Finite Group Databasebeta

← #10596175

Subgroups — #10596175

Order 576.

750 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#126C211yesmaximal, characteristic, complemented
2#64107C211yesmaximal, characteristic, complemented
3#64105C211yesmaximal, characteristic, complemented
4#64104C211yesmaximal, characteristic, complemented
5#64104C211yesmaximal, characteristic, complemented
6#64103C211yesmaximal, characteristic, complemented
7#64102C211yesmaximal, characteristic, complemented
8#64094C211yesmaximal, characteristic, complemented
9#64093C211yesmaximal, characteristic, complemented
10#64092C211yesmaximal, characteristic, complemented
11#64091C211yesmaximal, characteristic, complemented
12#64091C211yesmaximal, characteristic, complemented
13C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
14#64084C211yesmaximal, characteristic, complemented
15C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
16#5704—34nomaximal
17C23×C3⋊D3C2211yescharacteristic
18C2×#638C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C4⋊(C3⋊D3)C2211yescharacteristic
23C2×C4⋊(C3⋊D3)C2211yescharacteristic
24C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
25C2×C4×C3⋊D3C2211yescharacteristic
26C2×C4×C3⋊D3C2211yescharacteristic
27C2×C4×C3⋊D3C2211yescharacteristic, complemented
28C2×C4×C3⋊D3C2211yescharacteristic, complemented
29C2×#634C2211yescharacteristic, complemented
30C32×C4⋊C4C2211yescharacteristic, complemented
31C32×C4⋊C4C2211yescharacteristic, complemented
32C32×C4⋊C4C2211yescharacteristic, complemented
33C32×C22⋊C4C2211yescharacteristic
34C32×C22⋊C4C2211yescharacteristic
35C32×C22⋊C4C2211yescharacteristic
36C32×C42C2211yescharacteristic, complemented
37#3757C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40#3752C2211yescharacteristic
41#3752C2211yescharacteristic
42#3752C2211yescharacteristic
43#3752C2211yescharacteristic
44#3752C2211yescharacteristic, complemented
45#3752C2211yescharacteristic, complemented
46#3752C2211yescharacteristic, complemented
47#3752C2211yescharacteristic, complemented
48C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
49#3750C2211yescharacteristic, complemented
50#3750C2211yescharacteristic, complemented
51C4×C3⋊Dic3C2211yescharacteristic, complemented
52C3×#126D314yescomplemented
53#898—34no—
54#896—34no—
55#895—34no—
56#895—34no—
57#894—34no—
58#893—34no—
59#885—34no—
60#884—34no—
61#883—34no—
62#882—34no—
63#882—34no—
64D3×C22⋊C4—34no—
65#875—34no—
66C4×D12—34no—
67C23×C32C2311yescharacteristic
68C22×C3⋊D3—22no—
69C22×C3⋊D3—41no—
70C22×C3⋊D3—41no—
71C22×C3⋊D3C2311yescharacteristic
72C22×C3⋊D3C2311yescharacteristic
73C22×C3⋊D3C2311yescharacteristic
74C22×C3⋊D3C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76C2×C32×C4C2311yescharacteristic
77C2×C32×C4C2311yescharacteristic
78C2×C32×C4C2311yescharacteristic
79C2×C32×C4C2311yescharacteristic
80C2×C32×C4C2311yescharacteristic
81#638—41no—
82#638—41no—
83C2×C3⋊Dic3C2311yescharacteristic
84C2×C3⋊Dic3C2311yescharacteristic
