Finite Group Databasebeta

← #13088958

Subgroups — #13088958

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624224C211yesmaximal, characteristic, complemented
2#10624199C211yesmaximal, characteristic, complemented
3#10624198C211yesmaximal, characteristic, complemented
4C5×#3799C211yesmaximal, characteristic, complemented
5#10624151C211yesmaximal, characteristic, complemented
6C3×D15⋊Q8C211yesmaximal, characteristic, complemented
7C3×D3×Dic10C211yesmaximal, characteristic, complemented
8#10624054C212yesmaximal, complemented
9#10624051C212yesmaximal, complemented
10#10624049C212yesmaximal, complemented
11#10624039C212yesmaximal, complemented
12#92030—31nomaximal
13#92004—31nomaximal
14C4×C3⋊D15C2211yescharacteristic
15C5×C4⋊(C3⋊D3)C2211yescharacteristic
16C3×C4×D15C2211yescharacteristic
17C3×Dic30C2211yescharacteristic
18C3×C4×C5×D3C2211yescharacteristic
19C3×C5×Dic6C2211yescharacteristic
20C32×Dic10C2211yescharacteristic
21#67156C2212yes—
22#67155C2212yes—
23#67153C2211yescharacteristic
24#67153C2212yes—
25#67152C2211yescharacteristic
26#67151C2211yescharacteristic
27#67150C2211yescharacteristic
28#67149C2212yes—
29C5×#626C2212yes—
30C5×#624C2212yes—
31#67140C2212yes—
32C3⋊D3×Dic5C2212yes—
33C3×#1204C2212yes—
34C3×#1204C2212yes—
35C3×#1200C2212yes—
36C3×D3×Dic5C2212yes—
37#64316—51nomaximal, Hall
38Q8×D15—31no—
39#6817—31no—
40C5×#239—31no—
41C5×#235—31no—
42C3×Q8×D5—31no—
43C2×C3×Dic10—31no—
44#6782—32no—
45#6782—32no—
46#6772—31no—
47#6772—32no—
48#6771—31no—
49#6771—32no—
50D15⋊Q8—31no—
51D15⋊Q8D311yescharacteristic, complemented
52#6769—32no—
53#6768—31no—
54#6768—31no—
55#6768—31no—
56#6768—32no—
57#6768—61no—
58D3×Dic10—32no—
59D3×Dic10—32no—
60D3×Dic10D311yescharacteristic, complemented
61C2×C3⋊D15C2311yescharacteristic
62C2×C5×C3⋊D3C2312yes—
63C2×C3×D15C2311yescharacteristic
64C2×C3×C5×D3C2311yescharacteristic
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2311yescharacteristic
67C3×Dic15C2312yes—
68C3×Dic15C2311yescharacteristic
69C3×C5×Dic3C2312yes—
70C3×C5×Dic3C2311yescharacteristic
71C32×Dic5C2312yes—
72#4150—91noHall
73#3832—51no—
74C3×D3×Q8—51no—
75C3×D3×Q8—51no—
76#3799—51no—
77#3799—52no—
78#3799—52no—
79#3799D511yescharacteristic, complemented
80#3798—51no—
81#3798—52no—
82#3797—51no—
83#3797—52no—
84C3×C5×Q8—31no—
85C2×C3×C4×C5—31no—
86#1220—31no—
87#1220—31no—
88D60—31no—
89C4×D15—31no—
90C4×D15—31no—
91C4×D15—32no—
92C4×D15—61no—
93C4×D15C2×D311yescharacteristic
94Dic30—31no—
95Dic30—32no—
96Dic30C2×D311yescharacteristic
97C5×#67—32no—
98C5×D12—32no—
99C5×D12—31no—
100C5×D12—31no—
101C5×D12—61no—
102C4×C5×D3—32no—
103C4×C5×D3—32no—
104C4×C5×D3C2×D311yescharacteristic
105C5×Dic6C2×D311yescharacteristic
106C2×C3×Dic5—32no—
107C3×C4×D5—31no—
108C3×C4×D5—32no—
109C3×Dic10—21no—
110C3×Dic10—31no—
111C3×Dic10—32no—
112C3×Dic10—32no—
113C3×Dic10C2×D311yescharacteristic
114C3×Dic10C2×D311yescharacteristic
115#1204—31no—
116#1204—32no—
117#1204C2×D312yes—
118#1204C2×D312yes—
119#1203—32no—
120#1203—31no—
121#1203—32no—
122#1203—32no—
123#1203—62no—
124#1202—31no—
125#1201—31no—
126#1201—32no—
127#1200—32no—
128#1200C2×D312yes—
129D3×Dic5—32no—
130D3×Dic5—32no—
131D3×Dic5—32no—
132D3×Dic5—32no—
133D3×Dic5—62no—
134D3×Dic5C2×D312yes—
