Finite Group Databasebeta

← #10596167

Subgroups — #10596167

Order 576.

576 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#125C211yesmaximal, characteristic, complemented
2#64107C211yesmaximal, characteristic
3#64107C211yesmaximal, characteristic
4#64107C211yesmaximal, characteristic
5#64105C211yesmaximal, characteristic
6#64105C211yesmaximal, characteristic
7#64101C211yesmaximal, characteristic
8#64099C211yesmaximal, characteristic, complemented
9#64098C211yesmaximal, characteristic, complemented
10#64098C211yesmaximal, characteristic, complemented
11#64098C211yesmaximal, characteristic, complemented
12#64097C211yesmaximal, characteristic, complemented
13#64096C211yesmaximal, characteristic, complemented
14#64085C211yesmaximal, characteristic
15C4×#634C211yesmaximal, characteristic, complemented
16#5696—34nomaximal
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×#634C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C32×C4⋊C4C2211yescharacteristic
21C32×C4⋊C4C2211yescharacteristic
22C32×C4⋊C4C2211yescharacteristic
23C32×C4⋊C4C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25C32×C4⋊C4C2211yescharacteristic
26C32×C42C2211yescharacteristic
27#3752C2211yescharacteristic
28#3752C2211yescharacteristic
29#3752C2211yescharacteristic
30#3752C2211yescharacteristic
31#3752C2211yescharacteristic
32#3752C2211yescharacteristic
33C4⋊(C3⋊Dic3)C2211yescharacteristic
34C4⋊(C3⋊Dic3)C2211yescharacteristic
35C4⋊(C3⋊Dic3)C2211yescharacteristic
36C4⋊(C3⋊Dic3)C2211yescharacteristic
37#3750C2211yescharacteristic
38#3750C2211yescharacteristic
39#3750C2211yescharacteristic
40#3750C2211yescharacteristic
41#3750C2211yescharacteristic
42#3750C2211yescharacteristic
43#3750C2211yescharacteristic
44#3750C2211yescharacteristic
45#3750C2211yescharacteristic
46#3750C2211yescharacteristic
47C4×C3⋊Dic3C2211yescharacteristic
48C4×C3⋊Dic3C2211yescharacteristic
49C4×C3⋊Dic3C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×#125D314yescomplemented
53#898—34no—
54#898—34no—
55#898—34no—
56#896—34no—
57#896—34no—
58#892—34no—
59#890—34no—
60#889—34no—
61#889—34no—
62#889—34no—
63#888—34no—
64#887—34no—
65#876—34no—
66C4×Dic6—34no—
67C22×C3⋊D3C2311yescharacteristic
68C2×C32×C4C2311yescharacteristic
69C2×C32×C4C2311yescharacteristic
70C2×C32×C4C2311yescharacteristic
71C2×C32×C4C2311yescharacteristic
72C2×C32×C4C2311yescharacteristic
73C2×C32×C4C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75C2×C3⋊Dic3C2311yescharacteristic
76C2×C3⋊Dic3C2311yescharacteristic
77C2×C3⋊Dic3C2311yescharacteristic
78C2×C3⋊Dic3C2311yescharacteristic
79C2×C3⋊Dic3C2311yescharacteristic
80C2×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊Dic3C2311yescharacteristic
82C4×C3⋊D3—22no—
83#634—22no—
84#634—22no—
85#541—91noSylow
86C2×C4×D3—34no—
87C2×Dic6—34no—
88C2×Dic6—34no—
89C3×C4⋊C4C2×D314yes—
90C3×C4⋊C4C2×D314yes—
91C3×C4⋊C4C2×D314yes—
92C3×C4⋊C4C2×D314yes—
93C3×C4⋊C4C2×D314yes—
94C3×C4⋊C4C2×D314yes—
95C3×C42C2×D314yes—
96(C2×D3)⋊C4—34no—
97(C2×D3)⋊C4—34no—
98(C2×D3)⋊C4—34no—
99(C2×D3)⋊C4—34no—
100(C2×D3)⋊C4—34no—
101(C2×D3)⋊C4—34no—
102C4⋊Dic3—34no—
103C4⋊Dic3—34no—
104C4⋊Dic3—34no—
105C4⋊Dic3—34no—
106Dic3⋊C4—34no—
107Dic3⋊C4—34no—
108Dic3⋊C4—34no—
109Dic3⋊C4—34no—
110Dic3⋊C4—34no—
111Dic3⋊C4—34no—
112Dic3⋊C4—34no—
113Dic3⋊C4—34no—
114Dic3⋊C4—34no—
115Dic3⋊C4—34no—
116C4×Dic3—34no—
117C4×Dic3—34no—
118C4×Dic3—34no—
119C4×Dic3—34no—
120C4×Dic3—34no—
121C22×C32C2411yescharacteristic
