Finite Group Databasebeta

← C2×#76466 #11429932

Subgroups — C2×#76466 #11429932

Order 768.

3934 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#883C212yesmaximal, complemented
2C22×#884C212yesmaximal, complemented
3C2×C3×#390C211yesmaximal, characteristic, complemented
4C2×#5223C212yesmaximal, complemented
5C2×#4989C212yesmaximal, complemented
6C2×#4948C212yesmaximal, complemented
7C2×#4961C212yesmaximal, complemented
8C2×#4939C211yesmaximal, characteristic, complemented
9C2×#4928C211yesmaximal, characteristic, complemented
10#76466C2116yesmaximal, direct product factor
11C2×#2682—31nomaximal, Sylow
12C23×C3×D4C2211yescharacteristic
13C23×#67C2212yes—
14C23×#67C2212yes—
15C23×D12C2211yescharacteristic
16C23×C4×D3C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C22×#217C2212yes—
22C22×#217C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×C4⋊Dic3C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2212yes—
31C2×#884—216no—
32C2×#884C22116yescomplemented
33C2×#883—216no—
34C2×#883C22116yescomplemented
35C3×#390C2218yescomplemented
36#5223C22116yescomplemented
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4989C22116yescomplemented
40#4961C22116yescomplemented
41#4948C22116yescomplemented
42#4939C2218yescomplemented
43#4928C2218yescomplemented
44C22×#123—32no—
45C22×#121—32no—
46#2682—316no—
47C2×#396—32no—
48C2×#394—32no—
49C2×#392—32no—
50C2×#390—31no—
51C2×#390D311yescharacteristic, complemented
52C2×#389—31no—
53C2×#385—32no—
54C25×C3C2312yes—
55C24×D3C2312yes—
56C22×C3×D4—24no—
57C22×C3×D4—42no—
58C22×C3×D4—42no—
59C22×C3×D4C2314yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—28no—
65C22×#67—28no—
66C22×#67—28no—
67C22×#67—44no—
68C22×#67—44no—
69C22×#67—44no—
70C22×#67C2318yes—
71C22×#67C2318yes—
72C23×Dic3C2312yes—
73C23×Dic3C2312yes—
74C23×Dic3C2312yes—
75C22×D12—42no—
76C22×D12—42no—
77C22×D12D418yescomplemented
78C22×D12C2314yes—
79C22×C4×D3—28no—
80C22×C4×D3—28no—
81C22×C4×D3—44no—
82C22×C4×D3C2318yes—
83C2×C3×C4⋊C4—42no—
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C22⋊C4—28no—
86C2×C3×C22⋊C4—28no—
87C2×C3×C22⋊C4C2318yes—
88C2×C3×C22⋊C4C2318yes—
89C2×C3×C42—42no—
90C2×C3×C42C2314yes—
91C2×#217—28no—
92C2×#217—44no—
93C2×#217C2318yes—
94C2×#217C2318yes—
95C2×(C2×D3)⋊C4—28no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—28no—
98C2×(C2×D3)⋊C4C2318yes—
99C2×(C2×D3)⋊C4C2318yes—
100C2×(C2×D3)⋊C4C2318yes—
101C2×C4⋊Dic3—42no—
102C2×C4⋊Dic3—42no—
103C2×C4⋊Dic3D418yescomplemented
104C2×C4⋊Dic3C2314yes—
105C2×C4⋊Dic3C2314yes—
106C2×C4⋊Dic3C2314yes—
107C2×Dic3⋊C4—28no—
108C2×Dic3⋊C4—28no—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111#884—264no—
112#883—264no—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C23×D4—32no—
116C23×D4—32no—
117C23×D4—31no—
118C23×D4C2×D311yescharacteristic
119C24×C4—32no—
120C2×#123—316no—
121C2×#123—616no—
122C2×#121—316no—
123C2×#121—616no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4C2×D311yescharacteristic
130C22×C22⋊C4—32no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4—32no—
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4C2×D312yes—
136C22×C22⋊C4C2×D312yes—
137C22×C42C2×D311yescharacteristic
138#396—316no—
139#394—316no—
140#392—316no—
141#390—38no—
142#390C2×D318yescomplemented
143#389—38no—
144#385—316no—
145C2×#95—32no—
146C2×#95—32no—
147C24×C3—24no—
148C24×C3—24no—
149C24×C3—24no—
150C24×C3—24no—
151C24×C3—24no—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3C2414yes—
155C24×C3C2411yescharacteristic
156C23×D3—24no—
157C23×D3—24no—
158C23×D3—24no—
159C23×D3—24no—
160C23×D3—24no—
161C23×D3—24no—
162C23×D3C2×D418yes—
163C23×D3C2414yes—
164C2×C3×D4—28no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4—416no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D414yes—
179C22×C3×C4#4114yes—
