Finite Group Databasebeta

← C2×#77676 #10964949

Subgroups — C2×#77676 #10964949

Order 768.

4166 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#892C211yesmaximal, characteristic, complemented
2C22×D3×C22⋊C4C211yesmaximal, characteristic
3C2×C3×#397C211yesmaximal, characteristic, complemented
4C2×#4992C211yesmaximal, characteristic, complemented
5C2×#4955C212yesmaximal
6C2×#4949C211yesmaximal, characteristic, complemented
7C2×#4951C212yesmaximal, complemented
8#77676C2116yesmaximal, direct product factor
9C2×#4677C211yesmaximal, characteristic, complemented
10C2×#4668C212yesmaximal, complemented
11C2×#4664C211yesmaximal, characteristic, complemented
12C2×#4669C212yesmaximal, complemented
13C2×#2482—31nomaximal, Sylow
14C25×D3C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic, complemented
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C22×#217C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×C4⋊Dic3C2211yescharacteristic, complemented
29C22×C4⋊Dic3C2212yescomplemented
30C22×Dic3⋊C4C2212yescomplemented
31C22×C4×Dic3C2212yescomplemented
32C2×#892—28no—
33C2×#892C2218yescomplemented
34C2×D3×C22⋊C4—28no—
35C2×D3×C22⋊C4C2218yes—
36C3×#397C2218yescomplemented
37C2×C3×#95C2212yescomplemented
38C2×C3×#95C2211yescharacteristic, complemented
39C2×C3⋊#95C2211yescharacteristic, complemented
40C2×C3⋊#95C2212yescomplemented
41C2×C3⋊#95C2212yescomplemented
42#4992C2218yescomplemented
43#4955C22116yes—
44#4951C22116yescomplemented
45#4949C2218yescomplemented
46#4677C2218yescomplemented
47#4669C22116yescomplemented
48#4668C22116yescomplemented
49#4664C2218yescomplemented
50C22×#117—31no—
51C23×C22⋊C4—31no—
52#2482—316no—
53C2×#399—31no—
54C2×#399—31no—
55C2×#397—31no—
56C2×#397D311yescharacteristic, complemented
57C2×#388—32no—
58C2×#388—32no—
59C2×#385—32no—
60C2×#381—31no—
61C2×#379—32no—
62C25×C3C2311yescharacteristic
63C24×D3—24no—
64C24×D3—24no—
65C24×D3—24no—
66C24×D3—28no—
67C24×D3—44no—
68C24×D3C2311yescharacteristic
69C24×D3C2314yes—
70C24×D3C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2312yes—
73C23×C3×C4C2312yes—
74C23×C3×C4C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2312yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C22×C4×D3—28no—
80C22×C4×D3—24no—
81C22×C4×D3—44no—
82C22×C4×D3D418yes—
83C22×C4×D3C2314yes—
84C22×C4×D3C2314yes—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4C2314yescomplemented
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4C2314yes—
90C2×C3×C22⋊C4C2318yes—
91C2×#217—24no—
92C2×#217—44no—
93C2×#217C2314yes—
94C2×#217C2318yes—
95C2×(C2×D3)⋊C4—28no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—28no—
98C2×(C2×D3)⋊C4—24no—
99C2×(C2×D3)⋊C4—24no—
100C2×(C2×D3)⋊C4—44no—
101C2×(C2×D3)⋊C4C2318yes—
102C2×(C2×D3)⋊C4C2318yes—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×(C2×D3)⋊C4C2314yes—
106C2×(C2×D3)⋊C4C2318yes—
107C2×C4⋊Dic3—24no—
108C2×C4⋊Dic3—44no—
109C2×C4⋊Dic3C2314yescomplemented
110C2×C4⋊Dic3C2318yescomplemented
111C2×Dic3⋊C4—44no—
112C2×Dic3⋊C4C2318yescomplemented
113C2×C4×Dic3—28no—
114C2×C4×Dic3C2318yescomplemented
115#892—232no—
116D3×C22⋊C4—232no—
117C3×#95C2314yescomplemented
118C3×#95C2318yescomplemented
119C3⋊#95C2318yescomplemented
120C3⋊#95C2314yescomplemented
