Finite Group Databasebeta

← #10596176

Subgroups — #10596176

Order 576.

744 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#126C211yesmaximal, characteristic, complemented
2#64106C211yesmaximal, characteristic, complemented
3#64105C211yesmaximal, characteristic, complemented
4#64103C211yesmaximal, characteristic, complemented
5#64103C211yesmaximal, characteristic, complemented
6#64102C211yesmaximal, characteristic, complemented
7#64101C211yesmaximal, characteristic, complemented
8#64094C211yesmaximal, characteristic, complemented
9#64092C211yesmaximal, characteristic, complemented
10#64091C211yesmaximal, characteristic
11C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
12C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
13#64088C211yesmaximal, characteristic, complemented
14C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
15#64081C211yesmaximal, characteristic, complemented
16#5705—34nomaximal
17C23×C3⋊D3C2211yescharacteristic
18C2×#638C2211yescharacteristic
19C2×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic
23C2×C4×C3⋊D3C2211yescharacteristic
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×C4×C3⋊D3C2211yescharacteristic, complemented
26C2×#634C2211yescharacteristic, complemented
27C32×C4⋊C4C2211yescharacteristic, complemented
28C32×C4⋊C4C2211yescharacteristic, complemented
29C32×C4⋊C4C2211yescharacteristic, complemented
30C32×C22⋊C4C2211yescharacteristic
31C32×C22⋊C4C2211yescharacteristic
32C32×C22⋊C4C2211yescharacteristic
33C32×C42C2211yescharacteristic, complemented
34#3757C2211yescharacteristic
35#3757C2211yescharacteristic
36#3752C2211yescharacteristic
37#3752C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40#3752C2211yescharacteristic
41#3752C2211yescharacteristic
42#3752C2211yescharacteristic, complemented
43#3752C2211yescharacteristic, complemented
44#3752C2211yescharacteristic, complemented
45C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
46C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
47#3750C2211yescharacteristic, complemented
48#3750C2211yescharacteristic, complemented
49#3750C2211yescharacteristic, complemented
50C4×C3⋊Dic3C2211yescharacteristic, complemented
51C4×C3⋊Dic3C2211yescharacteristic, complemented
52C3×#126D314yescomplemented
53#897—34no—
54#896—34no—
55#894—34no—
56#894—34no—
57#893—34no—
58#892—34no—
59#885—34no—
60#883—34no—
61#882—34no—
62D3×C22⋊C4—34no—
63D3×C22⋊C4—34no—
64#879—34no—
65C4×D12—34no—
66#872—34no—
67C23×C32C2311yescharacteristic
68C22×C3⋊D3—22no—
69C22×C3⋊D3—22no—
70C22×C3⋊D3—41no—
71C22×C3⋊D3C2311yescharacteristic
72C22×C3⋊D3C2311yescharacteristic
73C22×C3⋊D3C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76C2×C32×C4C2311yescharacteristic
77C2×C32×C4C2311yescharacteristic
78C2×C32×C4C2311yescharacteristic
79C2×C32×C4C2311yescharacteristic
80#638—41no—
81C2×C3⋊Dic3C2311yescharacteristic
82C2×C3⋊Dic3C2311yescharacteristic
83C2×C3⋊Dic3C2311yescharacteristic
