Finite Group Databasebeta

← #10712659

Subgroups — #10712659

Order 768.

486 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 123 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#83799C211yesmaximal, characteristic
2#83780C211yesmaximal, characteristic, complemented
3C3×#1813C211yesmaximal, characteristic, complemented
4C2×#4495C212yesmaximal, complemented
5#67735C212yesmaximal, complemented
6#13079—31nomaximal, Sylow
7C2×C3×#131C2211yescharacteristic
8#5830—21no—
9C22×D24C2211yescharacteristic
10C2×C3×#108C2212yes—
11C3×#413—21no—
12#5140—24no—
13#5135C414yes—
14#5131—21no—
15#5115—22no—
16#5106C2211yescharacteristic
17C2×#812—22no—
18C2×#812C2212yes—
19C3×#338—21no—
20#4495—24no—
21#3407—31no—
22#3058—31no—
23C2×#361—32no—
24#1813D311yescharacteristic, complemented
25#1389—32no—
26C2×C3×#41D411yescharacteristic
27C22×D12—21no—
28C3×#131—22no—
29C3×#131—22no—
30C2×C3×M16—22no—
31C2×C3×M16C2×C412yes—
32C2×C3×M16D411yescharacteristic
33C22×C3×C8D412yes—
34C22×C3×C8C2311yescharacteristic
35#947—41no—
36#940—41no—
37#937—42no—
38#930—24no—
39#926—21no—
40C2×D24—22no—
41C2×D24—22no—
42C2×D24—21no—
43C2×D24C2×C412yes—
44C3×#108—22no—
45C3×#108—22no—
46C3×#101—41no—
47C3×#100—41no—
48#821—24no—
49#818—21no—
50#817C2×C412yes—
51#812—24no—
52#812—22no—
53#812D414yescomplemented
54C2×C3⋊C16—24no—
55C2×C3⋊C16—22no—
56C22×D8—31no—
57C2×#131C2×D311yescharacteristic
58#535—61no—
59#504—64no—
60C2×M32—32no—
61C2×M32—62no—
62#469—34no—
63#466—62no—
64C2×#108C2×D312yes—
65#427—31no—
66#418—61no—
67#413—21no—
68#361—64no—
69#338—21no—
70C23×D3—41no—
71C3×#41—22no—
72C3×#41—22no—
73C2×C3×Q8D811yescharacteristic
74C2×C3×D4D811yescharacteristic
75C2×C3×D4SD1612yes—
76C22×C3×C4C22⋊C412yes—
77C22×C3×C4C2×D411yescharacteristic
78C2×#67—44no—
79C2×D12—22no—
80C2×D12—42no—
81C2×D12—42no—
82C2×D12—41no—
83C3×M16—22no—
84C3×M16—24no—
85C3×M16—42no—
86C3×M16—42no—
87C2×C3×C8—22no—
88C2×C3×C8—22no—
89C2×C3×C8—42no—
90C2×C3×C8C22⋊C412yes—
91C2×C3×C8C22×C411yescharacteristic
92C2×C3×C8C2×D411yescharacteristic
93C2×C3×C8C2×D412yes—
94C2×C3×C8#4112yes—
95#217—41no—
96(C2×D3)⋊C4—44no—
97C4⋊Dic3—22no—
98C4×Dic3—41no—
99D24—24no—
100D24—41no—
101D24—41no—
102C3⋊C16—24no—
103C2×#41#6711yescharacteristic
104C22×D4—61no—
105C2×D8—32no—
106C2×D8—61no—
107C2×D8—62no—
108C2×D8—62no—
109#131—22no—
110#131—22no—
111C2×M16—22no—
112C2×M16C4×D312yes—
113C2×M16#6711yescharacteristic
114C22×C8D1212yes—
115C22×C8C22×D311yescharacteristic
116#127—121no—
117#124—121no—
118#121—64no—
119#120—122no—
120#117—61no—
121M32—34no—
122M32—64no—
123M32—62no—
124C2×C16—64no—
125C2×C16—62no—
126#108—22no—
127#108—22no—
128#107—61no—
129#106—32no—
130#102—64no—
131#101—41no—
132#100—41no—
133C23×C3#10212yes—
134C23×C3C2×SD1611yescharacteristic
135C22×D3—42no—
136C22×D3—82no—
137C22×D3—162no—
138C3×Q8—22no—
139C3×D4—24no—
