Finite Group Databasebeta

← #13089363

Subgroups — #13089363

Order 1440.

710 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊D15C211yesmaximal, characteristic, complemented
2C3×#6817C211yesmaximal, characteristic, complemented
3C3×C5×#235C211yesmaximal, characteristic, complemented
4#10624218C212yesmaximal, complemented
5#10624208C212yesmaximal, complemented
6#10624205C211yesmaximal, characteristic, complemented
7C4×D3×D15C212yesmaximal, complemented
8#10624199C212yesmaximal, complemented
9#10624198C211yesmaximal, characteristic, complemented
10#10624196C211yesmaximal, characteristic, complemented
11#10624194C211yesmaximal, characteristic, complemented
12#41×D15—31nomaximal
13#92016—31nomaximal
14C22×C3⋊D15C2211yescharacteristic
15C2×D3×D15C2212yes—
16C2×C32×C4×C5C2211yescharacteristic, complemented
17C2×C3⋊Dic15C2211yescharacteristic, complemented
18C4×C3⋊D15C2211yescharacteristic
19C4×C3⋊D15C2211yescharacteristic
20C4×C3⋊D15C2211yescharacteristic, complemented
21C4×C3⋊D15C2211yescharacteristic, complemented
22C3×#1220C2212yescomplemented
23C3×D60C2211yescharacteristic, complemented
24C3×C4×D15C2212yescomplemented
25C3×Dic30C2211yescharacteristic, complemented
26C3×C5×#67C2212yescomplemented
27C3×C5×D12C2211yescharacteristic, complemented
28C3×C4×C5×D3C2212yescomplemented
29C3×C5×Dic6C2211yescharacteristic, complemented
30#67153C2212yescomplemented
31#67152C2212yescomplemented
32#67151C2212yescomplemented
33#67150C2212yescomplemented
34#67149C2212yes—
35D3×Dic15C2212yescomplemented
36Dic3×D15C2212yescomplemented
37#64306—51nomaximal, Hall
38C3×C5×#41—31no—
39#6821—31no—
40Q8×D15—31no—
41#6819—32no—
42#6819—31no—
43D4×D15—32no—
44D4×D15—31no—
45#6817—32no—
46#6817D311yescharacteristic, complemented
47C2×C4×D15—31no—
48C2×C4×D15—31no—
49C2×C4×D15—32no—
50C2×C4×D15—61no—
51C5×#235D311yescharacteristic, complemented
52C3×#709—31no—
53#6790—32no—
54#6780—32no—
55#6777—31no—
56C4×D3×D5—32no—
57#6771—32no—
58D15⋊Q8—31no—
59#6768—31no—
60#6766—31no—
61C22×C32×C5C2311yescharacteristic
62C2×C3⋊D15—22no—
63C2×C3⋊D15C2311yescharacteristic
64C2×C3⋊D15C2311yescharacteristic
65C2×C3×D15C2312yes—
66C2×C3×C5×D3C2312yes—
67D3×D15—24no—
68C32×C4×C5C2311yescharacteristic
69C32×C4×C5C2311yescharacteristic
70C3⋊Dic15C2311yescharacteristic
71C3⋊Dic15C2311yescharacteristic
72C3×Dic15C2312yes—
73C3×C5×Dic3C2312yes—
74D5×#41—91noHall
75C2×C4×C3⋊D3—51no—
76C3×#235—51no—
77C3×#235D511yescharacteristic, complemented
78#3811—52no—
79#3805—52no—
80#3802—51no—
81C4×D32—52no—
82#3798—52no—
83#3797—51no—
84#3796—51no—
85#3796—51no—
86C22×D15—31no—
87C22×D15—31no—
88C22×D15—32no—
89C22×D15—61no—
90C2×D3×D5—32no—
91C3×C5×Q8—31no—
92C3×C5×D4—32no—
93C3×C5×D4—31no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—32no—
96C2×C3×C4×C5C2×D311yescharacteristic, complemented
97C2×C3×C4×C5C2×D311yescharacteristic, complemented
98#1220—32no—
99#1220—32no—
100#1220C2×D312yescomplemented
101C2×Dic15—31no—
102C2×Dic15—31no—
103C2×Dic15—32no—
104C2×Dic15—61no—
105D60—32no—
106D60C2×D311yescharacteristic, complemented
107C4×D15—32no—
108C4×D15—31no—
109C4×D15—31no—
