Finite Group Databasebeta

← #11665171

Subgroups — #11665171

Order 768.

1998 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86257C212yesmaximal, complemented
2C3×#3160C211yesmaximal, characteristic, complemented
3#83334C212yesmaximal, complemented
4#81936C214yesmaximal, complemented
5#81911C218yesmaximal, complemented
6#81889C212yesmaximal, complemented
7#81888C212yesmaximal, complemented
8#79181C214yesmaximal, complemented
9#79166C214yesmaximal, complemented
10#78467C211yesmaximal, characteristic, complemented
11C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
12#60693—31nomaximal, Sylow
13C3×#533C2212yes—
14C2×#904C2214yes—
15C22×ΓL(1,𝔽25)C2212yes—
16#5728C2214yes—
17#5725C2214yes—
18#5691C2218yes—
19#5627—24no—
20#5624—24no—
21#5580—24no—
22C4×#239—24no—
23#5524C2214yes—
24#5522C2214yes—
25#5477C2212yes—
26C2×C42×D3C2211yescharacteristic
27C3×#500C2212yescomplemented
28C3×#499C2214yescomplemented
29C3×#498C2212yescomplemented
30C3×#427C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C2×#815C2212yes—
33#5093C2214yescomplemented
34#5088C2214yescomplemented
35#5068C2218yescomplemented
36#5002C2218yes—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4889C22116yescomplemented
39#4886C22116yes—
40#4869C2218yescomplemented
41#4867C2218yes—
42#4865C2218yes—
43#4853C2218yescomplemented
44#4699C2211yescharacteristic, complemented
45#4696C2214yescomplemented
46#4695C2211yescharacteristic, complemented
47#4693C2212yescomplemented
48C4×ΓL(1,𝔽25)C2212yes—
49C4×ΓL(1,𝔽25)C2212yes—
50C4×ΓL(1,𝔽25)C2214yes—
51#3444—32no—
52#3185—32no—
53#3184—32no—
54#3183—38no—
55#3182—34no—
56#3160D311yescharacteristic, complemented
57#3147—31no—
58#3131—32no—
59#3054—34no—
60#2943—34no—
61C2×C4×M16—31no—
62C2×#239—24no—
63C2×#235C2314yes—
64C22×C4×D3D412yes—
65C22×C4×D3C2311yescharacteristic
66C22×C3×C8C2312yes—
67C3×C4⋊Q8C2314yes—
68C3×#125C2314yes—
69C3×C22⋊Q8—24no—
70C3×C4×Q8—24no—
71C3×#117C2314yes—
72C2×C3×C42C2311yescharacteristic
73#947—24no—
74#946—24no—
75Q8×Dic3—24no—
76#930—24no—
77C2×C4×Dic3D412yes—
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2312yes—
80C2×Dic12C2314yes—
81#904—216no—
82C2×D24C2314yes—
83C2×#204C2318yes—
84C2×ΓL(1,𝔽25)—28no—
85C2×ΓL(1,𝔽25)C2314yes—
86C2×ΓL(1,𝔽25)C2314yes—
87#898—28no—
88#897C2318yes—
89#895—24no—
90#895C2318yes—
91#894—28no—
92#893—28no—
93#893—24no—
94#893C2318yes—
95#892—24no—
96#892—28no—
97#890—24no—
98#887C2318yes—
99#886—24no—
100#882—24no—
101#67⋊C4—28no—
102#875C2314yes—
103#874C2312yes—
104C4×D12—28no—
105C4×D12—24no—
106#872—24no—
107C42×D3—24no—
108C42×D3D414yes—
109C42×D3C2312yes—
110C42×D3C2312yes—
111#869C2312yes—
112C3×#107C2318yes—
113C3×#106C2318yes—
114C3×C4×C8C2311yescharacteristic
115C3×C4×C8C2311yescharacteristic
116C3×C4×C8C2312yes—
117#815C2314yes—
118#815C2314yes—
119#810C23116yes—
120#809C23116yes—
121C4×C3⋊C8C2312yes—
122C4×C3⋊C8C2312yes—
123C2×#135—34no—
124C22×M16—32no—
125#533—34no—
126#533C2×D312yes—
127#532—34no—
128#532—32no—
129#529—38no—
130#529—64no—
131#529—64no—
132#525—34no—
133#525—64no—
134C4×#41—34no—
135C4×#41—64no—
136C22×C42—31no—
137#500C2×D312yescomplemented
138#499C2×D314yescomplemented
139#498C2×D312yescomplemented
