Finite Group Databasebeta

← #10594231

Subgroups — #10594231

Order 576.

523 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 130 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#64090C211yesmaximal, characteristic, complemented
2C3×#67⋊C4C212yesmaximal, complemented
3#63980C211yesmaximal, characteristic, complemented
4#63978C212yesmaximal, complemented
5C2×#3724C211yesmaximal, characteristic, complemented
6#63912C211yesmaximal, characteristic, complemented
7C4×#625C211yesmaximal, characteristic, complemented
8#63874C211yesmaximal, characteristic, complemented
9#63870C211yesmaximal, characteristic, complemented
10#63859C212yesmaximal, complemented
11#63850C212yesmaximal, complemented
12#5540—32nomaximal
13C2×#638C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#67C2212yes—
17C22×C3×Dic3C2212yes—
18C2×C3×C4×D3C2212yes—
19C2×#626C2212yes—
20C2×#625C2211yescharacteristic
21C32×C22⋊C4C2211yescharacteristic
22#3752C2211yescharacteristic
23#3750C2211yescharacteristic
24C4×C3⋊Dic3C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×Dic3⋊C4C2212yes—
27C3×C4×Dic3C2212yes—
28#3724C2212yes—
29#3724C2211yescharacteristic
30#3724C2211yescharacteristic
31#3724C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36C3×C4×D4—32no—
37#939—32no—
38#939—32no—
39#930—32no—
40#929—32no—
41C4×#67—32no—
42C2×Dic3⋊C4—32no—
43#894—32no—
44#889—32no—
45#884—32no—
46#67⋊C4—32no—
47#67⋊C4—61no—
48#67⋊C4D312yescomplemented
49#879—32no—
50#876—32no—
51#872—32no—
52C23×C32C2311yescharacteristic
53C22×C3⋊D3C2311yescharacteristic
54C22×C3×D3C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57#638—22no—
58C2×C3⋊Dic3—22no—
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2311yescharacteristic
62C4×C3⋊D3—22no—
63C3×#67—24no—
64C2×C3×Dic3—42no—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C3×C4×D3—42no—
69#626—42no—
70#625—22no—
71#553—91noSylow
72C2×C3×D4—32no—
73C22×C3×C4—32no—
74C22×C3×C4—32no—
75C2×#67—32no—
76C2×#67—32no—
77C2×#67—32no—
78C2×#67—61no—
79C2×#67C2×D312yes—
80C22×Dic3—32no—
81C22×Dic3—61no—
82C22×Dic3C2×D312yes—
83C2×D12—32no—
84C2×C4×D3—32no—
85C2×C4×D3—61no—
86C2×C4×D3C2×D312yes—
87C3×C4⋊C4—32no—
88C3×C22⋊C4—21no—
89C3×C22⋊C4—32no—
90C3×C22⋊C4C2×D312yes—
91C3×C42—32no—
92#217—32no—
93#217—32no—
94(C2×D3)⋊C4—32no—
95(C2×D3)⋊C4—32no—
96(C2×D3)⋊C4—32no—
97(C2×D3)⋊C4—32no—
98(C2×D3)⋊C4—32no—
99(C2×D3)⋊C4—61no—
100(C2×D3)⋊C4C2×D312yes—
101C4⋊Dic3—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—32no—
109Dic3⋊C4—61no—
110Dic3⋊C4C2×D312yes—
111C4×Dic3—32no—
112C4×Dic3—61no—
113C4×Dic3C2×D312yes—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32#4112yes—
117C22×C32C2411yescharacteristic
118C2×C3⋊D3—21no—
119C2×C3⋊D3—21no—
120C2×C3⋊D3—21no—
121C2×C3×D3—22no—
122C2×C3×D3—22no—
123C2×C3×D3—22no—
124C32×C4—21no—
125C32×C4—21no—
126C3⋊Dic3—21no—
127C3⋊Dic3—21no—
128C3⋊Dic3#4112yes—
129C3×Dic3—22no—
130C3×Dic3—22no—
131C3×Dic3—22no—
132#126—92no—
133#125—91no—
134#123—92no—
135#121—91no—
136#121—92no—
137#121—91no—
138C4×D4—32no—
139C4×D4—91no—
140C4×D4—91no—
141#117—91no—
142C2×C4⋊C4—91no—
143C23×C3—21no—
144C23×C3—32no—
145C23×C3C22×D312yes—
146C22×D3—32no—
147C22×D3—61no—