180C22×C3×C4C2414yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2412yes—
183C22×C3×C4C2412yes—
184C2×#67—216no—
185C2×#67—216no—
186C2×#67—216no—
187C2×#67—48no—
188C2×#67—48no—
189C2×#67—48no—
190C2×#67—432no—
191C2×#67—432no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3C2×D418yes—
201C22×Dic3C2414yes—
202C22×Dic3C2414yes—
203C22×Dic3C2414yes—
204C2×D12—416no—
205C2×D12—44no—
206C2×D12—44no—
207C2×D12C2×D4116yescomplemented
208C2×C4×D3—216no—
209C2×C4×D3—216no—
210C2×C4×D3—48no—
211C2×C4×D3—432no—
212C3×C4⋊C4—44no—
213C3×C22⋊C4—216no—
214C3×C22⋊C4—216no—
215C3×C42—44no—
216#217—216no—
217#217—48no—
218(C2×D3)⋊C4—216no—
219(C2×D3)⋊C4—216no—
220(C2×D3)⋊C4—216no—
221C4⋊Dic3—44no—
222C4⋊Dic3—44no—
223C4⋊Dic3C2×D4116yescomplemented
224Dic3⋊C4—216no—
225Dic3⋊C4—216no—
226C25—32no—
227C25C22×D312yes—
228C22×D4—24no—
229C22×D4—38no—
230C22×D4—38no—
231C22×D4—38no—
232C22×D4—34no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4—68no—
236C22×D4—68no—
237C22×D4—68no—
238C22×D4—122no—
239C22×D4—122no—
240C22×D4—124no—
241C22×D4—124no—
242C22×D4—124no—
243C22×D4C22×D314yes—
244C23×C4—32no—
245C23×C4—32no—
246C23×C4—32no—
247C23×C4—38no—
248C23×C4—68no—
249C23×C4—68no—
250C23×C4—124no—
251C23×C4C22×D312yes—
252C23×C4C22×D311yescharacteristic
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255#123—664no—
256#121—664no—
257C2×C4⋊C4—38no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—38no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—68no—
265C2×C4⋊C4—68no—
266C2×C4⋊C4—122no—
267C2×C4⋊C4—122no—
268C2×C4⋊C4C22×D314yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—38no—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—38no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—68no—
277C2×C22⋊C4—68no—
278C2×C22⋊C4—68no—
279C2×C22⋊C4—68no—
280C2×C22⋊C4—124no—
281C2×C22⋊C4C22×D318yes—
282C2×C22⋊C4C22×D318yes—
283C2×C42—42no—
284C2×C42C22×D314yes—
285#95—38no—
286#95—38no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—48no—
295C23×C3—48no—
296C23×C3—48no—
297C23×C3—48no—
298C23×C3—48no—
299C23×C3—48no—
300C23×C3—48no—
301C23×C3C22×D412yes—
302C23×C3C22×D412yes—
303C23×C3C2×#4112yes—
304C23×C3H(5,𝔽2)12yes—
305C23×C3H(5,𝔽2)12yes—
306C23×C3H(5,𝔽2)12yes—
307C23×C32−1+412yes—
308C23×C3C2511yescharacteristic
309C22×D3—216no—
310C22×D3—216no—
311C22×D3—216no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—48no—
319C22×D3—48no—
320C22×D3—48no—
321C22×D3—48no—
322C22×D3C22×D418yes—
323C3×D4—416no—
324C2×C3×C4—28no—
325C2×C3×C4—28no—
326C2×C3×C4—28no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—48no—
337C2×C3×C4C22×D414yes—
338C2×C3×C4C22×D414yes—
339C2×C3×C4C2×#4114yes—
340#67—432no—
341#67—432no—
342C2×Dic3—216no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—48no—
352C2×Dic3—48no—
353C2×Dic3C22×D418yes—
354D12—416no—
355C4×D3—432no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—38no—
364C24—34no—
365C24—64no—
366C24—64no—
367C24—64no—
368C24—64no—
369C24—64no—
370C24—64no—
371C24C23×D314yes—
372C24C23×D311yescharacteristic
373C2×D4—28no—
374C2×D4—316no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—416no—
378C2×D4—616no—
379C2×D4—616no—
380C2×D4—616no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—1216no—
384C2×D4—128no—
385C2×D4—128no—
386C2×D4—128no—
387C2×D4—1232no—
388C2×D4—1232no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—38no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—34no—
402C22×C4—616no—
403C22×C4—616no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—64no—
412C22×C4—128no—
413C22×C4—1232no—
414C22×C4#23514yes—
415C22×C4D3×D414yes—
416C22×C4D3×D414yes—
417C22×C4C23×D314yes—
418C22×C4C23×D312yes—
419C22×C4C23×D312yes—
420C22×C4C23×D312yes—
421C4⋊C4—316no—
422C4⋊C4—44no—
423C4⋊C4—616no—
424C4⋊C4—616no—
425C4⋊C4—124no—
426C4⋊C4—124no—
427C22⋊C4—216no—
428C22⋊C4—216no—
429C22⋊C4—616no—
