Finite Group Databasebeta

← #10596134

Subgroups — #10596134

Order 576.

720 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#123C211yesmaximal, characteristic, complemented
2#64149C211yesmaximal, characteristic, complemented
3#64148C211yesmaximal, characteristic, complemented
4#64139C211yesmaximal, characteristic, complemented
5C4×#638C211yesmaximal, characteristic, complemented
6#64104C211yesmaximal, characteristic, complemented
7#64103C211yesmaximal, characteristic, complemented
8#64101C211yesmaximal, characteristic, complemented
9#64098C211yesmaximal, characteristic, complemented
10#64095C211yesmaximal, characteristic, complemented
11#64093C211yesmaximal, characteristic, complemented
12#64093C211yesmaximal, characteristic, complemented
13#64093C211yesmaximal, characteristic, complemented
14#64092C211yesmaximal, characteristic, complemented
15#64090C211yesmaximal, characteristic, complemented
16#5663—34nomaximal
17C2×C32×D4C2211yescharacteristic
18C22×C32×C4C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×#638C2211yescharacteristic
22C2×#638C2211yescharacteristic
23C2×#638C2211yescharacteristic
24C2×#638C2211yescharacteristic
25C22×C3⋊Dic3C2211yescharacteristic
26C2×C4⋊(C3⋊D3)C2211yescharacteristic
27C2×C4⋊(C3⋊D3)C2211yescharacteristic
28C2×C4⋊(C3⋊D3)C2211yescharacteristic
29C2×C4×C3⋊D3C2211yescharacteristic
30C2×C4×C3⋊D3C2211yescharacteristic
31C2×C4×C3⋊D3C2211yescharacteristic
32C32×C4⋊C4C2211yescharacteristic
33C32×C4⋊C4C2211yescharacteristic
34C32×C22⋊C4C2211yescharacteristic
35C32×C22⋊C4C2211yescharacteristic
36C32×C22⋊C4C2211yescharacteristic
37#3757C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40#3752C2211yescharacteristic
41#3752C2211yescharacteristic
42#3752C2211yescharacteristic
43#3752C2211yescharacteristic
44C4⋊(C3⋊Dic3)C2211yescharacteristic
45C4⋊(C3⋊Dic3)C2211yescharacteristic
46#3750C2211yescharacteristic
47#3750C2211yescharacteristic
48#3750C2211yescharacteristic
49#3750C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×#123D314yescomplemented
53#940—34no—
54#939—34no—
55#930—34no—
56C4×#67—34no—
57#895—34no—
58#894—34no—
59#892—34no—
60#889—34no—
61#886—34no—
62#884—34no—
63#884—34no—
64#884—34no—
65#883—34no—
66#67⋊C4—34no—
67C23×C32C2311yescharacteristic
68C23×C32C2311yescharacteristic
69C22×C3⋊D3C2311yescharacteristic
70C22×C3⋊D3C2311yescharacteristic
71C22×C3⋊D3C2311yescharacteristic
72C32×D4—41no—
73C2×C32×C4—41no—
74C2×C32×C4C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76C2×C32×C4C2311yescharacteristic
77C2×C32×C4C2311yescharacteristic
78C2×C32×C4C2311yescharacteristic
79#638—22no—
80#638—22no—
81#638—41no—
82#638—41no—
83#638—41no—
84#638—41no—
85C2×C3⋊Dic3—41no—
86C2×C3⋊Dic3C2311yescharacteristic
87C2×C3⋊Dic3C2311yescharacteristic
88C2×C3⋊Dic3C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C4⋊(C3⋊D3)—41no—
92C4⋊(C3⋊D3)—41no—
93C4⋊(C3⋊D3)—41no—
94C4×C3⋊D3—22no—
95C4×C3⋊D3—41no—
96C4×C3⋊D3—41no—
97#540—91noSylow
98C2×C3×D4C2×D314yes—
99C22×C3×C4C2×D314yes—
100C2×#67—34no—
101C2×#67—34no—
102C2×#67—34no—
103C2×#67—34no—
104C2×#67—34no—
105C2×#67—34no—
106C22×Dic3—34no—
107C2×D12—34no—
108C2×D12—34no—
109C2×D12—34no—
110C2×C4×D3—34no—
111C2×C4×D3—34no—
112C2×C4×D3—34no—
113C3×C4⋊C4C2×D314yes—
114C3×C4⋊C4C2×D314yes—
115C3×C22⋊C4C2×D314yes—
116C3×C22⋊C4C2×D314yes—
117C3×C22⋊C4C2×D314yes—
118#217—34no—
119(C2×D3)⋊C4—34no—
120(C2×D3)⋊C4—34no—
121(C2×D3)⋊C4—34no—
122(C2×D3)⋊C4—34no—
123(C2×D3)⋊C4—34no—
124(C2×D3)⋊C4—34no—
125C4⋊Dic3—34no—
126C4⋊Dic3—34no—
127Dic3⋊C4—34no—
128Dic3⋊C4—34no—
129Dic3⋊C4—34no—
130Dic3⋊C4—34no—
131C4×Dic3—34no—
132C4×Dic3—34no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32C2411yescharacteristic
140C2×C3⋊D3—21no—
141C2×C3⋊D3—21no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3—21no—
144C2×C3⋊D3—21no—
145C2×C3⋊D3—21no—
146C2×C3⋊D3—21no—
147C2×C3⋊D3—21no—
148C2×C3⋊D3—21no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4—21no—
154C3⋊Dic3—21no—
155C3⋊Dic3—21no—
156C3⋊Dic3—21no—
157C3⋊Dic3—21no—
158C3⋊Dic3#4112yes—
