Finite Group Databasebeta

← C2×#76449 #10964635

Subgroups — C2×#76449 #10964635

Order 768.

3158 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#938C211yesmaximal, characteristic
2C22×#890C211yesmaximal, characteristic, complemented
3C2×C3×#390C211yesmaximal, characteristic, complemented
4C2×#5221C211yesmaximal, characteristic, complemented
5C2×#4987C211yesmaximal, characteristic, complemented
6C2×#4952C212yesmaximal, complemented
7C2×#4961C212yesmaximal, complemented
8C2×#4957C212yesmaximal, complemented
9C2×#4951C212yesmaximal, complemented
10C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
12#76449C2116yesmaximal, direct product factor
13C2×#2490—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×#67C2212yes—
16C23×C4×D3C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C22×#217C2212yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C2×#938—28no—
25C2×#938C2218yes—
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×(C2×D3)⋊C4C2211yescharacteristic
28C22×(C2×D3)⋊C4C2212yes—
29C22×C4⋊Dic3C2211yescharacteristic
30C22×C4⋊Dic3C2212yes—
31C22×C4⋊Dic3C2211yescharacteristic
32C22×Dic3⋊C4C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C22×C4×Dic3C2212yes—
35C2×#890—28no—
36C2×#890C2218yescomplemented
37C3×#390C2218yescomplemented
38#5221C2218yescomplemented
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2211yescharacteristic
42C2×C3⋊#95C2212yes—
43#4987C2218yescomplemented
44#4961C22116yescomplemented
45#4957C22116yescomplemented
46#4952C22116yescomplemented
47#4951C22116yescomplemented
48C4×(C2×D3)⋊C4C2218yescomplemented
49C4×C4⋊Dic3C2218yescomplemented
50C22×#125—31no—
51C22×#121—31no—
52#2490—316no—
53C2×#397—32no—
54C2×#394—32no—
55C2×#390—31no—
56C2×#390—32no—
57C2×#390D311yescharacteristic, complemented
58C2×#388—32no—
59C2×#380—31no—
60C2×C4×C4⋊C4—31no—
61C2×C4×C22⋊C4—31no—
62C25×C3C2312yes—
63C24×D3C2311yescharacteristic
64C22×C3×D4—24no—
65C22×C3×D4—24no—
66C22×C3×D4—24no—
67C22×C3×D4C2314yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C22×#67—28no—
73C22×#67—28no—
74C22×#67—44no—
75C22×#67C2318yes—
76C23×Dic3C2311yescharacteristic
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2312yes—
81C22×C4×D3—24no—
82C22×C4×D3—24no—
83C22×C4×D3—24no—
84C22×C4×D3C2314yes—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4—44no—
89C2×C3×C22⋊C4C2318yes—
90C2×C3×C22⋊C4C2318yes—
91C2×C3×C42—24no—
92C2×C3×C42C2314yes—
93C2×#217—28no—
94C2×#217—44no—
95C2×#217—44no—
96C2×#217C2318yes—
97C2×#217C2318yes—
98C2×#217C2318yes—
99#938—232no—
100C2×(C2×D3)⋊C4—24no—
101C2×(C2×D3)⋊C4—24no—
102C2×(C2×D3)⋊C4—44no—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2314yes—
105C2×(C2×D3)⋊C4C2318yes—
106C2×C4⋊Dic3—24no—
107C2×C4⋊Dic3—28no—
108C2×C4⋊Dic3D418yes—
109C2×C4⋊Dic3C2314yes—
110C2×C4⋊Dic3C2318yes—
111C2×C4⋊Dic3C2314yes—
112C2×Dic3⋊C4—28no—
113C2×Dic3⋊C4C2318yes—
114C2×C4×Dic3—24no—
115C2×C4×Dic3—44no—
116C2×C4×Dic3C2314yes—
117C2×C4×Dic3C2318yes—
118#890—232no—
119C3⋊#95C2318yes—
120C3⋊#95C2314yes—
121C3⋊#95C2314yes—
122C3⋊#95C2318yes—