135#1007—151no—
136#1007—151no—
137C2×C32×C5C2411yescharacteristic
138C3⋊D15—21no—
139C5×C3⋊D3—22no—
140C3×D15—21no—
141C3×C5×D3—21no—
142C5×#41—91no—
143Q8×D5—31no—
144Q8×D5—91no—
145#711—91no—
146#711—92no—
147#711—92no—
148#711—91no—
149#709—92no—
150#709—91no—
151C2×Dic10—31no—
152C2×Dic10—92no—
153C32×Q8—51no—
154C4⋊(C3⋊D3)—52no—
155C4⋊(C3⋊D3)C2×D511yescharacteristic
156C4×C3⋊D3—52no—
157C4×C3⋊D3—51no—
158C3×C4×D3—52no—
159C3×C4×D3—51no—
160C3×C4×D3—52no—
161C3×C4×D3C2×D511yescharacteristic
162C3×Dic6—51no—
163C3×Dic6—52no—
164C3×Dic6—52no—
165C3×Dic6C2×D511yescharacteristic
166#627—52no—
167#627—51no—
168#626—52no—
169#626—51no—
170#626—51no—
171#626C2×D512yes—
172#625—51no—
173#624—52no—
174#624—52no—
175#624C2×D512yes—
176C22×C3×C5—31no—
177C2×D15—31no—
178C2×D15—31no—
179C2×D15—61no—
180C2×D15C22×D311yescharacteristic
181C2×C5×D3—32no—
182C2×C5×D3—32no—
183C2×C5×D3—62no—
184C2×C5×D3C22×D311yescharacteristic
185C2×C3×D5—31no—
186C3×C4×C5—21no—
187C3×C4×C5—31no—
188C3×C4×C5—32no—
189C3×C4×C5C22×D311yescharacteristic
190C3×C4×C5C22×D311yescharacteristic
191Dic15—31no—
192Dic15—31no—
193Dic15—61no—
194Dic15C22×D311yescharacteristic
195Dic15C22×D312yes—
196C3×Dic5—22no—
197C3×Dic5—31no—
198C3×Dic5—32no—
199C3×Dic5C22×D312yes—
200C3×Dic5C22×D312yes—
201C5×Dic3C22×D312yes—
202C5×Dic3C22×D311yescharacteristic
203C2×C3×Q8—151no—
204C2×C3×Q8—151no—
205#239—31no—
206#239—152no—
207#239—152no—
208#239—151no—
209#239—151no—
210#239—151no—
211#239—301no—
212D3×Q8—51no—
213D3×Q8—51no—
214D3×Q8—152no—
215D3×Q8—152no—
216D3×Q8—151no—
217D3×Q8—151no—
218#235—31no—
219#235—152no—
220#235—151no—
221#235—152no—
222#235—152no—
223#235—152no—
224#235—151no—
225#235—151no—
226C32×C52−1+411yescharacteristic, complemented, Hall
227C5×Q8—31no—
228C5×D4—92no—
229C5×D4—91no—
230C2×C4×C5—31no—
231C2×C4×C5—92no—
232C22⋊D5—92no—
233C22⋊D5—91no—
234C22⋊D5—91no—
235C22⋊D5—92no—
236C2×Dic5—32no—
237C2×Dic5—92no—
238C2×Dic5—92no—
239D20—91no—
240C4×D5—31no—
241C4×D5—32no—
242C4×D5—92no—
243C4×D5—91no—
244Dic10—31no—
245Dic10—32no—
246Dic10—32no—
247Dic10—91no—
248Dic10—92no—
249Dic10D3211yescharacteristic
250C2×C3⋊D3—51no—
251C2×C3⋊D3C22×D512yes—
252C2×C3×D3—51no—
253C2×C3×D3C22×D511yescharacteristic
254C32×C4—52no—
255C32×C4C22×D511yescharacteristic
256C3⋊Dic3—51no—
257C3×Dic3—52no—
258C3×Dic3—51no—
259C3×Dic3C22×D512yes—
260C3×Dic3C22×D511yescharacteristic
2612−1+4—451noSylow
262C2×C3×C5—21no—
263C2×C3×C5—61no—
264C2×C3×C5C23×D311yescharacteristic
265C2×C3×C5C23×D311yescharacteristic
266D15—21no—
267D15—61no—
268D15—61no—
269D15—121no—
270C3×D5—61no—
271C5×D3—21no—
272C5×D3—62no—
273C5×D3—62no—
274C5×D3—122no—
275C3×Q8—31no—
276C3×Q8—51no—
277C3×Q8—51no—
278C3×Q8—101no—
279C3×Q8—152no—
280C3×Q8—152no—
281C3×Q8—151no—
282C2×C3×C4—31no—
283C2×C3×C4—152no—
284C2×C3×C4—152no—
285C2×C3×C4—151no—
286#67—32no—
287#67—152no—
288#67—151no—
289#67—151no—
290#67—151no—
291#67—151no—
292D12—32no—
293D12—31no—
294D12—31no—
295D12—61no—
296D12—151no—
297D12—152no—
298D12—152no—
299D12—152no—
300D12—151no—
301D12—302no—
302C4×D3—32no—
303C4×D3—32no—
304C4×D3—52no—
305C4×D3—51no—
306C4×D3—52no—
307C4×D3—152no—
308C4×D3—152no—
309C4×D3—152no—
310C4×D3—152no—
311C4×D3—152no—
312C4×D3—151no—
