Finite Group Databasebeta

← #11665167

Subgroups — #11665167

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#86219C212yesmaximal, complemented
2C3×#3147C211yesmaximal, characteristic, complemented
3#83915C214yesmaximal, complemented
4#83897C214yesmaximal, complemented
5#83281C212yesmaximal, complemented
6#81812C218yesmaximal, complemented
7#81794C212yesmaximal, complemented
8#81777C212yesmaximal, complemented
9#81754C214yesmaximal, complemented
10C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
11#78442C211yesmaximal, characteristic, complemented
12#60693—31nomaximal, Sylow
13C2×C3×#135C2214yes—
14C3×#532C2212yes—
15#5828C2214yes—
16Dic3×#41C2214yes—
17C22×ΓL(1,𝔽25)C2212yes—
18#5728C2214yes—
19#5716C2214yes—
20#5680C2218yes—
21#5606—24no—
22#5600—24no—
23#5557—24no—
24C4×#237—24no—
25C2×C42×D3C2211yescharacteristic
26#5470C2212yes—
27C3×#495C2214yescomplemented
28C3×#494C2211yescharacteristic, complemented
29C3×#493C2211yescharacteristic, complemented
30C3×#492C2212yescomplemented
31C2×C3×C4×C8C2211yescharacteristic
32#5233C2218yes—
33#5189C2218yes—
34#5184C2218yescomplemented
35#5174C22116yes—
36#5164C22116yescomplemented
37#5159C2218yes—
38#5152C2218yescomplemented
39#5106C2214yes—
40C2×#815C2212yes—
41#5079C2214yescomplemented
42#5070C2214yescomplemented
43#5056C2218yescomplemented
44C2×C4×C3⋊C8C2211yescharacteristic
45C4×ΓL(1,𝔽25)C2212yes—
46C4×ΓL(1,𝔽25)C2212yes—
47C4×ΓL(1,𝔽25)C2214yes—
48#4683C2214yescomplemented
49#4682C2212yescomplemented
50#4680C2212yescomplemented
51#3444—32no—
52#3185—32no—
53#3184—32no—
54#3183—38no—
55#3182—34no—
56#3160—31no—
57#3147D311yescharacteristic, complemented
58#3131—32no—
59#3054—34no—
60#2943—34no—
61C2×C4×M16—31no—
62C2×C3×#41C2314yes—
63C2×#237—24no—
64C22×C4×D3D412yes—
65C22×C4×D3C2311yescharacteristic
66C3×#135—216no—
67C2×C3×Q16C2314yes—
68C2×C3×SD16C2318yes—
69C2×C3×D8C2314yes—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8C2312yes—
72C3×#127C2312yes—
73C3×#124C2314yes—
74C3×#121—24no—
75C3×C4×D4—24no—
76C2×C3×C42C2311yescharacteristic
77#946C2318yes—
78Q8×Dic3C2318yes—
79#938—24no—
80#938C2318yes—
81#937—24no—
82#936—24no—
83#935—24no—
84D4×Dic3—28no—
85D4×Dic3—24no—
86D4×Dic3C2318yes—
87C4×#67—28no—
88#926—24no—
89#926C2314yes—
90#924—24no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3D412yes—
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95C2×ΓL(1,𝔽25)—28no—
96C2×ΓL(1,𝔽25)C2314yes—
97C2×ΓL(1,𝔽25)C2314yes—
98#897—24no—
99#892—24no—
100#890—24no—
101#887—24no—
102#883—28no—
103#67⋊C4—28no—
104#879—28no—
105#877—28no—
106C42×D3D414yes—
107C42×D3C2312yes—
108C42×D3C2312yes—
109#870C2314yes—
110#869C2314yes—
111C4×Dic6—24no—
112C3×C4×C8C2311yescharacteristic
113C3×C4×C8C2311yescharacteristic
114C3×C4×C8C2312yes—
115#835C23116yes—
116#832C23116yes—
117#818C2318yes—
118#817C2318yes—
119#815C2314yes—
120#815C2314yes—
121C4×C3⋊C8C2312yes—
122C4×C3⋊C8C2312yes—
123C2×#135C2×D314yes—
124C22×M16—32no—
125#533—34no—
126#533—32no—
127#532—34no—
128#532C2×D312yes—
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#500—32no—
138#499—34no—
139#498—32no—