123C23×D4—32no—
124C23×D4C2×D311yescharacteristic
125C24×C4—31no—
126C2×#125—38no—
127C2×#125—68no—
128C2×#121—38no—
129C2×#121—68no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C22⋊C4—32no—
136C22×C22⋊C4—31no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4C2×D312yes—
142C22×C22⋊C4C2×D312yes—
143C22×C42—31no—
144C22×C42—32no—
145C22×C42C2×D311yescharacteristic
146#397—316no—
147#394—316no—
148#390—38no—
149#390—316no—
150#390C2×D318yescomplemented
151#388—316no—
152#380—38no—
153C4×C4⋊C4—38no—
154C4×C22⋊C4—38no—
155C2×#95—32no—
156C2×#95—31no—
157C2×#95—32no—
158C2×#95—31no—
159C24×C3—24no—
160C24×C3—24no—
161C24×C3—24no—
162C24×C3—24no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3C2414yes—
167C24×C3C2411yescharacteristic
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—22no—
171C23×D3—22no—
172C23×D3—22no—
173C23×D3—22no—
174C23×D3#4114yes—
175C23×D3C2412yes—
176C2×C3×D4—28no—
177C2×C3×D4—28no—
178C2×C3×D4—28no—
179C2×C3×D4—416no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4C2×D414yes—
189C22×C3×C4#4114yes—
190C22×C3×C4#4114yes—
191C22×C3×C4#4114yes—
192C22×C3×C4C2412yes—
193C22×C3×C4C2414yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2412yes—
196C2×#67—216no—
197C2×#67—216no—
198C2×#67—48no—
199C2×#67—432no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—24no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3C2×D418yes—
215C22×Dic3C2412yes—
216C22×Dic3C2414yes—
217C22×Dic3C2412yes—
218C22×Dic3C2412yes—
219C22×Dic3C2414yes—
220C2×C4×D3—28no—
221C2×C4×D3—28no—
222C2×C4×D3—28no—
223C2×C4×D3—416no—
224C3×C4⋊C4—28no—
225C3×C22⋊C4—216no—
226C3×C22⋊C4—48no—
227C3×C42—28no—
228#217—216no—
229#217—48no—
230#217—48no—
231(C2×D3)⋊C4—28no—
232(C2×D3)⋊C4—28no—
233(C2×D3)⋊C4—48no—
234C4⋊Dic3—28no—
235C4⋊Dic3—216no—
236C4⋊Dic3C2×D4116yes—
237Dic3⋊C4—216no—
238C4×Dic3—28no—
239C4×Dic3—48no—
240C25—31no—
241C25C22×D312yes—
242C22×D4—24no—
243C22×D4—24no—
244C22×D4—24no—
245C22×D4—38no—
246C22×D4—68no—
247C22×D4—68no—
248C22×D4—124no—
249C22×D4C22×D314yes—
250C23×C4—32no—
251C23×C4—31no—
252C23×C4—31no—
253C23×C4—32no—
254C23×C4—31no—
255C23×C4—34no—
256C23×C4—64no—
257C23×C4—64no—
258C23×C4—64no—
259C23×C4C22×D312yes—
260C23×C4C22×D311yescharacteristic
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D311yescharacteristic
263#125—632no—
264#121—632no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—38no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—38no—
269C2×C4⋊C4—38no—
270C2×C4⋊C4—34no—
271C2×C4⋊C4—68no—
272C2×C4⋊C4—68no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4C22×D314yes—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—34no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—38no—
281C2×C22⋊C4—38no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—68no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—124no—
287C2×C22⋊C4—124no—
288C2×C22⋊C4—124no—
289C2×C22⋊C4C22×D318yes—
290C2×C22⋊C4C22×D318yes—
291C2×C42—24no—
292C2×C42—34no—
293C2×C42—38no—
294C2×C42—64no—
295C2×C42—124no—
