Finite Group Databasebeta

← #11664761

Subgroups — #11664761

Order 768.

3274 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3500C211yesmaximal, characteristic, complemented
2C2×#5837C212yesmaximal, complemented
3C2×#5835C212yesmaximal, complemented
4#85846C211yesmaximal, characteristic, complemented
5#83402C216yesmaximal, complemented
6#83382C216yesmaximal, complemented
7#83287C213yesmaximal, complemented
8#83286C213yesmaximal, complemented
9#82943C213yesmaximal, complemented
10C2×#5058C213yesmaximal, complemented
11#82511C211yesmaximal, characteristic, complemented
12#60650—31nomaximal, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C2×C3×H(5,𝔽2)C2211yescharacteristic
15C2×#1009C2212yes—
16C3×#548—22no—
17C3×#546—26no—
18C3×#536C2213yescomplemented
19C3×#535C2213yescomplemented
20C3×#532C2213yescomplemented
21C2×C3×#124C2213yescomplemented
22C3×C4×#41C2211yescharacteristic
23#5837—216no—
24#5835—216no—
25C2×#950C2216yescomplemented
26C2×#950C2216yescomplemented
27C2×#949C2216yescomplemented
28C2×#949C2216yescomplemented
29C2×#948C2212yes—
30C2×#942C2216yescomplemented
31C22×#215C2216yescomplemented
32C2×#932C2216yescomplemented
33C22×#213C2216yescomplemented
34#5577—22no—
35D4×D12—26no—
36#5525C2216yescomplemented
37#5477C2213yescomplemented
38C2×#874C2213yescomplemented
39#5086C2216yescomplemented
40#5085C2216yescomplemented
41#5074C2216yescomplemented
42#5073C2216yescomplemented
43#5060C2211yescharacteristic, complemented
44#5060C2213yescomplemented
45#5059C22112yescomplemented
46#5058C2213yescomplemented
47#5058C2213yescomplemented
48#5058C2216yescomplemented
49#5058C22112yescomplemented
50#4997C2216yescomplemented
51C4×#208C2213yescomplemented
52C2×#803C2213yescomplemented
53C2×#577—32no—
54C2×#576—32no—
55#3500D311yescharacteristic, complemented
56#3498—31no—
57#3155—36no—
58#3154—36no—
59#3153—33no—
60#3151—33no—
61#3150—33no—
62C2×#496—33no—
63#2875—31no—
64C3×2−1+4—28no—
65C3×H(5,𝔽2)—28no—
66C2×C3×#41—21no—
67C2×C3×#41—23no—
68C2×C3×#41C2313yes—
69C2×C3×#41C2313yes—
70C2×C3×#41C2311yescharacteristic
71C22×C3×Q8—21no—
72C22×C3×Q8C2313yes—
73C22×C3×D4—23no—
74C22×C3×D4C2313yes—
75#1009—28no—
76#1009—28no—
77C2×#239—22no—
78C2×D3×D4—26no—
79C2×#235C2316yes—
80C22×D12—26no—
81C22×D12C2316yes—
82C3×C4⋊Q8C2313yes—
83C3×#127C2313yes—
84C3×#124—26no—
85C3×#124C2313yes—
86C3×#124C2311yescharacteristic
87C3×#124C2316yes—
88C3×#123—26no—
89C3×C22⋊Q8—26no—
90C3×#121—26no—
91C3×#121—26no—
92C3×#120—26no—
93C3×C4×Q8D412yes—
94C3×C4×D4—26no—
95C3×C4×D4D416yes—
96C3×#117C2313yes—
97C2×C3×C22⋊C4—26no—
98C2×C3×C42C2313yes—
99#950—224no—
100#950—224no—
101#949—224no—
102#949—224no—
103#948D4116yescomplemented
104C2×#216C2316yes—
105C2×#216C2312yes—
106#942—224no—
107C2×#215—224no—
108C2×#215C2316yes—
109C2×#215C23112yes—
110C2×#215C2316yes—
111C2×#214C2316yes—
112C2×#214C2312yes—
113#932—224no—
114C2×#213—224no—
115C2×#213C2316yes—
116C2×#213C23112yes—
117C2×#213C2316yes—
118#930—26no—
119C2×#208C2316yes—
120C22×C3⋊C8C2316yes—
121#895—212no—
122#882—212no—
123#875C2316yes—
124#874—26no—
125#874C2313yes—
126#874C2316yes—
127C4×D12—26no—
128#869C2311yescharacteristic
129#803C2311yescharacteristic
130#803C2313yes—
131#803C2316yes—
132C4×C3⋊C8C2316yes—
133C2×2−1+4C2×D311yescharacteristic
134C2×H(5,𝔽2)—32no—
135C2×H(5,𝔽2)C2×D311yescharacteristic
136#577—616no—
137#576—616no—