110C4×D15—31no—
111C4×D15—31no—
112C4×D15—31no—
113C4×D15—31no—
114C4×D15—31no—
115C4×D15—31no—
116C4×D15—32no—
117C4×D15—61no—
118C4×D15—61no—
119C4×D15—61no—
120C4×D15—61no—
121C4×D15C2×D312yescomplemented
122Dic30—32no—
123Dic30C2×D311yescharacteristic, complemented
124C5×#67C2×D312yescomplemented
125C5×D12C2×D311yescharacteristic, complemented
126C4×C5×D3C2×D312yescomplemented
127C5×Dic6C2×D311yescharacteristic, complemented
128C3×C22⋊D5—32no—
129C3×D20—31no—
130C3×C4×D5—32no—
131C3×Dic10—31no—
132#1204—32no—
133#1203—32no—
134#1202—32no—
135#1201—32no—
136#1200—32no—
137D3×Dic5—32no—
138D5×Dic3—32no—
139D3×#41—151no—
140D3×#41—151no—
141C2×C32×C5—21no—
142C2×C32×C5C2411yescharacteristic
143C3⋊D15—21no—
144C3⋊D15#4112yescomplemented
145C3×D15—22no—
146C3×C5×D3—22no—
147C5×#41—31no—
148#713—91no—
149Q8×D5—91no—
150#711—91no—
151#711—92no—
152D4×D5—91no—
153D4×D5—92no—
154#709—31no—
155#709—92no—
156C2×C4×D5—92no—
157C2×C4×D5—91no—
158C22×C3⋊D3—51no—
159C2×D32—52no—
160C2×C32×C4C2×D511yescharacteristic, complemented
161C2×C3⋊Dic3—51no—
162C4×C3⋊D3—51no—
163C4×C3⋊D3—51no—
164C4×C3⋊D3—51no—
165C4×C3⋊D3—51no—
166C3×#67—52no—
167C3×#67C2×D512yescomplemented
168C3×D12—51no—
169C3×D12C2×D511yescharacteristic, complemented
170C3×C4×D3—52no—
171C3×C4×D3C2×D512yescomplemented
172C3×Dic6—51no—
173C3×Dic6C2×D511yescharacteristic, complemented
174#627—52no—
175#626—52no—
176#626—52no—
177#625—52no—
178#624—52no—
179D3×Dic3—52no—
180D3×Dic3—52no—
181C22×C3×C5—21no—
182C22×C3×C5—32no—
183C22×C3×C5C22×D311yescharacteristic
184C22×C3×C5C22×D311yescharacteristic
185C2×D15—31no—
186C2×D15—31no—
187C2×D15—31no—
188C2×D15—31no—
189C2×D15—61no—
190C2×D15—61no—
191C2×D15—64no—
192C2×D15—62no—
193C2×D15—62no—
194C2×D15—64no—
195C2×D15C22×D312yes—
196C2×C5×D3C22×D312yes—
197C2×C3×D5—32no—
198D3×D5—64no—
199C3×C4×C5—21no—
200C3×C4×C5—21no—
201C3×C4×C5—32no—
202C3×C4×C5C22×D311yescharacteristic
203C3×C4×C5C22×D311yescharacteristic
204C3×C4×C5C22×D311yescharacteristic
205C3×C4×C5C22×D311yescharacteristic
206Dic15—31no—
207Dic15—31no—
208Dic15—31no—
209Dic15—31no—
210Dic15—61no—
211Dic15—61no—
212Dic15C22×D312yes—
213C3×Dic5—32no—
214C5×Dic3C22×D312yes—
215C3×#41—31no—
216C3×#41—151no—
217#239—151no—
218#239—151no—
219D3×Q8—151no—
220D3×Q8—151no—
221#237—151no—
222#237—152no—
223#237—152no—
224#237—151no—
225D3×D4—151no—
226D3×D4—152no—
227D3×D4—152no—
228D3×D4—151no—
229#235—51no—
230#235—152no—
231#235—152no—
232#235D1511yescharacteristic, complemented
233C2×C4×D3—152no—
234C2×C4×D3—151no—
235C2×C4×D3—151no—
236C2×C4×D3—152no—
237C2×C4×D3—301no—
238C32×C5C2×#4111yescharacteristic, complemented, Hall
239C22×D5—92no—
240C22×D5—91no—
241C5×Q8—31no—
242C5×D4—32no—
243C5×D4—31no—
244C2×C4×C5—32no—
245C2×C4×C5D3211yescharacteristic, complemented
246C22⋊D5—32no—
247C22⋊D5—92no—
248C22⋊D5—92no—
249C2×Dic5—92no—
250C2×Dic5—91no—
251D20—31no—
252D20—92no—
253C4×D5—32no—
254C4×D5—92no—
255C4×D5—92no—
256C4×D5—91no—
257C4×D5—91no—
258C4×D5—91no—
259C4×D5—91no—
260Dic10—31no—
261Dic10—92no—
262C22×C32C22×D511yescharacteristic
263C2×C3⋊D3—51no—
264C2×C3⋊D3—51no—
