Finite Group Databasebeta

← #10595293

Subgroups — #10595293

Order 576.

903 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×D4C211yesmaximal, characteristic, complemented
2C2×D3×#67C212yesmaximal, complemented
3C2×#3802C211yesmaximal, characteristic, complemented
4#64147C211yesmaximal, characteristic, complemented
5C3×#940C211yesmaximal, characteristic, complemented
6#63983C212yesmaximal, complemented
7#63976C212yesmaximal, complemented
8#63970C212yesmaximal, complemented
9#63908C211yesmaximal, characteristic, complemented
10C4×#626C211yesmaximal, characteristic, complemented
11Dic3×D12C211yesmaximal, characteristic, complemented
12D4×#67—31nomaximal
13#5717—31nomaximal
14C23×C3×D3C2212yescomplemented
15C22×D32C2212yescomplemented
16C2×C32×D4C2211yescharacteristic, complemented
17C2×#638C2212yescomplemented
18C2×C4×C3⋊D3C2211yescharacteristic, complemented
19C2×C3×#67C2212yescomplemented
20C2×C3×#67C2212yescomplemented
21C2×C3×#67C2212yescomplemented
22C3×D3×D4—24no—
23C2×C3×D12C2211yescharacteristic
24C2×C3×D12C2211yescharacteristic, complemented
25C2×C3×C4×D3C2211yescharacteristic, complemented
26D3×#67—28no—
27C2×#626C2211yescharacteristic
28C2×#625C2212yescomplemented
29C2×#625C2212yescomplemented
30C2×D3×Dic3C2212yescomplemented
31#3802—24no—
32#3757C2212yescomplemented
33C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
34C3×C4×Dic3C2211yescharacteristic, complemented
35Dic3⋊Dic3C2211yescharacteristic, complemented
36#3722C2211yescharacteristic, complemented
37(C2×D3)⋊Dic3C2212yescomplemented
38(C2×D3)⋊Dic3C2211yescharacteristic, complemented
39(C2×D3)⋊Dic3C2212yescomplemented
40C22×C3×D4—31no—
41C22×#67—32no—
42C2×D3×D4—32no—
43C2×D3×D4—31no—
44C2×D3×D4—31no—
45C2×D3×D4D311yescharacteristic, complemented
46C3×#127—31no—
47#953—32no—
48#940D311yescharacteristic, complemented
49#939—32no—
50#938—31no—
51#938—31no—
52#938—32no—
53#938—31no—
54#938—61no—
55#937—32no—
56#937—32no—
57#937—32no—
58D4×Dic3—31no—
59#930—31no—
60C4×#67—31no—
61C4×D12—31no—
62C4×D12—31no—
63C23×C32C2312yes—
64C22×C3⋊D3C2311yescharacteristic
65C22×C3×D3—24no—
66C22×C3×D3—24no—
67C22×C3×D3—42no—
68C22×C3×D3C2312yes—
69C22×C3×D3C2312yes—
70C22×C3×D3C2311yescharacteristic
71C2×D32—24no—
72C2×D32—24no—
73C2×D32—42no—
74C32×D4—22no—
75C2×C32×C4C2311yescharacteristic
76#638—24no—
77C2×C3⋊Dic3C2312yes—
78C2×C3⋊Dic3C2311yescharacteristic
79C4×C3⋊D3—22no—
80C3×#67—24no—
81C3×#67—24no—
82C3×#67—42no—
83C2×C3×Dic3C2311yescharacteristic
84C2×C3×Dic3C2311yescharacteristic
85C2×C3×Dic3C2311yescharacteristic
86C3×D12—22no—
87C3×D12D414yescomplemented
88C3×C4×D3—22no—
89#626D414yescomplemented
90#625—24no—
91#625—24no—
92D3×Dic3—24no—
93D42—91noSylow
94C24×C3—32no—
95C23×D3—32no—
96C23×D3—32no—
97C23×D3C2×D312yescomplemented
98C2×C3×D4—21no—
99C2×C3×D4—32no—
100C2×C3×D4—32no—
101C2×C3×D4—32no—
102C2×C3×D4—31no—
103C2×C3×D4—31no—
104C2×C3×D4—64no—
105C2×C3×D4C2×D311yescharacteristic, complemented
106C2×C3×D4C2×D311yescharacteristic, complemented
107C22×C3×C4—31no—
108C2×#67—32no—
109C2×#67—32no—
110C2×#67—32no—
111C2×#67—32no—
112C2×#67—32no—
113C2×#67—32no—
114C2×#67—31no—
115C2×#67—62no—
116C2×#67—68no—
117C2×#67C2×D312yescomplemented
118C2×#67C2×D312yescomplemented
119C2×#67C2×D312yescomplemented
120C22×Dic3—32no—
121D3×D4—24no—
122D3×D4—64no—
123D3×D4—68no—
124D3×D4—64no—
125C2×D12—31no—
126C2×D12C2×D311yescharacteristic
127C2×D12C2×D311yescharacteristic, complemented
128C2×C4×D3—32no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—61no—
132C2×C4×D3C2×D311yescharacteristic, complemented
