Finite Group Databasebeta

← C2×#74075 #10964238

Subgroups — C2×#74075 #10964238

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#929C211yesmaximal, characteristic
2C23×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C22×#886C211yesmaximal, characteristic
4C22×#67⋊C4C211yesmaximal, characteristic
5C2×C3×#381C211yesmaximal, characteristic, complemented
6C2×#4943C211yesmaximal, characteristic, complemented
7C2×#4942C211yesmaximal, characteristic, complemented
8#74075C2116yesmaximal, direct product factor
9C2×#4675C212yesmaximal, complemented
10C2×#4667C212yesmaximal, complemented
11C2×#4657C212yesmaximal, complemented
12C2×#4647C212yesmaximal, complemented
13C2×#2486—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C23×C4×D3C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×#217C2211yescharacteristic
22C2×#929—28no—
23C2×#929C2218yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×C4⋊Dic3C2212yes—
28C22×Dic3⋊C4—28no—
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2211yescharacteristic
33C22×Dic3⋊C4C2212yes—
34C22×Dic3⋊C4C2218yescomplemented
35C22×C4×Dic3C2211yescharacteristic
36C2×#886—28no—
37C2×#886C2218yes—
38C2×#67⋊C4—28no—
39C2×#67⋊C4C2218yes—
40C3×#381C2218yescomplemented
41C2×C3×#95C2212yes—
42C2×C3×#95C2212yes—
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45C2×C3⋊#95C2212yes—
46#4943C2218yescomplemented
47#4942C2218yescomplemented
48#4675C22116yescomplemented
49#4667C22116yescomplemented
50#4657C22116yescomplemented
51#4647C22116yescomplemented
52C22×#123—31no—
53C22×#123—31no—
54C22×C4×D4—31no—
55C23×C4⋊C4—31no—
56#2486—316no—
57C2×#398—32no—
58C2×#397—32no—
59C2×#389—32no—
60C2×#381—31no—
61C2×#381D311yescharacteristic, complemented
62C2×#380—32no—
63C2×#380—31no—
64C25×C3C2311yescharacteristic
65C24×D3C2311yescharacteristic
66C23×C3×C4—24no—
67C23×C3×C4—24no—
68C23×C3×C4—24no—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2312yes—
71C23×C3×C4C2314yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2312yes—
74C22×#67—24no—
75C22×#67—24no—
76C22×#67D418yes—
77C22×#67C2314yes—
78C23×Dic3—28no—
79C23×Dic3—24no—
80C23×Dic3—24no—
81C23×Dic3—24no—
82C23×Dic3D418yes—
83C23×Dic3C2314yes—
84C23×Dic3C2312yes—
85C23×Dic3C2312yes—
86C23×Dic3C2314yes—
87C23×Dic3C2311yescharacteristic
88C23×Dic3C2311yescharacteristic
89C23×Dic3C2311yescharacteristic
90C22×C4×D3—24no—
91C22×C4×D3—28no—
92C22×C4×D3C2314yes—
93C2×C3×C22⋊C4—24no—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4C2314yes—
96C2×C3×C22⋊C4C2314yes—
97C2×#217—24no—
98C2×#217C2314yes—
99#929—232no—
100C2×(C2×D3)⋊C4—28no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—24no—
103C2×(C2×D3)⋊C4C2318yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×(C2×D3)⋊C4C2314yes—
106C2×C4⋊Dic3—28no—
107C2×C4⋊Dic3C2318yes—
108C2×Dic3⋊C4—24no—
109C2×Dic3⋊C4—28no—
110C2×Dic3⋊C4—28no—
111C2×Dic3⋊C4—232no—
112C2×Dic3⋊C4—28no—
113C2×Dic3⋊C4—28no—
114C2×Dic3⋊C4C2318yes—
115C2×Dic3⋊C4C2318yes—
116C2×Dic3⋊C4C2318yes—
117C2×Dic3⋊C4C2314yes—
118C2×Dic3⋊C4C2318yes—
119C2×C4×Dic3—24no—