140#495—34no—
141#494—31no—
142#493—31no—
143#492—32no—
144#488—38no—
145#487—34no—
146#486—34no—
147#470—38no—
148#469—38no—
149#468—316no—
150#442—38no—
151#441—38no—
152#440—316no—
153#427C2×D314yes—
154#416—38no—
155C4×M16—32no—
156C4×M16—34no—
157C4×M16—32no—
158C2×#97—32no—
159C2×C4×C8—31no—
160C2×C4×C8C2×D311yescharacteristic
161C23×D3C2×D411yescharacteristic
162C2×C3×Q8—24no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2411yescharacteristic
165C2×#67—24no—
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D412yes—
168#239—416no—
169#235—216no—
170C2×D12—28no—
171C2×D12—24no—
172C2×D12C2414yes—
173C2×C4×D3—24no—
174C2×C4×D3—24no—
175C2×C4×D3—28no—
176C2×C4×D3—44no—
177C2×C4×D3C2×D412yes—
178C2×C4×D3C2×D414yes—
179C2×C4×D3C2×D414yes—
180C2×C4×D3C2412yes—
181C2×Dic6C2414yes—
182C2×C3×C8#4118yescomplemented
183C2×C3×C8C2412yes—
184C2×C3×C8C2412yes—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4C2418yes—
188C3×C22⋊C4—24no—
189C3×C42—24no—
190C3×C42C2×D412yes—
191C3×C42C2411yescharacteristic
192C3×C42C2411yescharacteristic
193(C2×D3)⋊C4—28no—
194(C2×D3)⋊C4—28no—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4—28no—
197C4⋊Dic3—24no—
198C4⋊Dic3—28no—
199Dic3⋊C4—28no—
200C4×Dic3—28no—
201C4×Dic3C2×D412yes—
202C4×Dic3C2×D414yes—
203C4×Dic3C2×D414yes—
204C4×Dic3C2412yes—
205C2×C3⋊C8—24no—
206C2×C3⋊C8C2414yes—
207Dic12—28no—
208D24—28no—
209#204—216no—
210ΓL(1,𝔽25)—28no—
211ΓL(1,𝔽25)—28no—
212C2×#41—34no—
213C2×#41—64no—
214C23×C4—31no—
215C23×C4—32no—
216#135—616no—
217C2×Q16—34no—
218C2×SD16—38no—
219C2×D8—34no—
220C2×M16—34no—
221C2×M16—34no—
222C2×M16—68no—
223C22×C8—32no—
224C22×C8C22×D312yes—
225C4⋊Q8—32no—
226C4⋊Q8C22×D314yes—
227#127—32no—
228#126—68no—
229#125—64no—
230#125C22×D314yes—
231#124—34no—
232#124—64no—
233#123—64no—
234#123—68no—
235C22⋊Q8—24no—
236C22⋊Q8—38no—
237C22⋊Q8—64no—
238#121—38no—
239#121—64no—
240#121—64no—
241#120—64no—
242C4×Q8—24no—
243C4×Q8—38no—
244C4×Q8—64no—
245C4×D4—38no—
246C4×D4—68no—
247C4×D4—64no—
248C4×D4—68no—
249C4×D4—68no—
250C4×D4—64no—
251#117—64no—
252#117—68no—
253#117—64no—
254#117C22×D314yes—
255C2×C42—32no—
256C2×C42—31no—
257C2×C42—34no—
258C2×C42—32no—
259C2×C42—32no—
260C2×C42—64no—
261C2×C42C22×D311yescharacteristic
262#107C22×D318yes—
263#106C22×D318yes—
264#103—316no—
265#102—316no—
266#97—34no—
267#97—34no—
268C4×C8—32no—
269C4×C8—32no—
270C4×C8C22×D311yescharacteristic
271C4×C8C22×D311yescharacteristic
272C4×C8C22×D312yes—
273C23×C3C22×D411yescharacteristic
274C22×D3—22no—
275C22×D3—24no—
276C22×D3—42no—
277C22×D3C22×D412yes—
278C3×Q8—48no—
279C2×C3×C4—24no—
280C2×C3×C4—28no—
281C2×C3×C4C22×D412yes—
282C2×C3×C4C22×D412yes—
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4114yes—
285C2×C3×C4C2511yescharacteristic
286#67—48no—
287C2×Dic3—22no—
288C2×Dic3—24no—
289C2×Dic3—42no—
290C2×Dic3C22×D414yes—
291C2×Dic3C22×D412yes—
292D12—28no—
293D12—416no—
294D12—48no—
295C4×D3—24no—
296C4×D3—28no—
297C4×D3—24no—
298C4×D3—44no—
299C4×D3—416no—
300C4×D3—48no—
301Dic6—28no—
302C3×C8C2×#4114yes—
303C3×C8C2×#4114yes—
304C3⋊C8—24no—
305C24—31no—
306#41—616no—
307#41—1216no—
308C2×Q8—24no—
309C2×Q8—34no—
310C2×D4—34no—
311C2×D4—68no—
312C2×D4—64no—
