Finite Group Databasebeta

← #13089373

Subgroups — #13089373

Order 1440.

718 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C3⋊D15C211yesmaximal, characteristic, complemented
2C3×D4×D15C211yesmaximal, characteristic, complemented
3C3×C5×#237C211yesmaximal, characteristic, complemented
4#10624218C212yesmaximal, complemented
5#67×D15C212yesmaximal, complemented
6C2×#67151C212yesmaximal, complemented
7#10624207C212yesmaximal, complemented
8D3×D60C211yesmaximal, characteristic, complemented
9#10624201C211yesmaximal, characteristic, complemented
10#10624199C211yesmaximal, characteristic, complemented
11#10624194C211yesmaximal, characteristic, complemented
12#92100—31nomaximal
13#92026—31nomaximal
14C22×C3⋊D15C2212yescomplemented
15C22×C3×D15C2212yescomplemented
16C2×D3×D15C2212yescomplemented
17C32×C5×D4C2211yescharacteristic, complemented
18#67184C2212yescomplemented
19C4⋊(C3⋊D15)C2211yescharacteristic, complemented
20C4×C3⋊D15C2211yescharacteristic, complemented
21C3×#1220C2212yescomplemented
22C3×D60C2211yescharacteristic, complemented
23C3×C4×D15C2211yescharacteristic, complemented
24C3×C5×#67C2212yescomplemented
25C2×C3×C5×Dic3C2212yescomplemented
26C3×C4×C5×D3C2211yescharacteristic, complemented
27C3×C5×Dic6C2211yescharacteristic, complemented
28#67153C2211yescharacteristic, complemented
29#67152C2211yescharacteristic, complemented
30#67151C2212yescomplemented
31#67151C2211yescharacteristic, complemented
32#67151C2212yescomplemented
33#67151C2212yescomplemented
34#67150C2211yescharacteristic, complemented
35#67149C2212yescomplemented
36Dic3×D15C2212yescomplemented
37#64314—51nomaximal, Hall
38C3×C5×#41—31no—
39#6821—31no—
40#6821—31no—
41D4×D15—32no—
42D4×D15—31no—
43D4×D15—31no—
44D4×D15—32no—
45D4×D15—61no—
46D4×D15D311yescharacteristic, complemented
47#6817—32no—
48#6817—31no—
49C2×D60—32no—
50C2×D60—31no—
51C5×#237D311yescharacteristic, complemented
52C3×D4×D5—31no—
53#6790—32no—
54D5×#67—32no—
55C2×#1202—32no—
56#6779—32no—
57D3×D20—31no—
58#6773—31no—
59#6771—31no—
60#6766—31no—
61C22×C32×C5C2312yes—
62C2×C3⋊D15—42no—
63C2×C3⋊D15C2312yes—
64C2×C3⋊D15C2311yescharacteristic
65C2×C3×D15—42no—
66C2×C3×D15C2312yes—
67C2×C3×D15C2311yescharacteristic
68C2×C3×C5×D3C2311yescharacteristic
69D3×D15—42no—
70C32×C4×C5C2311yescharacteristic
71C3⋊Dic15C2311yescharacteristic
72C3×Dic15C2311yescharacteristic
73C3×C5×Dic3C2312yes—
74C3×C5×Dic3C2311yescharacteristic
75#4149—91noHall
76D4×C3⋊D3—51no—
77C3×#237D511yescharacteristic, complemented
78C3×D3×D4—51no—
79#3811—52no—
80D3×#67—52no—
81C2×#626—52no—
82#3804—52no—
83D3×D12—51no—
84#3799—51no—
85#3798—51no—
86#3796—51no—
87C22×D15—32no—
88C22×D15—32no—
89C22×D15—32no—
90C22×D15—62no—
91C22×D15C2×D312yescomplemented
92C22×C3×D5—32no—
93C2×D3×D5—32no—
94C3×C5×Q8—31no—
95C3×C5×D4—21no—
96C3×C5×D4—32no—
97C3×C5×D4C2×D311yescharacteristic, complemented
98C3×C5×D4C2×D311yescharacteristic, complemented
99C2×C3×C4×C5—32no—
100C2×C3×C4×C5—31no—
101#1220—32no—
102#1220—32no—
103#1220—31no—
104#1220—31no—
105#1220—62no—
106#1220C2×D312yescomplemented
107D60—32no—