122C2×C3⋊D3—21no—
123C2×C3⋊D3—21no—
124C2×C3⋊D3—21no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—21no—
128C32×C4—21no—
129C32×C4—21no—
130C32×C4—21no—
131C32×C4—21no—
132C3⋊Dic3—21no—
133C3⋊Dic3—21no—
134C3⋊Dic3—21no—
135C3⋊Dic3—21no—
136C3⋊Dic3—21no—
137C3⋊Dic3—21no—
138C3⋊Dic3#4112yes—
139C4⋊Q8—91no—
140#126—91no—
141#126—91no—
142#126—91no—
143#126—91no—
144#125—91no—
145#125—91no—
146#125—91no—
147#125—91no—
148#125C3⋊D311yescharacteristic, complemented
149C22⋊Q8—91no—
150C22⋊Q8—91no—
151C4×Q8—91no—
152C4×Q8—91no—
153#117—91no—
154C22×D3—34no—
155C2×C3×C4C22×D314yes—
156C2×C3×C4C22×D314yes—
157C2×C3×C4C22×D314yes—
158C2×C3×C4C22×D314yes—
159C2×C3×C4C22×D314yes—
160C2×C3×C4C22×D314yes—
161C2×C3×C4C22×D314yes—
162C2×Dic3—34no—
163C2×Dic3—34no—
164C2×Dic3—34no—
165C2×Dic3—34no—
166C2×Dic3—34no—
167C2×Dic3—34no—
168C2×Dic3—34no—
169C4×D3—68no—
170Dic6—68no—
171Dic6—68no—
172C2×C32C2×#4111yescharacteristic
173C2×C322−1+411yescharacteristic
174C2×C322−1+411yescharacteristic
175C3⋊D3—41no—
176C2×Q8—91no—
177C2×Q8—91no—
178C22×C4—91no—
179C4⋊C4—91no—
180C4⋊C4—91no—
181C4⋊C4—91no—
182C4⋊C4—91no—
183C4⋊C4—91no—
184C4⋊C4—91no—
185C4⋊C4—91no—
186C4⋊C4—91no—
187C4⋊C4—91no—
188C4⋊C4—91no—
189C4⋊C4—91no—
190C4⋊C4—91no—
191C4⋊C4—91no—
192C4⋊C4—91no—
193C4⋊C4C2×C3⋊D311yescharacteristic
194C4⋊C4C2×C3⋊D311yescharacteristic
195C4⋊C4C2×C3⋊D311yescharacteristic
196C4⋊C4C2×C3⋊D311yescharacteristic
197C4⋊C4C2×C3⋊D311yescharacteristic
198C4⋊C4C2×C3⋊D311yescharacteristic
199C22⋊C4—91no—
200C22⋊C4—91no—
201C22⋊C4—91no—
202C22⋊C4—91no—
203C22⋊C4—91no—
204C22⋊C4—91no—
205C42—91no—
206C42—91no—
207C42—91no—
208C42—91no—
209C42—91no—
210C42C2×C3⋊D311yescharacteristic
211C22×C3C23×D314yes—
212C2×D3—64no—
213C2×D3—64no—
214C2×D3—64no—
215C3×C4—24no—
216C3×C4—24no—
217C3×C4—24no—
218C3×C4—24no—
219C3×C4—24no—
220C3×C4—24no—
221C3×C4—24no—
222Dic3—38no—
223Dic3—64no—
224Dic3—64no—
225Dic3—64no—
226Dic3—64no—
227Dic3—64no—
228Dic3—64no—
229C32#54111yescharacteristic, complemented, Sylow
230C23—91no—
231Q8—182no—
232Q8—182no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—182no—
241C2×C4C22×C3⋊D311yescharacteristic
242C2×C4C22×C3⋊D311yescharacteristic
243C2×C4C22×C3⋊D311yescharacteristic
244C2×C4C22×C3⋊D311yescharacteristic
245C2×C4C22×C3⋊D311yescharacteristic
246C2×C4C22×C3⋊D311yescharacteristic
247C2×C4C22×C3⋊D311yescharacteristic
248C2×C3#100714yes—
249C2×C3D3×#4114yes—
250C2×C3#101014yes—
251D3—124no—
252C22—181no—
253C22—181no—
254C22—181no—
255C22C23×C3⋊D311yescharacteristic
256C4—21no—
257C4—21no—
258C4—21no—
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—92no—
264C4—181no—
265C4—181no—
266C4—181no—
267C4—181no—
268C4—181no—
269C4—181no—
270C3#569614yescomplemented
271C2—361no—
272C2#6436411yescharacteristic
273C2#41×C3⋊D311yescharacteristic
274C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#125C2
prime radicalC22×C3⋊D3C23
hypercenter#125C3⋊D3
hypocenterC32#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C32×#125C2semidirect
8. #64099C2semidirect
9. #64098C2semidirect
10. #64098C2semidirect
11. #64098C2semidirect
12. #64097C2semidirect
13. #64096C2semidirect
15. C4×#634C2semidirect
52. C3×#125D3semidirect
148. #125C3⋊D3semidirect
229. C32#541semidirect
270. C3#5696semidirect

Maximal subgroups

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

subgroupindex
C32×#1252
#641072
#641052
#641012
#640992
#640982
#640972
#640962
#640852
C4×#6342
#56963