Finite Group Databasebeta

← #11753050

Subgroups — #11753050

Order 960.

484 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91767C211yesmaximal, characteristic, complemented
2C5×D3×C22⋊C4C211yesmaximal, characteristic
3C3×#4024C211yesmaximal, characteristic, complemented
4#91576C211yesmaximal, characteristic, complemented
5#91574C211yesmaximal, characteristic, complemented
6D3×C22⋊Dic5C211yesmaximal, characteristic
7C2×(C2×D3)⋊Dic5C211yesmaximal, characteristic
8#91436C211yesmaximal, characteristic, complemented
9#91432C211yesmaximal, characteristic, complemented
10#91417C211yesmaximal, characteristic, complemented
11#91410C211yesmaximal, characteristic, complemented
12#91369C211yesmaximal, characteristic, complemented
13#91365C211yesmaximal, characteristic, complemented
14#91352C211yesmaximal, characteristic, complemented
15#91352C211yesmaximal, characteristic, complemented
16#65817—31nomaximal, Hall
17C23×C5×D3C2211yescharacteristic
18C22×Dic15C2211yescharacteristic
19C2×Dic30C2211yescharacteristic, complemented
20C2×C4×C5×D3C2211yescharacteristic
21C2×C4×C5×D3C2211yescharacteristic
22C22×C3×Dic5C2211yescharacteristic
23C2×C3×Dic10C2211yescharacteristic, complemented
24C2×#1204C2211yescharacteristic, complemented
25C2×#1204C2211yescharacteristic, complemented
26C2×D3×Dic5C2211yescharacteristic
27C2×D3×Dic5C2211yescharacteristic
28C3×C5×C22⋊C4C2211yescharacteristic
29#6719C2211yescharacteristic
30C4⋊Dic15C2211yescharacteristic, complemented
31#6712C2211yescharacteristic, complemented
32#6712C2211yescharacteristic, complemented
33C5×#217C2211yescharacteristic
34C5×(C2×D3)⋊C4C2211yescharacteristic
35C5×(C2×D3)⋊C4C2211yescharacteristic
36C3×C22⋊Dic5C2211yescharacteristic
37C3×C4⋊Dic5C2211yescharacteristic, complemented
38C3×#683C2211yescharacteristic, complemented
39C3×#683C2211yescharacteristic, complemented
40Dic3⋊Dic5C2211yescharacteristic, complemented
41Dic3⋊Dic5C2211yescharacteristic, complemented
42#6669C2211yescharacteristic, complemented
43#6669C2211yescharacteristic, complemented
44#6668C2211yescharacteristic, complemented
45#6668C2211yescharacteristic, complemented
46(C2×D3)⋊Dic5C2211yescharacteristic
47(C2×D3)⋊Dic5C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49(C2×D3)⋊Dic5C2211yescharacteristic
50(C2×D3)⋊Dic5C2211yescharacteristic
51(C2×D3)⋊Dic5C2211yescharacteristic
52#5634—51nomaximal, Hall
53C2×C5×C22⋊C4—31no—
54C2×C22⋊Dic5—31no—
55C2×C22⋊Dic5—31no—
56#4070—31no—
57#4070—31no—
58#4070—31no—
59#4070—31no—
60#4024—31no—
61#4024—31no—
62#4024—31no—
63#4024—31no—
64#4024—31no—
65#4024—31no—
66#4024—31no—
67#4024D311yescharacteristic, complemented
68C23×C3×C5C2311yescharacteristic
69C22×C5×D3—41no—
70C22×C5×D3—41no—
71C22×C5×D3Q814yescomplemented
72C22×C5×D3C2311yescharacteristic
73C22×C5×D3C2311yescharacteristic
74C2×C3×C4×C5C2311yescharacteristic