120C2×C4×Dic3C2314yes—
121#886—232no—
122#67⋊C4—232no—
123C3×#95C2318yes—
124C3×#95C2318yes—
125C3⋊#95C2318yes—
126C3⋊#95C2318yes—
127C3⋊#95C2318yes—
128C23×D4—31no—
129C24×C4—31no—
130C24×C4—31no—
131C24×C4—31no—
132C24×C4C2×D311yescharacteristic
133C2×#123—38no—
134C2×#123—38no—
135C2×#123—68no—
136C2×#123—68no—
137C2×C4×D4—38no—
138C2×C4×D4—68no—
139C22×C4⋊C4—32no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4—32no—
142C22×C4⋊C4—32no—
143C22×C4⋊C4—32no—
144C22×C4⋊C4—38no—
145C22×C4⋊C4—32no—
146C22×C4⋊C4—68no—
147C22×C22⋊C4—31no—
148C22×C22⋊C4—31no—
149C22×C22⋊C4—32no—
150C22×C22⋊C4—32no—
151C22×C22⋊C4C2×D311yescharacteristic
152C22×C22⋊C4C2×D311yescharacteristic
153C22×C42—31no—
154#398—316no—
155#397—316no—
156#389—316no—
157#381—38no—
158#381C2×D318yescomplemented
159#380—38no—
160#380—316no—
161C2×#95—32no—
162C2×#95—32no—
163C2×#95—32no—
164C2×#95C2×D312yes—
165C2×#95C2×D312yes—
166C24×C3—24no—
167C24×C3—24no—
168C24×C3C2×D414yes—
169C24×C3C2×D414yes—
170C24×C3#4114yes—
171C24×C3C2412yes—
172C24×C3C2411yescharacteristic
173C23×D3—22no—
174C23×D3—24no—
175C23×D3—22no—
176C23×D3—24no—
177C23×D3C2×D414yes—
178C23×D3C2412yes—
179C22×C3×C4—22no—
180C22×C3×C4—28no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—28no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—28no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—416no—
193C22×C3×C4C2412yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2414yes—
196C22×C3×C4C2414yes—
197C2×#67—232no—
198C2×#67—28no—
199C2×#67—28no—
200C2×#67C2×D4116yes—
201C22×Dic3—22no—
202C22×Dic3—24no—
203C22×Dic3—22no—
204C22×Dic3—232no—
205C22×Dic3—216no—
206C22×Dic3—28no—
207C22×Dic3—28no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—28no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—416no—
216C22×Dic3C2×D4116yes—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×D414yes—
219C22×Dic3C2×D414yes—
220C22×Dic3#4118yes—
221C22×Dic3C2414yes—
222C22×Dic3C2414yes—
223C22×Dic3C2412yes—
224C22×Dic3C2412yes—
225C22×Dic3C2412yes—
226C2×C4×D3—28no—
227C2×C4×D3—216no—
228C2×C4×D3—416no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—28no—
231#217—28no—
232(C2×D3)⋊C4—216no—
233(C2×D3)⋊C4—216no—
234(C2×D3)⋊C4—28no—
235C4⋊Dic3—216no—
236Dic3⋊C4—28no—
237Dic3⋊C4—216no—
238Dic3⋊C4—216no—
239Dic3⋊C4—216no—
240Dic3⋊C4—216no—
241C4×Dic3—28no—
242C25—31no—
243C25C22×D311yescharacteristic
244C22×D4—38no—
245C22×D4—34no—
246C22×D4—64no—
247C22×D4—64no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—31no—
252C23×C4—38no—
253C23×C4—34no—
254C23×C4—31no—
255C23×C4—31no—
256C23×C4—34no—
257C23×C4—32no—
258C23×C4—32no—
259C23×C4—34no—
260C23×C4—64no—
261C23×C4—68no—
262C23×C4—68no—
263C23×C4—64no—
264C23×C4—64no—
265C23×C4—64no—
266C23×C4C22×D314yes—
267C23×C4C22×D312yes—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D312yes—
270C23×C4C22×D311yescharacteristic
271#123—632no—
272#123—632no—
273C4×D4—632no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—38no—
276C2×C4⋊C4—34no—
277C2×C4⋊C4—38no—
278C2×C4⋊C4—38no—
279C2×C4⋊C4—38no—
