Finite Group Databasebeta

← C2×#78029 #10964354

Subgroups — C2×#78029 #10964354

Order 768.

3374 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×#67⋊C4C211yesmaximal, characteristic
3C2×C3×#399C211yesmaximal, characteristic, complemented
4C2×#4989C212yesmaximal, complemented
5C2×#4985C211yesmaximal, characteristic, complemented
6C2×#4976C212yesmaximal, complemented
7C2×#4956C212yesmaximal
8C2×#4949C211yesmaximal, characteristic, complemented
9C2×#4942C212yesmaximal, complemented
10#78029C2116yesmaximal, direct product factor
11C2×#4667C211yesmaximal, characteristic, complemented
12C2×#4659C211yesmaximal, characteristic, complemented
13C2×#2487—31nomaximal, Sylow
14C23×#67C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C23×C4×D3C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×#217C2212yes—
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2212yes—
25C22×C4⋊Dic3C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×C4×Dic3C2211yescharacteristic
31C2×D3×C4⋊C4—28no—
32C2×D3×C4⋊C4C2218yescomplemented
33C2×#67⋊C4—28no—
34C2×#67⋊C4C2218yes—
35C3×#399C2218yescomplemented
36C2×C3×#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42#4989C22116yescomplemented
43#4985C2218yescomplemented
44#4976C22116yescomplemented
45#4956C22116yes—
46#4949C2218yescomplemented
47#4942C22116yescomplemented
48#4667C2218yescomplemented
49#4659C2218yescomplemented
50C22×C4×D4—31no—
51C23×C4⋊C4—31no—
52#2487—316no—
53C2×#400—31no—
54C2×#400—31no—
55C2×#399—31no—
56C2×#399D311yescharacteristic, complemented
57C2×#389—32no—
58C2×#386—32no—
59C2×#385—32no—
60C2×#380—31no—
61C2×#380—32no—
62C25×C3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C22×#67—24no—
69C22×#67—24no—
70C22×#67Q818yes—
71C22×#67C2314yes—
72C23×Dic3—24no—
73C23×Dic3—28no—
74C23×Dic3C2312yes—
75C23×Dic3C2314yes—
76C23×Dic3C2311yescharacteristic
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2311yescharacteristic
80C22×C4×D3—28no—
81C22×C4×D3—28no—
82C22×C4×D3—44no—
83C22×C4×D3—44no—
84C22×C4×D3D418yes—
85C22×C4×D3C2314yes—
86C22×C4×D3C2318yes—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4—44no—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C4⋊C4C2318yes—
91C2×C3×C22⋊C4—24no—
92C2×C3×C22⋊C4—44no—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2318yes—
95C2×#217—44no—
96C2×#217C2318yes—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—28no—
99C2×(C2×D3)⋊C4C2318yes—
100C2×(C2×D3)⋊C4C2314yes—
101C2×C4⋊Dic3—24no—
102C2×C4⋊Dic3C2314yes—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—28no—
106C2×Dic3⋊C4—44no—
107C2×Dic3⋊C4C2314yes—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111C2×C4×Dic3—24no—
112C2×C4×Dic3C2314yes—
113D3×C4⋊C4—232no—
114#67⋊C4—232no—
115C3×#95C2314yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2314yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C23×D4—31no—
122C24×C4—31no—
123C24×C4—31no—
124C24×C4—32no—
125C2×C4×D4—38no—
126C2×C4×D4—68no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—38no—
133C22×C4⋊C4—68no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D312yes—
136C22×C22⋊C4—31no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4C2×D311yescharacteristic
140C22×C22⋊C4C2×D312yes—
141C22×C42—31no—
142#400—38no—
143#400—38no—
144#399—38no—
145#399C2×D318yescomplemented
146#389—316no—
147#386—316no—
148#385—316no—
149#380—316no—
150#380—38no—
151C2×#95—31no—
152C2×#95—32no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95C2×D311yescharacteristic
157C24×C3—22no—
158C24×C3—24no—
159C24×C3—22no—
160C24×C3—24no—
161C24×C3C2×Q814yes—
162C24×C3C2412yes—
163C24×C3C2411yescharacteristic
164C23×D3—24no—
165C23×D3—24no—
166C23×D3C2×D414yes—