85C2×C3⋊Dic3C2311yescharacteristic
86C2×C3⋊Dic3C2311yescharacteristic
87C4⋊(C3⋊D3)—22no—
88C4⋊(C3⋊D3)—22no—
89C4⋊(C3⋊D3)—41no—
90C4⋊(C3⋊D3)—41no—
91C4⋊(C3⋊D3)—41no—
92C4×C3⋊D3—41no—
93C4×C3⋊D3—41no—
94C4×C3⋊D3—41no—
95C4×C3⋊D3—41no—
96#634—41no—
97#538—91noSylow
98C23×D3—34no—
99C2×#67—34no—
100C2×#67—34no—
101C2×D12—34no—
102C2×D12—34no—
103C2×D12—34no—
104C2×D12—34no—
105C2×D12—34no—
106C2×C4×D3—34no—
107C2×C4×D3—34no—
108C2×C4×D3—34no—
109C2×C4×D3—34no—
110C2×Dic6—34no—
111C3×C4⋊C4C2×D314yescomplemented
112C3×C4⋊C4C2×D314yescomplemented
113C3×C4⋊C4C2×D314yescomplemented
114C3×C22⋊C4C2×D314yes—
115C3×C22⋊C4C2×D314yes—
116C3×C22⋊C4C2×D314yes—
117C3×C42C2×D314yescomplemented
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—
125(C2×D3)⋊C4—34no—
126(C2×D3)⋊C4—34no—
127(C2×D3)⋊C4—34no—
128(C2×D3)⋊C4—34no—
129C4⋊Dic3—34no—
130Dic3⋊C4—34no—
131Dic3⋊C4—34no—
132C4×Dic3—34no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32C2411yescharacteristic
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3—21no—
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—44no—
149C2×C3⋊D3#4112yes—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—21no—
155C32×C4—21no—
156C3⋊Dic3—21no—
157C3⋊Dic3—21no—
158C3⋊Dic3—21no—
159C3⋊Dic3—21no—
160#126—91no—
161#126C3⋊D311yescharacteristic, complemented
162#124—91no—
163#124—91no—
164#123—91no—
165#123—91no—
166C22⋊Q8—91no—
167#121—91no—
168#121—91no—
169#121—91no—
170#120—91no—
171#120—91no—
172C4×D4—91no—
173C4×D4—91no—
174C2×C22⋊C4—91no—
175C23×C3C22×D314yes—
176C22×D3—34no—
177C22×D3—34no—
178C22×D3—34no—
179C22×D3—34no—
180C22×D3—68no—
181C22×D3—124no—
182C22×D3—124no—
183C2×C3×C4C22×D314yes—
184C2×C3×C4C22×D314yes—
185C2×C3×C4C22×D314yes—
186C2×C3×C4C22×D314yes—
187C2×C3×C4C22×D314yes—
188C2×C3×C4C22×D314yes—
189#67—124no—
190#67—124no—
191C2×Dic3—34no—
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—34no—
195D12—68no—
196D12—68no—
197D12—124no—
198D12—124no—
199D12—124no—
200C4×D3—124no—
201C4×D3—124no—
202C4×D3—124no—
203C4×D3—124no—
204Dic6—124no—
205C2×C32—41no—
206C2×C32C2×#4111yescharacteristic
207C2×C32H(5,𝔽2)11yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C3⋊D3—22no—
210C3⋊D3—41no—
211C3⋊D3—41no—
212C3⋊D3—41no—
213C24—91no—
214C2×Q8—91no—
215C2×D4—91no—
216C2×D4—91no—
217C2×D4—91no—
218C2×D4—91no—
219C2×D4—91no—
220C2×D4—91no—
221C2×D4—91no—
222C22×C4—91no—
223C22×C4—91no—
224C22×C4—91no—
225C22×C4—91no—
226C4⋊C4—91no—
227C4⋊C4—91no—
228C4⋊C4—91no—
229C4⋊C4C2×C3⋊D311yescharacteristic, complemented
230C4⋊C4C2×C3⋊D311yescharacteristic, complemented
231C4⋊C4C2×C3⋊D311yescharacteristic, complemented
232C22⋊C4—91no—
233C22⋊C4—91no—
234C22⋊C4—91no—
235C22⋊C4—91no—
236C22⋊C4—91no—
237C22⋊C4—91no—
238C22⋊C4—91no—
239C22⋊C4—91no—
240C22⋊C4—91no—
241C22⋊C4—91no—
242C22⋊C4—91no—
243C22⋊C4C2×C3⋊D311yescharacteristic
244C22⋊C4C2×C3⋊D311yescharacteristic
245C22⋊C4C2×C3⋊D311yescharacteristic
246C42—91no—
247C42C2×C3⋊D311yescharacteristic, complemented
