Finite Group Databasebeta

← #13088914

Subgroups — #13088914

Order 1440.

694 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624221C211yesmaximal, characteristic, complemented
2#10624205C211yesmaximal, characteristic, complemented
3C4×D3×D15C211yesmaximal, characteristic, complemented
4C5×#3795C211yesmaximal, characteristic, complemented
5#10624153C211yesmaximal, characteristic, complemented
6C3×#6769C211yesmaximal, characteristic, complemented
7C3×#6768C211yesmaximal, characteristic, complemented
8#10624053C212yesmaximal, complemented
9#10624050C212yesmaximal, complemented
10#10624035C212yesmaximal, complemented
11D3×#1200C212yesmaximal, complemented
12#92024—31nomaximal
13#92016—31nomaximal
14C2×D3×D15C2212yes—
15C2×D3×D15C2211yescharacteristic
16C4×C3⋊D15C2211yescharacteristic
17C5×#634C2211yescharacteristic, complemented
18C3×D60C2211yescharacteristic, complemented
19C3×C4×D15C2211yescharacteristic, complemented
20C3×C5×D12C2211yescharacteristic, complemented
21C3×C4×C5×D3C2211yescharacteristic, complemented
22C32×Dic10C2211yescharacteristic, complemented
23#67157C2212yescomplemented
24#67154C2212yescomplemented
25#67150C2212yescomplemented
26#67149C2211yescharacteristic
27D3×Dic15C2211yescharacteristic, complemented
28Dic3×D15C2211yescharacteristic, complemented
29C5×#625C2212yescomplemented
30C5×D3×Dic3C2212yescomplemented
31#67141C2212yescomplemented
32#67137C2212yes—
33C3×#1203C2212yescomplemented
34C3×#1203C2212yescomplemented
35C3×#1200C2212yescomplemented
36C3×D3×Dic5C2212yescomplemented
37#64319—51nomaximal, Hall
38D4×D15—31no—
39C2×C4×D15—31no—
40C5×#237—31no—
41C5×#235—31no—
42C3×#711—31no—
43C3×#709—31no—
44#6790—32no—
45#6790—32no—
46C2×#1200—32no—
47#6782—32no—
48#6780—32no—
49#6779—32no—
50#6777—31no—
51C4×D3×D5—31no—
52C4×D3×D5—32no—
53#6773—31no—
54#6771—32no—
55D15⋊Q8—31no—
56D15⋊Q8—31no—
57D15⋊Q8—61no—
58#6769D311yescharacteristic, complemented
59#6768D311yescharacteristic, complemented
60#6766—31no—
61C2×C3⋊D15C2311yescharacteristic
62C2×C3×D15C2312yes—
63C2×C3×D15C2311yescharacteristic
64C2×C3×C5×D3C2312yes—
65C2×C3×C5×D3C2311yescharacteristic
66D3×D15—24no—
67D3×D15—22no—
68C32×C4×C5C2311yescharacteristic
69C3⋊Dic15C2311yescharacteristic
70C5×C3⋊Dic3C2312yes—
71C3×Dic15C2311yescharacteristic
72C3×C5×Dic3C2311yescharacteristic
73C32×Dic5C2312yes—
74D5×#41—91noHall
75Q8×C3⋊D3—51no—
76C3×#239—51no—
77C3×#239—51no—
78#3802—51no—
79#3802—52no—
80C4×D32—51no—
81C4×D32—52no—
82#3795—51no—
83#3795—52no—
84#3795—52no—
85#3795D511yescharacteristic, complemented
86C22×D15—32no—
87C22×D15—31no—
88C2×D3×D5—32no—
89C2×D3×D5—31no—
90C3×C5×D4—31no—
91C2×C3×C4×C5—31no—
92#1220—32no—
93C2×Dic15—31no—
94D60C2×D311yescharacteristic, complemented
95C4×D15—31no—
96C4×D15—31no—
97C4×D15—31no—
98C4×D15—31no—
99C4×D15—61no—
100C4×D15C2×D311yescharacteristic, complemented
101C5×#67—32no—
102C5×#67—32no—