140#495C2×D314yescomplemented
141#494C2×D311yescharacteristic, complemented
142#493C2×D311yescharacteristic, complemented
143#492C2×D312yescomplemented
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#427—34no—
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
162C3×#41—216no—
163C2×C3×Q8C2414yes—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4C2414yes—
167C22×C3×C4—24no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2411yescharacteristic
170C2×#67—28no—
171C22×Dic3—28no—
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D412yes—
174#237—416no—
175C2×C4×D3—24no—
176C2×C4×D3—44no—
177C2×C4×D3C2×D412yes—
178C2×C4×D3C2×D414yes—
179C2×C4×D3C2×D414yes—
180C2×C4×D3C2412yes—
181C2×Dic6—24no—
182C3×Q16—28no—
183C3×SD16—216no—
184C3×D8—28no—
185C2×C3×C8#4118yescomplemented
186C2×C3×C8C2412yes—
187C2×C3×C8C2412yes—
188C3×C4⋊C4—24no—
189C3×C22⋊C4—28no—
190C3×C42C2×D412yes—
191C3×C42C2411yescharacteristic
192C3×C42C2411yescharacteristic
193#217—24no—
194#217—28no—
195#217—28no—
196(C2×D3)⋊C4—28no—
197C4⋊Dic3—24no—
198C4⋊Dic3—28no—
199C4⋊Dic3C2418yes—
200Dic3⋊C4—28no—
201Dic3⋊C4—28no—
202C4×Dic3—24no—
203C4×Dic3—28no—
204C4×Dic3C2×D412yes—
205C4×Dic3C2×D414yes—
206C4×Dic3C2×D414yes—
207C4×Dic3C2412yes—
208C2×C3⋊C8—24no—
209C2×C3⋊C8C2414yes—
210ΓL(1,𝔽25)—28no—
211ΓL(1,𝔽25)—28no—
212C2×#41—64no—
213C2×#41C22×D314yes—
214C23×C4—31no—
215C23×C4—32no—
216#135—216no—
217C2×Q16C22×D314yes—
218C2×SD16C22×D318yes—
219C2×D8C22×D314yes—
220C2×M16—34no—
221C2×M16—34no—
222C2×M16—68no—
223C22×C8—32no—
224C22×C8C22×D312yes—
225C4⋊Q8—34no—
226C4⋊Q8C22×D312yes—
227#127C22×D312yes—
228#126—68no—
229#125—34no—
230#125—64no—
231#124—64no—
232#124C22×D314yes—
233#123—68no—
234#123—64no—
235C22⋊Q8—38no—
236C22⋊Q8—64no—
237C22⋊Q8—64no—
238#121—24no—
239#121—38no—
240#121—64no—
241#120—64no—
242C4×Q8—38no—
243C4×Q8—64no—
244C4×Q8—64no—
245C4×D4—24no—
246C4×D4—38no—
247C4×D4—64no—
248C4×D4—68no—
249C4×D4—68no—
250C4×D4—68no—
251#117—34no—
252#117—68no—
253#117—64no—
254#117—64no—
255C2×C42—32no—
256C2×C42—31no—
257C2×C42—34no—
258C2×C42—32no—
259C2×C42—32no—
260C2×C42—64no—
261C2×C42C22×D311yescharacteristic
262#107—38no—
263#106—38no—
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×C3—24no—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C22×D3—42no—
277C22×D3C22×D412yes—
278C3×Q8—28no—
279C3×D4—28no—
280C3×D4—48no—
281C3×D4—48no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—48no—
285C2×C3×C4C22×D412yes—
286C2×C3×C4C22×D412yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4114yes—
289C2×C3×C4C2511yescharacteristic
290#67—416no—
291C2×Dic3—22no—
292C2×Dic3—28no—
293C2×Dic3—42no—
294C2×Dic3—416no—
295C2×Dic3C22×D414yes—
296C2×Dic3C22×D412yes—
297C4×D3—24no—
298C4×D3—28no—
299C4×D3—24no—
300C4×D3—44no—
301Dic6—48no—
302C3×C8C2×#4114yes—
303C3×C8C2×#4114yes—
304C3⋊C8—24no—
305C24—31no—
306#41—216no—
307#41—1216no—
308C2×Q8—64no—
309C2×Q8C23×D314yes—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—68no—
313C2×D4C23×D314yes—
314C22×C4—24no—
315C22×C4—31no—
316C22×C4—32no—
317C22×C4—34no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—34no—