138C2×#137—36no—
139C2×AGL(1,ℤ/8)—36no—
140C2×AGL(1,ℤ/8)—36no—
141C2×AGL(1,ℤ/8)—36no—
142C2×#135—36no—
143C2×#135—36no—
144C22×SD16—36no—
145C22×D8—36no—
146C2×#131—32no—
147#550—62no—
148#548—22no—
149#546—26no—
150D42—66no—
151#536C2×D313yescomplemented
152#535—36no—
153#535C2×D313yescomplemented
154#532—33no—
155#532C2×D313yescomplemented
156C2×#127—33no—
157C2×#124C2×D313yescomplemented
158C4×#41C2×D311yescharacteristic
159#497—36no—
160#497—31no—
161#497—33no—
162#496—36no—
163#496—33no—
164#496—36no—
165#496—312no—
166#496—33no—
167#495—312no—
168#493—36no—
169#492—36no—
170#405—36no—
171C4×M16—33no—
172C2×#97—33no—
173C24×C3—23no—
174C23×D3—26no—
175C3×#41—212no—
176C3×#41—212no—
177C3×#41—44no—
178C3×#41—412no—
179C3×#41C2×D418yes—
180C2×C3×Q8—23no—
181C2×C3×Q8—212no—
182C2×C3×Q8—44no—
183C2×C3×Q8C2×D411yescharacteristic
184C2×C3×Q8C2413yes—
185C2×C3×Q8C2411yescharacteristic
186C2×C3×D4—23no—
187C2×C3×D4—23no—
188C2×C3×D4—212no—
189C2×C3×D4—23no—
190C2×C3×D4—26no—
191C2×C3×D4—412no—
192C2×C3×D4C2×D413yes—
193C2×C3×D4C2413yes—
194C2×C3×D4C2411yescharacteristic
195C22×C3×C4—23no—
196C22×C3×C4—23no—
197C22×C3×C4—23no—
198C22×C3×C4C2×D416yes—
199C22×C3×C4C2413yes—
200C2×#67—26no—
201#239—48no—
202D3×D4—424no—
203#235—212no—
204#235—212no—
205C2×D12—212no—
206C2×D12—212no—
207C2×D12—26no—
208C2×D12—26no—
209C2×D12—212no—
210C2×D12—812no—
211C2×D12C2416yes—
212C2×C4×D3—26no—
213C2×Dic6C2412yes—
214C3×C4⋊C4—26no—
215C3×C4⋊C4C2×D416yes—
216C3×C22⋊C4—26no—
217C3×C22⋊C4—212no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4C2×D416yes—
220C3×C42C2×D416yes—
221C3×C42C2413yes—
222C3×C42C2411yescharacteristic
223#216—212no—
224#216—24no—
225#215—212no—
226#215—224no—
227#215—212no—
228#214—212no—
229#214—24no—
230#213—212no—
231#213—224no—
232#213—212no—
233(C2×D3)⋊C4—212no—
234C4⋊Dic3—22no—
235#208C2×D4124yescomplemented
236C2×C3⋊C8C2×D4124yescomplemented
237C2×C3⋊C8C2416yes—
238C2×C3⋊C8C2412yes—
2392−1+4—28no—
240H(5,𝔽2)—28no—
241H(5,𝔽2)—68no—
242H(5,𝔽2)—68no—
243C2×#41—21no—
244C2×#41—23no—
245C2×#41—36no—
246C2×#41—62no—
247C2×#41C22×D313yes—
248C2×#41C22×D313yes—
249C2×#41C22×D311yescharacteristic
250C22×Q8—21no—
251C22×Q8C22×D313yes—
252C22×D4—23no—
253C22×D4—36no—
254C22×D4—66no—
255C22×D4—66no—
256C22×D4C22×D313yes—
257#137—624no—
258AGL(1,ℤ/8)—624no—
259AGL(1,ℤ/8)—624no—
260AGL(1,ℤ/8)—624no—
261#135—624no—
262#135—624no—
263C2×Q16—32no—
264C2×Q16—36no—
265C2×SD16—36no—
266C2×SD16—312no—
267C2×SD16—36no—
268C2×SD16—32no—
269C2×SD16—36no—
270C2×SD16—624no—
271C2×D8—36no—
272C2×D8—312no—
273C2×D8—36no—
274C2×D8—624no—
275#131—316no—
276C2×M16—36no—
277C22×C8—36no—
278C4⋊Q8—31no—
279C4⋊Q8C22×D313yes—
280#127—33no—
281#127—36no—
282#127—66no—
283#127C22×D313yes—
284#124—26no—
285#124—36no—
286#124C22×D313yes—
287#124C22×D316yes—
288#124C22×D311yescharacteristic
289#123—26no—
290C22⋊Q8—26no—
291#121—26no—
292#121—26no—
293#121—66no—
294#121—612no—
295#120—26no—
296#120—612no—
297C4×Q8#6712yes—
298C4×D4—26no—
299C4×D4—66no—
300C4×D4#6716yes—
301#117C22×D313yes—
302C2×C22⋊C4—26no—
303C2×C42C22×D313yes—
304#97—31no—
305#97—33no—
306#97—36no—
307C4×C8—36no—
308C23×C3—23no—
309C23×C3—21no—
310C23×C3—46no—
311C23×C3—86no—
312C23×C3C22×D413yes—
313C22×D3—26no—
314C22×D3—412no—
315C22×D3—812no—
316C3×Q8—26no—
317C3×Q8—22no—
318C3×Q8—46no—
319C3×Q8#12714yes—
320C3×D4—26no—