75C2×C3×C4×C5C2311yescharacteristic
76C2×Dic15—41no—
77C2×Dic15C2311yescharacteristic
78C2×Dic15C2311yescharacteristic
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81Dic30—41no—
82C2×C5×Dic3C2311yescharacteristic
83C2×C5×Dic3C2311yescharacteristic
84C4×C5×D3—41no—
85C4×C5×D3—41no—
86C2×C3×Dic5—41no—
87C2×C3×Dic5C2311yescharacteristic
88C2×C3×Dic5C2311yescharacteristic
89C2×C3×Dic5C2311yescharacteristic
90C2×C3×Dic5C2311yescharacteristic
91C3×Dic10—41no—
92#1204—41no—
93#1204—41no—
94D3×Dic5—41no—
95D3×Dic5—41no—
96C3×C22⋊Q8—51no—
97#946—51no—
98#946—51no—
99C2×(C2×D3)⋊C4—51no—
100#924—51no—
101#897—51no—
102#897—51no—
103#896—51no—
104#896—51no—
105#896—51no—
106#896—51no—
107D3×C22⋊C4—51no—
108D3×C22⋊C4D511yescharacteristic
109#878—51no—
110#878—51no—
111C24×C5—31no—
112C22×C4×C5—31no—
113C22×C4×C5—31no—
114C22×Dic5—31no—
115C22×Dic5—31no—
116C22×Dic5—31no—
117C22×Dic5C2×D311yescharacteristic
118C2×Dic10—31no—
119C2×Dic10—31no—
120C2×Dic10—31no—
121C2×Dic10C2×D311yescharacteristic, complemented
122C5×C22⋊C4—31no—
123C5×C22⋊C4—31no—
124C5×C22⋊C4—31no—
125C5×C22⋊C4C2×D311yescharacteristic
126C22⋊Dic5—31no—
127C22⋊Dic5—31no—
128C22⋊Dic5—31no—
129C22⋊Dic5—31no—
130C22⋊Dic5—31no—
131C22⋊Dic5—31no—
132C22⋊Dic5—31no—
133C22⋊Dic5C2×D311yescharacteristic
134C4⋊Dic5—31no—
135C4⋊Dic5—31no—
136C4⋊Dic5—31no—
137C4⋊Dic5C2×D311yescharacteristic, complemented
138#683—31no—
139#683—31no—
140#683—31no—
141#683—31no—
142#683—31no—
143#683—31no—
144#683C2×D311yescharacteristic, complemented
145#683C2×D311yescharacteristic, complemented
146#543—151noSylow
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5C2×Q812yes—
150C22×C3×C5C2411yescharacteristic
151C2×C5×D3—21no—
152C2×C5×D3—21no—
153C2×C5×D3—21no—
154C2×C5×D3—21no—
155C2×C5×D3—44no—
156C2×C5×D3C2×Q812yes—
157C2×C5×D3C2×Q812yes—
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160Dic15—21no—
161Dic15—21no—
162Dic15—21no—
163Dic15—21no—
164C3×Dic5—21no—
165C3×Dic5—21no—
166C3×Dic5—21no—
167C3×Dic5—21no—
168C5×Dic3—21no—
169C5×Dic3—21no—
170C23×D3C2×D511yescharacteristic
171C2×C3×Q8—51no—
172C22×C3×C4—51no—
173C22×Dic3—51no—
174C2×C4×D3—51no—
175C2×C4×D3—51no—
176C2×C4×D3C2×D511yescharacteristic
177C2×C4×D3C2×D511yescharacteristic
178C2×Dic6—51no—
179C2×Dic6—51no—
180C2×Dic6—51no—
181C3×C4⋊C4—51no—
182C3×C4⋊C4—51no—
183C3×C4⋊C4—51no—
184C3×C22⋊C4—51no—
185C3×C22⋊C4C2×D511yescharacteristic
186#217—51no—
187#217C2×D511yescharacteristic
188(C2×D3)⋊C4—51no—
189(C2×D3)⋊C4—51no—
190(C2×D3)⋊C4—51no—
191(C2×D3)⋊C4—51no—
192(C2×D3)⋊C4—51no—
193(C2×D3)⋊C4—51no—
194(C2×D3)⋊C4C2×D511yescharacteristic
195(C2×D3)⋊C4C2×D511yescharacteristic
196C4⋊Dic3—51no—
197C4⋊Dic3—51no—
198C4⋊Dic3—51no—
199Dic3⋊C4—51no—
200Dic3⋊C4—51no—
201Dic3⋊C4—51no—
202Dic3⋊C4—51no—
203Dic3⋊C4—51no—
204Dic3⋊C4—51no—
205C23×C5—31no—
206C23×C5—31no—
207C23×C5—34no—
208C23×C5—121no—
