Finite Group Databasebeta

← C2×#80331 #11662000

Subgroups — C2×#80331 #11662000

Order 768.

2874 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#912C212yesmaximal, complemented
2C2×#5716C211yesmaximal, characteristic, complemented
3C2×C4×#237C211yesmaximal, characteristic, complemented
4C2×C3×#459C211yesmaximal, characteristic, complemented
5C2×#5153C212yesmaximal, complemented
6C2×#5075C211yesmaximal, characteristic, complemented
7C2×#5019C211yesmaximal, characteristic, complemented
8#80331C2116yesmaximal, direct product factor
9C2×#4849C211yesmaximal, characteristic, complemented
10C2×D3×C4⋊C8C211yesmaximal, characteristic, complemented
11C2×#4767C212yesmaximal, complemented
12C2×#4772C212yesmaximal, complemented
13C2×#3035—31nomaximal, Sylow
14C22×#237C2212yes—
15C22×#237C2211yescharacteristic
16C22×C3×D8C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#938—21no—
20C2×#938—21no—
21C2×#936C2212yes—
22C2×D4×Dic3—21no—
23C2×D4×Dic3—21no—
24C2×D4×Dic3C2211yescharacteristic
25C22×#214C2212yes—
26C22×#214C2212yes—
27C2×C4×#67—21no—
28C22×C4×Dic3—21no—
29C2×#912—216no—
30C2×#912C22116yescomplemented
31C22×Dic12C2212yes—
32C22×D3×C8C2212yes—
33#5716—24no—
34#5716C2218yescomplemented
35C4×#237—24no—
36C4×#237C2218yescomplemented
37C2×#892C2211yescharacteristic
38C2×#887C2211yescharacteristic
39C2×#67⋊C4—21no—
40C2×#877—21no—
41C2×C42×D3C2211yescharacteristic
42C2×#869C2211yescharacteristic
43C2×C4×Dic6C2211yescharacteristic
44C3×#459C2218yescomplemented
45C2×C3×C4⋊C8C2211yescharacteristic
46C2×C3×#102C2212yes—
47C2×#832C2212yes—
48#5153C22116yescomplemented
49C2×#816C2212yes—
50C2×Dic3⋊C8C2212yes—
51#5075C2218yescomplemented
52#5019C2218yescomplemented
53C2×#810C2212yes—
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55#4849C2218yescomplemented
56D3×C4⋊C8C2218yescomplemented
57#4772C22116yescomplemented
58#4767C22116yescomplemented
59C22×#135—32no—
60C2×#532—31no—
61C2×C4×#41—31no—
62#3035—316no—
63C2×#464—32no—
64C2×#462—31no—
65C2×#460—31no—
66C2×#463—32no—
67C2×#461—31no—
68C2×#459D311yescharacteristic, complemented
69C22×C4⋊C8—31no—
70C2×#416—32no—
71C22×C3×D4—21no—
72C22×C3×D4—21no—
73C22×C3×D4C2312yes—
74C22×C3×D4C2311yescharacteristic
75C23×C3×C4—21no—
76C22×#67—21no—
77C22×#67—22no—
78C23×Dic3—21no—
79C23×Dic3—22no—
80C23×Dic3—21no—
81C2×#237—22no—
82C2×#237—28no—
83C2×#237—24no—
84C2×#237—24no—
85C2×#237—28no—
86C2×#237D418yescomplemented
87C2×#237C2318yes—
88C2×#237C2314yes—
89C22×C4×D3D411yescharacteristic
90C22×C4×D3D411yescharacteristic
91C22×C4×D3C2311yescharacteristic
92C22×Dic6C2312yes—
93C22×Dic6C2311yescharacteristic
94C2×C3×D8—28no—
95C2×C3×D8C2318yes—
96C22×C3×C8C2312yes—
97C3×#127—22no—
98C3×#127C2314yes—
99C3×C4×D4—22no—
100C3×C4×D4C2314yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#217—22no—
106C2×#217—22no—
107#938—28no—
108#938—28no—
109#936—24no—
110#936C2318yes—
111D4×Dic3—28no—
112D4×Dic3—28no—
113D4×Dic3—22no—
114D4×Dic3C2314yes—
115C2×#214—28no—
116C2×#214—28no—
117C2×#214C2318yes—
118C2×#214C2318yes—
119C4×#67—28no—
120C2×(C2×D3)⋊C4—21no—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2311yescharacteristic
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3—28no—
130C2×C4×Dic3D411yescharacteristic
131C2×C4×Dic3D411yescharacteristic
132C2×C4×Dic3C2311yescharacteristic
133C22×C3⋊C8C2312yes—
134#912—264no—
135C2×Dic12—28no—
136C2×Dic12C2318yes—
137C2×D3×C8—28no—
138C2×D3×C8C2318yes—
139#892—22no—
140#892C2314yes—
141#887—22no—
142#887C2314yes—
143#67⋊C4—28no—
144#877—28no—
145C42×D3D414yes—
