Finite Group Databasebeta

← C2×#81565 #11664473

Subgroups — C2×#81565 #11664473

Order 768.

2398 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
1C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#5716C211yesmaximal, characteristic
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C2×#5093C211yesmaximal, characteristic, complemented
5C2×#5070C211yesmaximal, characteristic, complemented
6#81565C2116yesmaximal, direct product factor
7C2×#4771C214yesmaximal, complemented
8C2×#4765C214yesmaximal, complemented
9C2×#4681C211yesmaximal, characteristic, complemented
10C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
11C2×#3141—31nomaximal, Sylow
12C22×D3×Q8—21no—
13C22×#237C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×#944—21no—
17C2×#938—21no—
18C2×#938—21no—
19C2×#936C2212yes—
20C2×D4×Dic3—21no—
21C2×D4×Dic3—21no—
22C22×ΓL(1,𝔽25)C2212yes—
23D3×C4⋊Q8—24no—
24D3×C4⋊Q8C2218yescomplemented
25#5716—24no—
26#5716C2218yes—
27C2×D3×C4⋊C4—21no—
28C2×D3×C4⋊C4C2212yes—
29C2×#888—21no—
30C2×#888C2212yes—
31C2×C42×D3C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×#869C2211yescharacteristic
34C3×#486C2218yescomplemented
35C2×C3×#102C2214yes—
36C2×C3×C4×C8C2211yescharacteristic
37C2×#832C2214yes—
38C2×#816C2214yes—
39C2×#815C2212yes—
40#5093C2218yescomplemented
41#5070C2218yescomplemented
42C2×#810C2214yes—
43C2×C4×C3⋊C8C2211yescharacteristic
44#4771C22132yescomplemented
45#4765C22132yescomplemented
46C4×ΓL(1,𝔽25)C2218yescomplemented
47#4681C2218yescomplemented
48C2×#532—31no—
49C22×C4⋊Q8—31no—
50#3141—316no—
51C2×#489—34no—
52C2×#487—31no—
53C2×#487—31no—
54C2×#486—31no—
55C2×#486D311yescharacteristic, complemented
56C2×#417—34no—
57C2×C4×M16—31no—
58C22×C3×Q8—21no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C22×#67—22no—
63C23×Dic3—22no—
64C2×D3×Q8—28no—
65C2×D3×Q8—48no—
66C2×#237—24no—
67C2×#237—216no—
68C2×#237C2318yes—
69C22×C4×D3—22no—
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6—21no—
74C22×Dic6—22no—
75C22×Dic6C2312yes—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8C2314yes—
79C3×#127—22no—
80C3×#127C2314yes—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C42C2311yescharacteristic
84C2×#217—22no—
85C2×#217—22no—
86#944—28no—
87#938—28no—
88#938—28no—
89#936—24no—
90#936C2318yes—
91D4×Dic3—28no—
92D4×Dic3—28no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2312yes—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3D411yescharacteristic
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8C2312yes—
103C2×ΓL(1,𝔽25)—28no—
104C2×ΓL(1,𝔽25)C2318yes—
105D3×C4⋊C4—24no—
106D3×C4⋊C4—28no—
107D3×C4⋊C4C2318yes—
108#888—24no—
109#888—28no—
110#888C2318yes—
111C42×D3D414yes—
112C42×D3C2314yes—
113#869—22no—
114#869—22no—
115#869C2314yes—
116#869C2314yes—
117C3×#102C23116yes—
118C3×C4×C8C2314yes—
119#832C23116yes—
120#816C23116yes—
121#815C2318yes—
122#810C23116yes—
123C4×C3⋊C8C2314yes—
124C22×#41—32no—
125C23×Q8—61no—
126C22×M16—32no—
127#532—38no—
128#532—64no—
129C2×C4⋊Q8—38no—
130C2×C4⋊Q8—32no—
131C2×C4⋊Q8—31no—
132C2×C4⋊Q8—31no—
133C2×C4⋊Q8—61no—
134C2×C4⋊Q8—64no—
135C2×C4⋊Q8—61no—
136C2×C4⋊Q8C2×D311yescharacteristic
137C2×#127C2×D311yescharacteristic
138C2×#124—32no—
139C2×#121—61no—
140C2×#121—61no—
141C2×C4×D4—61no—
142C2×C4×D4—61no—
143C22×C4⋊C4—32no—
144C22×C4⋊C4—61no—
145C22×C42—31no—
146#489—332no—
147#487—38no—
148#487—38no—
149#486—38no—
150#486C2×D318yescomplemented
151#417—332no—
152C2×#103—34no—
153C2×#103—34no—
154C2×#102—34no—
155C2×#102C2×D314yes—
156C4×M16—38no—
157C2×#97—32no—
158C2×C4×C8—31no—
159C2×C4×C8C2×D311yescharacteristic