133C3×C42—31no—
134#217—32no—
135#217—32no—
136#217—31no—
137#217—32no—
138#217—32no—
139#217—62no—
140(C2×D3)⋊C4—31no—
141(C2×D3)⋊C4—32no—
142(C2×D3)⋊C4—32no—
143(C2×D3)⋊C4—31no—
144(C2×D3)⋊C4—31no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—61no—
149Dic3⋊C4—31no—
150C4×Dic3C2×D311yescharacteristic, complemented
151C22×C32—22no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32C2411yescharacteristic
155C2×C3⋊D3—21no—
156C2×C3⋊D3—21no—
157C2×C3⋊D3C2×D412yescomplemented
158C2×C3×D3—22no—
159C2×C3×D3—21no—
160C2×C3×D3—22no—
161C2×C3×D3—21no—
162C2×C3×D3—22no—
163C2×C3×D3—22no—
164C2×C3×D3—22no—
165C2×C3×D3—48no—
166C2×C3×D3C2×D412yescomplemented
167C2×C3×D3C2×D414yescomplemented
168D32—48no—
169C32×C4C2×D412yescomplemented
170C3⋊Dic3—22no—
171C3⋊Dic3—21no—
172C3×Dic3—21no—
173C3×Dic3—21no—
174C3×Dic3C2×D412yescomplemented
175C22×D4—31no—
176C22×D4—91no—
177C22×D4—92no—
178#127—31no—
179#121—91no—
180#121—92no—
181#121—91no—
182#120—92no—
183#120—92no—
184C4×D4—91no—
185C4×D4—91no—
186C23×C3—22no—
187C23×C3—32no—
188C23×C3—31no—
189C23×C3—32no—
190C23×C3—64no—
191C23×C3—64no—
192C23×C3—122no—
193C23×C3C22×D312yes—
194C23×C3C22×D312yes—
195C22×D3—24no—
196C22×D3—24no—
197C22×D3—31no—
198C22×D3—31no—
199C22×D3—42no—
200C22×D3—61no—
201C22×D3—64no—
202C22×D3—64no—
203C22×D3—64no—
204C22×D3—64no—
205C22×D3—122no—
206C22×D3—122no—
207C22×D3C22×D312yes—
208C22×D3C22×D311yescharacteristic
209C22×D3C22×D312yes—
210C3×D4—22no—
211C3×D4—22no—
212C3×D4—34no—
213C3×D4—42no—
214C3×D4—64no—
215C3×D4—64no—
216C3×D4—62no—
217C3×D4—122no—
218C2×C3×C4—21no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—62no—
223C2×C3×C4C22×D311yescharacteristic
224C2×C3×C4C22×D311yescharacteristic
225#67—24no—
226#67—24no—
227#67—34no—
228#67—42no—
229#67—64no—
230#67—64no—
231#67—64no—
232#67—64no—
233#67—64no—
234#67—64no—
235#67—124no—
236C2×Dic3—31no—
237C2×Dic3—32no—
238C2×Dic3—31no—
239C2×Dic3—32no—
240C2×Dic3—62no—
241C2×Dic3—61no—
242C2×Dic3—64no—
243C2×Dic3C22×D311yescharacteristic
244C2×Dic3C22×D311yescharacteristic
245C2×Dic3C22×D311yescharacteristic
246D12—22no—
247D12—34no—
248D12#6714yescomplemented
249C4×D3—22no—
250C4×D3—64no—
251C4×D3—62no—
252C4×D3—62no—
253C4×D3—122no—
254C2×C32—42no—
255C2×C32C22×D411yescharacteristic
256C2×C32C22×D411yescharacteristic
257C2×C32H(5,𝔽2)11yescharacteristic
258C3⋊D3—22no—
259C3×D3—24no—
260C3×D3—22no—
261C3×D3—42no—
262C24—32no—
263C24—92no—
264C2×D4—32no—
265C2×D4—32no—
266C2×D4—32no—
267C2×D4—31no—
268C2×D4—31no—
269C2×D4—64no—
270C2×D4—91no—
271C2×D4—92no—
272C2×D4—92no—
273C2×D4—92no—
274C2×D4—184no—
275C2×D4—188no—
276C2×D4D3211yescharacteristic, complemented
277C22×C4—31no—
278C22×C4—92no—
279C22×C4—91no—
280C4⋊C4—91no—
281C4⋊C4—91no—
282C22⋊C4—91no—
283C22⋊C4—91no—
284C22⋊C4—92no—
285C22⋊C4—92no—
286C22⋊C4—92no—
287C42—31no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—32no—
296C22×C3—34no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—42no—
300C22×C3—62no—
301C22×C3—61no—
302C22×C3—62no—
303C22×C3—61no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—62no—
307C22×C3—128no—
308C22×C3C23×D311yescharacteristic
309C22×C3C23×D311yescharacteristic
310C2×D3—22no—
311C2×D3—21no—
312C2×D3—22no—
313C2×D3—21no—
314C2×D3—22no—
315C2×D3—22no—
316C2×D3—22no—
317C2×D3—32no—
318C2×D3—32no—
319C2×D3—48no—
320C2×D3—62no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—128no—
328C2×D3—128no—
329C2×D3D3×D412yescomplemented
330C2×D3C2×#6714yescomplemented
331C3×C4—22no—
332C3×C4—32no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4D3×D412yescomplemented