121C3⋊#95C2318yescomplemented
122C26—31no—
123C24×C4—31no—
124C24×C4—31no—
125C2×#117—38no—
126C2×#117—68no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4C2×D311yescharacteristic, complemented
131C22×C22⋊C4—31no—
132C22×C22⋊C4—31no—
133C22×C22⋊C4—32no—
134C22×C22⋊C4—31no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—38no—
140C22×C22⋊C4—68no—
141C22×C22⋊C4C2×D311yescharacteristic
142C22×C22⋊C4C2×D312yes—
143C22×C42—32no—
144#399—38no—
145#399—38no—
146#397—38no—
147#397C2×D318yescomplemented
148#388—316no—
149#388—316no—
150#385—316no—
151#381—38no—
152#379—316no—
153C2×#95—31no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95C2×D311yescharacteristic, complemented
157C2×#95C2×D312yescomplemented
158C24×C3—22no—
159C24×C3—22no—
160C24×C3—24no—
161C24×C3—24no—
162C24×C3#4114yes—
163C24×C3C2412yes—
164C24×C3C2411yescharacteristic
165C23×D3—22no—
166C23×D3—22no—
167C23×D3—24no—
168C23×D3—232no—
169C23×D3—22no—
170C23×D3—28no—
171C23×D3—22no—
172C23×D3—24no—
173C23×D3—24no—
174C23×D3—28no—
175C23×D3—28no—
176C23×D3—216no—
177C23×D3—48no—
178C23×D3—432no—
179C23×D3—432no—
180C23×D3—416no—
181C23×D3—416no—
182C23×D3C2×D414yes—
183C23×D3#4114yes—
184C23×D3#4118yes—
185C23×D3C2412yes—
186C23×D3C2412yes—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4C2×D414yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2414yes—
199C22×C3×C4C2412yes—
200C22×C3×C4C2412yes—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—22no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3C2×D414yes—
211C22×Dic3C2414yes—
212C22×Dic3C2412yes—
213C22×Dic3C2414yes—
214C22×Dic3C2412yes—
215C2×C4×D3—232no—
216C2×C4×D3—216no—
217C2×C4×D3—28no—
218C2×C4×D3—48no—
219C2×C4×D3—416no—
220C2×C4×D3C2×D4116yes—
221C3×C4⋊C4—28no—
222C3×C22⋊C4—216no—
223C3×C22⋊C4—28no—
224#217—28no—
225#217—48no—
226(C2×D3)⋊C4—216no—
227(C2×D3)⋊C4—216no—
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—28no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—48no—
232C4⋊Dic3—28no—
233C4⋊Dic3—48no—
234Dic3⋊C4—48no—
235C4×Dic3—216no—
236C25—31no—
237C25—31no—
238C25—34no—
239C25—64no—
240C25—64no—
241C25—68no—
242C25—64no—
243C25—124no—
244C25C22×D311yescharacteristic
245C23×C4—31no—
246C23×C4—38no—
247C23×C4—34no—
248C23×C4—32no—
249C23×C4—32no—
250C23×C4—31no—
251C23×C4—34no—
252C23×C4—68no—
253C23×C4—64no—
254C23×C4—124no—
255C23×C4C22×D312yes—
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259#117—632no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—38no—
263C2×C4⋊C4—38no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—124no—
266C2×C4⋊C4—124no—
267C2×C4⋊C4C22×D314yescomplemented
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—34no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—34no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—38no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—632no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—68no—
281C2×C22⋊C4—68no—
282C2×C22⋊C4—68no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—124no—
286C2×C22⋊C4—124no—
287C2×C22⋊C4C22×D314yes—
288C2×C22⋊C4C22×D318yes—
289C2×C42—38no—
290C2×C42—68no—
291#95—38no—
292#95—34no—
293#95—38no—
294#95C22×D314yescomplemented
295#95C22×D318yescomplemented
296C23×C3—28no—