313C2×D4—64no—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—34no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—34no—
320C22×C4—68no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4—124no—
324C22×C4D3×D412yes—
325C22×C4C23×D311yescharacteristic
326Q16—68no—
327SD16—616no—
328D8—68no—
329M16—68no—
330M16—68no—
331C2×C8—34no—
332C2×C8—64no—
333C2×C8#23918yescomplemented
334C2×C8C23×D312yes—
335C2×C8C23×D312yes—
336C4⋊C4—24no—
337C4⋊C4—28no—
338C4⋊C4—68no—
339C4⋊C4—68no—
340C4⋊C4—64no—
341C4⋊C4C23×D318yes—
342C22⋊C4—24no—
343C22⋊C4—68no—
344C22⋊C4—68no—
345C22⋊C4—68no—
346C22⋊C4—68no—
347C42—24no—
348C42—34no—
349C42—34no—
350C42—32no—
351C42—32no—
352C42—68no—
353C42D3×D412yes—
354C42C23×D311yescharacteristic
355C42C23×D311yescharacteristic
356C22×C3—22no—
357C22×C3#53212yes—
358C22×C3C23×D411yescharacteristic
359C2×D3—22no—
360C2×D3—24no—
361C2×D3—44no—
362C2×D3—44no—
363C2×D3—48no—
364C3×C4—22no—
365C3×C4—48no—
366C3×C4#53212yes—
367C3×C4#57514yes—
368C3×C4C22×#4112yes—
369Dic3—22no—
370Dic3—24no—
371Dic3—44no—
372C23—32no—
373C23—62no—
374C23—64no—
375C23—122no—
376C23C2×D3×D411yescharacteristic
377Q8—48no—
378Q8—68no—
379D4—68no—
380D4—1216no—
381D4—128no—
382D4—128no—
383C2×C4—24no—
384C2×C4—28no—
385C2×C4—34no—
386C2×C4—32no—
387C2×C4—62no—
388C2×C4—64no—
389C2×C4—64no—
390C2×C4—68no—
391C2×C4—64no—
392C2×C4—124no—
393C2×C4—1216no—
394C2×C4—122no—
395C2×C4—128no—
396C2×C4C2×D3×D412yes—
397C2×C4C2×D3×D412yes—
398C2×C4C2×D3×D411yescharacteristic
399C2×C4C2×#23914yes—
400C2×C4C24×D311yescharacteristic
401C8—64no—
402C8C2×#23914yes—
403C8C2×#23914yes—
404C2×C3—22no—
405C2×C3C2×#53211yescharacteristic
406C2×C3C2×#57512yes—
407D3—42no—
408D3—84no—
409C22—22no—
410C22—64no—
411C22—62no—
412C22—124no—
413C22—128no—
414C22—124no—
415C22#572512yes—
416C22C22×D3×D411yescharacteristic
417C4—22no—
418C4—48no—
419C4—64no—
420C4—62no—
421C4—124no—
422C4#572512yes—
423C4#576814yes—
424C4C22×#23912yes—
425C3#6069311yescharacteristic, complemented, Sylow
426C2—22no—
427C2—122no—
428C2—244no—
429C2C2×#572511yescharacteristic
430C2C2×#576812yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3160C2
socleC22×C3C23×D4
hypercenter#3160D3
hypocenterC3#60693
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
1. #86257C2semidirect
2. C3×#3160C2semidirect
3. #83334C2semidirect
4. #81936C2semidirect
5. #81911C2semidirect
6. #81889C2semidirect
7. #81888C2semidirect
8. #79181C2semidirect
9. #79166C2semidirect
10. #78467C2semidirect
11. C2×C4×ΓL(1,𝔽25)C2semidirect
27. C3×#500C22semidirect
28. C3×#499C22semidirect
29. C3×#498C22semidirect
33. #5093C22semidirect
34. #5088C22semidirect
35. #5068C22semidirect
38. #4889C22semidirect
40. #4869C22semidirect
43. #4853C22semidirect
44. #4699C22semidirect
45. #4696C22semidirect
46. #4695C22semidirect
47. #4693C22semidirect
56. #3160D3semidirect
137. #500C2×D3semidirect
138. #499C2×D3semidirect
139. #498C2×D3semidirect
182. C2×C3×C8#41semidirect
333. C2×C8#239semidirect
425. C3#60693semidirect

Maximal subgroups

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

subgroupindex
#862572
C3×#31602
#833342
#819362
#819112
#818892
#818882
#791812
#791662
#784672
C2×C4×ΓL(1,𝔽25)2
#606933