103C2×C5×Dic3—32no—
104C5×D12C2×D311yescharacteristic, complemented
105C4×C5×D3—32no—
106C4×C5×D3C2×D311yescharacteristic, complemented
107C5×Dic6—31no—
108C5×Dic6—31no—
109C5×Dic6—61no—
110C3×C22⋊D5—32no—
111C3×C22⋊D5—32no—
112C2×C3×Dic5—32no—
113C3×D20—31no—
114C3×C4×D5—31no—
115C3×C4×D5—32no—
116C3×Dic10—21no—
117C3×Dic10C2×D311yescharacteristic, complemented
118C3×Dic10C2×D311yescharacteristic, complemented
119#1204—32no—
120#1204—32no—
121#1204—62no—
122#1203C2×D312yescomplemented
123#1203C2×D312yescomplemented
124#1202—32no—
125#1202—32no—
126#1201—32no—
127#1200—32no—
128#1200—32no—
129#1200—32no—
130#1200—31no—
131#1200—62no—
132#1200C2×D312yescomplemented
133D3×Dic5—31no—
134D3×Dic5C2×D312yescomplemented
135D5×Dic3—32no—
136D5×Dic3—31no—
137D3×#41—151no—
138D3×#41—151no—
139C2×C32×C5C2411yescharacteristic
140C3⋊D15#4112yescomplemented
141C3×D15—22no—
142C3×D15—21no—
143C3×C5×D3—22no—
144C3×C5×D3—21no—
145C5×#41—91no—
146#713—91no—
147Q8×D5—91no—
148#711—31no—
149#711—92no—
150D4×D5—91no—
151D4×D5—92no—
152#709—31no—
153#709—92no—
154C2×C4×D5—92no—
155C2×C4×D5—91no—
156C2×D32—51no—
157C2×D32—52no—
158C32×Q8—51no—
159C4×C3⋊D3—51no—
160C4×C3⋊D3—52no—
161#634—52no—
162#634C2×D511yescharacteristic, complemented
163C3×D12—52no—
164C3×D12—52no—
165C3×D12—51no—
166C3×D12C2×D511yescharacteristic, complemented
167C3×C4×D3—52no—
168C3×C4×D3—51no—
169C3×C4×D3—52no—
170C3×C4×D3C2×D511yescharacteristic, complemented
171#625—52no—
172#625—52no—
173#625C2×D512yescomplemented
174#624—51no—
175D3×Dic3—51no—
176D3×Dic3—52no—
177D3×Dic3—51no—
178D3×Dic3C2×D512yescomplemented
179C22×C3×C5—32no—
180C22×C3×C5—31no—
181C2×D15—31no—
182C2×D15—31no—
183C2×D15—61no—
184C2×D15—62no—
185C2×D15—64no—
186C2×D15C22×D312yes—
187C2×D15C22×D311yescharacteristic
188C2×C5×D3C22×D311yescharacteristic
189C2×C5×D3C22×D312yes—
190C2×C3×D5—32no—
191C2×C3×D5—31no—
192D3×D5—64no—
193D3×D5—62no—
194C3×C4×C5—21no—
195C3×C4×C5—31no—
196C3×C4×C5C22×D311yescharacteristic
197C3×C4×C5C22×D311yescharacteristic
198Dic15—31no—
199Dic15—31no—
200Dic15—61no—
201Dic15C22×D311yescharacteristic
202C3×Dic5—22no—
203C3×Dic5—31no—
204C3×Dic5C22×D312yes—
205C3×Dic5C22×D312yes—
206C5×Dic3—32no—
207C5×Dic3—32no—
208C5×Dic3—62no—
209C5×Dic3C22×D311yescharacteristic
210C3×#41—151no—
211C3×#41—151no—
212#239—51no—
213#239—51no—
214D3×Q8—151no—
215D3×Q8—151no—
216D3×Q8—301no—
217#237—31no—
218#237—151no—
219#237—152no—
220#237—152no—
221D3×D4—152no—
222D3×D4—152no—
223D3×D4—151no—
224D3×D4—151no—
225#235—31no—
226#235—151no—
227#235—152no—
228#235—152no—
229C2×C4×D3—152no—
230C2×C4×D3—152no—
231C2×C4×D3—151no—
232C2×C4×D3—151no—
233C32×C5C2×#4111yescharacteristic, complemented, Hall
234C22×D5—92no—
235C22×D5—91no—
236C5×Q8—91no—
237C5×D4—31no—
238C5×D4—92no—
239C2×C4×C5—31no—
240C2×C4×C5—92no—
241C22⋊D5—32no—
242C22⋊D5—32no—
243C22⋊D5—92no—
244C2×Dic5—32no—