297C23×C3—216no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—44no—
303C23×C3—44no—
304C23×C3—48no—
305C23×C3C22×D412yes—
306C23×C3C2×#4114yes—
307C23×C3C2×#4112yes—
308C23×C3C2×#4114yes—
309C23×C3C2×#4112yes—
310C23×C3H(5,𝔽2)14yes—
311C23×C3C2511yescharacteristic
312C22×D3—28no—
313C22×D3—28no—
314C22×D3—216no—
315C22×D3—216no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—24no—
319C22×D3—232no—
320C22×D3—28no—
321C22×D3—216no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—432no—
327C22×D3—432no—
328C22×D3—416no—
329C22×D3—4128no—
330C22×D3—44no—
331C22×D3—44no—
332C22×D3—48no—
333C22×D3—416no—
334C22×D3#123116yes—
335C22×D3C22×D414yes—
336C22×D3C2×#4114yes—
337C22×D3C2×#4118yes—
338C2×C3×C4—28no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—48no—
349C2×C3×C4—44no—
350C2×C3×C4—48no—
351C2×C3×C4C22×D414yes—
352C2×Dic3—28no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—44no—
363C2×Dic3—48no—
364C2×Dic3—48no—
365C2×Dic3C22×D414yes—
366C4×D3—232no—
367C4×D3—416no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—24no—
372C24—34no—
373C24—38no—
374C24—32no—
375C24—34no—
376C24—32no—
377C24—68no—
378C24—68no—
379C24—616no—
380C24—632no—
381C24—62no—
382C24—62no—
383C24—64no—
384C24—62no—
385C24—68no—
386C24—62no—
387C24—64no—
388C24—64no—
389C24—128no—
390C24—1232no—
391C24—1216no—
392C24—1232no—
393C24—1216no—
394C24#23714yes—
395C24C23×D312yes—
396C24C23×D311yescharacteristic
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—34no—
407C22×C4—32no—
408C22×C4—316no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—34no—
412C22×C4—632no—
413C22×C4—616no—
414C22×C4—68no—
415C22×C4—64no—
416C22×C4—64no—
417C22×C4—62no—
418C22×C4—64no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—128no—
425C22×C4—1216no—
426C22×C4D3×D414yes—
427C22×C4C23×D314yes—
428C22×C4C23×D314yes—
429C22×C4C23×D312yes—
430C22×C4C23×D312yes—
431C4⋊C4—28no—
432C4⋊C4—68no—
433C4⋊C4—128no—
434C4⋊C4—128no—
435C22⋊C4—216no—
436C22⋊C4—28no—
437C22⋊C4—68no—
438C22⋊C4—616no—
439C22⋊C4—616no—
440C22⋊C4—616no—
441C22⋊C4—68no—
442C22⋊C4—68no—
443C22⋊C4—128no—
444C22⋊C4—128no—
445C42—616no—
446C22×C3—24no—
447C22×C3—28no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—44no—
451C22×C3—416no—
452C22×C3C2×#12314yes—
453C22×C3#52514yes—
454C22×C3#53514yes—
455C22×C3#55118yes—
456C22×C3#55118yes—
457C22×C3C23×D411yescharacteristic
458C22×C3C22×#4111yescharacteristic
459C22×C3C22×#4112yes—
460C22×C3C22×#4111yescharacteristic
461C22×C3C2×H(5,𝔽2)12yes—
462C2×D3—24no—
463C2×D3—24no—
464C2×D3—28no—
465C2×D3—28no—
466C2×D3—232no—
467C2×D3—24no—
468C2×D3—28no—
469C2×D3—464no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—44no—
473C2×D3—416no—
474C2×D3C2×#12318yes—
475C3×C4—24no—
476C3×C4—44no—
477C3×C4—42no—
478C3×C4—44no—
479Dic3—24no—
480Dic3—42no—
481Dic3—44no—
482Dic3—44no—
483C23—216no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—316no—
490C23—34no—
491C23—38no—
492C23—34no—
493C23—44no—
494C23—44no—
495C23—48no—
496C23—632no—
497C23—616no—
498C23—68no—
499C23—62no—
500C23—616no—
501C23—62no—
502C23—64no—
503C23—68no—
504C23—62no—
505C23—62no—
506C23—616no—
507C23—64no—
508C23—64no—
509C23—68no—
510C23—1232no—
511C23—1216no—
512C23—1232no—
513C23—1216no—