108D60—31no—
109D60—32no—
110D60—32no—
111D60—31no—
112D60—31no—
113D60—61no—
114D60C2×D311yescharacteristic, complemented
115C4×D15—32no—
116C4×D15—31no—
117C4×D15—31no—
118C4×D15—32no—
119C4×D15—61no—
120C4×D15C2×D311yescharacteristic, complemented
121Dic30—31no—
122C5×#67C2×D312yescomplemented
123C2×C5×Dic3C2×D312yescomplemented
124C4×C5×D3C2×D311yescharacteristic, complemented
125C5×Dic6C2×D311yescharacteristic, complemented
126C3×C22⋊D5—32no—
127C3×D20—31no—
128C3×C4×D5—31no—
129#1204—31no—
130#1203—31no—
131#1202—32no—
132#1202—31no—
133#1202—32no—
134#1202—32no—
135#1201—31no—
136#1200—32no—
137D5×Dic3—32no—
138#1009—151no—
139#1004—151no—
140C2×C32×C5—22no—
141C2×C32×C5C2411yescharacteristic
142C3⋊D15—21no—
143C3⋊D15—22no—
144C3×D15—22no—
145C3×D15—21no—
146C3×C5×D3—21no—
147C5×#41—31no—
148#713—91no—
149#713—91no—
150D4×D5—31no—
151D4×D5—92no—
152D4×D5—92no—
153D4×D5—91no—
154#709—91no—
155#709—92no—
156C2×D20—91no—
157C2×D20—92no—
158C22×C3⋊D3—52no—
159C22×C3×D3—52no—
160C2×D32—52no—
161C32×D4C2×D511yescharacteristic, complemented
162#638—52no—
163C4⋊(C3⋊D3)—51no—
164C4×C3⋊D3—51no—
165C3×#67—52no—
166C3×#67C2×D512yescomplemented
167C2×C3×Dic3C2×D512yescomplemented
168C3×D12—51no—
169C3×C4×D3—51no—
170C3×C4×D3C2×D511yescharacteristic, complemented
171C3×Dic6C2×D511yescharacteristic, complemented
172#627—51no—
173#626—51no—
174#626—52no—
175#626—51no—
176#626—52no—
177#626—52no—
178#625—51no—
179#624—52no—
180D3×Dic3—52no—
181C22×C3×C5—22no—
182C22×C3×C5—31no—
183C22×C3×C5C22×D312yes—
184C22×C3×C5C22×D312yes—
185C2×D15—32no—
186C2×D15—31no—
187C2×D15—32no—
188C2×D15—31no—
189C2×D15—42no—
190C2×D15—61no—
191C2×D15—62no—
192C2×D15—122no—
193C2×D15—122no—
194C2×D15—122no—
195C2×D15—124no—
196C2×D15C22×D312yes—
197C2×D15C22×D311yescharacteristic
198C2×C5×D3C22×D311yescharacteristic
199C2×C3×D5—32no—
200C2×C3×D5—31no—
201C2×C3×D5—122no—
202D3×D5—122no—
203C3×C4×C5—21no—
204C3×C4×C5—31no—
205C3×C4×C5—32no—
206C3×C4×C5C22×D311yescharacteristic
207C3×C4×C5C22×D311yescharacteristic
208Dic15—31no—
209Dic15—31no—
210Dic15—61no—
211Dic15C22×D311yescharacteristic
212C3×Dic5—31no—
213C5×Dic3C22×D312yes—
214C5×Dic3C22×D311yescharacteristic
215C3×#41—31no—
216C2×C3×D4—151no—
217C2×#67—152no—
218C2×#67—152no—
219#239—151no—
220#239—151no—
221#237—152no—
222#237—151no—
223#237D1511yescharacteristic, complemented
224D3×D4—51no—
225D3×D4—152no—
226D3×D4—152no—
227D3×D4—151no—
228D3×D4—151no—
229D3×D4—152no—
230D3×D4—151no—
231D3×D4—301no—
232#235—151no—
233#235—152no—
234#235—151no—
235#235—151no—
236C2×D12—151no—
237C2×D12—152no—
238C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
239C22×D5—32no—
240C22×D5—92no—
241C22×D5—92no—
242C5×Q8—31no—
243C5×D4—32no—
244C5×D4D3211yescharacteristic, complemented
245C2×C4×C5—32no—
246C2×C4×C5—31no—
247C22⋊D5—32no—
248C22⋊D5—91no—
249C22⋊D5—91no—
250C22⋊D5—92no—
251D20—31no—
252D20—91no—
253D20—92no—
254D20—92no—
255D20—92no—
256D20—91no—
257C4×D5—31no—
258C4×D5—92no—
259C4×D5—92no—
260C4×D5—91no—
261Dic10—91no—
262C22×C32C22×D512yes—
263C2×C3⋊D3—52no—
264C2×C3⋊D3—51no—