84C2×C3⋊Dic3C2311yescharacteristic
85C2×C3⋊Dic3C2311yescharacteristic
86C4⋊(C3⋊D3)—22no—
87C4⋊(C3⋊D3)—22no—
88C4×C3⋊D3—22no—
89C4×C3⋊D3—22no—
90C4×C3⋊D3—41no—
91C4×C3⋊D3—41no—
92C4×C3⋊D3—41no—
93#634—41no—
94#551—91noSylow
95C23×D3—34no—
96C2×#67—34no—
97C2×D12—34no—
98C2×D12—34no—
99C2×C4×D3—34no—
100C2×C4×D3—34no—
101C2×C4×D3—34no—
102C2×C4×D3—34no—
103C2×C4×D3—34no—
104C2×Dic6—34no—
105C3×C4⋊C4C2×D314yescomplemented
106C3×C4⋊C4C2×D314yescomplemented
107C3×C4⋊C4C2×D314yescomplemented
108C3×C22⋊C4C2×D314yes—
109C3×C22⋊C4C2×D314yes—
110C3×C22⋊C4C2×D314yes—
111C3×C42C2×D314yescomplemented
112#217—34no—
113#217—34no—
114(C2×D3)⋊C4—34no—
115(C2×D3)⋊C4—34no—
116(C2×D3)⋊C4—34no—
117(C2×D3)⋊C4—34no—
118(C2×D3)⋊C4—34no—
119(C2×D3)⋊C4—34no—
120(C2×D3)⋊C4—34no—
121(C2×D3)⋊C4—34no—
122(C2×D3)⋊C4—34no—
123C4⋊Dic3—34no—
124C4⋊Dic3—34no—
125Dic3⋊C4—34no—
126Dic3⋊C4—34no—
127Dic3⋊C4—34no—
128C4×Dic3—34no—
129C4×Dic3—34no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32C2411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3—21no—
140C2×C3⋊D3—21no—
141C2×C3⋊D3—44no—
142C2×C3⋊D3#4112yes—
143C2×C3⋊D3#4112yes—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—21no—
150C3⋊Dic3—21no—
151C3⋊Dic3—21no—
152C3⋊Dic3—21no—
153C3⋊Dic3—21no—
154C3⋊Dic3—21no—
155#126—91no—
156#126C3⋊D311yescharacteristic, complemented
157#124—91no—
158#123—91no—
159#123—91no—
160#123—91no—
161C22⋊Q8—91no—
162C22⋊Q8—91no—
163#120—91no—
164C4×D4—91no—
165C4×D4—91no—
166#117—91no—
167#117—91no—
168C2×C22⋊C4—91no—
169C2×C22⋊C4—91no—
170C23×C3C22×D314yes—
171C22×D3—34no—
172C22×D3—34no—
173C22×D3—34no—
174C22×D3—68no—
175C22×D3—68no—
176C22×D3—124no—
177C2×C3×C4C22×D314yes—
178C2×C3×C4C22×D314yes—
179C2×C3×C4C22×D314yes—
180C2×C3×C4C22×D314yes—
181C2×C3×C4C22×D314yes—
182C2×C3×C4C22×D314yes—
183#67—124no—
184C2×Dic3—34no—
185C2×Dic3—34no—
186C2×Dic3—34no—
187C2×Dic3—34no—
188C2×Dic3—34no—
189D12—68no—
190D12—68no—
191C4×D3—68no—
192C4×D3—68no—
193C4×D3—124no—
194C4×D3—124no—
195C4×D3—124no—
196Dic6—124no—
197C2×C32—41no—
198C2×C32C2×#4111yescharacteristic
199C2×C32C2×#4111yescharacteristic
200C2×C32H(5,𝔽2)11yescharacteristic
201C3⋊D3—22no—
202C3⋊D3—22no—
203C3⋊D3—41no—
204C24—91no—
205C2×Q8—91no—
206C2×D4—91no—
207C2×D4—91no—
208C2×D4—91no—
209C22×C4—91no—
210C22×C4—91no—
211C22×C4—91no—
212C22×C4—91no—
213C22×C4—91no—
214C4⋊C4—91no—
215C4⋊C4—91no—
216C4⋊C4—91no—
217C4⋊C4—91no—
218C4⋊C4—91no—
219C4⋊C4C2×C3⋊D311yescharacteristic, complemented
220C4⋊C4C2×C3⋊D311yescharacteristic, complemented
221C4⋊C4C2×C3⋊D311yescharacteristic, complemented
222C22⋊C4—91no—
223C22⋊C4—91no—
224C22⋊C4—91no—
225C22⋊C4—91no—
226C22⋊C4—91no—
227C22⋊C4—91no—
228C22⋊C4—91no—
229C22⋊C4—91no—
230C22⋊C4—91no—
231C22⋊C4—91no—
232C22⋊C4—91no—
233C22⋊C4C2×C3⋊D311yescharacteristic
234C22⋊C4C2×C3⋊D311yescharacteristic
235C22⋊C4C2×C3⋊D311yescharacteristic
236C42—91no—
237C42—91no—
238C42C2×C3⋊D311yescharacteristic, complemented
239C22×C3—24no—
240C22×C3—24no—
241C22×C3—24no—