430C22⋊C4—616no—
431C22⋊C4—616no—
432C22⋊C4—616no—
433C22⋊C4—128no—
434C42—44no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—816no—
443C22×C3#53414yes—
444C22×C3#53714yes—
445C22×C3#53814yes—
446C22×C3D4214yes—
447C22×C3#54714yes—
448C22×C3#55014yes—
449C22×C3#56214yes—
450C22×C3C23×D411yescharacteristic
451C22×C3C23×D411yescharacteristic
452C22×C3C22×#4111yescharacteristic
453C22×C3C2×H(5,𝔽2)11yescharacteristic
454C22×C3C2×H(5,𝔽2)11yescharacteristic
455C22×C3C2×H(5,𝔽2)11yescharacteristic
456C22×C3C2×2−1+411yescharacteristic
457C2×D3—28no—
458C2×D3—28no—
459C2×D3—28no—
460C2×D3—44no—
461C2×D3—432no—
462C2×D3—44no—
463C2×D3—44no—
464C2×D3—44no—
465C3×C4—24no—
466C3×C4—24no—
467C3×C4—24no—
468C3×C4—44no—
469Dic3—28no—
470Dic3—44no—
471Dic3—44no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—38no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—48no—
487C23—64no—
488C23—64no—
489C23—64no—
490C23—64no—
491C23—64no—
492C23—64no—
493C23—616no—
494C23—616no—
495C23—616no—
496C23—128no—
497C23—128no—
498C23—128no—
499C23—128no—
500C23C2×#23512yes—
501C23C2×D3×D412yes—
502C23C2×D3×D412yes—
503C23#100412yes—
504C23#100412yes—
505C23#100912yes—
506C23#101012yes—
507C23C24×D311yescharacteristic
508D4—416no—
509D4—1216no—
510D4—1232no—
511D4—1232no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—38no—
525C2×C4—48no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—616no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—1232no—
536C2×C4—128no—
537C2×C4—128no—
538C2×C4C2×#23514yes—
539C2×C4C2×D3×D414yes—
540C2×C4C2×D3×D414yes—
541C2×C3—84no—
542C2×C3#268218yescomplemented
543C2×C3C2×#53411yescharacteristic
544C2×C3C2×#53711yescharacteristic
545C2×C3C2×#53811yescharacteristic
546C2×C3C2×D4211yescharacteristic
547C2×C3C2×#54711yescharacteristic
548C2×C3C2×#55011yescharacteristic
549C2×C3C2×#56211yescharacteristic
550D3—48no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—68no—
559C22—68no—
560C22—68no—
561C22—816no—
562C22—124no—
563C22—124no—
564C22—1232no—
565C22—124no—
566C22—124no—
567C22#549414yes—
568C22#555114yes—
569C22#555314yes—
570C22#559714yes—
571C22#560514yes—
572C22#568414yes—
573C22#571714yes—
574C22C22×#23511yescharacteristic
575C22C22×D3×D411yescharacteristic
576C22C22×D3×D411yescharacteristic
577C22C2×#100411yescharacteristic
578C22C2×#100411yescharacteristic
579C22C2×#100911yescharacteristic
580C22C2×#101011yescharacteristic
581C4—24no—
582C4—24no—
583C4—24no—
584C4—44no—
585C4—68no—
586C4—124no—
587C4—124no—
588C3C2×#268211yescharacteristic, complemented, Sylow
589C2—84no—
590C2—128no—
591C2#7646618yesdirect product factor
592C2C2×#549411yescharacteristic
593C2C2×#555311yescharacteristic
594C2C2×#555111yescharacteristic
595C2C2×#559711yescharacteristic
596C2C2×#560511yescharacteristic
597C2C2×#568411yescharacteristic
598C2C2×#571711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#390C2
socleC24×C3C24
hypercenterC2×#390D3
hypocenterC3C2×#2682
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×#883C2semidirect
2. C22×#884C2semidirect
3. C2×C3×#390C2semidirect
4. C2×#5223C2semidirect
5. C2×#4989C2semidirect
6. C2×#4948C2semidirect
7. C2×#4961C2semidirect
8. C2×#4939C2semidirect
9. C2×#4928C2semidirect
10. #76466C2direct (N × H)
32. C2×#884C22semidirect
34. C2×#883C22semidirect
35. C3×#390C22semidirect
36. #5223C22semidirect
39. #4989C22semidirect
40. #4961C22semidirect
41. #4948C22semidirect
42. #4939C22semidirect
43. #4928C22semidirect
51. C2×#390D3semidirect
77. C22×D12D4semidirect
103. C2×C4⋊Dic3D4semidirect
142. #390C2×D3semidirect
207. C2×D12C2×D4semidirect
223. C4⋊Dic3C2×D4semidirect
542. C2×C3#2682semidirect
588. C3C2×#2682semidirect
591. C2#76466direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8832
C22×#8842
C2×C3×#3902
C2×#52232
C2×#49892
C2×#49482
C2×#49612
C2×#49392
C2×#49282
#764662
C2×#26823