209C23×C5—121no—
210C23×C5C22×D311yescharacteristic
211C2×C4×C5—31no—
212C2×C4×C5—31no—
213C2×C4×C5—121no—
214C2×C4×C5—121no—
215C2×C4×C5C22×D311yescharacteristic
216C2×C4×C5C22×D311yescharacteristic
217C2×Dic5—31no—
218C2×Dic5—31no—
219C2×Dic5—31no—
220C2×Dic5—31no—
221C2×Dic5—41no—
222C2×Dic5—121no—
223C2×Dic5—121no—
224C2×Dic5—121no—
225C2×Dic5C22×D311yescharacteristic
226C2×Dic5C22×D311yescharacteristic
227C2×Dic5C22×D311yescharacteristic
228C2×Dic5C22×D311yescharacteristic
229Dic10—41no—
230Dic10—121no—
231Dic10—121no—
232Dic10—121no—
233C22⋊Q8—51no—
234C22⋊Q8—151no—
235C22⋊Q8—151no—
236C22⋊Q8—151no—
237C22⋊Q8—151no—
238C22⋊Q8—151no—
239C22⋊Q8—151no—
240C22⋊Q8—151no—
241C22⋊Q8—151no—
242C22⋊Q8—151no—
243C22⋊Q8—151no—
244C22⋊Q8—151no—
245C2×C22⋊C4—31no—
246C2×C22⋊C4—151no—
247C2×C22⋊C4—151no—
248C2×C3×C5—22no—
249C2×C3×C5C22×Q811yescharacteristic
250C2×C3×C5H(5,𝔽2)11yescharacteristic
251C2×C3×C5H(5,𝔽2)11yescharacteristic
252C5×D3—22no—
253C5×D3—22no—
254C23×C3C22×D511yescharacteristic
255C22×D3—41no—
256C22×D3—41no—
257C22×D3Dic1014yescomplemented
258C22×D3C22×D511yescharacteristic
259C22×D3C22×D511yescharacteristic
260C3×Q8—201no—
261C2×C3×C4—51no—
262C2×C3×C4—51no—
263C2×C3×C4—51no—
264C2×C3×C4—51no—
265C2×C3×C4—201no—
266C2×C3×C4C22×D511yescharacteristic
267C2×C3×C4C22×D511yescharacteristic
268C2×Dic3—51no—
269C2×Dic3—51no—
270C2×Dic3—51no—
271C2×Dic3—51no—
272C2×Dic3—201no—
273C2×Dic3C22×D511yescharacteristic
274C2×Dic3C22×D511yescharacteristic
275C4×D3—41no—
276C4×D3—41no—
277C4×D3—201no—
278C4×D3—201no—
279Dic6—201no—
280Dic6—201no—
281Dic6—201no—
282C22×C5—21no—
283C22×C5—21no—
284C22×C5—32no—
285C22×C5—32no—
286C22×C5—61no—
287C22×C5—61no—
288C22×C5—61no—
289C22×C5—61no—
290C22×C5—124no—
291C22×C5D3×Q812yes—
292C22×C5C23×D311yescharacteristic
293C4×C5—21no—
294C4×C5—21no—
295C4×C5—61no—
296C4×C5—61no—
297Dic5—21no—
298Dic5—21no—
299Dic5—21no—
300Dic5—21no—
301Dic5—61no—
302Dic5—61no—
303Dic5—61no—
304Dic5—61no—
305C24—31no—
306C2×Q8—51no—
307C2×Q8—151no—
308C2×Q8—151no—
309C2×Q8—151no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—51no—
313C22×C4—151no—
314C22×C4—151no—
315C22×C4—151no—
316C4⋊C4—51no—
317C4⋊C4—51no—
318C4⋊C4—51no—
319C4⋊C4—151no—
320C4⋊C4—151no—
321C4⋊C4—151no—
322C4⋊C4—151no—
323C4⋊C4—151no—
324C4⋊C4—151no—
325C4⋊C4—151no—
326C4⋊C4—151no—
327C4⋊C4—151no—
328C22⋊C4—31no—
329C22⋊C4—31no—
330C22⋊C4—31no—
331C22⋊C4—51no—
332C22⋊C4—151no—
333C22⋊C4—151no—
334C22⋊C4—151no—
335C22⋊C4—151no—
336C22⋊C4—151no—
337C22⋊C4—151no—
338C22⋊C4—151no—
339C22⋊C4D3×D511yescharacteristic
340C3×C5#54311yescharacteristic, complemented, Hall
341C22×C3—21no—
342C22×C3—21no—
343C22×C3C2×Dic1012yes—
344C22×C3C23×D511yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—44no—
350C2×D3C2×Dic1012yes—
351C2×D3C2×Dic1012yes—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—101no—