140C3×D4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4#10212yes—
144C2×C3×C4C2×C22⋊C411yescharacteristic
145C2×C3×C4#12112yes—
146C2×C3×C4C2×D811yescharacteristic
147C2×C3×C4#13512yes—
148#67—164no—
149C2×Dic3—42no—
150D12—42no—
151D12—42no—
152D12—82no—
153C3×C8—22no—
154C3×C8—82no—
155C3×C8C4×D412yes—
156C3×C8#12011yescharacteristic
157C3×C8#12311yescharacteristic
158C24—121no—
159#41—22no—
160#41—22no—
161C2×Q8#21311yescharacteristic
162C2×D4—62no—
163C2×D4—124no—
164C2×D4—121no—
165C2×D4—122no—
166C2×D4—122no—
167C2×D4#21311yescharacteristic
168C2×D4#21512yes—
169C22×C4(C2×D3)⋊C412yes—
170C22×C4C2×#6711yescharacteristic
171D8—64no—
172D8—121no—
173D8—121no—
174M16—22no—
175M16—24no—
176M16—42no—
177M16—42no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—42no—
181C2×C8(C2×D3)⋊C412yes—
182C2×C8C2×C4×D311yescharacteristic
183C2×C8C2×D1211yescharacteristic
184C2×C8D3×D412yes—
185C2×C8#23712yes—
186C4⋊C4—62no—
187C22⋊C4—121no—
188C22⋊C4—124no—
189C42—121no—
190C16—64no—
191C22×C3—22no—
192C22×C3—41no—
193C22×C3—42no—
194C22×C3C2×#10211yescharacteristic
195C22×C3#46512yes—
196C2×D3—84no—
197C2×D3—82no—
198C2×D3—164no—
199C3×C4—22no—
200C3×C4#38611yescharacteristic
201C3×C4#41611yescharacteristic
202C3×C4#46612yes—
203Dic3—82no—
204C23—122no—
205C23—242no—
206C23—482no—
207C23#80912yes—
208C23C2×#21511yescharacteristic
209Q8—22no—
210D4—24no—
211D4—22no—
212D4—122no—
213D4—122no—
214D4—242no—
215D4—484no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—122no—
219C2×C4#80912yes—
220C2×C4C2×(C2×D3)⋊C411yescharacteristic
221C2×C4C2×#21311yescharacteristic
222C2×C4#93912yes—
223C2×C4#95012yes—
224C8—22no—
225C8—82no—
226C8#67⋊C412yes—
227C8#88211yescharacteristic
228C8#88611yescharacteristic
229C2×C3—21no—
230C2×C3—22no—
231C2×C3—42no—
232C2×C3#189311yescharacteristic
233D3—162no—
234C22—22no—
235C22—41no—
236C22—42no—
237C22—244no—
238C22—242no—
239C22—484no—
240C22C2×#80911yescharacteristic
241C22#504812yes—
242C4—22no—
243C4—242no—
244C4#495811yescharacteristic
245C4#500211yescharacteristic
246C4#503812yes—
247C3#1307911yescharacteristic, complemented, Sylow
248C2—21no—
249C2—22no—
250C2—42no—
251C2—482no—
252C2#6951511yescharacteristic
subgroupG/H
centerC2#69515
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#1813C2
socleC2×C3#1893
prime radical#83799C2
cosocleC22×C3×C8C23
hypercenter#1813D3
hypocenterC3#13079
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #1813

Complements

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

NH ≅ G/Ntype
2. #83780C2semidirect
3. C3×#1813C2semidirect
4. C2×#4495C2semidirect
5. #67735C2semidirect
24. #1813D3semidirect
53. #812D4semidirect
247. C3#13079semidirect

Maximal subgroups

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

subgroupindex
#837992
#837802
C3×#18132
C2×#44952
#677352
#130793