245C2×Dic5—91no—
246D20—31no—
247D20—92no—
248C4×D5—31no—
249C4×D5—32no—
250C4×D5—91no—
251C4×D5—92no—
252C4×D5—91no—
253C4×D5—91no—
254C4×D5—92no—
255Dic10—92no—
256Dic10D3211yescharacteristic, complemented
257C2×C3⋊D3—51no—
258C2×C3×D3—52no—
259C2×C3×D3—51no—
260C2×C3×D3C22×D511yescharacteristic
261C2×C3×D3C22×D512yes—
262D32—102no—
263D32—104no—
264C32×C4—52no—
265C32×C4C22×D511yescharacteristic
266C3⋊Dic3—51no—
267C3⋊Dic3C22×D512yes—
268C3×Dic3—51no—
269C3×Dic3C22×D511yescharacteristic
270C2×#41—451noSylow
271C2×C3×C5—21no—
272C2×C3×C5—62no—
273C2×C3×C5—61no—
274C2×C3×C5C23×D311yescharacteristic
275C2×C3×C5C23×D311yescharacteristic
276D15—22no—
277D15—21no—
278D15—32no—
279D15—32no—
280D15—62no—
281C3×D5—61no—
282C3×D5—62no—
283C5×D3—21no—
284C5×D3—22no—
285C22×D3—152no—
286C22×D3—151no—
287C22×D3—152no—
288C22×D3—151no—
289C3×Q8—51no—
290C3×Q8—51no—
291C3×Q8—101no—
292C3×D4—31no—
293C3×D4—152no—
294C3×D4—152no—
295C3×D4—151no—
296C2×C3×C4—31no—
297C2×C3×C4—152no—
298C2×C3×C4—152no—
299C2×C3×C4—151no—
300#67—32no—
301#67—32no—
302#67—152no—
303#67—152no—
304#67—152no—
305#67—152no—
306C2×Dic3—32no—
307C2×Dic3—152no—
308C2×Dic3—151no—
309C2×Dic3—151no—
310D12—52no—
311D12—52no—
312D12—51no—
313D12D3×D511yescharacteristic, complemented
314C4×D3—32no—
315C4×D3—52no—
316C4×D3—51no—
317C4×D3—52no—
318C4×D3—152no—
319C4×D3—152no—
320C4×D3—152no—
321C4×D3—151no—
322C4×D3—151no—
323C4×D3—151no—
324C4×D3—151no—
325C4×D3—151no—
326C4×D3—151no—
327C4×D3—302no—
328C4×D3—301no—
329C4×D3D3×D511yescharacteristic, complemented
330Dic6—31no—
331Dic6—31no—
332Dic6—61no—
333Dic6—152no—
334Dic6—152no—
335Dic6—302no—
336C22×C5—32no—
337C22×C5—31no—
338C2×D5—32no—
339C2×D5—31no—
340C2×D5—91no—
341C2×D5—184no—
342C2×D5—182no—
343C4×C5—31no—
344C4×C5—92no—
345C4×C5C2×D3211yescharacteristic
346Dic5—31no—
347Dic5—91no—
348Dic5C2×D3212yes—
349C2×C32C23×D511yescharacteristic
350C3⋊D3—52no—
351C3×D3—21no—
352C3×D3—22no—
353C3×D3—102no—
354C3×D3—101no—
355#41—91no—
356#41—151no—
357#41—151no—
358#41—452no—
359#41—452no—
360#41—451no—
361C2×Q8—451no—
362C2×D4—451no—
363C2×D4—452no—
364C22×C4—451no—
365C22×C4—452no—
366C3×C5—21no—
367C3×C5D3×#4111yescharacteristic, complemented
368C3×C5D3×#4111yescharacteristic, complemented
369C22×C3—32no—
370C22×C3—31no—
371C22×C3—151no—
372C22×C3—152no—
373C2×D3—52no—
374C2×D3—51no—
375C2×D3—151no—
376C2×D3—151no—
377C2×D3—301no—
378C2×D3—302no—
379C2×D3—304no—
380C2×D3—304no—
381C2×D3—302no—
382C2×D3C2×D3×D512yes—
383C2×D3C2×D3×D511yescharacteristic
384C3×C4—21no—
385C3×C4—31no—
386C3×C4—52no—
387C3×C4—52no—
388C3×C4—102no—
389C3×C4—151no—
390C3×C4C2×D3×D511yescharacteristic
391C3×C4C2×D3×D511yescharacteristic
392Dic3—32no—
393Dic3—32no—
394Dic3—51no—
395Dic3—62no—
396Dic3—151no—
397Dic3—151no—
398Dic3—301no—
399Dic3C2×D3×D511yescharacteristic
400C2×C5—62no—
401C2×C5—61no—
402C2×C5C22×D3211yescharacteristic