265C2×C3⋊D3—102no—
266C2×C3×D3—52no—
267C2×C3×D3C22×D512yes—
268D32—104no—
269C32×C4C22×D511yescharacteristic
270C32×C4C22×D511yescharacteristic
271C3⋊Dic3—51no—
272C3⋊Dic3—51no—
273C3×Dic3—52no—
274C3×Dic3C22×D512yes—
275C2×#41—451noSylow
276C2×C3×C5—22no—
277C2×C3×C5—21no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—62no—
281C2×C3×C5C23×D311yescharacteristic
282C2×C3×C5C23×D311yescharacteristic
283D15—22no—
284D15—32no—
285D15—32no—
286D15—62no—
287D15—62no—
288D15—61no—
289D15—61no—
290C3×D5—62no—
291C5×D3—22no—
292C22×D3—152no—
293C22×D3—151no—
294C22×D3—151no—
295C22×D3—152no—
296C22×D3—301no—
297C3×Q8—31no—
298C3×Q8—151no—
299C3×D4—32no—
300C3×D4—31no—
301C3×D4—152no—
302C3×D4—151no—
303C2×C3×C4—21no—
304C2×C3×C4—32no—
305C2×C3×C4—152no—
306C2×C3×C4D3×D511yescharacteristic, complemented
307C2×C3×C4C2×D1511yescharacteristic, complemented
308#67—52no—
309#67—152no—
310#67—152no—
311#67—152no—
312#67—152no—
313#67C2×D1512yescomplemented
314C2×Dic3—152no—
315C2×Dic3—151no—
316C2×Dic3—151no—
317C2×Dic3—152no—
318C2×Dic3—301no—
319D12—51no—
320D12—152no—
321D12—152no—
322D12C2×D1511yescharacteristic, complemented
323C4×D3—52no—
324C4×D3—152no—
325C4×D3—152no—
326C4×D3—152no—
327C4×D3—151no—
328C4×D3—151no—
329C4×D3—151no—
330C4×D3—151no—
331C4×D3—151no—
332C4×D3—151no—
333C4×D3—151no—
334C4×D3—151no—
335C4×D3—152no—
336C4×D3—301no—
337C4×D3—301no—
338C4×D3—301no—
339C4×D3—301no—
340C4×D3C2×D1512yescomplemented
341Dic6—51no—
342Dic6—152no—
343Dic6—152no—
344Dic6C2×D1511yescharacteristic, complemented
345C22×C5—32no—
346C22×C5C2×D3211yescharacteristic
347C2×D5—32no—
348C2×D5—91no—
349C2×D5—91no—
350C2×D5—184no—
351C2×D5—182no—
352C4×C5—32no—
353C4×C5C2×D3211yescharacteristic
354C4×C5C2×D3211yescharacteristic
355Dic5—32no—
356Dic5—91no—
357Dic5—91no—
358C2×C32—21no—
359C2×C32C23×D511yescharacteristic
360C3⋊D3—52no—
361C3⋊D3—101no—
362C3×D3—22no—
363C3×D3—102no—
364#41—31no—
365#41—151no—
366#41—452no—
367#41—452no—
368#41—451no—
369#41—451no—
370C2×Q8—451no—
371C2×D4—452no—
372C2×D4—451no—
373C22×C4—452no—
374C22×C4—451no—
375C3×C5—21no—
376C3×C5D3×#4111yescharacteristic, complemented
377C3×C5D3×#4111yescharacteristic, complemented
378C22×C3—21no—
379C22×C3—32no—
380C22×C3—152no—
381C22×C3C2×D3×D511yescharacteristic
382C22×C3C22×D1511yescharacteristic
383C2×D3—52no—
384C2×D3—151no—
385C2×D3—151no—
386C2×D3—151no—
387C2×D3—151no—
388C2×D3—301no—
389C2×D3—301no—
390C2×D3—304no—
391C2×D3—304no—
392C2×D3—302no—
393C2×D3—302no—
394C2×D3—304no—
395C2×D3C22×D1512yes—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—32no—
399C3×C4—152no—
400C3×C4C2×D3×D511yescharacteristic
401C3×C4C2×D3×D511yescharacteristic
402C3×C4C22×D1511yescharacteristic
403C3×C4C22×D1511yescharacteristic
404Dic3—52no—
405Dic3—151no—
406Dic3—151no—
407Dic3—151no—
408Dic3—151no—
409Dic3—301no—
410Dic3—301no—
411Dic3C22×D1512yes—
412C2×C5—21no—
413C2×C5—62no—
414C2×C5C22×D3211yescharacteristic
415D5—62no—
416D5—92no—
417D5—181no—
418C32D5×#4111yescharacteristic, complemented, Sylow
419C23—452no—
420C23—451no—
421Q8—31no—
422Q8—151no—
423Q8—452no—
424D4—32no—
