Finite Group Databasebeta

← #10594006

Subgroups — #10594006

Order 576.

698 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 144 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×C4×D32C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3#64081C211yesmaximal, characteristic, complemented
4C3×C4×D12C212yesmaximal, complemented
5#63922C212yesmaximal, complemented
6C4×#625C212yesmaximal, complemented
7#63899C212yesmaximal, complemented
8#63885C212yesmaximal, complemented
9#63854C212yesmaximal, complemented
10#5547—32nomaximal
11C22×D32C2211yescharacteristic
12C2×C4×C3⋊D3C2211yescharacteristic
13C2×C3×D12C2212yes—
14C2×C3×C4×D3C2212yes—
15C2×C3×C4×D3C2212yescomplemented
16C2×#627C2211yescharacteristic, complemented
17C2×#626C2212yes—
18C2×#625C2212yes—
19C2×#625C2211yescharacteristic, complemented
20C2×#624C2211yescharacteristic
21C2×D3×Dic3C2212yes—
22C4×D32—24no—
23#3798—24no—
24C32×C42C2211yescharacteristic, complemented
25#3752C2212yes—
26#3750C2212yescomplemented
27C4×C3⋊Dic3C2211yescharacteristic, complemented
28C3×(C2×D3)⋊C4C2212yes—
29C3×(C2×D3)⋊C4C2212yescomplemented
30C3×C4⋊Dic3C2212yescomplemented
31#3724C2212yescomplemented
32(C2×D3)⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yescomplemented
34C2×#235—32no—
35C22×C4×D3—32no—
36C3×C4×D4—32no—
37#939—32no—
38#937—32no—
39C4×#67—32no—
40C4×#67—32no—
41#896—32no—
42#894—32no—
43#884—32no—
44#883—32no—
45#882—32no—
46#878—32no—
47C4×D12D312yescomplemented
48#872—32no—
49#872—61no—
50C22×C3⋊D3C2311yescharacteristic
51C22×C3×D3C2312yes—
52C22×C3×D3C2312yes—
53C2×D32—22no—
54C2×D32—24no—
55C2×C32×C4C2312yes—
56C2×C32×C4C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2311yescharacteristic
59C4×C3⋊D3D414yescomplemented
60C2×C3×Dic3C2312yes—
61C2×C3×Dic3C2312yes—
62C3×D12—24no—
63C3×C4×D3—24no—
64C3×C4×D3—24no—
65#627—22no—
66#626—24no—
67#625—24no—
68#625—22no—
69#624—22no—
70D3×Dic3—24no—
71#525—91noSylow
72C23×D3—32no—
73C2×C3×D4—32no—
74C22×C3×C4—32no—
75C22×C3×C4—32no—
76C2×#67—32no—
77C2×#67—32no—
78C2×#67—32no—
79C2×#67—32no—
80C22×Dic3—32no—
81#235—68no—
82C2×D12—32no—
83C2×D12C2×D312yes—
84C2×C4×D3—32no—
85C2×C4×D3—32no—
86C2×C4×D3—32no—
87C2×C4×D3—61no—
88C2×C4×D3—68no—
89C2×C4×D3C2×D312yes—
90C2×C4×D3C2×D312yescomplemented
91C2×Dic6—32no—
92C3×C4⋊C4—32no—
93C3×C22⋊C4—32no—
94C3×C22⋊C4—32no—
95C3×C42—21no—
96C3×C42C2×D312yescomplemented
97#217—32no—
98#217—32no—
99(C2×D3)⋊C4—32no—
100(C2×D3)⋊C4—32no—
101(C2×D3)⋊C4—32no—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4—62no—
104(C2×D3)⋊C4C2×D312yes—
105(C2×D3)⋊C4C2×D312yescomplemented
106C4⋊Dic3C2×D312yescomplemented
107Dic3⋊C4—32no—
108Dic3⋊C4—32no—
109Dic3⋊C4—32no—
110Dic3⋊C4—32no—
111Dic3⋊C4—62no—
112C4×Dic3—32no—
113C4×Dic3—61no—
114C22×C32C2411yescharacteristic
115C2×C3⋊D3—22no—
116C2×C3⋊D3C2×D412yes—
117C2×C3×D3—22no—
118C2×C3×D3—22no—
119C2×C3×D3—22no—
120C2×C3×D3—22no—
121C2×C3×D3—22no—
122C2×C3×D3#4114yescomplemented
123D32—28no—
124C32×C4—22no—
125C32×C4C2×D412yes—
126C3⋊Dic3—22no—
127C3⋊Dic3C2×D412yes—
128C3×Dic3—22no—
129C3×Dic3—22no—
130C2×#41—91no—
131C23×C4—91no—
132#123—92no—
133C22⋊Q8—92no—
134#121—92no—
135#120—92no—
136C4×D4—32no—
137C4×D4—92no—
138#117—91no—
139C23×C3—32no—
140C23×C3—32no—
141C22×D3—32no—
142C22×D3—61no—
143C22×D3—64no—
144C22×D3—64no—
145C22×D3—64no—
146C22×D3C22×D312yes—
147C22×D3C22×D312yes—
148C3×D4—64no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—32no—
152C2×C3×C4—32no—