321C22×C4—68no—
322C22×C4—64no—
323C22×C4—124no—
324C22×C4D3×D412yes—
325C22×C4C23×D311yescharacteristic
326Q16—28no—
327SD16—216no—
328D8—28no—
329M16—68no—
330M16—68no—
331C2×C8—34no—
332C2×C8—64no—
333C2×C8#23718yescomplemented
334C2×C8C23×D312yes—
335C2×C8C23×D312yes—
336C4⋊C4—24no—
337C4⋊C4—38no—
338C4⋊C4—68no—
339C4⋊C4—68no—
340C4⋊C4—64no—
341C4⋊C4—68no—
342C22⋊C4—28no—
343C22⋊C4—68no—
344C22⋊C4—68no—
345C22⋊C4—68no—
346C22⋊C4—64no—
347C42—34no—
348C42—34no—
349C42—32no—
350C42—32no—
351C42—68no—
352C42—64no—
353C42D3×D412yes—
354C42C23×D311yescharacteristic
355C42C23×D311yescharacteristic
356C22×C3—22no—
357C22×C3—44no—
358C22×C3—48no—
359C22×C3#53212yes—
360C22×C3C23×D411yescharacteristic
361C2×D3—22no—
362C2×D3—24no—
363C2×D3—44no—
364C3×C4—22no—
365C3×C4—44no—
366C3×C4#53212yes—
367C3×C4#57514yes—
368C3×C4C22×#4112yes—
369Dic3—22no—
370Dic3—24no—
371Dic3—48no—
372C23—24no—
373C23—32no—
374C23—62no—
375C23—122no—
376C23C2×D3×D411yescharacteristic
377Q8—28no—
378Q8—128no—
379D4—28no—
380D4—48no—
381D4—48no—
382D4—1216no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—34no—
386C2×C4—32no—
387C2×C4—48no—
388C2×C4—62no—
389C2×C4—64no—
390C2×C4—64no—
391C2×C4—68no—
392C2×C4—68no—
393C2×C4—124no—
394C2×C4—122no—
395C2×C4—1216no—
396C2×C4C2×D3×D412yes—
397C2×C4C2×D3×D412yes—
398C2×C4C2×D3×D411yescharacteristic
399C2×C4C2×#23714yes—
400C2×C4C24×D311yescharacteristic
401C8—64no—
402C8C2×#23714yes—
403C8C2×#23714yes—
404C2×C3—22no—
405C2×C3—84no—
406C2×C3C2×#53211yescharacteristic
407C2×C3C2×#57512yes—
408D3—42no—
409C22—22no—
410C22—48no—
411C22—44no—
412C22—64no—
413C22—62no—
414C22—124no—
415C22#571612yes—
416C22C22×D3×D411yescharacteristic
417C4—22no—
418C4—44no—
419C4—64no—
420C4—62no—
421C4—128no—
422C4#571612yes—
423C4#576814yes—
424C4C22×#23712yes—
425C3#6069311yescharacteristic, complemented, Sylow
426C2—22no—
427C2—84no—
428C2—122no—
429C2C2×#571611yescharacteristic
430C2C2×#576812yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3147C2
socleC22×C3C23×D4
hypercenter#3147D3
hypocenterC3#60693
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #86219C2semidirect
2. C3×#3147C2semidirect
3. #83915C2semidirect
4. #83897C2semidirect
5. #83281C2semidirect
6. #81812C2semidirect
7. #81794C2semidirect
8. #81777C2semidirect
9. #81754C2semidirect
10. C2×C4×ΓL(1,𝔽25)C2semidirect
11. #78442C2semidirect
27. C3×#495C22semidirect
28. C3×#494C22semidirect
29. C3×#493C22semidirect
30. C3×#492C22semidirect
34. #5184C22semidirect
36. #5164C22semidirect
38. #5152C22semidirect
41. #5079C22semidirect
42. #5070C22semidirect
43. #5056C22semidirect
48. #4683C22semidirect
49. #4682C22semidirect
50. #4680C22semidirect
57. #3147D3semidirect
140. #495C2×D3semidirect
141. #494C2×D3semidirect
142. #493C2×D3semidirect
143. #492C2×D3semidirect
185. C2×C3×C8#41semidirect
333. C2×C8#237semidirect
425. C3#60693semidirect

Maximal subgroups

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

subgroupindex
#862192
C3×#31472
#839152
#838972
#832812
#818122
#817942
#817772
#817542
C2×C4×ΓL(1,𝔽25)2
#784422
#606933