159#127—91no—
160#125—91no—
161#123—91no—
162#123—91no—
163#123—91no—
164#123C3⋊D311yescharacteristic, complemented
165#121—91no—
166#121—91no—
167#121—91no—
168#121—91no—
169#121—91no—
170#121—91no—
171C4×D4—91no—
172C4×D4—91no—
173#117—91no—
174C23×C3C22×D314yes—
175C23×C3C22×D314yes—
176C22×D3—34no—
177C22×D3—34no—
178C22×D3—34no—
179C3×D4—44no—
180C2×C3×C4—44no—
181C2×C3×C4C22×D314yes—
182C2×C3×C4C22×D314yes—
183C2×C3×C4C22×D314yes—
184C2×C3×C4C22×D314yes—
185C2×C3×C4C22×D314yes—
186#67—68no—
187#67—68no—
188#67—124no—
189#67—124no—
190#67—124no—
191#67—124no—
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—34no—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—124no—
198D12—124no—
199D12—124no—
200D12—124no—
201C4×D3—68no—
202C4×D3—124no—
203C4×D3—124no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32C2×#4111yescharacteristic
207C2×C32H(5,𝔽2)11yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C3⋊D3—41no—
210C3⋊D3—41no—
211C3⋊D3—41no—
212C2×D4—91no—
213C2×D4—91no—
214C2×D4—91no—
215C2×D4—91no—
216C2×D4—91no—
217C2×D4—91no—
218C2×D4—91no—
219C2×D4—91no—
220C2×D4—91no—
221C2×D4C2×C3⋊D311yescharacteristic
222C22×C4—91no—
223C22×C4—91no—
224C22×C4—91no—
225C22×C4—91no—
226C22×C4C2×C3⋊D311yescharacteristic
227C4⋊C4—91no—
228C4⋊C4—91no—
229C4⋊C4—91no—
230C4⋊C4—91no—
231C4⋊C4—91no—
232C4⋊C4—91no—
233C4⋊C4C2×C3⋊D311yescharacteristic
234C4⋊C4C2×C3⋊D311yescharacteristic
235C22⋊C4—91no—
236C22⋊C4—91no—
237C22⋊C4—91no—
238C22⋊C4—91no—
239C22⋊C4—91no—
240C22⋊C4—91no—
241C22⋊C4—91no—
242C22⋊C4C2×C3⋊D311yescharacteristic
243C22⋊C4C2×C3⋊D311yescharacteristic
244C22⋊C4C2×C3⋊D311yescharacteristic
245C42—91no—
246C42—91no—
247C22×C3—24no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3—24no—
253C22×C3C23×D314yes—
254C2×D3—64no—
255C2×D3—64no—
256C2×D3—64no—
257C2×D3—64no—
258C2×D3—64no—
259C2×D3—64no—
260C2×D3—64no—
261C2×D3—64no—
262C2×D3—64no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—24no—
268Dic3—38no—
269Dic3—64no—
270Dic3—64no—
271Dic3—64no—
272Dic3—64no—
273C32#54011yescharacteristic, complemented, Sylow
274C23—91no—
275C23—91no—
276C23—91no—
277C23C22×C3⋊D311yescharacteristic
278C23C22×C3⋊D311yescharacteristic
279D4—41no—
280D4—182no—
281D4—182no—
282D4—361no—
283D4—361no—
284D4—361no—
285D4—361no—
286D4—361no—
287D4—361no—
288D4—361no—
289C2×C4—41no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—182no—
296C2×C4—361no—
297C2×C4—361no—
298C2×C4—361no—
299C2×C4C22×C3⋊D311yescharacteristic
300C2×C4C22×C3⋊D311yescharacteristic
301C2×C4C22×C3⋊D311yescharacteristic
302C2×C4C22×C3⋊D311yescharacteristic
303C2×C4C22×C3⋊D311yescharacteristic
304C2×C3—44no—
305C2×C3—44no—
306C2×C3#100414yes—
307C2×C3D3×#4114yes—
308C2×C3#100914yes—
309D3—124no—
310D3—124no—
311D3—124no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—181no—
319C22—181no—
320C22—181no—
321C22—181no—
322C22—181no—
323C22—181no—
324C22—181no—
325C22—181no—
326C22—181no—
327C22C23×C3⋊D311yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—92no—
334C4—181no—
335C4—181no—
336C4—181no—
337C4—181no—
338C3#566314yescomplemented
339C2—41no—
340C2—41no—
341C2—361no—
342C2—361no—
343C2—361no—
344C2#6436111yescharacteristic
345C2#41×C3⋊D311yescharacteristic
346C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#123C2
hypercenter#123C3⋊D3
hypocenterC32#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C32×#123C2semidirect
2. #64149C2semidirect
3. #64148C2semidirect
4. #64139C2semidirect
5. C4×#638C2semidirect
6. #64104C2semidirect
7. #64103C2semidirect
8. #64101C2semidirect
9. #64098C2semidirect
10. #64095C2semidirect
11. #64093C2semidirect
12. #64093C2semidirect
13. #64093C2semidirect
14. #64092C2semidirect
15. #64090C2semidirect
52. C3×#123D3semidirect
164. #123C3⋊D3semidirect
273. C32#540semidirect
338. C3#5663semidirect

Maximal subgroups

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

subgroupindex
C32×#1232
#641492
#641482
#641392
C4×#6382
#641042
#641032
#641012
#640982
#640952
#640932
#640922
#640902
#56633