Finite Group Databasebeta

← C2×#70176 #10822650

Subgroups — C2×#70176 #10822650

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#938C211yesmaximal, characteristic
2C22×C3×#102C211yesmaximal, characteristic, complemented
3C22×#832C211yesmaximal, characteristic, complemented
4C22×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
5C22×#817C211yesmaximal, characteristic, complemented
6C2×#4987C211yesmaximal, characteristic, complemented
7C2×#4529C211yesmaximal, characteristic, complemented
8#70176C218yesmaximal, direct product factor
9C2×#2034—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C23×#67—21no—
12C23×C4×D3C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×#217—21no—
16C2×#938—24no—
17C2×#938—22no—
18C2×#938—24no—
19C2×#938C2214yes—
20C22×(C2×D3)⋊C4—21no—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×C3×#102—24no—
25C2×C3×#102C2214yescomplemented
26C2×#832—24no—
27C2×#832C2214yescomplemented
28C2×C3⋊#95—21no—
29C2×(C2×D3)⋊C8—24no—
30C2×(C2×D3)⋊C8C2214yescomplemented
31C2×#817—24no—
32C2×#817C2214yescomplemented
33#4987—22no—
34#4987C2214yescomplemented
35#4529C2214yescomplemented
36C22×#121—31no—
37C22×#106—31no—
38C22×#102—31no—
39C22×#102D311yescharacteristic, complemented
40C22×C22⋊C8—31no—
41C2×#380—31no—
42#2034—38no—
43C2×#340—31no—
44C25×C3—21no—
45C24×D3D411yescharacteristic
46C22×C3×D4—22no—
47C22×C3×D4—21no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4D414yes—
53C22×C3×D4C2312yes—
54C23×C3×C4—21no—
55C23×C3×C4C2311yescharacteristic
56C22×#67—28no—
57C22×#67—24no—
58C22×#67—44no—
59C22×#67—44no—
60C23×Dic3—21no—
61C23×Dic3—21no—
62C23×Dic3D411yescharacteristic
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3D412yes—
70C22×C4×D3C2312yes—
71C22×C3×C8—22no—
72C22×C3×C8—22no—
73C22×C3×C8D414yescomplemented
74C22×C3×C8C2312yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2312yes—
79C2×#217—28no—
80C2×#217—24no—
81#938—216no—
82#938—28no—
83#938—28no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—44no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D414yes—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94C22×C3⋊C8—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8C2312yes—
98C3×#102—28no—
99#832—28no—
100C3⋊#95—24no—
101(C2×D3)⋊C8—28no—
102#817—28no—
103C23×D4—61no—
104C23×D4C2×D311yescharacteristic
105C24×C4—31no—
106C23×C8—31no—
107C23×C8C2×D311yescharacteristic
108C2×#121—34no—
109C2×#121—64no—
110C2×#121—62no—
111C2×#121—64no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C22⋊C4—61no—
116C22×C22⋊C4—61no—
117C2×#106—34no—
118C2×#106—64no—
119C2×#102—24no—
120C2×#102—34no—
121C2×#102—64no—
122C2×#102C2×D314yescomplemented
123C2×C22⋊C8—34no—
124C2×C22⋊C8—64no—
125#380—34no—
126#380—62no—
127C2×#95—61no—
128#340—34no—
129C24×C3—24no—
130C24×C3—22no—
131C24×C3—42no—
132C24×C3—42no—
133C24×C3—42no—
134C24×C3—42no—
135C24×C3—42no—
136C24×C3—42no—
137C24×C3C2×D411yescharacteristic
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3SD1614yes—
145C23×D3C2×D412yes—
146C2×C3×D4—24no—
147C2×C3×D4—28no—
148C2×C3×D4—28no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C2×C3×D4—44no—
156C2×C3×D4—42no—
157C2×C3×D4—44no—
158C2×C3×D4C2×D414yes—
159C22×C3×C4—22no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4112yes—
170C22×C3×C4C2411yescharacteristic
171C2×#67—216no—
172C2×#67—48no—
173C2×#67—432no—
174C2×#67—48no—
175C22×Dic3—24no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—42no—
182C22×Dic3—42no—
183C22×Dic3—42no—
184C22×Dic3—42no—
185C22×Dic3—42no—
186C22×Dic3C2×D412yes—
187C2×C4×D3—24no—
188C2×C4×D3—24no—
189C2×C4×D3—24no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—44no—
197C2×C4×D3—44no—
198C2×C4×D3—44no—
199C2×C4×D3—44no—
200C2×C3×C8—28no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8C2×D414yescomplemented