514C23—12128no—
515C23—124no—
516C23—128no—
517C23—124no—
518C23C2×D3×D412yes—
519C23C2×#23714yes—
520C23C2×#23712yes—
521C23D3×#4114yes—
522C23D3×#4112yes—
523C23#100914yes—
524C23C24×D311yescharacteristic
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—34no—
536C2×C4—48no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—632no—
540C2×C4—68no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—1216no—
551C2×C4—124no—
552C2×C4—128no—
553C2×C4—128no—
554C2×C4C2×D3×D414yes—
555C2×C3—44no—
556C2×C3#248218yescomplemented
557C2×C3C22×#12311yescharacteristic
558C2×C3C2×#52511yescharacteristic
559C2×C3C2×#53511yescharacteristic
560C2×C3C2×#55112yes—
561C2×C3C2×#55112yes—
562D3—28no—
563D3—44no—
564C22—24no—
565C22—28no—
566C22—38no—
567C22—42no—
568C22—44no—
569C22—416no—
570C22—42no—
571C22—64no—
572C22—68no—
573C22—632no—
574C22—64no—
575C22—68no—
576C22—68no—
577C22—64no—
578C22—1264no—
579C22—122no—
580C22—1216no—
581C22—122no—
582C22—124no—
583C22#560614yes—
584C22D3×#12314yes—
585C22#565818yes—
586C22#567414yes—
587C22#570518yes—
588C22C22×D3×D411yescharacteristic
589C22C22×#23711yescharacteristic
590C22C2×D3×#4111yescharacteristic
591C22C2×D3×#4112yes—
592C22C2×#100912yes—
593C4—24no—
594C4—42no—
595C4—44no—
596C4—44no—
597C4—64no—
598C4—122no—
599C4—124no—
600C4—124no—
601C3C2×#248211yescharacteristic, complemented, Sylow
602C2—44no—
603C2—68no—
604C2—124no—
605C2#7767618yesdirect product factor
606C2C2×#560611yescharacteristic
607C2C2×D3×#12311yescharacteristic
608C2C2×#565812yes—
609C2C2×#567411yescharacteristic
610C2C2×#570512yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#397C2
socleC24×C3C24
prime radicalC25×D3C22
hypercenterC2×#397D3
hypocenterC3C2×#2482
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×#892C2semidirect
3. C2×C3×#397C2semidirect
4. C2×#4992C2semidirect
6. C2×#4949C2semidirect
7. C2×#4951C2semidirect
8. #77676C2direct (N × H)
9. C2×#4677C2semidirect
10. C2×#4668C2semidirect
11. C2×#4664C2semidirect
12. C2×#4669C2semidirect
17. C22×C3×C4⋊C4C22semidirect
28. C22×C4⋊Dic3C22semidirect
29. C22×C4⋊Dic3C22semidirect
30. C22×Dic3⋊C4C22semidirect
31. C22×C4×Dic3C22semidirect
33. C2×#892C22semidirect
36. C3×#397C22semidirect
37. C2×C3×#95C22semidirect
38. C2×C3×#95C22semidirect
39. C2×C3⋊#95C22semidirect
40. C2×C3⋊#95C22semidirect
41. C2×C3⋊#95C22semidirect
42. #4992C22semidirect
44. #4951C22semidirect
45. #4949C22semidirect
46. #4677C22semidirect
47. #4669C22semidirect
48. #4668C22semidirect
49. #4664C22semidirect
56. C2×#397D3semidirect
86. C2×C3×C4⋊C4C23semidirect
109. C2×C4⋊Dic3C23semidirect
110. C2×C4⋊Dic3C23semidirect
112. C2×Dic3⋊C4C23semidirect
114. C2×C4×Dic3C23semidirect
117. C3×#95C23semidirect
118. C3×#95C23semidirect
119. C3⋊#95C23semidirect
120. C3⋊#95C23semidirect
121. C3⋊#95C23semidirect
130. C22×C4⋊C4C2×D3semidirect
147. #397C2×D3semidirect
156. C2×#95C2×D3semidirect
157. C2×#95C2×D3semidirect
267. C2×C4⋊C4C22×D3semidirect
294. #95C22×D3semidirect
295. #95C22×D3semidirect
556. C2×C3#2482semidirect
601. C3C2×#2482semidirect
605. C2#77676direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8922
C22×D3×C22⋊C42
C2×C3×#3972
C2×#49922
C2×#49552
C2×#49492
C2×#49512
#776762
C2×#46772
C2×#46682
C2×#46642
C2×#46692
C2×#24823