296C2×C42C22×D314yes—
297#95—34no—
298#95—34no—
299#95—38no—
300#95—38no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—48no—
309C23×C3—48no—
310C23×C3—48no—
311C23×C3—48no—
312C23×C3—48no—
313C23×C3—48no—
314C23×C3—48no—
315C23×C3C22×D412yes—
316C23×C3C2×#4112yes—
317C23×C3C2×#4112yes—
318C23×C3C2×#4112yes—
319C23×C3C2×#4112yes—
320C23×C3H(5,𝔽2)12yes—
321C23×C3H(5,𝔽2)12yes—
322C23×C3C2511yescharacteristic
323C22×D3—28no—
324C22×D3—28no—
325C22×D3—28no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—44no—
333C22×D3—44no—
334C22×D3—44no—
335C22×D3—44no—
336C22×D3C2×#4114yes—
337C3×D4—416no—
338C2×C3×C4—28no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—48no—
348C2×C3×C4—44no—
349C2×C3×C4#124116yes—
350C2×C3×C4C22×D414yes—
351C2×C3×C4C2×#4114yes—
352C2×C3×C4C2×#4114yes—
353C2×C3×C4C2×#4114yes—
354#67—432no—
355C2×Dic3—216no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—24no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—44no—
371C2×Dic3—48no—
372C2×Dic3—44no—
373C2×Dic3—44no—
374C2×Dic3C22×D418yes—
375C4×D3—416no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—34no—
384C24—32no—
385C24—62no—
386C24—62no—
387C24—62no—
388C24—62no—
389C24—62no—
390C24—62no—
391C24C23×D314yes—
392C24C23×D311yescharacteristic
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—416no—
397C2×D4—616no—
398C2×D4—616no—
399C2×D4—128no—
400C2×D4—1232no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—38no—
410C22×C4—34no—
411C22×C4—32no—
412C22×C4—34no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—64no—
416C22×C4—64no—
417C22×C4—64no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—64no—
428C22×C4—62no—
429C22×C4—68no—
430C22×C4—68no—
431C22×C4—68no—
432C22×C4—1216no—
433C22×C4#23514yes—
434C22×C4D3×D414yes—
435C22×C4#23714yes—
436C22×C4#23714yes—
437C22×C4C23×D312yes—
438C22×C4C23×D314yes—
439C22×C4C23×D312yes—
440C22×C4C23×D312yes—
441C4⋊C4—28no—
442C4⋊C4—316no—
443C4⋊C4—616no—
444C4⋊C4—616no—
445C4⋊C4—68no—
446C22⋊C4—216no—
447C22⋊C4—48no—
448C22⋊C4—68no—
449C22⋊C4—616no—
450C22⋊C4—68no—
451C22⋊C4—128no—
452C22⋊C4—128no—
453C22⋊C4—128no—
454C42—28no—
455C42—68no—
456C42—128no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—816no—
465C22×C3C2×#12414yes—
466C22×C3#52914yes—
467C22×C3#54014yes—
468C22×C3#54614yes—
469C22×C3#55014yes—
470C22×C3#55314yes—
471C22×C3#55314yes—
472C22×C3C23×D411yescharacteristic
473C22×C3C22×#4111yescharacteristic
474C22×C3C22×#4111yescharacteristic
475C22×C3C22×#4111yescharacteristic
476C22×C3C22×#4111yescharacteristic
477C22×C3C2×H(5,𝔽2)11yescharacteristic
478C22×C3C2×H(5,𝔽2)11yescharacteristic
479C2×D3—24no—
480C2×D3—24no—
481C2×D3—24no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—416no—
487C3×C4—24no—
488C3×C4—44no—
489C3×C4—42no—
490C3×C4C2×#12418yes—
491Dic3—28no—
492Dic3—42no—
493Dic3—44no—
494Dic3—42no—
495Dic3—42no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—34no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—68no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—62no—
517C23—62no—
