Finite Group Databasebeta

← #10596149

Subgroups — #10596149

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×#124C211yesmaximal, characteristic, complemented
2#64156C211yesmaximal, characteristic, complemented
3#64155C211yesmaximal, characteristic, complemented
4#64148C211yesmaximal, characteristic, complemented
5#64146C211yesmaximal, characteristic, complemented
6#64095C211yesmaximal, characteristic, complemented
7#64093C211yesmaximal, characteristic, complemented
8#64093C211yesmaximal, characteristic, complemented
9#64092C211yesmaximal, characteristic, complemented
10#64091C211yesmaximal, characteristic, complemented
11#64090C211yesmaximal, characteristic, complemented
12C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
13#64088C211yesmaximal, characteristic, complemented
14#64085C211yesmaximal, characteristic, complemented
15C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
16#5678—34nomaximal
17C23×C3⋊D3C2211yescharacteristic
18C2×C32×Q8C2211yescharacteristic, complemented
19C2×C32×D4C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C2×#638C2211yescharacteristic
22C2×#638C2211yescharacteristic
23C2×#638C2211yescharacteristic, complemented
24C22×C3⋊Dic3C2211yescharacteristic, complemented
25C2×C4⋊(C3⋊D3)C2211yescharacteristic
26C2×C4⋊(C3⋊D3)C2211yescharacteristic
27C2×C4×C3⋊D3C2211yescharacteristic
28C2×C4×C3⋊D3C2211yescharacteristic
29C2×C4×C3⋊D3C2211yescharacteristic, complemented
30C32×C22⋊C4C2211yescharacteristic
31C32×C22⋊C4C2211yescharacteristic
32C32×C22⋊C4C2211yescharacteristic, complemented
33C32×C22⋊C4C2211yescharacteristic, complemented
34C32×C42C2211yescharacteristic, complemented
35#3757C2211yescharacteristic
36#3757C2211yescharacteristic
37#3752C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40#3752C2211yescharacteristic
41#3752C2211yescharacteristic
42#3752C2211yescharacteristic
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
50#3750C2211yescharacteristic, complemented
51C4×C3⋊Dic3C2211yescharacteristic, complemented
52C3×#124D314yescomplemented
53#947—34no—
54#946—34no—
55#939—34no—
56#937—34no—
57#886—34no—
58#884—34no—
59#884—34no—
60#883—34no—
61#882—34no—
62#67⋊C4—34no—
63D3×C22⋊C4—34no—
64#879—34no—
65#876—34no—
66C4×D12—34no—
67C23×C32C2311yescharacteristic
68C23×C32C2311yescharacteristic
69C22×C3⋊D3—22no—
70C22×C3⋊D3—41no—
71C22×C3⋊D3—41no—
72C22×C3⋊D3C2311yescharacteristic
73C22×C3⋊D3C2311yescharacteristic
74C22×C3⋊D3C2311yescharacteristic
75C32×Q8—41no—
76C32×D4—41no—
77C2×C32×C4C2311yescharacteristic
78C2×C32×C4C2311yescharacteristic
79C2×C32×C4C2311yescharacteristic
80C2×C32×C4C2311yescharacteristic
81C2×C32×C4C2311yescharacteristic
82#638—22no—
83#638—41no—
84#638—41no—
85#638—41no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3C2311yescharacteristic
88C2×C3⋊Dic3C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C4⋊(C3⋊D3)—22no—
93C4⋊(C3⋊D3)—41no—
94C4×C3⋊D3—41no—
95C4×C3⋊D3—41no—
96C4×C3⋊D3—41no—
97#538—91noSylow
98C23×D3—34no—
99C2×C3×Q8C2×D314yescomplemented
100C2×C3×D4C2×D314yes—
101C2×#67—34no—
102C2×#67—34no—
103C2×#67—34no—
104C2×#67—34no—
105C22×Dic3—34no—
106C2×D12—34no—
107C2×D12—34no—
108C2×C4×D3—34no—
109C2×C4×D3—34no—
110C2×C4×D3—34no—
111C3×C22⋊C4C2×D314yes—
112C3×C22⋊C4C2×D314yes—
113C3×C22⋊C4C2×D314yescomplemented
114C3×C22⋊C4C2×D314yescomplemented
115C3×C42C2×D314yescomplemented
116#217—34no—
117#217—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—
123(C2×D3)⋊C4—34no—
124(C2×D3)⋊C4—34no—
125(C2×D3)⋊C4—34no—
126C4⋊Dic3—34no—
127C4⋊Dic3—34no—
128Dic3⋊C4—34no—
129Dic3⋊C4—34no—
130Dic3⋊C4—34no—
131Dic3⋊C4—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—44no—
149C2×C3⋊D3#4112yes—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—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#124C3⋊D311yescharacteristic, complemented
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—