403D5—62no—
404D5—61no—
405D5—92no—
406C32D5×#4111yescharacteristic, complemented, Sylow
407C23—451no—
408C23—452no—
409Q8—51no—
410Q8—91no—
411Q8—452no—
412D4—31no—
413D4—92no—
414D4—152no—
415D4—152no—
416D4—151no—
417D4—452no—
418D4—452no—
419C2×C4—31no—
420C2×C4—92no—
421C2×C4—152no—
422C2×C4—152no—
423C2×C4—151no—
424C2×C4—451no—
425C2×C4—451no—
426C2×C4—452no—
427C2×C4—452no—
428C2×C4—451no—
429C2×C4—451no—
430C2×C3—21no—
431C2×C3—62no—
432C2×C3—61no—
433C2×C3—301no—
434C2×C3—302no—
435C2×C3C22×D3×D511yescharacteristic
436C2×C3C22×D3×D511yescharacteristic
437D3—21no—
438D3—22no—
439D3—102no—
440D3—101no—
441D3—152no—
442D3—152no—
443D3—302no—
444C5#6431911yescharacteristic, complemented, Sylow
445C22—31no—
446C22—32no—
447C22—152no—
448C22—151no—
449C22—451no—
450C22—902no—
451C22—904no—
452C4—31no—
453C4—52no—
454C4—92no—
455C4—151no—
456C4—451no—
457C4D32×D511yescharacteristic
458C3—21no—
459C3#9201611yescharacteristic, complemented
460C3#9202411yescharacteristic, complemented
461C2—61no—
462C2—62no—
463C2—302no—
464C2—301no—
465C2—452no—
466C2C2×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. #10624221C2semidirect
2. #10624205C2semidirect
3. C4×D3×D15C2semidirect
4. C5×#3795C2semidirect
5. #10624153C2semidirect
6. C3×#6769C2semidirect
7. C3×#6768C2semidirect
8. #10624053C2semidirect
9. #10624050C2semidirect
10. #10624035C2semidirect
11. D3×#1200C2semidirect
17. C5×#634C22semidirect
18. C3×D60C22semidirect
19. C3×C4×D15C22semidirect
20. C3×C5×D12C22semidirect
21. C3×C4×C5×D3C22semidirect
22. C32×Dic10C22semidirect
23. #67157C22semidirect
24. #67154C22semidirect
25. #67150C22semidirect
27. D3×Dic15C22semidirect
28. Dic3×D15C22semidirect
29. C5×#625C22semidirect
30. C5×D3×Dic3C22semidirect
31. #67141C22semidirect
33. C3×#1203C22semidirect
34. C3×#1203C22semidirect
35. C3×#1200C22semidirect
36. C3×D3×Dic5C22semidirect
58. #6769D3semidirect
59. #6768D3semidirect
85. #3795D5semidirect
94. D60C2×D3semidirect
100. C4×D15C2×D3semidirect
104. C5×D12C2×D3semidirect
106. C4×C5×D3C2×D3semidirect
117. C3×Dic10C2×D3semidirect
118. C3×Dic10C2×D3semidirect
122. #1203C2×D3semidirect
123. #1203C2×D3semidirect
132. #1200C2×D3semidirect
134. D3×Dic5C2×D3semidirect
140. C3⋊D15#41semidirect
162. #634C2×D5semidirect
166. C3×D12C2×D5semidirect
170. C3×C4×D3C2×D5semidirect
173. #625C2×D5semidirect
178. D3×Dic3C2×D5semidirect
233. C32×C5C2×#41semidirect
256. Dic10D32semidirect
313. D12D3×D5semidirect
329. C4×D3D3×D5semidirect
367. C3×C5D3×#41semidirect
368. C3×C5D3×#41semidirect
406. C32D5×#41semidirect
444. C5#64319semidirect
459. C3#92016semidirect
460. C3#92024semidirect

Maximal subgroups

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

subgroupindex
#106242212
#106242052
C4×D3×D152
C5×#37952
#106241532
C3×#67692
C3×#67682
#106240532
#106240502
#106240352
D3×#12002
#920243
#920163
#643195