Finite Group Databasebeta

← #10812901

Subgroups — #10812901

Order 768.

766 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#427C211yesmaximal, characteristic
2#82239C212yesmaximal
3#71970C211yesmaximal, characteristic
4#71967C211yesmaximal, characteristic
5#71575C212yesmaximal
6#70613C212yesmaximal, complemented
7#70605C212yesmaximal, complemented
8#70557C214yesmaximal, complemented
9#22619—31nomaximal, Sylow
10C23×C3×C8C2211yescharacteristic
11C2×C3×#117C2212yes—
12C3×#427C418yescomplemented
13C2×C3×#107C2211yescharacteristic
14C2×C3×#107C2211yescharacteristic
15C2×C3×#106C2212yes—
16#5212C2212yes—
17C4×#217C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#817C2212yes—
20C2×#815C2212yes—
21#4984C2214yes—
22Dic3×C4⋊C4C2214yes—
23#4951—24no—
24#4936—22no—
25#4932—22no—
26#4551C2212yes—
27#4550C2212yes—
28#4529C2214yes—
29#4529C2214yes—
30C2×#427D311yescharacteristic
31#2346—32no—
32#1943—32no—
33#1919—34no—
34#1825—32no—
35#1807—31no—
36#1776—32no—
37#1752—31no—
38C23×C3×C4C2311yescharacteristic
39C22×C3×C8—24no—
40C22×C3×C8C2×C414yes—
41C22×C3×C8C2311yescharacteristic
42C22×C3×C8C2311yescharacteristic
43C3×#117—24no—
44C3×#117C2×C418yes—
45C2×C3×C4⋊C4C2312yes—
46C2×C3×C4⋊C4C2312yes—
47C2×C3×C22⋊C4—22no—
48C2×C3×C42—22no—
49C2×#217—22no—
50C2×#217D412yes—
51C2×C4⋊Dic3—24no—
52C2×C4⋊Dic3C2314yes—
53C2×C4×Dic3—24no—
54C2×C4×Dic3C2312yes—
55C22×C3⋊C8C2312yes—
56C3×#107C2×C414yes—
57C3×#107C2×C414yes—
58C3×#106C2×C418yes—
59C3⋊#95—22no—
60C3⋊#95—24no—
61C3⋊#95—24no—
62C3⋊#95D412yes—
63#830—22no—
64#817—24no—
65#815—24no—
66C23×C8C2×D311yescharacteristic
67C2×#117C2×D312yes—
68#427Dic318yescomplemented
69C2×#107C2×D311yescharacteristic
70C2×#107C2×D311yescharacteristic
71C2×#106—32no—
72C2×#106C2×D312yes—
73C2×C22⋊C8—31no—
74C2×#97—32no—
75#389—34no—
76#388—64no—
77#382—62no—
78#380—32no—
79C4×C4⋊C4—34no—
80C4×C22⋊C4—31no—
81#376—62no—
82#340—32no—
83#340—34no—
84#340—34no—
85#336—32no—
86C24×C3C2×D411yescharacteristic
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4C22×C412yes—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4112yes—
94C22×C3×C4C2411yescharacteristic
95C22×Dic3—24no—
96C22×Dic3C2×D414yes—
97C2×C3×C8—22no—
98C2×C3×C8—28no—
99C2×C3×C8C22×C412yes—
100C2×C3×C8C22×C412yes—
101C2×C3×C8#4114yes—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4C22×C414yes—
105C3×C4⋊C4C22×C414yes—
106C3×C22⋊C4—28no—
107C3×C42—28no—
108#217—42no—
109#217—82no—
110C4⋊Dic3—24no—
111C4⋊Dic3—24no—
112C4⋊Dic3—84no—
113C4×Dic3—22no—
114C4×Dic3—22no—
115C4×Dic3—48no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—22no—
118C2×C3⋊C8—22no—
119C23×C4C22×D311yescharacteristic
120C22×C8—24no—
121C22×C8—32no—
122C22×C8C2×Dic314yes—
123C22×C8C22×D311yescharacteristic
124C22×C8C22×D311yescharacteristic
125#117—24no—
126#117C2×Dic318yes—
127C2×C4⋊C4—34no—
128C2×C4⋊C4—64no—
129C2×C4⋊C4C22×D312yes—
130C2×C4⋊C4C22×D312yes—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—32no—
133C2×C22⋊C4—62no—
134C2×C42—22no—
135C2×C42—32no—
136C2×C42—64no—
137#107C2×Dic314yes—