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207#217—216no—
208(C2×D3)⋊C4—48no—
209C4⋊Dic3—24no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—42no—
214C4⋊Dic3C2×D414yes—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—44no—
219C25—21no—
220C25—31no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—64no—
228C22×D4—68no—
229C22×D4—124no—
230C22×D4—124no—
231C22×D4D1214yes—
232C22×D4C22×D312yes—
233C23×C4—21no—
234C23×C4—31no—
235C23×C4—32no—
236C23×C4—32no—
237C23×C4—61no—
238C23×C4—62no—
239C23×C4—62no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4—62no—
243C23×C4—62no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—32no—
249C22×C8—62no—
250C22×C8—62no—
251C22×C8—62no—
252C22×C8#6714yescomplemented
253C22×C8C22×D312yes—
254#121—68no—
255#121—616no—
256#121—68no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—34no—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4C22×D312yes—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—68no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—124no—
273#106—68no—
274#102—28no—
275#102—68no—
276C22⋊C8—68no—
277#95—64no—
278C23×C3—24no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—48no—
286C23×C3—48no—
287C23×C3—48no—
288C23×C3—84no—
289C23×C3—84no—
290C23×C3—84no—
291C23×C3—84no—
292C23×C3#12012yes—
293C23×C3#12112yes—
294C23×C3#12312yes—
295C23×C3C2×SD1612yes—
296C23×C3#13512yes—
297C23×C3AGL(1,ℤ/8)12yes—
298C23×C3AGL(1,ℤ/8)12yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—28no—
307C22×D3—28no—
308C22×D3—28no—
309C22×D3—44no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3—44no—
313C22×D3C2×SD1614yes—
314C3×D4—24no—
315C3×D4—24no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—48no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—84no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12114yes—
325C2×C3×C4#12112yes—
326C2×C3×C4#12114yes—
327C2×C3×C4#12312yes—
328C2×C3×C4#12314yes—
329C2×C3×C4#12414yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336#67—432no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—44no—
344C2×Dic3—48no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—84no—
349C4×D3—42no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—48no—
354C3×C8—24no—
355C3⋊C8—42no—
356C24—24no—
357C24—22no—
358C24—34no—
359C24—32no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—62no—
367C24—62no—
368C24—62no—
369C24—62no—
370C24—62no—
371C24—62no—
372C24D3×D411yescharacteristic
373C2×D4—24no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—616no—
386C2×D4—128no—
387C2×D4—128no—
388C2×D4—1232no—
389C2×D4C2×D1214yes—
390C22×C4—22no—
391C22×C4—32no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—62no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4—124no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4C2×D1212yes—
420C22×C4#23512yes—
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D412yes—
423C22×C4#23712yes—
424C22×C4#23712yes—
425C22×C4C2×#6712yes—
426C22×C4C23×D311yescharacteristic
427C2×C8—28no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—62no—
431C2×C8—62no—
432C2×C8—62no—
433C2×C8—124no—
434C2×C8C2×#6714yescomplemented
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—34no—
438C4⋊C4—42no—
439C4⋊C4—64no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—122no—
444C22⋊C4—616no—
445C22⋊C4—128no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—816no—
454C22×C3#39414yes—
455C22×C3#44914yes—
456C22×C3#45014yes—
457C22×C3#46514yes—
458C22×C3#46914yes—
459C22×C3#48114yes—
460C22×C3#48214yes—
461C22×C3C2×#12011yescharacteristic
462C22×C3C2×#12111yescharacteristic
463C22×C3C2×#12311yescharacteristic
464C22×C3C22×SD1611yescharacteristic
465C22×C3C2×#13511yescharacteristic
466C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
467C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
