Finite Group Databasebeta

← #13088937

Subgroups — #13088937

Order 1440.

696 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624226C211yesmaximal, characteristic, complemented
2#10624193C211yesmaximal, characteristic, complemented
3Dic6×D15C211yesmaximal, characteristic, complemented
4C5×#3799C211yesmaximal, characteristic, complemented
5#10624155C211yesmaximal, characteristic, complemented
6C3×C4×D3×D5C211yesmaximal, characteristic, complemented
7C3×#6773C211yesmaximal, characteristic, complemented
8#10624049C212yesmaximal, complemented
9#10624042C212yesmaximal, complemented
10#10624028C212yesmaximal, complemented
11#10624022C212yesmaximal, complemented
12#92034—31nomaximal
13D5×#235—31nomaximal
14C2×(C3⋊D3)⋊D5C2212yes—
15C2×C3×D3×D5C2211yescharacteristic
16#67180C2211yescharacteristic, complemented
17C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C3×D60C2211yescharacteristic, complemented
19C3×C4×D15C2211yescharacteristic
20C3×C4×C5×D3C2211yescharacteristic, complemented
21C3×C5×Dic6C2211yescharacteristic, complemented
22C32×C4×D5C2211yescharacteristic, complemented
23#67156C2212yescomplemented
24#67153C2212yescomplemented
25#67150C2212yescomplemented
26Dic3×D15C2212yes—
27Dic3×D15C2212yescomplemented
28C5×#626C2212yescomplemented
29C5×#624C2212yescomplemented
30#67138C2212yescomplemented
31C3⋊D3×Dic5C2212yescomplemented
32C3×#1202C2212yescomplemented
33C3×#1200C2211yescharacteristic
34C3×#1200C2212yescomplemented
35C3×D3×Dic5C2211yescharacteristic, complemented
36C3×D5×Dic3C2211yescharacteristic, complemented
37D3×#235—51nomaximal, Hall
38Q8×D15—31no—
39#6817—31no—
40C5×#239—31no—
41C5×#235—31no—
42C3×#713—31no—
43C2×C3×C4×D5—31no—
44D5×#67—32no—
45#6782—32no—
46#6777—32no—
47#6777—31no—
48D5×D12—31no—
49C4×D3×D5—32no—
50C4×D3×D5—32no—
51C4×D3×D5D311yescharacteristic, complemented
52#6773D311yescharacteristic, complemented
53#6772—31no—
54#6772—31no—
55#6772—32no—
56#6772—61no—
57#6771—32no—
58#6765—31no—
59#6765—32no—
60D5×Dic6—31no—
61C2×C5×C3⋊D3C2312yes—
62C2×C3×D15C2312yes—
63C2×C3×D15C2311yescharacteristic
64C2×C3×C5×D3C2311yescharacteristic
65C2×C32×D5C2311yescharacteristic
66(C3⋊D3)⋊D5—24no—
67C3×D3×D5—22no—
68C32×C4×C5C2311yescharacteristic
69C3⋊Dic15C2312yes—
70C3×Dic15C2311yescharacteristic
71C3×C5×Dic3C2311yescharacteristic
72C3×C5×Dic3C2312yes—
73C32×Dic5C2311yescharacteristic
74D5×#41—91noHall
75#3828—51no—
76C3×#235—51no—
77C2×C3×C4×D3—51no—
78D3×#67—52no—
79#3804—52no—
80D3×D12—51no—
81C4×D32—52no—
82#3799D511yescharacteristic, complemented
83#3798—52no—
84#3795—51no—
85D3×Dic6—51no—
86C22×C3×D5—31no—
87C2×D3×D5—32no—
88C2×D3×D5—32no—
89C2×D3×D5C2×D311yescharacteristic
90C3×C5×Q8—31no—
91C2×C3×C4×C5—31no—
92#1220—32no—
93D60C2×D311yescharacteristic, complemented
94C4×D15—32no—
95C4×D15—32no—
96C4×D15C2×D311yescharacteristic
97Dic30—31no—
98Dic30—31no—
99Dic30—32no—
100Dic30—61no—
101C5×#67—32no—
102C5×D12—32no—
103C5×D12—31no—
104C5×D12—31no—