167C23×D3C2×Q814yes—
168C23×D3#4114yes—
169C23×D3C2412yes—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2414yes—
181C22×C3×C4C2414yes—
182C22×C3×C4C2412yes—
183C22×C3×C4C2412yes—
184C2×#67—232no—
185C2×#67—28no—
186C2×#67—28no—
187C2×#67C2×Q8116yes—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—28no—
199C22×Dic3—216no—
200C22×Dic3—416no—
201C22×Dic3C2×D414yes—
202C22×Dic3C2×Q814yes—
203C22×Dic3#4118yes—
204C22×Dic3C2412yes—
205C22×Dic3C2414yes—
206C22×Dic3C2414yes—
207C22×Dic3C2412yes—
208C22×Dic3C2412yes—
209C2×C4×D3—232no—
210C2×C4×D3—216no—
211C2×C4×D3—216no—
212C2×C4×D3—48no—
213C2×C4×D3—48no—
214C2×C4×D3—432no—
215C2×C4×D3C2×D4116yes—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—48no—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—48no—
220#217—48no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4—216no—
223C4⋊Dic3—28no—
224Dic3⋊C4—28no—
225Dic3⋊C4—216no—
226Dic3⋊C4—216no—
227Dic3⋊C4—48no—
228C4×Dic3—28no—
229C25—31no—
230C25C22×D311yescharacteristic
231C22×D4—38no—
232C22×D4—34no—
233C22×D4—64no—
234C22×D4—64no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—38no—
238C23×C4—34no—
239C23×C4—34no—
240C23×C4—32no—
241C23×C4—32no—
242C23×C4—31no—
243C23×C4—38no—
244C23×C4—68no—
245C23×C4—68no—
246C23×C4—64no—
247C23×C4—68no—
248C23×C4—124no—
249C23×C4—124no—
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D312yes—
252C23×C4C22×D312yes—
253C23×C4C22×D311yescharacteristic
254C4×D4—632no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—38no—
257C2×C4⋊C4—34no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—38no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—632no—
263C2×C4⋊C4—68no—
264C2×C4⋊C4—68no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—124no—
268C2×C4⋊C4C22×D314yes—
269C2×C4⋊C4C22×D318yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—68no—
277C2×C22⋊C4—124no—
278C2×C22⋊C4C22×D314yes—
279C2×C22⋊C4C22×D318yes—
280C2×C42—34no—
281C2×C42—64no—
282#95—38no—
283#95—38no—
284#95—34no—
285#95—38no—
286#95—38no—
287#95C22×D314yes—
288C23×C3—28no—
289C23×C3—216no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—24no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3—48no—
297C23×C3C22×D412yes—
298C23×C3C22×Q814yes—
299C23×C3C22×Q812yes—
300C23×C3C2×#4112yes—
301C23×C3C2×#4114yes—
302C23×C3H(5,𝔽2)14yes—
303C23×C3C2511yescharacteristic
304C22×D3—28no—
305C22×D3—28no—
306C22×D3—216no—
307C22×D3—216no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3C22⋊Q8116yes—
311C22×D3C22×D414yes—
312C22×D3C22×Q814yes—
313C22×D3C2×#4114yes—
314C2×C3×C4—28no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—48no—
325C2×C3×C4—44no—
326C2×C3×C4—48no—
327C2×C3×C4C22×D414yes—
328#67—232no—
329C2×Dic3—28no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—44no—
341C2×Dic3—48no—
342C2×Dic3—48no—
343C2×Dic3—416no—
344C2×Dic3#123116yes—
345C2×Dic3C22×D414yes—
346C2×Dic3C22×Q814yes—
347C2×Dic3C2×#4118yes—
348C4×D3—232no—
349C4×D3—432no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—34no—
355C24—34no—
356C24—34no—
357C24—32no—
358C24—64no—
359C24—64no—
360C24D3×Q814yes—
361C24C23×D312yes—
362C24C23×D311yescharacteristic
363C2×D4—316no—
364C2×D4—68no—
365C2×D4—68no—
366C2×D4—632no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—34no—
377C22×C4—38no—
378C22×C4—32no—
379C22×C4—34no—
380C22×C4—32no—
381C22×C4—316no—
382C22×C4—32no—
383C22×C4—34no—
384C22×C4—34no—
385C22×C4—632no—
386C22×C4—616no—
387C22×C4—616no—
388C22×C4—68no—
389C22×C4—616no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—64no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4—128no—
401C22×C4—128no—
402C22×C4—1216no—
403C22×C4—1232no—