468C2×D3—24no—
469C2×D3—24no—
470C2×D3—24no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—416no—
476C3×C4—82no—
477C3×C4#39414yes—
478C3×C4C2×#12111yescharacteristic
479C3×C4C2×#12111yescharacteristic
480C3×C4C2×#12311yescharacteristic
481C3×C4C2×#12411yescharacteristic
482Dic3—42no—
483Dic3—44no—
484Dic3—82no—
485C23—24no—
486C23—34no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—48no—
494C23—42no—
495C23—48no—
496C23—68no—
497C23—62no—
498C23—62no—
499C23—62no—
500C23—62no—
501C23—62no—
502C23—68no—
503C23—62no—
504C23—68no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—124no—
510C23—124no—
511C23—124no—
512C23—124no—
513C23#88212yes—
514C23#88312yes—
515C23#91112yes—
516C23#91112yes—
517C23D3×SD1612yes—
518C23#91612yes—
519C23#93812yes—
520C23C2×D3×D411yescharacteristic
521D4—24no—
522D4—24no—
523D4—48no—
524D4—42no—
525D4—42no—
526D4—1232no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—62no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—84no—
535C2×C4—124no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—128no—
539C2×C4—128no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—244no—
548C2×C4#88212yes—
549C2×C4#88312yes—
550C2×C4#88514yes—
551C2×C4#88614yes—
552C2×C4#93014yes—
553C2×C4#93812yes—
554C2×C4#93914yes—
555C2×C4C22×D1211yescharacteristic
556C2×C4C2×#23511yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4C2×#23711yescharacteristic
560C2×C4C22×#6711yescharacteristic
561C8—24no—
562C8—122no—
563C2×C3—84no—
564C2×C3#203418yescomplemented
565C2×C3C2×#39411yescharacteristic
566C2×C3C2×#45011yescharacteristic
567C2×C3C2×#44911yescharacteristic
568C2×C3C2×#46511yescharacteristic
569C2×C3C2×#46911yescharacteristic
570C2×C3C2×#48211yescharacteristic
571C2×C3C2×#48111yescharacteristic
572D3—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—64no—
577C22—64no—
578C22—64no—
579C22—816no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—1216no—
585C22—122no—
586C22—122no—
587C22—122no—
588C22—122no—
589C22#477614yes—
590C22#477714yes—
591C22#477914yes—
592C22#478114yes—
593C22#496114yes—
594C22#515914yes—
595C22#517114yes—
596C22C2×#88211yescharacteristic
597C22C2×#88311yescharacteristic
598C22C2×#91111yescharacteristic
599C22C2×#91111yescharacteristic
600C22C2×D3×SD1611yescharacteristic
601C22C2×#91611yescharacteristic
602C22C2×#93811yescharacteristic
603C4—82no—
604C4—124no—
605C4—122no—
606C4—242no—
607C4#496114yes—
608C4C2×#88511yescharacteristic
609C4C2×#88611yescharacteristic
610C4C2×#93011yescharacteristic
611C4C2×#93911yescharacteristic
612C3C2×#203411yescharacteristic, complemented, Sylow
613C2—84no—
614C2—124no—
615C2#7017618yesdirect product factor
616C2C2×#477911yescharacteristic
617C2C2×#477611yescharacteristic
618C2C2×#478111yescharacteristic
619C2C2×#477711yescharacteristic
620C2C2×#496111yescharacteristic
621C2C2×#515911yescharacteristic
622C2C2×#517111yescharacteristic
subgroupG/H
centerC24D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#102C2
socleC24×C3C2×D4
prime radicalC22×#938C2
hypercenterC22×#102D3
hypocenterC3C2×#2034
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C3×#102C2semidirect
3. C22×#832C2semidirect
4. C22×(C2×D3)⋊C8C2semidirect
5. C22×#817C2semidirect
6. C2×#4987C2semidirect
7. C2×#4529C2semidirect
8. #70176C2direct (N × H)
25. C2×C3×#102C22semidirect
27. C2×#832C22semidirect
30. C2×(C2×D3)⋊C8C22semidirect
32. C2×#817C22semidirect
34. #4987C22semidirect
35. #4529C22semidirect
39. C22×#102D3semidirect
73. C22×C3×C8D4semidirect
122. C2×#102C2×D3semidirect
203. C2×C3×C8C2×D4semidirect
252. C22×C8#67semidirect
434. C2×C8C2×#67semidirect
564. C2×C3#2034semidirect
612. C3C2×#2034semidirect
615. C2#70176direct (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×C3×#1022
C22×#8322
C22×(C2×D3)⋊C82
C22×#8172
C2×#49872
C2×#45292
#701762
C2×#20343