Finite Group Databasebeta

← #10596064

Subgroups — #10596064

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×#120C211yesmaximal, characteristic, complemented
2#64162C211yesmaximal, characteristic, complemented
3C2×#3757C211yesmaximal, characteristic, complemented
4#64148C211yesmaximal, characteristic, complemented
5#64148C211yesmaximal, characteristic, complemented
6#64147C211yesmaximal, characteristic, complemented
7#64145C211yesmaximal, characteristic, complemented
8#64144C211yesmaximal, characteristic, complemented
9D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
10#64094C211yesmaximal, characteristic, complemented
11#64092C211yesmaximal, characteristic, complemented
12#64090C211yesmaximal, characteristic, complemented
13#64088C211yesmaximal, characteristic, complemented
14#64088C211yesmaximal, characteristic, complemented
15#64087C211yesmaximal, characteristic, complemented
16#5593—34nomaximal
17C24×C32C2211yescharacteristic
18C2×C32×D4C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C2×C32×D4C2211yescharacteristic
21C2×#638C2211yescharacteristic
22C2×#638C2211yescharacteristic
23C2×#638C2211yescharacteristic
24C2×#638C2211yescharacteristic, complemented
25C22×C3⋊Dic3C2211yescharacteristic
26C22×C3⋊Dic3C2211yescharacteristic
27C22×C3⋊Dic3C2211yescharacteristic, complemented
28C2×C4×C3⋊D3C2211yescharacteristic, complemented
29C2×#634C2211yescharacteristic, complemented
30C32×C22⋊C4C2211yescharacteristic
31C32×C22⋊C4C2211yescharacteristic
32C32×C22⋊C4C2211yescharacteristic
33#3757C2211yescharacteristic
34#3757C2211yescharacteristic
35#3757C2211yescharacteristic
36#3757C2211yescharacteristic
37#3757C2211yescharacteristic
38#3757C2211yescharacteristic
39#3757C2211yescharacteristic
40#3757C2211yescharacteristic, complemented
41#3757C2211yescharacteristic, complemented
42#3752C2211yescharacteristic, complemented
43#3752C2211yescharacteristic, complemented
44C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
45C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
46#3750C2211yescharacteristic, complemented
47#3750C2211yescharacteristic, complemented
48#3750C2211yescharacteristic, complemented
49#3750C2211yescharacteristic, complemented
50C4×C3⋊Dic3C2211yescharacteristic, complemented
51C4×C3⋊Dic3C2211yescharacteristic, complemented
52C3×#120D314yescomplemented
53#953—34no—
54C2×#217—34no—
55#939—34no—
56#939—34no—
57#938—34no—
58#936—34no—
59#935—34no—
60D4×Dic3—34no—
61#885—34no—
62#883—34no—
63#67⋊C4—34no—
64#879—34no—
65#879—34no—
66#878—34no—
67C23×C32—22no—
68C23×C32—41no—
69C23×C32—41no—
70C23×C32C2311yescharacteristic
71C23×C32C2311yescharacteristic
72C23×C32C2311yescharacteristic
73C23×C32C2311yescharacteristic
74C22×C3⋊D3C2311yescharacteristic
75C32×D4—22no—
76C32×D4—41no—
77C32×D4—41no—
78C2×C32×C4C2311yescharacteristic
79C2×C32×C4C2311yescharacteristic
80C2×C32×C4C2311yescharacteristic
81#638—22no—
82#638—41no—
83#638—41no—
84#638—41no—
85C2×C3⋊Dic3—41no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3—41no—
88C2×C3⋊Dic3C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C4×C3⋊D3—41no—
96#634—41no—
97#538—91noSylow
98C24×C3C2×D314yes—
99C2×C3×D4C2×D314yes—
100C2×C3×D4C2×D314yes—
101C2×C3×D4C2×D314yes—
102C2×#67—34no—
103C2×#67—34no—
104C2×#67—34no—
105C2×#67—34no—
106C22×Dic3—34no—
107C22×Dic3—34no—
108C22×Dic3—34no—
109C2×C4×D3—34no—
110C2×Dic6—34no—
111C3×C22⋊C4C2×D314yes—
112C3×C22⋊C4C2×D314yes—
113C3×C22⋊C4C2×D314yes—
114#217—34no—
115#217—34no—
116#217—34no—
117#217—34no—
118#217—34no—
119#217—34no—
120#217—34no—
121#217—34no—
122#217—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×C32—21no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—44no—
145C22×C32#4112yes—
146C22×C32C2411yescharacteristic
147C2×C3⋊D3—21no—
148C2×C3⋊D3—21no—
149C2×C3⋊D3—21no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—21no—
153C3⋊Dic3—21no—
154C3⋊Dic3—21no—
155C3⋊Dic3—21no—
156C3⋊Dic3—21no—
157C3⋊Dic3—21no—
158C3⋊Dic3—21no—
159C3⋊Dic3—21no—
160#126—91no—
161#126—91no—