321C3×D4—22no—
322C3×D4—46no—
323C3×D4—46no—
324C3×D4—46no—
325C3×D4—86no—
326C3×D4#127112yes—
327C2×C3×C4—23no—
328C2×C3×C4—21no—
329C2×C3×C4—46no—
330C2×C3×C4—46no—
331C2×C3×C4—46no—
332C2×C3×C4—46no—
333C2×C3×C4C22×D4112yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×D416yes—
336C2×C3×C4C22×D413yes—
337C2×C3×C4C2511yescharacteristic
338#67—412no—
339C2×Dic3—22no—
340D12—26no—
341D12—26no—
342D12—412no—
343D12—412no—
344D12—812no—
345C4×D3—412no—
346Dic6—22no—
347Dic6—22no—
348C3⋊C8C22×D414yes—
349C3⋊C8C22×D4112yes—
350C24—23no—
351C24—66no—
352#41—212no—
353#41—212no—
354#41—44no—
355#41—412no—
356#41—612no—
357#41—612no—
358#41—128no—
359#41D3×D418yes—
360C2×Q8—23no—
361C2×Q8—212no—
362C2×Q8—32no—
363C2×Q8—44no—
364C2×Q8C2×#6711yescharacteristic
365C2×Q8C23×D313yes—
366C2×Q8C23×D311yescharacteristic
367C2×D4—23no—
368C2×D4—23no—
369C2×D4—212no—
370C2×D4—23no—
371C2×D4—26no—
372C2×D4—36no—
373C2×D4—412no—
374C2×D4—66no—
375C2×D4—612no—
376C2×D4—612no—
377C2×D4—66no—
378C2×D4—66no—
379C2×D4—612no—
380C2×D4—1224no—
381C2×D4—2412no—
382C2×D4C2×#6713yes—
383C2×D4C23×D313yes—
384C2×D4C23×D311yescharacteristic
385C22×C4—23no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—66no—
389C22×C4C2×#6716yes—
390C22×C4C23×D313yes—
391Q16—64no—
392Q16—612no—
393SD16—64no—
394SD16—612no—
395SD16—612no—
396SD16—624no—
397SD16—612no—
398D8—612no—
399D8—624no—
400D8—612no—
401M16—324no—
402C2×C8—324no—
403C2×C8—36no—
404C2×C8—32no—
405C4⋊C4—26no—
406C4⋊C4—62no—
407C4⋊C4C2×#6716yes—
408C22⋊C4—26no—
409C22⋊C4—212no—
410C22⋊C4—22no—
411C22⋊C4—612no—
412C22⋊C4C2×#6716yes—
413C42C2×#6716yes—
414C42C23×D313yes—
415C42C23×D311yescharacteristic
416C22×C3—26no—
417C22×C3—42no—
418C22×C3—46no—
419C22×C3—41no—
420C22×C3—43no—
421C22×C3—166no—
422C22×C3C2×#12716yes—
423C22×C3C23×D411yescharacteristic
424C2×D3—412no—
425C2×D3—46no—
426C2×D3—1612no—
427C3×C4—22no—
428C3×C4—26no—
429C3×C4—41no—
430C3×C4—43no—
431C3×C4C2×#12716yes—
432C3×C4C23×D412yes—
433Dic3—42no—
434C23—23no—
435C23—21no—
436C23—46no—
437C23—66no—
438C23—86no—
439C23—1212no—
440C23—2412no—
441C23C22×#6713yes—
442Q8—26no—
443Q8—22no—
444Q8—46no—
445Q8—62no—
446Q8—62no—
447Q8#94014yes—
448D4—26no—
449D4—22no—
450D4—46no—
451D4—46no—
452D4—46no—
453D4—66no—
454D4—66no—
455D4—86no—
456D4—1212no—
457D4—1212no—
458D4—1212no—
459D4—2412no—
460D4#940112yes—
461C2×C4—23no—
462C2×C4—21no—
463C2×C4—46no—
464C2×C4—46no—
465C2×C4—46no—
466C2×C4—46no—
467C2×C4—62no—
468C2×C4—1212no—
469C2×C4C2×D3×D4112yes—
470C2×C4C22×#6712yes—
471C2×C4C22×#6716yes—
472C2×C4C22×#6713yes—
473C2×C4C24×D311yescharacteristic
474C8—34no—
475C8—312no—
476C2×C3—26no—
477C2×C3—81no—
478C2×C3—83no—
479C2×C3C22×#12711yescharacteristic
480C2×C3#358512yes—
481D3—86no—
482C22—26no—
483C22—42no—
484C22—46no—
485C22—41no—
486C22—43no—
487C22—1212no—
488C22—126no—
489C22—166no—
490C22—4812no—
491C22C2×#94016yes—
492C22C23×#6711yescharacteristic
493C4—22no—
494C4—26no—
495C4—41no—
496C4—43no—
497C4—122no—
498C4C2×#94016yes—
499C4C22×D3×D412yes—
500C3#6065011yescharacteristic, complemented, Sylow
501C2—26no—
502C2—83no—
503C2—81no—
504C2—246no—
505C2C22×#94011yescharacteristic
506C2#8729812yes—
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3500C2
socleC22×C3C23×D4
hypercenter#3500D3
hypocenterC3#60650
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3500