148C22×D3C22×D312yes—
149C3×D4—64no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—32no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—122no—
156C2×C3×C4—122no—
157C2×C3×C4C22×D312yes—
158C2×C3×C4C22×D312yes—
159#67—24no—
160#67—64no—
161#67—64no—
162#67—64no—
163#67—122no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—32no—
167C2×Dic3—42no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—64no—
172C2×Dic3—122no—
173C2×Dic3C22×D312yes—
174C2×Dic3C22×D312yes—
175C2×Dic3C22×D312yes—
176D12—122no—
177C4×D3—42no—
178C4×D3—64no—
179C4×D3—122no—
180C2×C32—22no—
181C2×C32C2×#4111yescharacteristic
182C2×C32C2×#4111yescharacteristic
183C2×C32H(5,𝔽2)11yescharacteristic
184C3⋊D3—41no—
185C3×D3—42no—
186C2×D4—32no—
187C2×D4—92no—
188C2×D4—91no—
189C2×D4—91no—
190C22×C4—32no—
191C22×C4—32no—
192C22×C4—91no—
193C22×C4—91no—
194C4⋊C4—32no—
195C4⋊C4—91no—
196C4⋊C4—91no—
197C4⋊C4—92no—
198C4⋊C4—91no—
199C4⋊C4—92no—
200C4⋊C4—91no—
201C22⋊C4—32no—
202C22⋊C4—92no—
203C22⋊C4—92no—
204C22⋊C4—92no—
205C22⋊C4—91no—
206C22⋊C4D3211yescharacteristic
207C42—32no—
208C42—91no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—21no—
212C22×C3—22no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—62no—
216C22×C3—62no—
217C22×C3—62no—
218C22×C3#23514yes—
219C22×C3C23×D312yes—
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—22no—
223C2×D3—62no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—121no—
228C2×D3—121no—
229C3×C4—22no—
230C3×C4—22no—
231C3×C4—41no—
232C3×C4—41no—
233C3×C4—62no—
234C3×C4—62no—
235C3×C4—62no—
236Dic3—22no—
237Dic3—22no—
238Dic3—22no—
239Dic3—34no—
240Dic3—62no—
241Dic3—62no—
242Dic3—62no—
243Dic3—62no—
244Dic3—121no—
245C32#55311yescharacteristic, complemented, Sylow
246C23—32no—
247C23—91no—
248C23C2×D3211yescharacteristic
249D4—64no—
250D4—182no—
251D4—182no—
252D4—362no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—182no—
262C2×C4—182no—
263C2×C4C2×D3211yescharacteristic
264C2×C4C2×D3211yescharacteristic
265C2×C3—24no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—42no—
270C2×C3—122no—
271C2×C3C2×#23512yes—
272C2×C3#100412yes—
273C2×C3D3×#4112yes—
274D3—42no—
275D3—122no—
276D3—122no—
277C22—21no—
278C22—21no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22—181no—
283C22—181no—
284C22—181no—
285C22#379812yes—
286C22C22×D3211yescharacteristic
287C4—21no—
288C4—21no—
289C4—62no—
290C4—62no—
291C4—62no—
292C4—92no—
293C4—181no—
294C4—181no—
295C3—21no—
296C3#554012yescomplemented
297C2—22no—
298C2—122no—
299C2—361no—
300C2C2×#379811yescharacteristic
301C2#6431211yescharacteristic
302C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64090C2semidirect
2. C3×#67⋊C4C2semidirect
3. #63980C2semidirect
4. #63978C2semidirect
5. C2×#3724C2semidirect
6. #63912C2semidirect
7. C4×#625C2semidirect
8. #63874C2semidirect
9. #63870C2semidirect
10. #63859C2semidirect
11. #63850C2semidirect
48. #67⋊C4D3semidirect
245. C32#553semidirect
296. C3#5540semidirect

Maximal subgroups

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

subgroupindex
#640902
C3×#67⋊C42
#639802
#639782
C2×#37242
#639122
C4×#6252
#638742
#638702
#638592
#638502
#55403