280C2×C4⋊C4—68no—
281C2×C4⋊C4—68no—
282C2×C4⋊C4—632no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—68no—
286C2×C4⋊C4—68no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—34no—
290C2×C22⋊C4—34no—
291C2×C22⋊C4—38no—
292C2×C22⋊C4—38no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—68no—
295C2×C22⋊C4—68no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4C22×D314yes—
298C2×C22⋊C4C22×D314yes—
299C2×C42—34no—
300C2×C42—64no—
301#95—38no—
302#95—38no—
303#95—38no—
304#95C22×D318yes—
305#95C22×D318yes—
306C23×C3—28no—
307C23×C3—28no—
308C23×C3—216no—
309C23×C3—216no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3#121116yes—
313C23×C3C22×D414yes—
314C23×C3C22×D414yes—
315C23×C3C22×D412yes—
316C23×C3C22×D412yes—
317C23×C3C2×#4114yes—
318C23×C3C2×#4112yes—
319C23×C3C2×#4114yes—
320C23×C32−1+414yes—
321C23×C3C2511yescharacteristic
322C22×D3—28no—
323C22×D3—216no—
324C22×D3—22no—
325C22×D3—24no—
326C22×D3—22no—
327C22×D3—24no—
328C22×D3—44no—
329C22×D3—44no—
330C22×D3—48no—
331C22×D3C22×D414yes—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—416no—
343C2×C3×C4—48no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—48no—
347#67—232no—
348C2×Dic3—28no—
349C2×Dic3—28no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—232no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—416no—
361C2×Dic3—48no—
362C2×Dic3—48no—
363C2×Dic3#123116yescomplemented
364C2×Dic3C22×D414yes—
365C2×Dic3C22×D414yes—
366C2×Dic3C22×D414yes—
367C2×Dic3C2×#4118yes—
368C4×D3—416no—
369C24—24no—
370C24—24no—
371C24—34no—
372C24—32no—
373C24—62no—
374C24—62no—
375C24—64no—
376C24—64no—
377C24#23514yes—
378C24D3×D414yes—
379C24D3×D414yes—
380C24C23×D312yes—
381C24C23×D311yescharacteristic
382C2×D4—316no—
383C2×D4—632no—
384C2×D4—68no—
385C2×D4—68no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—34no—
400C22×C4—38no—
401C22×C4—32no—
402C22×C4—34no—
403C22×C4—34no—
404C22×C4—316no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—34no—
408C22×C4—34no—
409C22×C4—416no—
410C22×C4—68no—
411C22×C4—616no—
412C22×C4—632no—
413C22×C4—616no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—64no—
417C22×C4—68no—
418C22×C4—64no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—68no—
422C22×C4—64no—
423C22×C4—68no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—1216no—
427C22×C4—1216no—
428C22×C4C23×D312yes—
429C22×C4C23×D312yes—
430C22×C4C23×D314yes—
431C22×C4C23×D314yes—
432C4⋊C4—616no—
433C4⋊C4—616no—
434C4⋊C4—68no—
435C4⋊C4—616no—
436C4⋊C4—616no—
437C4⋊C4—616no—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—68no—
441C22⋊C4—616no—
442C22⋊C4—616no—
443C22⋊C4—68no—
444C42—68no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—28no—
448C22×C3—28no—
449C22×C3—232no—
450C22×C3C2×#12118yes—
451C22×C3C2×#12114yes—
452C22×C3C2×#12314yes—
453C22×C3#53614yes—
454C22×C3#54718yes—
455C22×C3#55218yes—
456C22×C3C23×D411yescharacteristic
457C22×C3C23×D411yescharacteristic
458C22×C3C22×#4111yescharacteristic
459C22×C3C22×#4112yes—
460C22×C3C2×2−1+412yes—
461C2×D3—24no—