176C23×C3C22×D314yes—
177C22×D3—34no—
178C22×D3—34no—
179C22×D3—34no—
180C22×D3—68no—
181C22×D3—124no—
182C22×D3—124no—
183C3×Q8—44no—
184C3×D4—44no—
185C2×C3×C4C22×D314yes—
186C2×C3×C4C22×D314yes—
187C2×C3×C4C22×D314yes—
188C2×C3×C4C22×D314yes—
189C2×C3×C4C22×D314yes—
190#67—68no—
191#67—124no—
192#67—124no—
193#67—124no—
194C2×Dic3—34no—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—34no—
199C2×Dic3—124no—
200D12—68no—
201D12—124no—
202C4×D3—124no—
203C4×D3—124no—
204C4×D3—124no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32C2×#4111yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C2×C32H(5,𝔽2)11yescharacteristic
210C3⋊D3—22no—
211C3⋊D3—41no—
212C3⋊D3—41no—
213C24—91no—
214C2×Q8C2×C3⋊D311yescharacteristic, complemented
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—
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⋊C4C2×C3⋊D311yescharacteristic
243C22⋊C4C2×C3⋊D311yescharacteristic
244C22⋊C4C2×C3⋊D311yescharacteristic, complemented
245C22⋊C4C2×C3⋊D311yescharacteristic, complemented
246C42—91no—
247C42C2×C3⋊D311yescharacteristic, complemented
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3—24no—
253C22×C3—24no—
254C22×C3C23×D314yes—
255C2×D3—38no—
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—
270Dic3—64no—
271Dic3—64no—
272Dic3—64no—
273Dic3—64no—
274Dic3—64no—
275C32#53811yescharacteristic, complemented, Sylow
276C23—91no—
277C23—91no—
278C23—91no—
279C23—182no—
280C23—361no—
281C23—361no—
282C23C22×C3⋊D311yescharacteristic
283C23C22×C3⋊D311yescharacteristic
284Q8—41no—
285D4—41no—
286D4—182no—
287D4—182no—
288D4—361no—
289D4—361no—
290D4—361no—
291D4—361no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—361no—
298C2×C4—361no—
299C2×C4—361no—
300C2×C4—361no—
301C2×C4C22×C3⋊D311yescharacteristic
302C2×C4C22×C3⋊D311yescharacteristic
303C2×C4C22×C3⋊D311yescharacteristic
304C2×C4C22×C3⋊D311yescharacteristic
305C2×C4C22×C3⋊D311yescharacteristic
306C2×C3—44no—
307C2×C3—44no—
308C2×C3#100414yes—
309C2×C3D3×#4114yes—
310C2×C3#100914yes—
311D3—68no—
312D3—124no—
313D3—124no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—92no—
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—181no—
337C4—181no—
338C4—181no—
339C4—181no—
340C4—181no—
341C3#567814yescomplemented
342C2—41no—
343C2—41no—
344C2—182no—
345C2—361no—
346C2—361no—
347C2#6436111yescharacteristic
348C2#41×C3⋊D311yescharacteristic
349C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
hypercenter#124C3⋊D3
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C32×#124C2semidirect
2. #64156C2semidirect
3. #64155C2semidirect
4. #64148C2semidirect
5. #64146C2semidirect
6. #64095C2semidirect
7. #64093C2semidirect
8. #64093C2semidirect
9. #64092C2semidirect
10. #64091C2semidirect
11. #64090C2semidirect
12. C22⋊C4×C3⋊D3C2semidirect
13. #64088C2semidirect
14. #64085C2semidirect
15. C4×C4⋊(C3⋊D3)C2semidirect
18. C2×C32×Q8C22semidirect
23. C2×#638C22semidirect
24. C22×C3⋊Dic3C22semidirect
29. C2×C4×C3⋊D3C22semidirect
32. C32×C22⋊C4C22semidirect
33. C32×C22⋊C4C22semidirect
34. C32×C42C22semidirect
43. #3752C22semidirect
44. #3752C22semidirect
45. C4⋊(C3⋊Dic3)C22semidirect
46. C4⋊(C3⋊Dic3)C22semidirect
47. #3750C22semidirect
48. #3750C22semidirect
49. #3750C22semidirect
50. #3750C22semidirect
51. C4×C3⋊Dic3C22semidirect
52. C3×#124D3semidirect
99. C2×C3×Q8C2×D3semidirect
113. C3×C22⋊C4C2×D3semidirect
114. C3×C22⋊C4C2×D3semidirect
115. C3×C42C2×D3semidirect
163. #124C3⋊D3semidirect
214. C2×Q8C2×C3⋊D3semidirect
244. C22⋊C4C2×C3⋊D3semidirect
245. C22⋊C4C2×C3⋊D3semidirect
247. C42C2×C3⋊D3semidirect
275. C32#538semidirect
341. C3#5678semidirect

Maximal subgroups

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

subgroupindex
C32×#1242
#641562
#641552
#641482
#641462
#640952
#640932
#640922
#640912
#640902
C22⋊C4×C3⋊D32
#640882
#640852
C4×C4⋊(C3⋊D3)2
#56783