
255C22C2×A4⋊Q811yescharacteristic
256C4—31no—
257C4—31no—
258C4—32no—
259C4C4⋊SL(2,ℤ/4)12yes—
260C4C2×C4⋊S411yescharacteristic
261C4C22×SL(2,ℤ/4)11yescharacteristic
262C3—41noSylow
263C2—21no—
264C2—22no—
265C2—31no—
266C2—31no—
267C2—61no—
268C2—62no—
269C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC4C2×C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC23×M16D3
socleC23C2×C4⋊Dic3
prime radicalC23×A4Q8
cosocleC2×C4×A4C23
hypercenterC2×M16S4
hypocenterA4#430
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. #73105C2semidirect
182. A4#430semidirect
253. C22#5125semidirect

Maximal subgroups

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

subgroupindex
C2×A4×M162
C2×#54092
#731052
#206673
#51254