Finite Group Databasebeta

← #11754702

Subgroups — #11754702

Order 960.

524 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6819C211yesmaximal, characteristic, complemented
2#92016C211yesmaximal, characteristic, complemented
3C5×#932C211yesmaximal, characteristic, complemented
4C3×#4078C211yesmaximal, characteristic, complemented
5#91499C212yesmaximal, complemented
6#91497C212yesmaximal, complemented
7#91488C212yesmaximal, complemented
8#91482C212yesmaximal, complemented
9#91322C211yesmaximal, characteristic, complemented
10#91313C211yesmaximal, characteristic, complemented
11#91292C211yesmaximal, characteristic, complemented
12#66107—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic, complemented
14C2×#1220—21no—
15C22×Dic15—21no—
16#6819—21no—
17#6819—21no—
18#6819C2212yes—
19#6819C2212yescomplemented
20C2×C4×D15C2211yescharacteristic
21C2×Dic30C2211yescharacteristic, complemented
22C5×#235C2211yescharacteristic
23C3×#709C2211yescharacteristic
24#6790—21no—
25#6780—21no—
26#6777C2211yescharacteristic, complemented
27C4×D3×D5—21no—
28#6771—21no—
29D15⋊Q8C2211yescharacteristic, complemented
30#6768C2211yescharacteristic, complemented
31#6766C2211yescharacteristic, complemented
32C5×#214C2212yescomplemented
33C5×#213C2212yescomplemented
34C5×#208C2211yescharacteristic, complemented
35C3×#687C2212yescomplemented
36C3×#686C2212yescomplemented
37C3×#681C2211yescharacteristic, complemented
38#6663C2212yescomplemented
39#6662C2212yescomplemented
40#6659C2212yescomplemented
41#6657C2212yescomplemented
42#6651C2212yescomplemented
43#6648C2212yescomplemented
44#5773—51nomaximal, Hall
45D5×#41—31no—
46C2×#711—31no—
47C5×AGL(1,ℤ/8)—31no—
48#4097—31no—
49#4078D311yescharacteristic, complemented
50#4062—32no—
51#4061—32no—
52#4058—32no—
53#4057—32no—
54#4055—31no—
55D5×M16—31no—
56C23×C3×C5—21no—
57C22×D15D411yescharacteristic
58C2×D3×D5—21no—
59C3×C5×D4—21no—
60C3×C5×D4C2312yes—
61C2×C3×C4×C5C2311yescharacteristic
62#1220—22no—
63#1220—22no—
64C2×Dic15—21no—
65C2×Dic15—22no—
66C2×Dic15—22no—
67C2×Dic15D411yescharacteristic
68C4×D15D411yescharacteristic
69C4×D15D411yescharacteristic
70C4×D15C2311yescharacteristic
71C4×D15C2311yescharacteristic
72Dic30—21no—
73Dic30C2312yes—
74C5×#67—21no—
75C5×D12C2311yescharacteristic
76C4×C5×D3—21no—
77C5×Dic6C2311yescharacteristic
78C3×C22⋊D5—21no—
79C3×D20C2311yescharacteristic
80C3×C4×D5—21no—
81C3×Dic10C2311yescharacteristic
82#1204—21no—
83#1203—21no—
84#1202—21no—
85#1201—21no—
86#1200—21no—
87D3×Dic5—21no—
88D5×Dic3—21no—
89C3×#168C2312yes—
90C5×C3⋊C8C2312yes—
91D3×#41—51no—
92C2×#237—51no—
93C3×AGL(1,ℤ/8)—51no—
94#951—51no—
95#932D511yescharacteristic, complemented
96#916—52no—
97#915—52no—
98#912—52no—
99#911—52no—
100#909—51no—
101D3×M16—51no—
102C5×#41—31no—
103C2×C5×D4C2×D311yescharacteristic, complemented
104C2×C22⋊D5—61no—
105C22×Dic5—61no—
106#713—31no—
107Q8×D5—31no—
108#711—32no—
109#711—32no—
110#711—31no—
111#711—61no—
112#711—61no—
113#711—61no—
114D4×D5—31no—
115D4×D5—61no—
116#709—61no—
117#709C2×D311yescharacteristic
118C2×C4×D5—31no—
119C2×C4×D5—61no—
120C2×Dic10—31no—
121C5×SD16—32no—
122C5×D8—32no—
123C5×M16—31no—
124#689—32no—
125#687—32no—
126#687C2×D312yescomplemented
127#686C2×D312yescomplemented