160C24×C3—22no—
161C23×D3C2×D411yescharacteristic
162C2×C3×Q8—24no—
163C2×C3×Q8—44no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—28no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4C2414yes—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—28no—
176C2×#67—48no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—28no—
180C22×Dic3—48no—
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183C22×Dic3C2×D411yescharacteristic
184D3×Q8—432no—
185#237—232no—
186#237—416no—
187C2×C4×D3—24no—
188C2×C4×D3—24no—
189C2×C4×D3—28no—
190C2×C4×D3—48no—
191C2×C4×D3C2×D414yes—
192C2×C4×D3C2×D414yes—
193C2×C4×D3C2×D412yes—
194C2×C4×D3#4118yes—
195C2×C4×D3C2412yes—
196C2×Dic6—28no—
197C2×Dic6—22no—
198C2×Dic6—24no—
199C2×Dic6—28no—
200C2×Dic6—48no—
201C2×Dic6—44no—
202C2×Dic6C2414yes—
203C2×C3×C8—24no—
204C2×C3×C8C2414yes—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4C2414yes—
208C3×C42C2×D412yes—
209C3×C42C2412yes—
210#217—28no—
211#217—28no—
212C4⋊Dic3—22no—
213C4⋊Dic3—24no—
214C4⋊Dic3—24no—
215C4⋊Dic3—24no—
216C4⋊Dic3C2414yes—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219C4×Dic3C2×D412yes—
220C4×Dic3C2×D414yes—
221C4×Dic3C2×D414yes—
222C4×Dic3C2412yes—
223C2×C3⋊C8—24no—
224C2×C3⋊C8C2414yes—
225ΓL(1,𝔽25)—216no—
226C2×#41—38no—
227C2×#41—64no—
228C2×#41—616no—
229C22×Q8—21no—
230C22×Q8—32no—
231C22×Q8—62no—
232C22×Q8—68no—
233C22×Q8—61no—
234C22×Q8—128no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—62no—
238C22×D4C22×D312yes—
239C23×C4—31no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—62no—
243C23×C4—62no—
244C2×M16—38no—
245C2×M16—68no—
246C22×C8—32no—
247C22×C8C22×D312yes—
248C4⋊Q8—22no—
249C4⋊Q8—38no—
250C4⋊Q8—34no—
251C4⋊Q8—34no—
252C4⋊Q8—68no—
253C4⋊Q8—64no—
254C4⋊Q8—68no—
255C4⋊Q8—62no—
256C4⋊Q8—62no—
257C4⋊Q8C22×D314yes—
258#127—22no—
259#127C22×D314yes—
260#124—38no—
261#124—64no—
262#121—68no—
263#121—68no—
264C4×D4—68no—
265C4×D4—68no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—38no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—68no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C42—31no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—34no—
283C2×C42—34no—
284C2×C42C22×D311yescharacteristic
285#103—316no—
286#103—316no—
287#102—316no—
288#102C22×D3116yes—
289#97—38no—
290C4×C8—34no—
291C4×C8C22×D314yes—
292C23×C3—24no—
293C23×C3—44no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3C22×D411yescharacteristic
297C22×D3—22no—
298C22×D3—22no—
299C22×D3#12414yes—
300C22×D3C22×D412yes—
301C3×Q8—48no—
302C3×D4—28no—
303C3×D4—44no—
304C3×D4—48no—
305C3×D4—48no—
306C2×C3×C4—24no—
307C2×C3×C4—44no—
308C2×C3×C4AGL(1,ℤ/8)14yes—
309C2×C3×C4#13714yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D412yes—
312C2×C3×C4C22×D412yes—
313C2×C3×C4C2×#4114yes—
314C2×C3×C4C2511yescharacteristic
315#67—416no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—44no—
321C2×Dic3—44no—
322C2×Dic3—416no—
323C2×Dic3#12414yes—
324C2×Dic3C22×D412yes—
325C2×Dic3C22×D412yes—
326C2×Dic3C22×D412yes—
327C4×D3—28no—
328C4×D3—28no—
329C4×D3—416no—
330C4×D3#12418yes—
331C4×D3C2×#4118yes—
332Dic6—28no—
333Dic6—416no—
334Dic6—44no—
335Dic6—48no—
336C3×C8—24no—
337C3⋊C8—24no—
338C24—22no—
339C24—31no—
340#41—632no—
341#41—1216no—
342C2×Q8—24no—
343C2×Q8—34no—
344C2×Q8—44no—
345C2×Q8—68no—
346C2×Q8—62no—
347C2×Q8—64no—
348C2×Q8—68no—
349C2×Q8—128no—
350C2×Q8—1232no—
351C2×Q8—124no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—28no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—68no—
359C2×D4—128no—
360C2×D4C23×D314yes—
361C22×C4—22no—
362C22×C4—31no—