265C2×C3⋊D3—202no—
266C2×C3×D3—52no—
267C2×C3×D3—51no—
268C2×C3×D3—202no—
269C2×C3×D3C22×D511yescharacteristic
270D32—202no—
271C32×C4C22×D511yescharacteristic
272C3⋊Dic3—51no—
273C3×Dic3—51no—
274C3×Dic3C22×D512yes—
275C3×Dic3C22×D511yescharacteristic
276H(5,𝔽2)—451noSylow
277C2×C3×C5—22no—
278C2×C3×C5—22no—
279C2×C3×C5—21no—
280C2×C3×C5—42no—
281C2×C3×C5—61no—
282C2×C3×C5C23×D311yescharacteristic
283C2×C3×C5C23×D311yescharacteristic
284D15—21no—
285D15—22no—
286D15—62no—
287D15—61no—
288D15—62no—
289D15—62no—
290D15—61no—
291D15—122no—
292C3×D5—62no—
293C3×D5—61no—
294C5×D3—21no—
295C23×C3—152no—
296C22×D3—52no—
297C22×D3—152no—
298C22×D3—152no—
299C22×D3—152no—
300C22×D3—152no—
301C22×D3—302no—
302C3×Q8—31no—
303C3×D4—21no—
304C3×D4—32no—
305C3×D4—152no—
306C3×D4—151no—
307C3×D4D3×D511yescharacteristic, complemented
308C3×D4C2×D1511yescharacteristic, complemented
309C2×C3×C4—32no—
310C2×C3×C4—31no—
311C2×C3×C4—151no—
312#67—52no—
313#67—151no—
314#67—151no—
315#67—152no—
316#67—152no—
317#67—151no—
318#67—151no—
319#67—152no—
320#67—152no—
321#67—152no—
322#67—302no—
323#67C2×D1512yescomplemented
324C2×Dic3—152no—
325C2×Dic3C2×D1512yescomplemented
326D12—51no—
327D12—151no—
328D12—152no—
329D12—152no—
330D12—152no—
331D12—151no—
332D12—151no—
333D12—151no—
334D12—301no—
335C4×D3—51no—
336C4×D3—152no—
337C4×D3—152no—
338C4×D3—151no—
339C4×D3—151no—
340C4×D3—152no—
341C4×D3—301no—
342C4×D3C2×D1511yescharacteristic, complemented
343Dic6—151no—
344Dic6—151no—
345Dic6C2×D1511yescharacteristic, complemented
346C22×C5—31no—
347C22×C5C2×D3212yes—
348C2×D5—31no—
349C2×D5—32no—
350C2×D5—91no—
351C2×D5—92no—
352C2×D5—122no—
353C2×D5—362no—
354C2×D5—362no—
355C4×C5—31no—
356C4×C5—32no—
357C4×C5C2×D3211yescharacteristic
358Dic5—31no—
359Dic5—91no—
360C2×C32—22no—
361C2×C32C23×D511yescharacteristic
362C3⋊D3—101no—
363C3⋊D3—102no—
364C3×D3—21no—
365C3×D3—102no—
366C3×D3—101no—
367#41—31no—
368#41—451no—
369#41—452no—
370#41—451no—
371#41—451no—
372C2×D4—151no—
373C2×D4—451no—
374C2×D4—452no—
375C2×D4—452no—
376C2×D4—452no—
377C2×D4—451no—
378C3×C5—21no—
379C3×C5#100411yescharacteristic, complemented
380C3×C5#100911yescharacteristic, complemented
381C22×C3—22no—
382C22×C3—31no—
383C22×C3—152no—
384C22×C3—151no—
385C22×C3—602no—
386C22×C3C2×D3×D512yes—
387C22×C3C22×D1512yes—
388C2×D3—51no—
389C2×D3—52no—
390C2×D3—152no—
391C2×D3—151no—
392C2×D3—152no—
393C2×D3—151no—
394C2×D3—202no—
395C2×D3—301no—
396C2×D3—302no—
397C2×D3—602no—
398C2×D3—602no—
399C2×D3—602no—
400C2×D3—602no—
401C2×D3—604no—
402C2×D3C22×D1511yescharacteristic
403C3×C4—21no—
404C3×C4—31no—
405C3×C4—32no—
406C3×C4—151no—
407C3×C4C2×D3×D511yescharacteristic
408C3×C4C22×D1511yescharacteristic
409Dic3—51no—
410Dic3—151no—
411Dic3—151no—
412Dic3—301no—
413Dic3C22×D1511yescharacteristic
414Dic3C22×D1512yes—
415C2×C5—22no—
416C2×C5—61no—
417C2×C5C22×D3211yescharacteristic
418D5—61no—
419D5—62no—
420D5—181no—
421D5—182no—
422C32#414911yescharacteristic, complemented, Sylow
423C23—152no—
424C23—452no—
425C23—452no—
426Q8—31no—
427Q8—451no—
428D4—32no—
429D4—152no—