425D4—31no—
426D4—152no—
427D4—151no—
428D4—452no—
429D4—452no—
430D4—452no—
431C2×C4—32no—
432C2×C4—152no—
433C2×C4—452no—
434C2×C4—452no—
435C2×C4—452no—
436C2×C4—451no—
437C2×C4—451no—
438C2×C4—451no—
439C2×C4—451no—
440C2×C4—451no—
441C2×C4D3×D1511yescharacteristic, complemented
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—62no—
447C2×C3—302no—
448C2×C3C22×D3×D511yescharacteristic
449C2×C3C23×D1511yescharacteristic
450D3—22no—
451D3—102no—
452D3—152no—
453D3—152no—
454D3—302no—
455D3—302no—
456D3—301no—
457D3—301no—
458C5#6430611yescharacteristic, complemented, Sylow
459C22—32no—
460C22—152no—
461C22—451no—
462C22—451no—
463C22—902no—
464C22—904no—
465C22C2×D3×D1511yescharacteristic
466C4—32no—
467C4—152no—
468C4—451no—
469C4—451no—
470C4C2×D3×D1511yescharacteristic
471C4C2×D3×D1511yescharacteristic
472C3—21no—
473C3#9201611yescharacteristic, complemented
474C3#41×D1511yescharacteristic, complemented
475C2—21no—
476C2—62no—
477C2—302no—
478C2—452no—
479C2—901no—
480C2C22×D3×D1511yescharacteristic
subgroupG/H
centerC4C2×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FrattiniC2C22×D3×D15
FittingC2×C32×C4×C5C22
hypercenterC2×C4D3×D15
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C3⋊D15C2semidirect
2. C3×#6817C2semidirect
3. C3×C5×#235C2semidirect
4. #10624218C2semidirect
5. #10624208C2semidirect
6. #10624205C2semidirect
7. C4×D3×D15C2semidirect
8. #10624199C2semidirect
9. #10624198C2semidirect
10. #10624196C2semidirect
11. #10624194C2semidirect
16. C2×C32×C4×C5C22semidirect
17. C2×C3⋊Dic15C22semidirect
20. C4×C3⋊D15C22semidirect
21. C4×C3⋊D15C22semidirect
22. C3×#1220C22semidirect
23. C3×D60C22semidirect
24. C3×C4×D15C22semidirect
25. C3×Dic30C22semidirect
26. C3×C5×#67C22semidirect
27. C3×C5×D12C22semidirect
28. C3×C4×C5×D3C22semidirect
29. C3×C5×Dic6C22semidirect
30. #67153C22semidirect
31. #67152C22semidirect
32. #67151C22semidirect
33. #67150C22semidirect
35. D3×Dic15C22semidirect
36. Dic3×D15C22semidirect
46. #6817D3semidirect
51. C5×#235D3semidirect
77. C3×#235D5semidirect
96. C2×C3×C4×C5C2×D3semidirect
97. C2×C3×C4×C5C2×D3semidirect
100. #1220C2×D3semidirect
106. D60C2×D3semidirect
121. C4×D15C2×D3semidirect
123. Dic30C2×D3semidirect
124. C5×#67C2×D3semidirect
125. C5×D12C2×D3semidirect
126. C4×C5×D3C2×D3semidirect
127. C5×Dic6C2×D3semidirect
144. C3⋊D15#41semidirect
160. C2×C32×C4C2×D5semidirect
167. C3×#67C2×D5semidirect
169. C3×D12C2×D5semidirect
171. C3×C4×D3C2×D5semidirect
173. C3×Dic6C2×D5semidirect
232. #235D15semidirect
238. C32×C5C2×#41semidirect
245. C2×C4×C5D32semidirect
306. C2×C3×C4D3×D5semidirect
307. C2×C3×C4C2×D15semidirect
313. #67C2×D15semidirect
322. D12C2×D15semidirect
340. C4×D3C2×D15semidirect
344. Dic6C2×D15semidirect
376. C3×C5D3×#41semidirect
377. C3×C5D3×#41semidirect
418. C32D5×#41semidirect
441. C2×C4D3×D15semidirect
458. C5#64306semidirect
473. C3#92016semidirect
474. C3#41×D15semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊D152
C3×#68172
C3×C5×#2352
#106242182
#106242082
#106242052
C4×D3×D152
#106241992
#106241982
#106241962
#106241942
#41×D153
#920163
#643065