462C2×D3—28no—
463C2×D3—42no—
464C2×D3—42no—
465C2×D3—44no—
466C2×D3—416no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—44no—
471Dic3—24no—
472Dic3—24no—
473Dic3—44no—
474Dic3—44no—
475Dic3C2×#12318yescomplemented
476C23—28no—
477C23—28no—
478C23—216no—
479C23—24no—
480C23—24no—
481C23—216no—
482C23—34no—
483C23—616no—
484C23—62no—
485C23—62no—
486C23—64no—
487C23—64no—
488C23—68no—
489C23—124no—
490C23—124no—
491C23—128no—
492C23#884116yes—
493C23C2×#23514yes—
494C23C2×#23512yes—
495C23C2×D3×D414yes—
496C23C2×D3×D414yes—
497C23C2×D3×D412yes—
498C23C2×D3×D412yes—
499C23D3×#4114yes—
500C23#101014yes—
501C23C24×D311yescharacteristic
502D4—632no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—316no—
514C2×C4—34no—
515C2×C4—38no—
516C2×C4—34no—
517C2×C4—34no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—48no—
521C2×C4—416no—
522C2×C4—48no—
523C2×C4—632no—
524C2×C4—68no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—64no—
528C2×C4—68no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—1216no—
536C2×C4—128no—
537C2×C4—1216no—
538C2×C4—128no—
539C2×C3—28no—
540C2×C3#248618yescomplemented
541C2×C3C22×#12111yescharacteristic
542C2×C3C22×#12311yescharacteristic
543C2×C3C2×#53611yescharacteristic
544C2×C3C2×#54712yes—
545C2×C3C2×#55212yes—
546D3—44no—
547C22—232no—
548C22—28no—
549C22—24no—
550C22—24no—
551C22—28no—
552C22—64no—
553C22—68no—
554C22—1216no—
555C22—124no—
556C22—122no—
557C22—122no—
558C22C2×#88418yes—
559C22C2×#88414yes—
560C22#553118yes—
561C22#564814yes—
562C22D3×#12314yes—
563C22#565518yes—
564C22C22×#23511yescharacteristic
565C22C22×D3×D411yescharacteristic
566C22C22×D3×D411yescharacteristic
567C22C2×D3×#4112yes—
568C22C2×#101012yes—
569C4—38no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—44no—
574C4—64no—
575C4—64no—
576C4—124no—
577C4—124no—
578C3C2×#248611yescharacteristic, complemented, Sylow
579C2—28no—
580C2—124no—
581C2#7407518yesdirect product factor
582C2C22×#88411yescharacteristic
583C2C2×#553112yes—
584C2C2×#564811yescharacteristic
585C2C2×D3×#12311yescharacteristic
586C2C2×#565512yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#381C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#381D3
hypocenterC3C2×#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. C23×Dic3⋊C4C2semidirect
5. C2×C3×#381C2semidirect
6. C2×#4943C2semidirect
7. C2×#4942C2semidirect
8. #74075C2direct (N × H)
9. C2×#4675C2semidirect
10. C2×#4667C2semidirect
11. C2×#4657C2semidirect
12. C2×#4647C2semidirect
34. C22×Dic3⋊C4C22semidirect
40. C3×#381C22semidirect
46. #4943C22semidirect
47. #4942C22semidirect
48. #4675C22semidirect
49. #4667C22semidirect
50. #4657C22semidirect
51. #4647C22semidirect
61. C2×#381D3semidirect
158. #381C2×D3semidirect
363. C2×Dic3#123semidirect
475. Dic3C2×#123semidirect
540. C2×C3#2486semidirect
578. C3C2×#2486semidirect
581. C2#74075direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9292
C23×Dic3⋊C42
C22×#8862
C22×#67⋊C42
C2×C3×#3812
C2×#49432
C2×#49422
#740752
C2×#46752
C2×#46672
C2×#46572
C2×#46472
C2×#24863