105C5×D12—61no—
106C4×C5×D3—32no—
107C4×C5×D3—32no—
108C4×C5×D3C2×D311yescharacteristic, complemented
109C5×Dic6C2×D311yescharacteristic, complemented
110C2×C3×Dic5—31no—
111C3×D20—32no—
112C3×D20—31no—
113C3×C4×D5—21no—
114C3×C4×D5—31no—
115C3×C4×D5—31no—
116C3×C4×D5—31no—
117C3×C4×D5—32no—
118C3×C4×D5C2×D311yescharacteristic, complemented
119C3×C4×D5C2×D311yescharacteristic, complemented
120#1204—32no—
121#1203—32no—
122#1202C2×D312yescomplemented
123#1201—32no—
124#1201—32no—
125#1201—32no—
126#1201—32no—
127#1201—62no—
128#1200C2×D311yescharacteristic
129#1200C2×D312yescomplemented
130D3×Dic5—32no—
131D3×Dic5—32no—
132D3×Dic5—62no—
133D3×Dic5C2×D311yescharacteristic, complemented
134D5×Dic3—32no—
135D5×Dic3—32no—
136D5×Dic3C2×D311yescharacteristic, complemented
137D3×#41—151no—
138C2×#235—151no—
139C2×C32×C5C2411yescharacteristic
140C5×C3⋊D3—22no—
141C3×D15—22no—
142C3×D15#4112yescomplemented
143C3×C5×D3—21no—
144C32×D5—21no—
145C5×#41—91no—
146#713—31no—
147Q8×D5—91no—
148#711—92no—
149#711—91no—
150D4×D5—91no—
151D4×D5—92no—
152#709—92no—
153#709—91no—
154C2×C4×D5—31no—
155C2×C4×D5—92no—
156C22×C3×D3—51no—
157C2×D32—52no—
158C2×C32×C4—51no—
159#638—52no—
160C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
161C4×C3⋊D3—52no—
162#634—51no—
163C3×#67—52no—
164C2×C3×Dic3—51no—
165C3×D12—51no—
166C3×C4×D3—51no—
167C3×C4×D3—51no—
168C3×C4×D3—51no—
169C3×C4×D3—52no—
170C3×C4×D3C2×D511yescharacteristic, complemented
171C3×Dic6C2×D511yescharacteristic, complemented
172#627—52no—
173#626—52no—
174#626C2×D512yescomplemented
175#625—52no—
176#624C2×D512yescomplemented
177D3×Dic3—52no—
178D3×Dic3—52no—
179C22×C3×C5—31no—
180C2×D15C22×D312yes—
181C2×D15C22×D311yescharacteristic
182C2×C5×D3—32no—
183C2×C5×D3—32no—
184C2×C5×D3—62no—
185C2×C5×D3C22×D311yescharacteristic
186C2×C3×D5—21no—
187C2×C3×D5—31no—
188C2×C3×D5—32no—
189C2×C3×D5—62no—
190C2×C3×D5C22×D311yescharacteristic
191C2×C3×D5C22×D311yescharacteristic
192D3×D5—22no—
193D3×D5—64no—
194D3×D5—64no—
195C3×C4×C5—21no—
196C3×C4×C5—31no—
197C3×C4×C5—32no—
198C3×C4×C5C22×D311yescharacteristic
199C3×C4×C5C22×D311yescharacteristic
200Dic15—32no—
201Dic15—32no—
202Dic15—62no—
203Dic15C22×D311yescharacteristic
204C3×Dic5—21no—
205C3×Dic5—31no—
206C3×Dic5C22×D311yescharacteristic
207C3×Dic5C22×D311yescharacteristic
208C5×Dic3C22×D311yescharacteristic
209C5×Dic3C22×D312yes—
210C3×#41—151no—
211C22×C3×C4—151no—
212C2×#67—152no—
213#239—31no—
214D3×Q8—151no—
215#237—152no—
216#237—151no—
217D3×D4—151no—
218D3×D4—152no—
219#235—31no—
220#235—51no—
221#235—152no—
222#235—152no—
223#235—152no—
224#235—151no—
225#235—151no—
226#235—151no—
227#235—301no—
228C2×D12—151no—
229C2×C4×D3—51no—
230C2×C4×D3—152no—
231C2×C4×D3—152no—
232C2×Dic6—151no—
233C32×C5C2×#4111yescharacteristic, complemented, Hall
234C22×D5—31no—
235C22×D5—92no—
236C5×Q8—31no—
237C5×D4—92no—
238C5×D4—91no—
239C2×C4×C5—31no—
240C2×C4×C5—92no—
241C22⋊D5—92no—
242C22⋊D5—92no—
