Finite Group Databasebeta

← #10595022

Subgroups — #10595022

Order 576.

698 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 168 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×C3×#235C211yesmaximal, characteristic, complemented
2C22×C3×C4×D3C211yesmaximal, characteristic, complemented
3#64135C211yesmaximal, characteristic, complemented
4#63976C212yesmaximal, complemented
5#63961C212yesmaximal, complemented
6#63908C212yesmaximal, complemented
7C4×#625C214yesmaximal, complemented
8#63856C212yesmaximal, complemented
9#5828—31nomaximal
10#5791—31nomaximal
11C23×C3×D3C2211yescharacteristic
12C22×C32×C4C2211yescharacteristic
13C2×C3×#67C2212yes—
14C22×C3×Dic3C2211yescharacteristic
15C3×#235—24no—
16C2×C3×D12C2211yescharacteristic, complemented
17C2×C3×C4×D3—24no—
18C2×C3×C4×D3C2212yes—
19C2×C3×C4×D3C2212yes—
20C2×C3×C4×D3C2212yescomplemented
21C2×C3×Dic6C2211yescharacteristic, complemented
22C2×#625C2214yes—
23#3757C2212yes—
24C4⋊(C3⋊Dic3)C2212yescomplemented
25C4×C3⋊Dic3C2212yescomplemented
26#3724C2214yescomplemented
27(C2×D3)⋊Dic3C2214yes—
28(C2×D3)⋊Dic3C2214yescomplemented
29C2×C3×#41—31no—
30C23×C3×C4—31no—
31C2×#235D311yescharacteristic, complemented
32C22×C4×D3D311yescharacteristic, complemented
33#953—32no—
34#946—32no—
35#938—32no—
36#937—32no—
37#935—32no—
38#930—32no—
39#929—32no—
40C4×#67—34no—
41C4×#67—34no—
42#926—31no—
43#926—31no—
44#926—61no—
45#924—32no—
46C23×C32C2311yescharacteristic
47C22×C3×D3—22no—
48C22×C3×D3—24no—
49C22×C3×D3C2312yes—
50C22×C3×D3C2312yes—
51C2×C32×C4D414yescomplemented
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C3⋊Dic3C2314yes—
55C3×#67—24no—
56C2×C3×Dic3—22no—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C3×D12—22no—
60C3×C4×D3—24no—
61C3×C4×D3—24no—
62C3×C4×D3—24no—
63C3×Dic6—22no—
64#625—28no—
65#525—91noSylow
66C24×C3—31no—
67C23×D3C2×D311yescharacteristic
68C3×#41—64no—
69C2×C3×Q8—31no—
70C2×C3×D4—32no—
71C2×C3×D4—31no—
72C22×C3×C4—21no—
73C22×C3×C4—31no—
74C22×C3×C4—32no—
75C22×C3×C4—32no—
76C22×C3×C4—32no—
77C22×C3×C4—64no—
78C22×C3×C4C2×D311yescharacteristic
79C22×C3×C4C2×D311yescharacteristic
80C2×#67—34no—
81C2×#67—34no—
82C2×#67C2×D312yes—
83C22×Dic3C2×D311yescharacteristic
84#235—24no—
85C2×D12C2×D311yescharacteristic, complemented
86C2×C4×D3—24no—
87C2×C4×D3C2×D312yes—
88C2×C4×D3C2×D312yes—
89C2×C4×D3C2×D312yescomplemented
90C2×Dic6C2×D311yescharacteristic, complemented
91#217—32no—
92#217—32no—
93#217—34no—
94#217—34no—
95#217—62no—
96(C2×D3)⋊C4—34no—
97(C2×D3)⋊C4—34no—
98C4⋊Dic3—32no—
99C4⋊Dic3—32no—
100C4⋊Dic3—62no—
101Dic3⋊C4—34no—
102Dic3⋊C4—34no—
103C4×Dic3—32no—
104C4×Dic3—32no—
105C4×Dic3—62no—
106C22×C32—22no—
107C22×C32C2×D412yes—
108C22×C32C2411yescharacteristic
109C2×C3×D3—28no—
110C2×C3×D3—22no—
111C2×C3×D3—22no—
112C2×C3×D3—22no—
113C2×C3×D3—24no—
114C2×C3×D3#4114yescomplemented
115C32×C4C2×D412yes—
116C32×C4C2×D412yes—
117C3⋊Dic3—24no—
118C3×Dic3—22no—
119C3×Dic3—22no—
120C2×#41—31no—
121C23×C4—31no—
122#123—92no—
123C22⋊Q8—92no—
124#121—92no—
125#120—92no—
126C4×D4—94no—
127#117—91no—
128C23×C3—21no—
129C23×C3—32no—
130C23×C3—32no—
131C23×C3—64no—
132C23×C3—62no—
133C23×C3C22×D311yescharacteristic
134C23×C3C22×D311yescharacteristic
135C22×D3—22no—
136C22×D3—24no—
137C22×D3C22×D312yes—
138C22×D3C22×D312yes—
139C3×Q8—62no—
140C3×D4—64no—
141C3×D4—62no—
142C2×C3×C4—24no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—62no—
148C2×C3×C4—64no—
149C2×C3×C4—64no—
150C2×C3×C4—64no—
151C2×C3×C4#6714yescomplemented