242C22×C3C23×D314yes—
243C2×D3—38no—
244C2×D3—38no—
245C2×D3—64no—
246C2×D3—64no—
247C2×D3—64no—
248C2×D3—64no—
249C2×D3—64no—
250C2×D3—64no—
251C2×D3—64no—
252C2×D3—1216no—
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—24no—
256C3×C4—24no—
257C3×C4—24no—
258C3×C4—24no—
259Dic3—64no—
260Dic3—64no—
261Dic3—64no—
262Dic3—64no—
263Dic3—64no—
264C32#55111yescharacteristic, complemented, Sylow
265C23—91no—
266C23—91no—
267C23—91no—
268C23—182no—
269C23—182no—
270C23—361no—
271C23C22×C3⋊D311yescharacteristic
272Q8—361no—
273D4—182no—
274D4—182no—
275D4—361no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—182no—
282C2×C4—182no—
283C2×C4—361no—
284C2×C4—361no—
285C2×C4—361no—
286C2×C4C22×C3⋊D311yescharacteristic
287C2×C4C22×C3⋊D311yescharacteristic
288C2×C4C22×C3⋊D311yescharacteristic
289C2×C4C22×C3⋊D311yescharacteristic
290C2×C4C22×C3⋊D311yescharacteristic
291C2×C4C22×C3⋊D311yescharacteristic
292C2×C3—44no—
293C2×C3D3×#4114yes—
294C2×C3D3×#4114yes—
295C2×C3#100914yes—
296D3—68no—
297D3—68no—
298D3—124no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—92no—
303C22—92no—
304C22—181no—
305C22—181no—
306C22—181no—
307C22—181no—
308C22—181no—
309C22—181no—
310C22—181no—
311C22—364no—
312C22C23×C3⋊D311yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—181no—
320C4—181no—
321C4—181no—
322C4—181no—
323C4—181no—
324C3#570514yescomplemented
325C2—41no—
326C2—182no—
327C2—182no—
328C2—361no—
329C2#41×C3⋊D311yescharacteristic
330C2#41×C3⋊D311yescharacteristic
331C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#126C2
prime radical#64091C2
hypercenter#126C3⋊D3
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C32×#126C2semidirect
2. #64106C2semidirect
3. #64105C2semidirect
4. #64103C2semidirect
5. #64103C2semidirect
6. #64102C2semidirect
7. #64101C2semidirect
8. #64094C2semidirect
9. #64092C2semidirect
11. C22⋊C4×C3⋊D3C2semidirect
12. C22⋊C4×C3⋊D3C2semidirect
13. #64088C2semidirect
14. C4×C4⋊(C3⋊D3)C2semidirect
15. #64081C2semidirect
25. C2×C4×C3⋊D3C22semidirect
26. C2×#634C22semidirect
27. C32×C4⋊C4C22semidirect
28. C32×C4⋊C4C22semidirect
29. C32×C4⋊C4C22semidirect
33. C32×C42C22semidirect
42. #3752C22semidirect
43. #3752C22semidirect
44. #3752C22semidirect
45. C4⋊(C3⋊Dic3)C22semidirect
46. C4⋊(C3⋊Dic3)C22semidirect
47. #3750C22semidirect
48. #3750C22semidirect
49. #3750C22semidirect
50. C4×C3⋊Dic3C22semidirect
51. C4×C3⋊Dic3C22semidirect
52. C3×#126D3semidirect
105. C3×C4⋊C4C2×D3semidirect
106. C3×C4⋊C4C2×D3semidirect
107. C3×C4⋊C4C2×D3semidirect
111. C3×C42C2×D3semidirect
156. #126C3⋊D3semidirect
219. C4⋊C4C2×C3⋊D3semidirect
220. C4⋊C4C2×C3⋊D3semidirect
221. C4⋊C4C2×C3⋊D3semidirect
238. C42C2×C3⋊D3semidirect
264. C32#551semidirect
324. C3#5705semidirect

Maximal subgroups

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

subgroupindex
C32×#1262
#641062
#641052
#641032
#641022
#641012
#640942
#640922
#640912
C22⋊C4×C3⋊D32
#640882
C4×C4⋊(C3⋊D3)2
#640812
#57053