153C2×C3×C4—64no—
154C2×C3×C4—64no—
155C2×C3×C4C22×D312yes—
156C2×C3×C4C22×D312yes—
157C2×C3×C4C22×D312yes—
158#67—64no—
159#67—64no—
160#67—64no—
161#67—64no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—61no—
166C2×Dic3—62no—
167C2×Dic3—64no—
168C2×Dic3C22×D312yes—
169C2×Dic3C22×D312yes—
170D12—24no—
171D12—64no—
172C4×D3—24no—
173C4×D3—24no—
174C4×D3—38no—
175C4×D3—64no—
176C4×D3—64no—
177C4×D3—64no—
178Dic6—64no—
179C2×C32C22×D411yescharacteristic
180C2×C32C2×#4112yes—
181C3⋊D3—22no—
182C3×D3—24no—
183C3×D3—42no—
184C24—91no—
185#41—184no—
186C2×Q8—91no—
187C2×D4—32no—
188C2×D4—92no—
189C2×D4—92no—
190C2×D4—91no—
191C22×C4—32no—
192C22×C4—32no—
193C22×C4—91no—
194C22×C4—92no—
195C22×C4—91no—
196C22×C4—184no—
197C4⋊C4—32no—
198C4⋊C4—92no—
199C4⋊C4—92no—
200C22⋊C4—32no—
201C22⋊C4—32no—
202C22⋊C4—92no—
203C22⋊C4—92no—
204C22⋊C4—92no—
205C42—91no—
206C42D3211yescharacteristic, complemented
207C22×C3—21no—
208C22×C3—34no—
209C22×C3—62no—
210C22×C3—62no—
211C22×C3—62no—
212C22×C3—62no—
213C22×C3—62no—
214C22×C3C23×D312yes—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—22no—
218C2×D3—22no—
219C2×D3—22no—
220C2×D3—34no—
221C2×D3—62no—
222C2×D3—616no—
223C2×D3—62no—
224C2×D3—62no—
225C2×D3—122no—
226C2×D3#23514yescomplemented
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—22no—
230C3×C4—42no—
231C3×C4—62no—
232C3×C4—62no—
233C3×C4D3×D414yes—
234Dic3—22no—
235Dic3—22no—
236Dic3—34no—
237Dic3—62no—
238Dic3—62no—
239Dic3—62no—
240Dic3—122no—
241C32#52511yescharacteristic, complemented, Sylow
242C23—32no—
243C23—32no—
244C23—91no—
245C23—184no—
246C23—182no—
247Q8—182no—
248D4—64no—
249D4—184no—
250D4—182no—
251D4—184no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—64no—
255C2×C4—64no—
256C2×C4—91no—
257C2×C4—94no—
258C2×C4—92no—
259C2×C4—182no—
260C2×C4—184no—
261C2×C4C2×D3212yes—
262C2×C4C2×D3211yescharacteristic
263C2×C3—21no—
264C2×C3—22no—
265C2×C3—64no—
266C2×C3—122no—
267C2×C3C2×#23512yes—
268C2×C3C2×D3×D412yes—
269C2×C3D3×#4112yes—
270D3—24no—
271D3—42no—
272D3—64no—
273D3—122no—
274C22—34no—
275C22—62no—
276C22—62no—
277C22—62no—
278C22—62no—
279C22—62no—
280C22—92no—
281C22—188no—
282C22—182no—
283C22C22×D3211yescharacteristic
284C4—22no—
285C4—62no—
286C4—62no—
287C4—92no—
288C4—182no—
289C4#380212yes—
290C3—21no—
291C3#554712yescomplemented
292C2—64no—
293C2—122no—
294C2—182no—
295C2C2×#380211yescharacteristic
296C2D3×#23512yes—
subgroupG/H
centerC2×C4C2×D32
commutator, socle, cosocleC22×C32C24
FrattiniC22C22×D32
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×D32C2semidirect
2. C2×#3798C2semidirect
3. #64081C2semidirect
4. C3×C4×D12C2semidirect
5. #63922C2semidirect
6. C4×#625C2semidirect
7. #63899C2semidirect
8. #63885C2semidirect
9. #63854C2semidirect
15. C2×C3×C4×D3C22semidirect
16. C2×#627C22semidirect
19. C2×#625C22semidirect
24. C32×C42C22semidirect
26. #3750C22semidirect
27. C4×C3⋊Dic3C22semidirect
29. C3×(C2×D3)⋊C4C22semidirect
30. C3×C4⋊Dic3C22semidirect
31. #3724C22semidirect
33. (C2×D3)⋊Dic3C22semidirect
47. C4×D12D3semidirect
59. C4×C3⋊D3D4semidirect
90. C2×C4×D3C2×D3semidirect
96. C3×C42C2×D3semidirect
105. (C2×D3)⋊C4C2×D3semidirect
106. C4⋊Dic3C2×D3semidirect
122. C2×C3×D3#41semidirect
206. C42D32semidirect
226. C2×D3#235semidirect
241. C32#525semidirect
291. C3#5547semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D322
C2×#37982
#640812
C3×C4×D122
#639222
C4×#6252
#638992
#638852
#638542
#55473