518C23—68no—
519C23—68no—
520C23—124no—
521C23—124no—
522C23—124no—
523C23—124no—
524C23C2×#23512yes—
525C23C2×D3×D412yes—
526C23C2×#23712yes—
527C23C2×#23712yes—
528C23#100412yes—
529C23D3×#4112yes—
530C23#100912yes—
531C23C24×D311yescharacteristic
532D4—416no—
533D4—1232no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—38no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—616no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—64no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—64no—
560C2×C4—62no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4—128no—
565C2×C4—1216no—
566C2×C4#885116yes—
567C2×C4C2×#23514yes—
568C2×C4C2×D3×D414yes—
569C2×C4C2×#23714yes—
570C2×C4C2×#23714yes—
571C2×C3—84no—
572C2×C3#249018yescomplemented
573C2×C3C22×#12411yescharacteristic
574C2×C3C2×#52911yescharacteristic
575C2×C3C2×#54011yescharacteristic
576C2×C3C2×#54611yescharacteristic
577C2×C3C2×#55011yescharacteristic
578C2×C3C2×#55311yescharacteristic
579C2×C3C2×#55311yescharacteristic
580D3—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—64no—
589C22—64no—
590C22—64no—
591C22—816no—
592C22—122no—
593C22—1216no—
594C22—122no—
595C22—122no—
596C22—122no—
597C22C2×#88514yes—
598C22#555714yes—
599C22#556214yes—
600C22#561414yes—
601C22#561814yes—
602C22#567614yes—
603C22#571814yes—
604C22C22×#23511yescharacteristic
605C22C22×D3×D411yescharacteristic
606C22C22×#23711yescharacteristic
607C22C22×#23711yescharacteristic
608C22C2×#100411yescharacteristic
609C22C2×D3×#4111yescharacteristic
610C22C2×#100911yescharacteristic
611C4—24no—
612C4—44no—
613C4—42no—
614C4—68no—
615C4—124no—
616C4—122no—
617C4—122no—
618C4—122no—
619C4C2×#88518yes—
620C3C2×#249011yescharacteristic, complemented, Sylow
621C2—84no—
622C2—124no—
623C2#7644918yesdirect product factor
624C2C22×#88511yescharacteristic
625C2C2×#555711yescharacteristic
626C2C2×#556211yescharacteristic
627C2C2×#561811yescharacteristic
628C2C2×#561411yescharacteristic
629C2C2×#567611yescharacteristic
630C2C2×#571811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#390C2
socleC24×C3C24
prime radicalC22×#938C2
hypercenterC2×#390D3
hypocenterC3C2×#2490
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×#890C2semidirect
3. C2×C3×#390C2semidirect
4. C2×#5221C2semidirect
5. C2×#4987C2semidirect
6. C2×#4952C2semidirect
7. C2×#4961C2semidirect
8. C2×#4957C2semidirect
9. C2×#4951C2semidirect
10. C2×C4×C4⋊Dic3C2semidirect
11. C2×C4×(C2×D3)⋊C4C2semidirect
12. #76449C2direct (N × H)
36. C2×#890C22semidirect
37. C3×#390C22semidirect
38. #5221C22semidirect
43. #4987C22semidirect
44. #4961C22semidirect
45. #4957C22semidirect
46. #4952C22semidirect
47. #4951C22semidirect
48. C4×(C2×D3)⋊C4C22semidirect
49. C4×C4⋊Dic3C22semidirect
57. C2×#390D3semidirect
150. #390C2×D3semidirect
572. C2×C3#2490semidirect
620. C3C2×#2490semidirect
623. C2#76449direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9382
C22×#8902
C2×C3×#3902
C2×#52212
C2×#49872
C2×#49522
C2×#49612
C2×#49572
C2×#49512
C2×C4×C4⋊Dic32
C2×C4×(C2×D3)⋊C42
#764492
C2×#24903