243C22⋊D5—92no—
244C2×Dic5—31no—
245C2×Dic5—92no—
246D20—32no—
247D20—31no—
248C4×D5—31no—
249C4×D5—31no—
250C4×D5—31no—
251C4×D5—32no—
252C4×D5—92no—
253C4×D5—92no—
254C4×D5D3211yescharacteristic, complemented
255Dic10—92no—
256Dic10—91no—
257C22×C32—51no—
258C2×C3⋊D3C22×D512yes—
259C2×C3×D3—51no—
260C2×C3×D3—52no—
261C2×C3×D3—102no—
262C2×C3×D3C22×D511yescharacteristic
263D32—104no—
264C32×C4—51no—
265C32×C4C22×D511yescharacteristic
266C3⋊Dic3—52no—
267C3×Dic3—51no—
268C3×Dic3C22×D511yescharacteristic
269C3×Dic3C22×D512yes—
270C2×#41—451noSylow
271C2×C3×C5—21no—
272C2×C3×C5—61no—
273C2×C3×C5C23×D311yescharacteristic
274C2×C3×C5C23×D311yescharacteristic
275D15—22no—
276D15#23512yescomplemented
277C3×D5—21no—
278C3×D5—21no—
279C3×D5—32no—
280C3×D5—41no—
281C3×D5—62no—
282C5×D3—21no—
283C5×D3—62no—
284C5×D3—62no—
285C5×D3—122no—
286C23×C3—151no—
287C22×D3—51no—
288C22×D3—152no—
289C22×D3—152no—
290C3×Q8—31no—
291C3×D4—152no—
292C3×D4—151no—
293C2×C3×C4—31no—
294C2×C3×C4—51no—
295C2×C3×C4—51no—
296C2×C3×C4—101no—
297C2×C3×C4—151no—
298C2×C3×C4—152no—
299C2×C3×C4—151no—
300C2×C3×C4—151no—
301C2×C3×C4—151no—
302#67—32no—
303#67—52no—
304#67—152no—
305#67—152no—
306#67—152no—
307#67—152no—
308#67—152no—
309#67—302no—
310C2×Dic3—51no—
311C2×Dic3—152no—
312C2×Dic3—152no—
313D12—32no—
314D12—31no—
315D12—31no—
316D12—51no—
317D12—61no—
318D12—152no—
319C4×D3—32no—
320C4×D3—32no—
321C4×D3—51no—
322C4×D3—51no—
323C4×D3—51no—
324C4×D3—52no—
325C4×D3—152no—
326C4×D3—152no—
327C4×D3—152no—
328C4×D3—152no—
329C4×D3—302no—
330C4×D3D3×D511yescharacteristic, complemented
331Dic6—152no—
332Dic6—151no—
333Dic6—151no—
334Dic6—152no—
335Dic6—301no—
336Dic6D3×D511yescharacteristic, complemented
337C22×C5—31no—
338C22×C5—92no—
339C2×D5—31no—
340C2×D5—32no—
341C2×D5—62no—
342C2×D5—184no—
343C2×D5C2×D3211yescharacteristic
344C4×C5—32no—
345C4×C5—31no—
346C4×C5C2×D3211yescharacteristic
347Dic5—31no—
348Dic5—92no—
349Dic5C2×D3211yescharacteristic
350C2×C32—101no—
351C2×C32C23×D511yescharacteristic
352C3⋊D3—22no—
353C3×D3—21no—
354C3×D3—52no—
355C3×D3—102no—
356#41—91no—
357#41—151no—
358#41—451no—
359#41—451no—
360#41—452no—
361#41—452no—
362C2×Q8—451no—
363C2×D4—451no—
364C2×D4—452no—
365C22×C4—151no—
366C22×C4—452no—
367C3×C5—21no—
368C3×C5C2×#23511yescharacteristic, complemented
369C3×C5D3×#4111yescharacteristic, complemented
370C22×C3—31no—
371C22×C3—51no—
372C22×C3—51no—
373C22×C3—101no—
374C22×C3—151no—
375C22×C3—152no—
376C22×C3—302no—
377C2×D3—32no—
378C2×D3—32no—
379C2×D3—51no—
380C2×D3—52no—
381C2×D3—62no—
382C2×D3—102no—
383C2×D3—304no—
384C2×D3—304no—
385C2×D3C2×D3×D511yescharacteristic
386C3×C4—21no—
387C3×C4—32no—
388C3×C4—31no—
389C3×C4—51no—
390C3×C4—51no—
391C3×C4—101no—
392C3×C4—151no—
393C3×C4C2×D3×D511yescharacteristic
394C3×C4C2×D3×D511yescharacteristic
395Dic3—51no—
396Dic3—152no—
397Dic3—152no—
398Dic3—302no—
399Dic3C2×D3×D512yes—
400Dic3C2×D3×D511yescharacteristic
401C2×C5—61no—