404C22×C4D3×D414yes—
405C22×C4C23×D314yes—
406C22×C4C23×D314yes—
407C22×C4C23×D312yes—
408C22×C4C23×D312yes—
409C4⋊C4—28no—
410C4⋊C4—48no—
411C4⋊C4—616no—
412C4⋊C4—616no—
413C4⋊C4—68no—
414C4⋊C4—68no—
415C4⋊C4—128no—
416C22⋊C4—28no—
417C22⋊C4—48no—
418C22⋊C4—68no—
419C22⋊C4—616no—
420C22⋊C4—128no—
421C42—68no—
422C22×C3—24no—
423C22×C3—28no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—416no—
428C22×C3C2×C22⋊Q814yes—
429C22×C3C2×#12314yes—
430C22×C3#53514yes—
431C22×C3#55318yes—
432C22×C3#55418yes—
433C22×C3C23×D411yescharacteristic
434C22×C3C23×Q811yescharacteristic
435C22×C3C22×#4111yescharacteristic
436C22×C3C22×#4112yes—
437C22×C3C2×H(5,𝔽2)12yes—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—28no—
441C2×D3—28no—
442C2×D3—232no—
443C2×D3C2×C22⋊Q818yes—
444C3×C4—24no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—44no—
448Dic3—24no—
449Dic3—42no—
450Dic3—44no—
451Dic3—44no—
452Dic3C2×#12318yes—
453C23—216no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—28no—
459C23—34no—
460C23—34no—
461C23—34no—
462C23—316no—
463C23—44no—
464C23—48no—
465C23—44no—
466C23—616no—
467C23—68no—
468C23—616no—
469C23—64no—
470C23—64no—
471C23—68no—
472C23C2×D3×D412yes—
473C23#100414yes—
474C23C2×D3×Q814yes—
475C23C2×D3×Q812yes—
476C23D3×#4114yes—
477C23D3×#4112yes—
478C23C24×D311yescharacteristic
479D4—632no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—316no—
491C2×C4—34no—
492C2×C4—38no—
493C2×C4—34no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—632no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—68no—
501C2×C4—62no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—1216no—
510C2×C4—1232no—
511C2×C4—124no—
512C2×C4—128no—
513C2×C4—128no—
514C2×C4C2×D3×D414yes—
515C2×C3—44no—
516C2×C3#248718yescomplemented
517C2×C3C22×C22⋊Q811yescharacteristic
518C2×C3C22×#12311yescharacteristic
519C2×C3C2×#53511yescharacteristic
520C2×C3C2×#55312yes—
521C2×C3C2×#55412yes—
522D3—28no—
523C22—28no—
524C22—24no—
525C22—38no—
526C22—42no—
527C22—416no—
528C22—44no—
529C22—42no—
530C22—64no—
531C22—632no—
532C22—68no—
533C22—68no—
534C22—64no—
535C22D3×C22⋊Q814yes—
536C22#563618yes—
537C22D3×#12314yes—
538C22#566118yes—
539C22#571514yes—
540C22C22×D3×D411yescharacteristic
541C22C2×#100412yes—
542C22C22×D3×Q811yescharacteristic
543C22C2×D3×#4112yes—
544C22C2×D3×#4111yescharacteristic
545C4—24no—
546C4—38no—
547C4—44no—
548C4—42no—
549C4—44no—
550C4—64no—
551C4—124no—
552C4—124no—
553C4—122no—
554C3C2×#248711yescharacteristic, complemented, Sylow
555C2—44no—
556C2—68no—
557C2#7802918yesdirect product factor
558C2C2×#563612yes—
559C2C2×D3×C22⋊Q811yescharacteristic
560C2C2×#566112yes—
561C2C2×D3×#12311yescharacteristic
562C2C2×#571511yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#399C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#399D3
hypocenterC3C2×#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×D3×C4⋊C4C2semidirect
3. C2×C3×#399C2semidirect
4. C2×#4989C2semidirect
5. C2×#4985C2semidirect
6. C2×#4976C2semidirect
8. C2×#4949C2semidirect
9. C2×#4942C2semidirect
10. #78029C2direct (N × H)
11. C2×#4667C2semidirect
12. C2×#4659C2semidirect
32. C2×D3×C4⋊C4C22semidirect
35. C3×#399C22semidirect
42. #4989C22semidirect
43. #4985C22semidirect
44. #4976C22semidirect
46. #4949C22semidirect
47. #4942C22semidirect
48. #4667C22semidirect
49. #4659C22semidirect
56. C2×#399D3semidirect
145. #399C2×D3semidirect
516. C2×C3#2487semidirect
554. C3C2×#2487semidirect
557. C2#78029direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C4⋊C42
C22×#67⋊C42
C2×C3×#3992
C2×#49892
C2×#49852
C2×#49762
C2×#49562
C2×#49492
C2×#49422
#780292
C2×#46672
C2×#46592
C2×#24873