128#681C2×D311yescharacteristic, complemented
129Dic20—32no—
130#677—32no—
131#676—32no—
132C8×D5—32no—
133#575—151noSylow
134C22×C3×C5—22no—
135C22×C3×C5—42no—
136C22×C3×C5C2×D411yescharacteristic
137C2×D15—41no—
138C2×D15C2×D411yescharacteristic
139C2×D15C2×D411yescharacteristic
140C2×C5×D3—21no—
141C2×C3×D5—21no—
142D3×D5—42no—
143C3×C4×C5C2×D411yescharacteristic
144C3×C4×C5C2411yescharacteristic
145Dic15—22no—
146Dic15C2×D411yescharacteristic
147Dic15C2×D411yescharacteristic
148C3×Dic5—21no—
149C5×Dic3—21no—
150C3×#41—51no—
151C2×C3×D4C2×D511yescharacteristic, complemented
152C2×#67—101no—
153C22×Dic3—101no—
154#239—51no—
155D3×Q8—51no—
156#237—52no—
157#237—52no—
158#237—51no—
159#237—101no—
160#237—101no—
161#237—101no—
162D3×D4—51no—
163D3×D4—101no—
164#235—101no—
165#235C2×D511yescharacteristic
166C2×C4×D3—51no—
167C2×C4×D3—101no—
168C2×Dic6—51no—
169C3×SD16—52no—
170C3×D8—52no—
171C3×M16—51no—
172#216—52no—
173#214—52no—
174#214C2×D512yescomplemented
175#213C2×D512yescomplemented
176#208C2×D511yescharacteristic, complemented
177Dic12—52no—
178#204—52no—
179ΓL(1,𝔽25)—52no—
180D3×C8—52no—
181C23×C5—21no—
182C22×D5—31no—
183C22×D5—61no—
184C5×Q8—31no—
185C5×D4—21no—
186C5×D4—31no—
187C5×D4—61no—
188C5×D4C22×D312yes—
189C2×C4×C5—61no—
190C2×C4×C5C22×D311yescharacteristic
191C22⋊D5—21no—
192C22⋊D5—61no—
193C22⋊D5—61no—
194C22⋊D5—62no—
195C22⋊D5—62no—
196C2×Dic5—31no—
197C2×Dic5—61no—
198C2×Dic5—62no—
199C2×Dic5—62no—
200C2×Dic5—61no—
201D20—61no—
202D20C22×D311yescharacteristic
203C4×D5—21no—
204C4×D5—31no—
205C4×D5—31no—
206C4×D5—31no—
207C4×D5—31no—
208C4×D5—61no—
209C4×D5—61no—
210Dic10—32no—
211Dic10—61no—
212Dic10—61no—
213Dic10C22×D311yescharacteristic
214C5×C8—32no—
215#168C22×D312yes—
216C2×#41—151no—
217C2×#41—151no—
218#137—151no—
219#137—151no—
220#137—152no—
221AGL(1,ℤ/8)—31no—
222AGL(1,ℤ/8)—51no—
223AGL(1,ℤ/8)—152no—
224#135—152no—
225#135—152no—
226C2×M16—151no—
227C2×C3×C5—21no—
228C2×C3×C5—42no—
229C2×C3×C5C22×D411yescharacteristic
230D15—21no—
231D15—21no—
232C3×D5—41no—
233C5×D3—41no—
234C23×C3—21no—
235C22×D3—51no—
236C22×D3—101no—
237C3×Q8—51no—
238C3×D4—21no—
239C3×D4—51no—
240C3×D4—101no—
241C3×D4C22×D512yes—
242C2×C3×C4—101no—
243C2×C3×C4C22×D511yescharacteristic
244#67—21no—
245#67—101no—
246#67—101no—
247#67—102no—
248#67—102no—
249C2×Dic3—51no—
250C2×Dic3—101no—
251C2×Dic3—102no—
252C2×Dic3—102no—
253C2×Dic3—101no—
254D12—101no—
255D12C22×D511yescharacteristic
256C4×D3—21no—
257C4×D3—51no—
258C4×D3—51no—
259C4×D3—51no—
260C4×D3—51no—
261C4×D3—101no—
262C4×D3—101no—
263Dic6—52no—
264Dic6—101no—
265Dic6—101no—
266Dic6C22×D511yescharacteristic
267C3×C8—52no—
268C3⋊C8C22×D512yes—
269C22×C5—22no—
270C22×C5—42no—
271C22×C5—61no—
272C22×C5D3×D411yescharacteristic
273C2×D5—21no—
274C2×D5—31no—
275C2×D5—31no—
276C2×D5—122no—
277C2×D5—121no—
278C4×C5—61no—
279C4×C5D3×D411yescharacteristic
280C4×C5C23×D311yescharacteristic
281Dic5—21no—
282Dic5—31no—
283Dic5—31no—
284Dic5—62no—
285#41—31no—
286#41—51no—
287#41—151no—
288#41—151no—
289#41—152no—
290#41—152no—
291#41—301no—
292#41—301no—
293#41—301no—
294#41—301no—
295C2×Q8—151no—
296C2×Q8—151no—
297C2×D4—151no—
298C2×D4—301no—
299C2×D4—301no—
300C2×D4D3×D511yescharacteristic, complemented
