Finite Group Databasebeta

← #11711601

Subgroups — #11711601

Order 768.

255 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 35 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#85440C211yesmaximal, characteristic
2A4×M32C211yesmaximal, characteristic, complemented
3#73110C211yesmaximal, characteristic, complemented
4#13065—31nomaximal, Sylow
5#5912—21no—
6C2×C8×A4C2211yescharacteristic
7C4⋊SL(2,ℤ/4)—21no—
8#5402C411yescharacteristic, complemented
9#5401—21no—
10#5398C411yescharacteristic, complemented
11A4×C16—21no—
12#4518—41nomaximal
13C22×M32D311yescharacteristic, complemented
14#3050—31no—
15C2×#360—31no—
16C2×#360—31no—
17#1807—31no—
18#1386—31no—
19#1386—31no—
20C2×C4×A4D411yescharacteristic
21GL(2,ℤ/4)—41no—
22C2×SL(2,ℤ/4)—41no—
23C4⋊S4—21no—
24C4×S4—41no—
25A4⋊Q8—21no—
26#904—41no—
27C8×A4C2×C411yescharacteristic
28C8×A4D411yescharacteristic
29C3×M32—41no—
30#817—41no—
31C2×#135—31no—
32C23×C8C2×D311yescharacteristic
33#525—61no—
34C2×M32—31no—
35C2×M32—31no—
36C2×M32—61no—
37C22×C16—21no—
38#467—31no—
39#467—31no—
40#466—31no—
41#466—31no—
42#465—61no—
43#465—61no—
44#427—31no—
45C2×#106—31no—
46C2×#106—31no—
47#416—61no—
48#380—61no—
49#360—62no—
50#360—61no—
51#360—61no—
52#340—61no—
53C22×A4D811yescharacteristic
54C2×S4—41no—
55#235—81no—
56C4×A4C22⋊C411yescharacteristic
57C4×A4SD1611yescharacteristic
58SL(2,ℤ/4)—41no—
59SL(2,ℤ/4)—81no—
60C2×C3×C8—41no—
61C4⋊Dic3—81no—
62Dic12—41no—
63D24—41no—
64#204—81no—
65C3×C16—81no—
66C2×#41—61no—
67C23×C4D1211yescharacteristic
68#135—61no—
69#135—61no—
70#135—61no—
71#135—61no—
72C2×Q16—31no—
73C2×SD16—61no—
74C2×D8—31no—
75C22×C8—31no—
76C22×C8—31no—
77C22×C8—31no—
78C22×C8—31no—
79C22×C8C4×D311yescharacteristic
80C22×C8#6711yescharacteristic
81#123—121no—
82C22⋊Q8—61no—
83C22⋊Q8—61no—
84#121—61no—
85#121—61no—
86#120—121no—
87C4×D4—121no—
88C4×D4—121no—
89#117—61no—
90C2×C4⋊C4—61no—
91C2×C4⋊C4—61no—
92C2×C22⋊C4—121no—
93M32—32no—
94M32—31no—
95M32—61no—
96M32S411yescharacteristic, complemented
97C2×C16—61no—
98C2×C16—61no—
99#107—31no—
100#107—31no—
101#106—61no—
102#106—61no—
103#106—61no—
104#106—61no—
105#106—61no—
106#103—61no—
107#103—61no—
108#102—61no—
109#102—61no—
110#95—121no—
111C2×A4—21no—
112C2×A4#10211yescharacteristic
113S4—81no—
114C2×C3×C4—41no—
115#67—161no—
116C2×Dic3—161no—
117D12—81no—
118C4×D3—161no—
119Dic6—81no—
120C3×C8—41no—
121C3×C8—41no—
122C24D2411yescharacteristic
123#41—121no—
124#41—121no—
125#41—121no—
126#41—121no—
127C2×Q8—61no—
128C2×D4—61no—
129C2×D4—121no—
130C2×D4—121no—
131C2×D4—121no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—121no—
137C22×C4—121no—
138C22×C4—121no—
139C22×C4#20411yescharacteristic
140C22×C4(C2×D3)⋊C411yescharacteristic
141Q16—61no—
142Q16—61no—
143SD16—121no—
144SD16—121no—
145D8—61no—
146D8—61no—
147C2×C8—31no—
148C2×C8—31no—
149C2×C8—31no—
150C2×C8—31no—
151C2×C8—31no—
152C2×C8—31no—
153C2×C8—31no—
154C2×C8—61no—
155C2×C8C2×S411yescharacteristic
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4—121no—
159C4⋊C4—121no—
160C4⋊C4—121no—
161C4⋊C4—121no—
162C4⋊C4—121no—
163C4⋊C4—121no—
164C22⋊C4—121no—
165C22⋊C4—121no—
166C22⋊C4—121no—
167C22⋊C4—121no—
168C22⋊C4—121no—
169C22⋊C4—241no—
170C22⋊C4—241no—
171C42—121no—
172C16—21no—
173C16—61no—
174C22×C3—41no—
175C2×D3—161no—
176A4#36011yescharacteristic, complemented
177C3×C4—41no—
178C3×C4—41no—
179Dic3—161no—
180Dic3—321no—
181C23—21no—
182C23—31no—
183C23—31no—
184C23—61no—
185C23—121no—
186C23#82111yescharacteristic
187Q8—121no—
188Q8—121no—
189D4—121no—
190D4—121no—
191D4—241no—
192D4—241no—
193D4—242no—
194D4—241no—
195D4—241no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—61no—
204C2×C4—121no—
205C2×C4—121no—
206C2×C4—121no—
207C2×C4—241no—
208C2×C4—241no—
209C2×C4—241no—
210C2×C4—241no—
211C2×C4—241no—
212C2×C4—241no—
213C2×C4—241no—
214C2×C4—241no—
215C2×C4C4⋊S411yescharacteristic
216C8—31no—
217C8—31no—
218C8C4×S411yescharacteristic
219C8GL(2,ℤ/4)11yescharacteristic
220C2×C3—41no—
221C2×C3—81no—
222D3—321no—
223C22—31no—
224C22—31no—
225C22—31no—
226C22—61no—
227C22—61no—
228C22—61no—
229C22—61no—
230C22—241no—
231C22—241no—
232C22—241no—
233C22#451811yescharacteristic, complemented
234C22#540211yescharacteristic
235C4—31no—
236C4—31no—
237C4—241no—
238C4—241no—
239C4—241no—
240C4—481no—
241C4—481no—
242C4#540111yescharacteristic
243C4(C2×S4)⋊C411yescharacteristic
244C3—41noSylow
245C2—21no—
246C2—31no—
247C2—31no—
248C2—61no—
249C2—481no—
250C2#7311311yescharacteristic
subgroupG/H
centerC2#73113
commutatorC8×A4C2×C4
FrattiniC2×C8C2×S4
FittingC22×M32D3
socleC23#821
prime radical#85440C2
cosocleC2×C8×A4C22
hypercenterM32S4
hypocenterA4#360
LCSG ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

Complements

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

NH ≅ G/Ntype
2. A4×M32C2semidirect
3. #73110C2semidirect
8. #5402C4semidirect
10. #5398C4semidirect
13. C22×M32D3semidirect
96. M32S4semidirect
176. A4#360semidirect
233. C22#4518semidirect

Maximal subgroups

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

subgroupindex
#854402
A4×M322
#731102
#130653
#45184