Complements

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

NH ≅ G/Ntype
1. C3×#3500C2semidirect
2. C2×#5837C2semidirect
3. C2×#5835C2semidirect
4. #85846C2semidirect
5. #83402C2semidirect
6. #83382C2semidirect
7. #83287C2semidirect
8. #83286C2semidirect
9. #82943C2semidirect
10. C2×#5058C2semidirect
11. #82511C2semidirect
18. C3×#536C22semidirect
19. C3×#535C22semidirect
20. C3×#532C22semidirect
21. C2×C3×#124C22semidirect
25. C2×#950C22semidirect
26. C2×#950C22semidirect
27. C2×#949C22semidirect
28. C2×#949C22semidirect
30. C2×#942C22semidirect
31. C22×#215C22semidirect
32. C2×#932C22semidirect
33. C22×#213C22semidirect
36. #5525C22semidirect
37. #5477C22semidirect
38. C2×#874C22semidirect
39. #5086C22semidirect
40. #5085C22semidirect
41. #5074C22semidirect
42. #5073C22semidirect
43. #5060C22semidirect
44. #5060C22semidirect
45. #5059C22semidirect
46. #5058C22semidirect
47. #5058C22semidirect
48. #5058C22semidirect
49. #5058C22semidirect
50. #4997C22semidirect
51. C4×#208C22semidirect
52. C2×#803C22semidirect
55. #3500D3semidirect
103. #948D4semidirect
151. #536C2×D3semidirect
153. #535C2×D3semidirect
155. #532C2×D3semidirect
157. C2×#124C2×D3semidirect
235. #208C2×D4semidirect
236. C2×C3⋊C8C2×D4semidirect
500. C3#60650semidirect

Maximal subgroups

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

subgroupindex
C3×#35002
C2×#58372
C2×#58352
#858462
#834022
#833822
#832872
#832862
#829432
C2×#50582
#825112
#606503