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3C23×D314yes—
252C2×D3—38no—
253C2×D3—64no—
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—
263C2×D3—64no—
264C2×D3—1216no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—24no—
271Dic3—64no—
272Dic3—64no—
273Dic3—64no—
274Dic3—64no—
275C32#53811yescharacteristic, complemented, Sylow
276C23—91no—
277C23—91no—
278C23—91no—
279C23—91no—
280C23—182no—
281C23—361no—
282C23—361no—
283C23C22×C3⋊D311yescharacteristic
284Q8—361no—
285D4—182no—
286D4—182no—
287D4—361no—
288D4—361no—
289D4—361no—
290D4—361no—
291D4—361no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—361no—
297C2×C4—361no—
298C2×C4—361no—
299C2×C4—361no—
300C2×C4C22×C3⋊D311yescharacteristic
301C2×C4C22×C3⋊D311yescharacteristic
302C2×C4C22×C3⋊D311yescharacteristic
303C2×C4C22×C3⋊D311yescharacteristic
304C2×C4C22×C3⋊D311yescharacteristic
305C2×C4C22×C3⋊D311yescharacteristic
306C2×C3—44no—
307C2×C3D3×#4114yes—
308C2×C3#100914yes—
309C2×C3#100914yes—
310D3—68no—
311D3—124no—
312D3—124no—
313D3—124no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—92no—
318C22—181no—
319C22—181no—
320C22—181no—
321C22—181no—
322C22—181no—
323C22—181no—
324C22—181no—
325C22—181no—
326C22—181no—
327C22—181no—
328C22—181no—
329C22—364no—
330C22C23×C3⋊D311yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—181no—
338C4—181no—
339C4—181no—
340C4—181no—
341C3#570414yescomplemented
342C2—41no—
343C2—182no—
344C2—361no—
345C2—361no—
346C2—361no—
347C2#41×C3⋊D311yescharacteristic
348C2#6436611yescharacteristic
349C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#126C2
hypercenter#126C3⋊D3
hypocenterC32#538
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. #64107C2semidirect
3. #64105C2semidirect
4. #64104C2semidirect
5. #64104C2semidirect
6. #64103C2semidirect
7. #64102C2semidirect
8. #64094C2semidirect
9. #64093C2semidirect
10. #64092C2semidirect
11. #64091C2semidirect
12. #64091C2semidirect
13. C22⋊C4×C3⋊D3C2semidirect
14. #64084C2semidirect
15. C4×C4⋊(C3⋊D3)C2semidirect
24. C2×C4⋊(C3⋊D3)C22semidirect
27. C2×C4×C3⋊D3C22semidirect
28. C2×C4×C3⋊D3C22semidirect
29. C2×#634C22semidirect
30. C32×C4⋊C4C22semidirect
31. C32×C4⋊C4C22semidirect
32. C32×C4⋊C4C22semidirect
36. C32×C42C22semidirect
44. #3752C22semidirect
45. #3752C22semidirect
46. #3752C22semidirect
47. #3752C22semidirect
48. C4⋊(C3⋊Dic3)C22semidirect
49. #3750C22semidirect
50. #3750C22semidirect
51. C4×C3⋊Dic3C22semidirect
52. C3×#126D3semidirect
111. C3×C4⋊C4C2×D3semidirect
112. C3×C4⋊C4C2×D3semidirect
113. C3×C4⋊C4C2×D3semidirect
117. C3×C42C2×D3semidirect
161. #126C3⋊D3semidirect
229. C4⋊C4C2×C3⋊D3semidirect
230. C4⋊C4C2×C3⋊D3semidirect
231. C4⋊C4C2×C3⋊D3semidirect
247. C42C2×C3⋊D3semidirect
275. C32#538semidirect
341. C3#5704semidirect

Maximal subgroups

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

subgroupindex
C32×#1262
#641072
#641052
#641042
#641032
#641022
#640942
#640932
#640922
#640912
C22⋊C4×C3⋊D32
#640842
C4×C4⋊(C3⋊D3)2
#57043