152C2×C3×C4#6714yescomplemented
153C2×C3×C4C22×D311yescharacteristic
154C2×C3×C4C22×D311yescharacteristic
155C2×C3×C4C22×D311yescharacteristic
156C2×C3×C4C22×D311yescharacteristic
157#67—24no—
158#67—68no—
159#67—68no—
160C2×Dic3—22no—
161C2×Dic3—34no—
162C2×Dic3—34no—
163C2×Dic3—64no—
164C2×Dic3C22×D312yes—
165C2×Dic3C22×D312yes—
166D12—22no—
167C4×D3—24no—
168C4×D3—24no—
169C4×D3—24no—
170Dic6—22no—
171C2×C32—22no—
172C2×C32C22×D411yescharacteristic
173C2×C32C2×#4112yes—
174C3×D3—24no—
175C3×D3—42no—
176C24—31no—
177#41—64no—
178C2×Q8—31no—
179C2×D4—32no—
180C2×D4—31no—
181C2×D4—94no—
182C22×C4—31no—
183C22×C4—32no—
184C22×C4—32no—
185C22×C4—32no—
186C22×C4—64no—
187C22×C4D3211yescharacteristic
188C4⋊C4—94no—
189C4⋊C4—92no—
190C22⋊C4—94no—
191C22⋊C4—94no—
192C22⋊C4—92no—
193C42—92no—
194C22×C3—22no—
195C22×C3—22no—
196C22×C3—21no—
197C22×C3—22no—
198C22×C3—34no—
199C22×C3—42no—
200C22×C3—68no—
201C22×C3—62no—
202C22×C3—62no—
203C22×C3—64no—
204C22×C3—62no—
205C22×C3C2×#6712yes—
206C22×C3C2×#6712yes—
207C22×C3C23×D311yescharacteristic
208C22×C3C23×D311yescharacteristic
209C2×D3—28no—
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—24no—
213C2×D3—22no—
214C2×D3#23514yescomplemented
215C3×C4—22no—
216C3×C4—22no—
217C3×C4—62no—
218C3×C4—62no—
219C3×C4C2×#6712yes—
220C3×C4C2×#6712yes—
221C3×C4C2×#6712yes—
222C3×C4C2×#6712yes—
223Dic3—22no—
224Dic3—22no—
225Dic3—64no—
226Dic3—64no—
227Dic3—124no—
228C32#52511yescharacteristic, complemented, Sylow
229C23—32no—
230C23—32no—
231C23—64no—
232C23—62no—
233C23C2×D3211yescharacteristic
234Q8—62no—
235D4—64no—
236D4—62no—
237D4—188no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—62no—
241C2×C4—64no—
242C2×C4—64no—
243C2×C4—64no—
244C2×C4—94no—
245C2×C4#62514yescomplemented
246C2×C4C2×D3211yescharacteristic
247C2×C4C2×D3211yescharacteristic
248C2×C3—22no—
249C2×C3—22no—
250C2×C3—22no—
251C2×C3—21no—
252C2×C3—42no—
253C2×C3—64no—
254C2×C3—122no—
255C2×C3C2×#23512yes—
256C2×C3D3×#4112yes—
257C2×C3C22×#6711yescharacteristic
258C2×C3C22×#6711yescharacteristic
259D3—24no—
260D3—42no—
261C22—22no—
262C22—34no—
263C22—68no—
264C22—62no—
265C22—62no—
266C22—64no—
267C22—62no—
268C22C2×#62512yes—
269C22C22×D3211yescharacteristic
270C4—62no—
271C4—62no—
272C4—184no—
273C4C2×#62512yes—
274C4C2×#62512yes—
275C3—21no—
276C3#579111yescharacteristic, complemented
277C3#582811yescharacteristic, complemented
278C2—22no—
279C2—64no—
280C2—122no—
281C2D3×#23512yes—
282C2C22×#62511yescharacteristic
subgroupG/H
centerC2×C4C2×D32
commutator, socle, cosocleC22×C32C24
FrattiniC22C22×D32
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#235C2semidirect
2. C22×C3×C4×D3C2semidirect
3. #64135C2semidirect
4. #63976C2semidirect
5. #63961C2semidirect
6. #63908C2semidirect
7. C4×#625C2semidirect
8. #63856C2semidirect
16. C2×C3×D12C22semidirect
20. C2×C3×C4×D3C22semidirect
21. C2×C3×Dic6C22semidirect
24. C4⋊(C3⋊Dic3)C22semidirect
25. C4×C3⋊Dic3C22semidirect
26. #3724C22semidirect
28. (C2×D3)⋊Dic3C22semidirect
31. C2×#235D3semidirect
32. C22×C4×D3D3semidirect
51. C2×C32×C4D4semidirect
85. C2×D12C2×D3semidirect
89. C2×C4×D3C2×D3semidirect
90. C2×Dic6C2×D3semidirect
114. C2×C3×D3#41semidirect
151. C2×C3×C4#67semidirect
152. C2×C3×C4#67semidirect
214. C2×D3#235semidirect
228. C32#525semidirect
245. C2×C4#625semidirect
276. C3#5791semidirect
277. C3#5828semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2352
C22×C3×C4×D32
#641352
#639762
#639612
#639082
C4×#6252
#638562
#58283
#57913