313C4×D3—151no—
314C4×D3—152no—
315C4×D3—301no—
316C4×D3—302no—
317C4×D3D3×D511yescharacteristic
318Dic6—51no—
319Dic6—52no—
320Dic6—52no—
321Dic6—151no—
322Dic6—152no—
323Dic6—151no—
324Dic6—152no—
325Dic6D3×D511yescharacteristic
326C22×C5—31no—
327C22×C5—92no—
328C2×D5—31no—
329C2×D5—91no—
330C4×C5—31no—
331C4×C5—32no—
332C4×C5C2×D3211yescharacteristic
333Dic5—31no—
334Dic5—32no—
335Dic5—91no—
336Dic5C2×D3212yes—
337C2×C32C23×D511yescharacteristic
338C3⋊D3—22no—
339C3⋊D3—101no—
340C3×D3—21no—
341C3×D3—101no—
342#41—91no—
343#41—451no—
344#41—452no—
345#41—452no—
346#41—452no—
347#41—451no—
348#41—451no—
349C2×Q8—151no—
350C2×Q8—151no—
351C2×Q8—452no—
352C2×Q8—451no—
353C3×C5—21no—
354C3×C5#100711yescharacteristic, complemented
355C3×C5#100711yescharacteristic, complemented
356C22×C3—31no—
357C22×C3—151no—
358C2×D3—32no—
359C2×D3—32no—
360C2×D3—51no—
361C2×D3—62no—
362C2×D3—151no—
363C2×D3—151no—
364C2×D3—301no—
365C2×D3C2×D3×D511yescharacteristic
366C3×C4—21no—
367C3×C4—31no—
368C3×C4—32no—
369C3×C4—52no—
370C3×C4—52no—
371C3×C4—102no—
372C3×C4—151no—
373C3×C4—152no—
374C3×C4C2×D3×D511yescharacteristic
375C3×C4C2×D3×D511yescharacteristic
376Dic3—52no—
377Dic3—51no—
378Dic3—151no—
379Dic3—151no—
380Dic3—301no—
381Dic3C2×D3×D511yescharacteristic
382Dic3C2×D3×D512yes—
383C2×C5—61no—
384C2×C5—182no—
385C2×C5C22×D3211yescharacteristic
386D5—61no—
387D5—181no—
388C32#415011yescharacteristic, complemented, Sylow
389Q8—31no—
390Q8—51no—
391Q8—152no—
392Q8—152no—
393Q8—151no—
394Q8—451no—
395Q8—452no—
396D4—92no—
397D4—91no—
398D4—451no—
399D4—452no—
400D4—451no—
401D4—452no—
402D4—451no—
403C2×C4—31no—
404C2×C4—92no—
405C2×C4—152no—
406C2×C4—152no—
407C2×C4—151no—
408C2×C4—452no—
409C2×C4—452no—
410C2×C4—452no—
411C2×C4—451no—
412C2×C3—21no—
413C2×C3—61no—
414C2×C3—301no—
415C2×C3C22×D3×D511yescharacteristic
416C2×C3C22×D3×D511yescharacteristic
417D3—21no—
418D3—62no—
419D3—62no—
420D3—101no—
421D3—122no—
422D3—301no—
423D3—301no—
424D3—601no—
425C5#6431611yescharacteristic, complemented, Sylow
426C22—31no—
427C22—92no—
428C22—151no—
429C22—451no—
430C4—31no—
431C4—32no—
432C4—52no—
433C4—152no—
434C4—151no—
435C4—451no—
436C4D32×D511yescharacteristic
437C3—21no—
438C3#9200411yescharacteristic, complemented
439C3#9203011yescharacteristic, complemented
440C2—61no—
441C2—182no—
442C2—301no—
443C2—901no—
444C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C52−1+4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624224C2semidirect
2. #10624199C2semidirect
3. #10624198C2semidirect
4. C5×#3799C2semidirect
5. #10624151C2semidirect
6. C3×D15⋊Q8C2semidirect
7. C3×D3×Dic10C2semidirect
8. #10624054C2semidirect
9. #10624051C2semidirect
10. #10624049C2semidirect
11. #10624039C2semidirect
51. D15⋊Q8D3semidirect
60. D3×Dic10D3semidirect
79. #3799D5semidirect
226. C32×C52−1+4semidirect
354. C3×C5#1007semidirect
355. C3×C5#1007semidirect
388. C32#4150semidirect
425. C5#64316semidirect
438. C3#92004semidirect
439. C3#92030semidirect

Maximal subgroups

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

subgroupindex
#106242242
#106241992
#106241982
C5×#37992
#106241512
C3×D15⋊Q82
C3×D3×Dic102
#106240542
#106240512
#106240492
#106240392
#920303
#920043
#643165