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#120C3⋊D311yescharacteristic, complemented
172C4×D4—91no—
173C4×D4—91no—
174C2×C22⋊C4—91no—
175C23×C3—28no—
176C23×C3—44no—
177C23×C3—44no—
178C23×C3C22×D314yes—
179C23×C3C22×D314yes—
180C23×C3C22×D314yes—
181C23×C3C22×D314yes—
182C22×D3—34no—
183C3×D4—28no—
184C3×D4—44no—
185C3×D4—44no—
186C2×C3×C4C22×D314yes—
187C2×C3×C4C22×D314yes—
188C2×C3×C4C22×D314yes—
189#67—68no—
190#67—124no—
191#67—124no—
192#67—124no—
193C2×Dic3—34no—
194C2×Dic3—34no—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—34no—
199C2×Dic3—34no—
200C2×Dic3—124no—
201C2×Dic3—124no—
202C2×Dic3—124no—
203C4×D3—124no—
204Dic6—124no—
205C2×C32—22no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32C2×#4111yescharacteristic
210C2×C32H(5,𝔽2)11yescharacteristic
211C2×C32H(5,𝔽2)11yescharacteristic
212C3⋊D3—41no—
213C24C2×C3⋊D311yescharacteristic
214C2×Q8—91no—
215C2×D4—91no—
216C2×D4—91no—
217C2×D4—91no—
218C2×D4—91no—
219C2×D4C2×C3⋊D311yescharacteristic
220C2×D4C2×C3⋊D311yescharacteristic
221C2×D4C2×C3⋊D311yescharacteristic
222C22×C4—91no—
223C22×C4—91no—
224C22×C4—91no—
225C22×C4—91no—
226C4⋊C4—91no—
227C4⋊C4—91no—
228C4⋊C4—91no—
229C4⋊C4—91no—
230C4⋊C4—91no—
231C4⋊C4—91no—
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—
247C42—91no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3—24no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—24no—
256C22×C3—24no—
257C22×C3—24no—
258C22×C3—24no—
259C22×C3—416no—
260C22×C3#23718yes—
261C22×C3C23×D314yes—
262C2×D3—64no—
263C2×D3—64no—
264C2×D3—64no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—24no—
268Dic3—64no—
269Dic3—64no—
270Dic3—64no—
271Dic3—64no—
272Dic3—64no—
273Dic3—64no—
274Dic3—64no—
275C32#53811yescharacteristic, complemented, Sylow
276C23—22no—
277C23—41no—
278C23—41no—
279C23—91no—
280C23C22×C3⋊D311yescharacteristic
281C23C22×C3⋊D311yescharacteristic
282C23C22×C3⋊D311yescharacteristic
283C23C22×C3⋊D311yescharacteristic
284Q8—361no—
285D4—22no—
286D4—41no—
287D4—41no—
288D4—182no—
289D4—361no—
290D4—361no—
291D4—361no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—361no—
300C2×C4—361no—
301C2×C4—361no—
302C2×C4—361no—
303C2×C4C22×C3⋊D311yescharacteristic
304C2×C4C22×C3⋊D311yescharacteristic
305C2×C4C22×C3⋊D311yescharacteristic
306C2×C3—28no—
307C2×C3—44no—
308C2×C3—44no—
309C2×C3—44no—
310C2×C3C2×#23714yes—
311C2×C3#100414yes—
312C2×C3#100414yes—
313D3—124no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—44no—
326C22—181no—
327C22—181no—
328C22—181no—
329C22#383012yes—
330C22C23×C3⋊D311yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—181no—
335C4—181no—
336C4—181no—
337C4—181no—
338C4—181no—
339C4—181no—
340C4—181no—
341C3#559314yescomplemented
342C2—22no—
343C2—41no—
344C2—41no—
345C2—41no—
346C2—361no—
347C2C2×#383011yescharacteristic
348C2#6436111yescharacteristic
349C2#6436111yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#120C2
hypercenter#120C3⋊D3
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C32×#120C2semidirect
2. #64162C2semidirect
3. C2×#3757C2semidirect
4. #64148C2semidirect
5. #64148C2semidirect
6. #64147C2semidirect
7. #64145C2semidirect
8. #64144C2semidirect
9. D4×C3⋊Dic3C2semidirect
10. #64094C2semidirect
11. #64092C2semidirect
12. #64090C2semidirect
13. #64088C2semidirect
14. #64088C2semidirect
15. #64087C2semidirect
24. C2×#638C22semidirect
27. C22×C3⋊Dic3C22semidirect
28. C2×C4×C3⋊D3C22semidirect
29. C2×#634C22semidirect
40. #3757C22semidirect
41. #3757C22semidirect
42. #3752C22semidirect
43. #3752C22semidirect
44. C4⋊(C3⋊Dic3)C22semidirect
45. C4⋊(C3⋊Dic3)C22semidirect
46. #3750C22semidirect
47. #3750C22semidirect
48. #3750C22semidirect
49. #3750C22semidirect
50. C4×C3⋊Dic3C22semidirect
51. C4×C3⋊Dic3C22semidirect
52. C3×#120D3semidirect
171. #120C3⋊D3semidirect
275. C32#538semidirect
341. C3#5593semidirect

Maximal subgroups

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

subgroupindex
C32×#1202
#641622
C2×#37572
#641482
#641472
#641452
#641442
D4×C3⋊Dic32
#640942
#640922
#640902
#640882
#640872
#55933