402C2×C5—182no—
403C2×C5C22×D3211yescharacteristic
404D5—21no—
405D5—32no—
406D5—62no—
407C32D5×#4111yescharacteristic, complemented, Sylow
408C23—151no—
409C23—452no—
410Q8—31no—
411Q8—452no—
412Q8—451no—
413D4—92no—
414D4—91no—
415D4—152no—
416D4—151no—
417D4—452no—
418D4—452no—
419D4—452no—
420C2×C4—31no—
421C2×C4—51no—
422C2×C4—92no—
423C2×C4—151no—
424C2×C4—152no—
425C2×C4—151no—
426C2×C4—151no—
427C2×C4—151no—
428C2×C4—452no—
429C2×C4—452no—
430C2×C4—452no—
431C2×C3—21no—
432C2×C3—61no—
433C2×C3—101no—
434C2×C3—101no—
435C2×C3—152no—
436C2×C3—201no—
437C2×C3—302no—
438C2×C3C22×D3×D511yescharacteristic
439C2×C3C22×D3×D511yescharacteristic
440D3—21no—
441D3—52no—
442D3—62no—
443D3—62no—
444D3—102no—
445D3—122no—
446C5D3×#23511yescharacteristic, complemented, Sylow
447C22—31no—
448C22—51no—
449C22—92no—
450C22—151no—
451C22—152no—
452C22—302no—
453C22—904no—
454C4—31no—
455C4—32no—
456C4—51no—
457C4—151no—
458C4—452no—
459C4D32×D511yescharacteristic
460C3—21no—
461C3D5×#23511yescharacteristic, complemented
462C3#9203411yescharacteristic, complemented
463C2—61no—
464C2—101no—
465C2—152no—
466C2—182no—
467C2—302no—
468C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624226C2semidirect
2. #10624193C2semidirect
3. Dic6×D15C2semidirect
4. C5×#3799C2semidirect
5. #10624155C2semidirect
6. C3×C4×D3×D5C2semidirect
7. C3×#6773C2semidirect
8. #10624049C2semidirect
9. #10624042C2semidirect
10. #10624028C2semidirect
11. #10624022C2semidirect
16. #67180C22semidirect
17. C5×C4⋊(C3⋊D3)C22semidirect
18. C3×D60C22semidirect
20. C3×C4×C5×D3C22semidirect
21. C3×C5×Dic6C22semidirect
22. C32×C4×D5C22semidirect
23. #67156C22semidirect
24. #67153C22semidirect
25. #67150C22semidirect
27. Dic3×D15C22semidirect
28. C5×#626C22semidirect
29. C5×#624C22semidirect
30. #67138C22semidirect
31. C3⋊D3×Dic5C22semidirect
32. C3×#1202C22semidirect
34. C3×#1200C22semidirect
35. C3×D3×Dic5C22semidirect
36. C3×D5×Dic3C22semidirect
51. C4×D3×D5D3semidirect
52. #6773D3semidirect
82. #3799D5semidirect
93. D60C2×D3semidirect
108. C4×C5×D3C2×D3semidirect
109. C5×Dic6C2×D3semidirect
118. C3×C4×D5C2×D3semidirect
119. C3×C4×D5C2×D3semidirect
122. #1202C2×D3semidirect
129. #1200C2×D3semidirect
133. D3×Dic5C2×D3semidirect
136. D5×Dic3C2×D3semidirect
142. C3×D15#41semidirect
160. C4⋊(C3⋊D3)C2×D5semidirect
170. C3×C4×D3C2×D5semidirect
171. C3×Dic6C2×D5semidirect
174. #626C2×D5semidirect
176. #624C2×D5semidirect
233. C32×C5C2×#41semidirect
254. C4×D5D32semidirect
276. D15#235semidirect
330. C4×D3D3×D5semidirect
336. Dic6D3×D5semidirect
368. C3×C5C2×#235semidirect
369. C3×C5D3×#41semidirect
407. C32D5×#41semidirect
446. C5D3×#235semidirect
461. C3D5×#235semidirect
462. C3#92034semidirect

Maximal subgroups

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

subgroupindex
#106242262
#106241932
Dic6×D152
C5×#37992
#106241552
C3×C4×D3×D52
C3×#67732
#106240492
#106240422
#106240282
#106240222
#920343
D5×#2353
D3×#2355