355C3×C4—101no—
356C3×C4—101no—
357C3×C4—101no—
358Dic3—21no—
359Dic3—21no—
360Dic3—101no—
361Dic3—101no—
362Dic3—101no—
363Dic3—101no—
364C2×C5—22no—
365C2×C5—62no—
366C2×C5—62no—
367C2×C5#100411yescharacteristic
368C2×C5C2×D3×Q811yescharacteristic
369C2×C5#100911yescharacteristic
370C23—31no—
371C23—31no—
372C23—34no—
373C23—121no—
374C23—121no—
375C23C2×D3×D511yescharacteristic
376Q8—201no—
377Q8—601no—
378Q8—601no—
379Q8—601no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—51no—
383C2×C4—51no—
384C2×C4—51no—
385C2×C4—51no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—151no—
389C2×C4—151no—
390C2×C4—151no—
391C2×C4—151no—
392C2×C4—201no—
393C2×C4—601no—
394C2×C4—601no—
395C2×C4—601no—
396C2×C4C2×D3×D511yescharacteristic
397C2×C4C2×D3×D511yescharacteristic
398C2×C3—22no—
399C2×C3C22×Dic1011yescharacteristic
400C2×C3#414411yescharacteristic
401C2×C3#414411yescharacteristic
402D3—22no—
403D3—22no—
404C5#563411yescharacteristic, complemented, Sylow
405C22—21no—
406C22—21no—
407C22—32no—
408C22—32no—
409C22—61no—
410C22—61no—
411C22—61no—
412C22—61no—
413C22—124no—
414C22D3×Dic1012yes—
415C22C22×D3×D511yescharacteristic
416C4—21no—
417C4—21no—
418C4—61no—
419C4—61no—
420C4—101no—
421C4—101no—
422C4—101no—
423C4—101no—
424C4—301no—
425C4—301no—
426C4—301no—
427C4—301no—
428C3#6581711yescharacteristic, complemented, Sylow
429C2—22no—
430C2—62no—
431C2—62no—
432C2C2×D3×Dic1011yescharacteristic
433C2#9202511yescharacteristic
434C2#9202711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
prime radicalC23×C5×D3C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#543
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #91767C2semidirect
3. C3×#4024C2semidirect
4. #91576C2semidirect
5. #91574C2semidirect
8. #91436C2semidirect
9. #91432C2semidirect
10. #91417C2semidirect
11. #91410C2semidirect
12. #91369C2semidirect
13. #91365C2semidirect
14. #91352C2semidirect
15. #91352C2semidirect
19. C2×Dic30C22semidirect
23. C2×C3×Dic10C22semidirect
24. C2×#1204C22semidirect
25. C2×#1204C22semidirect
30. C4⋊Dic15C22semidirect
31. #6712C22semidirect
32. #6712C22semidirect
37. C3×C4⋊Dic5C22semidirect
38. C3×#683C22semidirect
39. C3×#683C22semidirect
40. Dic3⋊Dic5C22semidirect
41. Dic3⋊Dic5C22semidirect
42. #6669C22semidirect
43. #6669C22semidirect
44. #6668C22semidirect
45. #6668C22semidirect
67. #4024D3semidirect
71. C22×C5×D3Q8semidirect
121. C2×Dic10C2×D3semidirect
137. C4⋊Dic5C2×D3semidirect
144. #683C2×D3semidirect
145. #683C2×D3semidirect
257. C22×D3Dic10semidirect
340. C3×C5#543semidirect
404. C5#5634semidirect
428. C3#65817semidirect

Maximal subgroups

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

subgroupindex
#917672
C5×D3×C22⋊C42
C3×#40242
#915762
#915742
D3×C22⋊Dic52
C2×(C2×D3)⋊Dic52
#914362
#914322
#914172
#914102
#913692
#913652
#913522
#658173
#56345