336C3×C4C2×#6712yescomplemented
337Dic3—21no—
338Dic3—21no—
339Dic3—62no—
340Dic3—61no—
341Dic3—62no—
342Dic3—61no—
343Dic3—121no—
344Dic3—122no—
345Dic3D3×D412yescomplemented
346C32D4211yescharacteristic, complemented, Sylow
347C23—31no—
348C23—32no—
349C23—32no—
350C23—64no—
351C23—64no—
352C23—91no—
353C23—122no—
354C23—184no—
355C23—184no—
356C23—362no—
357C23C2×D3212yes—
358D4—22no—
359D4—34no—
360D4—64no—
361D4—64no—
362D4—62no—
363D4—94no—
364D4—122no—
365D4—184no—
366D4—184no—
367D4—184no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—62no—
372C2×C4—92no—
373C2×C4—91no—
374C2×C4—184no—
375C2×C4—182no—
376C2×C4C2×D3211yescharacteristic
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—42no—
381C2×C3—42no—
382C2×C3—64no—
383C2×C3—62no—
384C2×C3—82no—
385C2×C3—122no—
386C2×C3C2×D3×D411yescharacteristic
387C2×C3C2×D3×D411yescharacteristic
388C2×C3C2×D3×D411yescharacteristic
389C2×C3#100411yescharacteristic
390C2×C3#100411yescharacteristic
391C2×C3C22×#6711yescharacteristic
392D3—22no—
393D3—24no—
394D3—42no—
395D3—62no—
396D3—62no—
397D3—122no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—32no—
402C22—34no—
403C22—62no—
404C22—61no—
405C22—62no—
406C22—61no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—92no—
411C22—128no—
412C22—181no—
413C22—181no—
414C22—368no—
415C22C22×D3211yescharacteristic
416C4—32no—
417C4—61no—
418C4—61no—
419C4—182no—
420C4—181no—
421C4D3×#6712yescomplemented
422C3—21no—
423C3#571711yescharacteristic, complemented
424C3D4×#6711yescharacteristic, complemented
425C2—42no—
426C2—62no—
427C2—64no—
428C2—122no—
429C2—182no—
430C2D32×D411yescharacteristic
431C2#6431311yescharacteristic
432C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32D42
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×D4C2semidirect
2. C2×D3×#67C2semidirect
3. C2×#3802C2semidirect
4. #64147C2semidirect
5. C3×#940C2semidirect
6. #63983C2semidirect
7. #63976C2semidirect
8. #63970C2semidirect
9. #63908C2semidirect
10. C4×#626C2semidirect
11. Dic3×D12C2semidirect
14. C23×C3×D3C22semidirect
15. C22×D32C22semidirect
16. C2×C32×D4C22semidirect
17. C2×#638C22semidirect
18. C2×C4×C3⋊D3C22semidirect
19. C2×C3×#67C22semidirect
20. C2×C3×#67C22semidirect
21. C2×C3×#67C22semidirect
24. C2×C3×D12C22semidirect
25. C2×C3×C4×D3C22semidirect
28. C2×#625C22semidirect
29. C2×#625C22semidirect
30. C2×D3×Dic3C22semidirect
32. #3757C22semidirect
33. C4⋊(C3⋊Dic3)C22semidirect
34. C3×C4×Dic3C22semidirect
35. Dic3⋊Dic3C22semidirect
36. #3722C22semidirect
37. (C2×D3)⋊Dic3C22semidirect
38. (C2×D3)⋊Dic3C22semidirect
39. (C2×D3)⋊Dic3C22semidirect
45. C2×D3×D4D3semidirect
48. #940D3semidirect
87. C3×D12D4semidirect
89. #626D4semidirect
97. C23×D3C2×D3semidirect
105. C2×C3×D4C2×D3semidirect
106. C2×C3×D4C2×D3semidirect
117. C2×#67C2×D3semidirect
118. C2×#67C2×D3semidirect
119. C2×#67C2×D3semidirect
127. C2×D12C2×D3semidirect
132. C2×C4×D3C2×D3semidirect
150. C4×Dic3C2×D3semidirect
157. C2×C3⋊D3C2×D4semidirect
166. C2×C3×D3C2×D4semidirect
167. C2×C3×D3C2×D4semidirect
169. C32×C4C2×D4semidirect
174. C3×Dic3C2×D4semidirect
248. D12#67semidirect
276. C2×D4D32semidirect
329. C2×D3D3×D4semidirect
330. C2×D3C2×#67semidirect
335. C3×C4D3×D4semidirect
336. C3×C4C2×#67semidirect
345. Dic3D3×D4semidirect
346. C32D42semidirect
421. C4D3×#67semidirect
423. C3#5717semidirect
424. C3D4×#67semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×D42
C2×D3×#672
C2×#38022
#641472
C3×#9402
#639832
#639762
#639702
#639082
C4×#6262
Dic3×D122
D4×#673
#57173