138#107C2×Dic314yes—
139#106—64no—
140#106C2×Dic318yes—
141C22⋊C8—62no—
142#97—64no—
143#95—32no—
144#95—64no—
145#95—64no—
146#95—62no—
147C23×C3—22no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3C4×D412yes—
151C23×C3C22×D411yescharacteristic
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—44no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4C4×D412yes—
162C2×C3×C4C23×C411yescharacteristic
163C2×C3×C4C2×#4112yes—
164C2×C3×C4C2×#4111yescharacteristic
165C2×C3×C42−1+412yes—
166C2×Dic3—24no—
167C2×Dic3—24no—
168C2×Dic3—24no—
169C2×Dic3—44no—
170C2×Dic3—44no—
171C2×Dic3—44no—
172C3×C8—22no—
173C3×C8#11714yes—
174C3⋊C8—42no—
175C24D3×D411yescharacteristic
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—34no—
181C22×C4—64no—
182C22×C4D3×D411yescharacteristic
183C22×C4#23712yes—
184C22×C4C22×Dic312yes—
185C22×C4C23×D311yescharacteristic
186C2×C8—22no—
187C2×C8—28no—
188C2×C8—62no—
189C2×C8—62no—
190C2×C8—62no—
191C2×C8#23514yes—
192C2×C8C22×Dic312yes—
193C2×C8C22×Dic312yes—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—64no—
197C4⋊C4—64no—
198C4⋊C4—244no—
199C4⋊C4C22×Dic314yes—
200C4⋊C4C22×Dic314yes—
201C22⋊C4—28no—
202C22⋊C4—122no—
203C22⋊C4—242no—
204C42—28no—
205C42—62no—
206C42—62no—
207C42—128no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—44no—
212C22×C3C2×C4×D411yescharacteristic
213C22×C3#54712yes—
214C22×C3#57512yes—
215C22×C3#57712yes—
216C3×C4—22no—
217C3×C4—44no—
218C3×C4—44no—
219C3×C4C2×#11711yescharacteristic
220C3×C4#51911yescharacteristic
221C3×C4#55212yes—
222Dic3—44no—
223Dic3—84no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23D4×Dic312yes—
228C23C2×D3×D411yescharacteristic
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—64no—
239C2×C4—64no—
240C2×C4—64no—
241C2×C4—124no—
242C2×C4—124no—
243C2×C4—124no—
244C2×C4D4×Dic312yes—
245C2×C4C2×#23512yes—
246C2×C4C2×#23711yescharacteristic
247C2×C4#100712yes—
248C2×C4C23×Dic311yescharacteristic
249C8—22no—
250C8—122no—
251C8#92614yes—
252C2×C3—42no—
253C2×C3#234411yescharacteristic
254C2×C3#294311yescharacteristic
255C2×C3#295211yescharacteristic
256C2×C3#334814yes—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—44no—
261C22#550712yes—
262C22#576812yes—
263C22#577012yes—
264C22C2×D4×Dic311yescharacteristic
265C4—22no—
266C4—44no—
267C4—44no—
268C4—124no—
269C4—244no—
270C4#551312yes—
271C4C2×#92611yescharacteristic
272C4#581211yescharacteristic
273C3#2261911yescharacteristic, complemented, Sylow
274C2—42no—
275C2#7919314yes—
276C2#8215911yescharacteristic
277C2#8389711yescharacteristic
278C2#8389911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#427C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenterC2×#427D3
hypocenterC3#22619
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
6. #70613C2semidirect
7. #70605C2semidirect
8. #70557C2semidirect
12. C3×#427C4semidirect
68. #427Dic3semidirect
273. C3#22619semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4272
#822392
#719702
#719672
#715752
#706132
#706052
#705572
#226193