363C22×C4—32no—
364C22×C4—38no—
365C22×C4—32no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—34no—
369C22×C4—34no—
370C22×C4—68no—
371C22×C4—68no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—128no—
377C22×C4—128no—
378C22×C4D3×D411yescharacteristic
379C22×C4D3×D411yescharacteristic
380C22×C4C23×D311yescharacteristic
381M16—616no—
382C2×C8—24no—
383C2×C8—34no—
384C2×C8—64no—
385C2×C8C23×D314yes—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—34no—
389C4⋊C4—68no—
390C4⋊C4—68no—
391C4⋊C4—62no—
392C4⋊C4—64no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4C23×D314yes—
396C22⋊C4—68no—
397C22⋊C4—68no—
398C42—34no—
399C42—34no—
400C42—32no—
401C42—32no—
402C42D3×D412yes—
403C42C23×D312yes—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—88no—
408C22×C3C2×#12412yes—
409C22×C3C2×AGL(1,ℤ/8)12yes—
410C22×C3C2×#13712yes—
411C22×C3C23×D411yescharacteristic
412C2×D3—28no—
413C2×D3—22no—
414C2×D3—22no—
415C2×D3C2×#12414yes—
416C3×C4—44no—
417C3×C4C2×#12412yes—
418C3×C4C2×AGL(1,ℤ/8)14yes—
419C3×C4C2×#13714yes—
420C3×C4C22×#4112yes—
421Dic3—22no—
422Dic3—22no—
423Dic3—44no—
424Dic3—44no—
425Dic3C2×#12414yes—
426C23—24no—
427C23—34no—
428C23—32no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—62no—
433C23—62no—
434C23C2×D3×D411yescharacteristic
435Q8—48no—
436Q8—68no—
437Q8—1216no—
438Q8—124no—
439Q8—128no—
440D4—28no—
441D4—44no—
442D4—48no—
443D4—48no—
444D4—1216no—
445C2×C4—24no—
446C2×C4—34no—
447C2×C4—32no—
448C2×C4—38no—
449C2×C4—38no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—44no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—68no—
456C2×C4—68no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—1216no—
460C2×C4—1216no—
461C2×C4—124no—
462C2×C4—124no—
463C2×C4#91114yes—
464C2×C4#91514yes—
465C2×C4C2×D3×D411yescharacteristic
466C2×C4C2×D3×D412yes—
467C2×C4C2×D3×D412yes—
468C2×C4D3×#4114yes—
469C2×C4C24×D311yescharacteristic
470C8—24no—
471C8—64no—
472C2×C3—84no—
473C2×C3#314114yescomplemented
474C2×C3C22×#12411yescharacteristic
475C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
476C2×C3C22×#13711yescharacteristic
477D3—24no—
478C22—34no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—62no—
483C22—62no—
484C22—68no—
485C22—88no—
486C22D3×#12412yes—
487C22C2×#91112yes—
488C22C2×#91512yes—
489C22C22×D3×D411yescharacteristic
490C4—34no—
491C4—44no—
492C4—62no—
493C4—62no—
494C4—124no—
495C4—124no—
496C4D3×#12412yes—
497C4C2×#91114yes—
498C4C2×#91514yes—
499C4C2×D3×#4112yes—
500C3C2×#314111yescharacteristic, complemented, Sylow
501C2—64no—
502C2—84no—
503C2#8156514yesdirect product factor
504C2C2×D3×#12411yescharacteristic
505C2C22×#91111yescharacteristic
506C2C22×#91511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#486C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenterC2×#486D3
hypocenterC3C2×#3141
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊Q8C2semidirect
3. C2×C3×#486C2semidirect
4. C2×#5093C2semidirect
5. C2×#5070C2semidirect
6. #81565C2direct (N × H)
7. C2×#4771C2semidirect
8. C2×#4765C2semidirect
9. C2×#4681C2semidirect
10. C2×C4×ΓL(1,𝔽25)C2semidirect
24. D3×C4⋊Q8C22semidirect
34. C3×#486C22semidirect
40. #5093C22semidirect
41. #5070C22semidirect
44. #4771C22semidirect
45. #4765C22semidirect
46. C4×ΓL(1,𝔽25)C22semidirect
47. #4681C22semidirect
55. C2×#486D3semidirect
150. #486C2×D3semidirect
473. C2×C3#3141semidirect
500. C3C2×#3141semidirect
503. C2#81565direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#57162
C2×C3×#4862
C2×#50932
C2×#50702
#815652
C2×#47712
C2×#47652
C2×#46812
C2×C4×ΓL(1,𝔽25)2
C2×#31413