301C22×C4—151no—
302C22×C4—301no—
303C22×C4—301no—
304Q16—152no—
305Q16—152no—
306SD16—32no—
307SD16—52no—
308SD16—152no—
309SD16—152no—
310D8—32no—
311D8—52no—
312M16—31no—
313M16—51no—
314M16—152no—
315C2×C8—152no—
316C3×C5#57511yescharacteristic, complemented, Hall
317C22×C3—22no—
318C22×C3—42no—
319C22×C3—101no—
320C22×C3D4×D511yescharacteristic
321C2×D3—21no—
322C2×D3—51no—
323C2×D3—51no—
324C2×D3—202no—
325C2×D3—201no—
326C3×C4—101no—
327C3×C4D4×D511yescharacteristic
328C3×C4C23×D511yescharacteristic
329Dic3—21no—
330Dic3—51no—
331Dic3—51no—
332Dic3—102no—
333C2×C5—21no—
334C2×C5—42no—
335C2×C5—121no—
336C2×C5C2×D3×D411yescharacteristic
337D5—41no—
338D5—61no—
339D5—61no—
340C23—21no—
341C23—151no—
342C23—301no—
343Q8—31no—
344Q8—51no—
345Q8—152no—
346Q8—301no—
347Q8—301no—
348D4—21no—
349D4—31no—
350D4—51no—
351D4—61no—
352D4—101no—
353D4—301no—
354D4—301no—
355D4—301no—
356D4—302no—
357D4—302no—
358D4C2×D3×D512yes—
359C2×C4—61no—
360C2×C4—101no—
361C2×C4—151no—
362C2×C4—151no—
363C2×C4—151no—
364C2×C4—151no—
365C2×C4—151no—
366C2×C4—301no—
367C2×C4—301no—
368C2×C4—301no—
369C2×C4—302no—
370C2×C4—302no—
371C2×C4—301no—
372C2×C4C2×D3×D511yescharacteristic
373C8—32no—
374C8—52no—
375C2×C3—21no—
376C2×C3—42no—
377C2×C3—201no—
378C2×C3C2×D4×D511yescharacteristic
379D3—41no—
380D3—101no—
381D3—101no—
382C5#577311yescharacteristic, complemented, Sylow
383C22—22no—
384C22—42no—
385C22—61no—
386C22—101no—
387C22—151no—
388C22—151no—
389C22—601no—
390C22—602no—
391C22#679011yescharacteristic
392C4—61no—
393C4—101no—
394C4—151no—
395C4—151no—
396C4—302no—
397C4#679011yescharacteristic
398C4C22×D3×D511yescharacteristic
399C3#6610711yescharacteristic, complemented, Sylow
400C2—21no—
401C2—42no—
402C2—121no—
403C2—201no—
404C2—301no—
405C2—301no—
406C2C2×#679011yescharacteristic
subgroupG/H
centerC2C2×#6790
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC2×C3×C5×D4C22
socleC2×C3×C5C22×D4
hypercenterC2×D4D3×D5
hypocenterC3×C5#575
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#6819C2semidirect
2. #92016C2semidirect
3. C5×#932C2semidirect
4. C3×#4078C2semidirect
5. #91499C2semidirect
6. #91497C2semidirect
7. #91488C2semidirect
8. #91482C2semidirect
9. #91322C2semidirect
10. #91313C2semidirect
11. #91292C2semidirect
13. C2×C3×C5×D4C22semidirect
19. #6819C22semidirect
21. C2×Dic30C22semidirect
26. #6777C22semidirect
29. D15⋊Q8C22semidirect
30. #6768C22semidirect
31. #6766C22semidirect
32. C5×#214C22semidirect
33. C5×#213C22semidirect
34. C5×#208C22semidirect
35. C3×#687C22semidirect
36. C3×#686C22semidirect
37. C3×#681C22semidirect
38. #6663C22semidirect
39. #6662C22semidirect
40. #6659C22semidirect
41. #6657C22semidirect
42. #6651C22semidirect
43. #6648C22semidirect
49. #4078D3semidirect
95. #932D5semidirect
103. C2×C5×D4C2×D3semidirect
126. #687C2×D3semidirect
127. #686C2×D3semidirect
128. #681C2×D3semidirect
151. C2×C3×D4C2×D5semidirect
174. #214C2×D5semidirect
175. #213C2×D5semidirect
176. #208C2×D5semidirect
300. C2×D4D3×D5semidirect
316. C3×C5#575semidirect
382. C5#5773semidirect
399. C3#66107semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
#920162
C5×#9322
C3×#40782
#914992
#914972
#914882
#914822
#913222
#913132
#912922
#661073
#57735