430D4—151no—
431D4—451no—
432D4—452no—
433D4—452no—
434D4—452no—
435D4—451no—
436D4—451no—
437D4—452no—
438D4—451no—
439D4D3×D1511yescharacteristic, complemented
440C2×C4—32no—
441C2×C4—31no—
442C2×C4—151no—
443C2×C4—452no—
444C2×C4—452no—
445C2×C4—451no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—42no—
450C2×C3—61no—
451C2×C3—302no—
452C2×C3—301no—
453C2×C3C22×D3×D511yescharacteristic
454C2×C3C23×D1511yescharacteristic
455D3—21no—
456D3—101no—
457D3—102no—
458D3—302no—
459D3—301no—
460D3—302no—
461D3—301no—
462D3—302no—
463D3—602no—
464C5#6431411yescharacteristic, complemented, Sylow
465C22—31no—
466C22—151no—
467C22—152no—
468C22—451no—
469C22—452no—
470C22—602no—
471C22—1802no—
472C22—1802no—
473C22C2×D3×D1512yes—
474C4—31no—
475C4—32no—
476C4—151no—
477C4—451no—
478C4C2×D3×D1511yescharacteristic
479C3—21no—
480C3#9202611yescharacteristic, complemented
481C3#9210011yescharacteristic, complemented
482C2—22no—
483C2—61no—
484C2—301no—
485C2—302no—
486C2—901no—
487C2—902no—
488C2C22×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4D3×D15
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊D15C2semidirect
2. C3×D4×D15C2semidirect
3. C3×C5×#237C2semidirect
4. #10624218C2semidirect
5. #67×D15C2semidirect
6. C2×#67151C2semidirect
7. #10624207C2semidirect
8. D3×D60C2semidirect
9. #10624201C2semidirect
10. #10624199C2semidirect
11. #10624194C2semidirect
14. C22×C3⋊D15C22semidirect
15. C22×C3×D15C22semidirect
16. C2×D3×D15C22semidirect
17. C32×C5×D4C22semidirect
18. #67184C22semidirect
19. C4⋊(C3⋊D15)C22semidirect
20. C4×C3⋊D15C22semidirect
21. C3×#1220C22semidirect
22. C3×D60C22semidirect
23. C3×C4×D15C22semidirect
24. C3×C5×#67C22semidirect
25. C2×C3×C5×Dic3C22semidirect
26. C3×C4×C5×D3C22semidirect
27. C3×C5×Dic6C22semidirect
28. #67153C22semidirect
29. #67152C22semidirect
30. #67151C22semidirect
31. #67151C22semidirect
32. #67151C22semidirect
33. #67151C22semidirect
34. #67150C22semidirect
35. #67149C22semidirect
36. Dic3×D15C22semidirect
46. D4×D15D3semidirect
51. C5×#237D3semidirect
77. C3×#237D5semidirect
91. C22×D15C2×D3semidirect
97. C3×C5×D4C2×D3semidirect
98. C3×C5×D4C2×D3semidirect
106. #1220C2×D3semidirect
114. D60C2×D3semidirect
120. C4×D15C2×D3semidirect
122. C5×#67C2×D3semidirect
123. C2×C5×Dic3C2×D3semidirect
124. C4×C5×D3C2×D3semidirect
125. C5×Dic6C2×D3semidirect
161. C32×D4C2×D5semidirect
166. C3×#67C2×D5semidirect
167. C2×C3×Dic3C2×D5semidirect
170. C3×C4×D3C2×D5semidirect
171. C3×Dic6C2×D5semidirect
223. #237D15semidirect
238. C32×C5H(5,𝔽2)semidirect
244. C5×D4D32semidirect
307. C3×D4D3×D5semidirect
308. C3×D4C2×D15semidirect
323. #67C2×D15semidirect
325. C2×Dic3C2×D15semidirect
342. C4×D3C2×D15semidirect
345. Dic6C2×D15semidirect
379. C3×C5#1004semidirect
380. C3×C5#1009semidirect
422. C32#4149semidirect
439. D4D3×D15semidirect
464. C5#64314semidirect
480. C3#92026semidirect
481. C3#92100semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊D152
C3×D4×D152
C3×C5×#2372
#106242182
#67×D152
C2×#671512
#106242072
D3×D602
#106242012
#106241992
#106241942
#921003
#920263
#643145