146C42×D3C2314yes—
147#869—22no—
148#869C2314yes—
149C4×Dic6—22no—
150C4×Dic6C2314yes—
151C3×C4⋊C8C2314yes—
152C3×#102C2318yes—
153#832C2318yes—
154#816C2318yes—
155Dic3⋊C8C2318yes—
156#810C2318yes—
157C4⋊(C3⋊C8)C2314yes—
158C22×#41—31no—
159C22×#41—32no—
160C2×#135—316no—
161C2×#135—616no—
162C22×Q16—32no—
163C22×SD16—32no—
164C22×SD16—32no—
165C22×D8C2×D312yes—
166C23×C8—32no—
167#532—38no—
168#532—64no—
169C2×C4⋊Q8—31no—
170C2×#127C2×D311yescharacteristic
171C2×#124—32no—
172C2×#121—61no—
173C2×#121—61no—
174C4×#41—38no—
175C4×#41—64no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×D4—31no—
179C2×C4×D4—61no—
180C2×C4×D4—61no—
181C2×C4×D4—61no—
182C2×C4×D4—61no—
183C2×C4×D4C2×D311yescharacteristic
184C2×#117—31no—
185C2×#117—61no—
186C22×C42—31no—
187C22×C42—61no—
188#464—316no—
189#463—316no—
190#462—38no—
191#461—38no—
192#460—38no—
193#459C2×D318yescomplemented
194C2×C4⋊C8—32no—
195C2×C4⋊C8—31no—
196C2×C4⋊C8—38no—
197C2×C4⋊C8C2×D311yescharacteristic
198#416—316no—
199C2×#103—32no—
200C2×#103—32no—
201C2×#102—32no—
202C2×#102C2×D312yes—
203C24×C3—21no—
204C24×C3—22no—
205C23×D3C2×D411yescharacteristic
206C2×C3×D4—21no—
207C2×C3×D4—24no—
208C2×C3×D4—24no—
209C2×C3×D4—24no—
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C2×C3×D4—24no—
213C2×C3×D4—44no—
214C2×C3×D4—44no—
215C2×C3×D4C2×D414yes—
216C2×C3×D4C2414yes—
217C2×C3×D4C2412yes—
218C22×C3×C4—21no—
219C22×C3×C4—24no—
220C22×C3×C4—44no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C2×#67—28no—
225C2×#67—24no—
226C2×#67—28no—
227C2×#67—48no—
228C22×Dic3—21no—
229C22×Dic3—28no—
230C22×Dic3—24no—
231C22×Dic3—21no—
232C22×Dic3—22no—
233C22×Dic3—24no—
234C22×Dic3—28no—
235C22×Dic3—44no—
236C22×Dic3—48no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240#237—216no—
241#237—216no—
242#237—28no—
243#237—216no—
244#237—416no—
245#237C2×D4116yescomplemented
246C2×C4×D3—24no—
247C2×C4×D3—42no—
248C2×C4×D3C2×D414yes—
249C2×C4×D3C2×D414yes—
250C2×C4×D3C2×D412yes—
251C2×C4×D3C2×D414yes—
252C2×C4×D3#4114yes—
253C2×C4×D3C2412yes—
254C2×Dic6—21no—
255C2×Dic6—22no—
256C2×Dic6—24no—
257C2×Dic6—24no—
258C2×Dic6—22no—
259C2×Dic6C2×D414yes—
260C2×Dic6C2414yes—
261C2×Dic6C2412yes—
262C3×D8—216no—
263C2×C3×C8—24no—
264C2×C3×C8C2414yes—
265C3×C4⋊C4—21no—
266C3×C4⋊C4C2412yes—
267C3×C22⋊C4—24no—
268C3×C42C2×D412yes—
269C3×C42C2412yes—
270#217—24no—
271#217—28no—
272#217—28no—
273#214—216no—
274#214—216no—
275(C2×D3)⋊C4—24no—
276C4⋊Dic3—21no—
277C4⋊Dic3—24no—
278C4⋊Dic3—24no—
279C4⋊Dic3C2412yes—
280Dic3⋊C4—24no—
281Dic3⋊C4—24no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—24no—
285C4×Dic3C2×D412yes—
286C4×Dic3C2×D414yes—
287C4×Dic3C2×D414yes—
288C4×Dic3C2412yes—
289C2×C3⋊C8—24no—
290C2×C3⋊C8C2414yes—
291Dic12—216no—
292D3×C8—216no—
293C2×#41—38no—
294C2×#41—34no—
295C2×#41—38no—
296C2×#41—62no—
297C2×#41—64no—
298C2×#41—64no—
299C2×#41—68no—
300C2×#41—68no—
301C22×Q8—31no—
302C22×Q8—32no—
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—61no—
306C22×D4—62no—
307C22×D4C22×D312yes—
308C22×D4C22×D311yescharacteristic
309C23×C4—21no—
310C23×C4—31no—
311C23×C4—31no—
312C23×C4—31no—
313C23×C4—61no—
314C23×C4—61no—
315C23×C4—62no—
316#135—664no—
317C2×Q16—38no—
318C2×Q16—68no—
319C2×SD16—38no—
320C2×SD16—38no—
321C2×SD16—68no—
322C2×SD16—68no—
323C2×D8—28no—
324C2×D8C22×D318yes—
325C22×C8—32no—
326C22×C8—38no—
327C22×C8—68no—
328C22×C8C22×D312yes—
329C4⋊Q8—34no—
330C4⋊Q8—62no—
331#127—22no—
332#127C22×D314yes—
333#124—38no—
334#124—64no—
335#121—68no—
336#121—68no—
337C4×Q8—34no—
338C4×Q8—34no—
339C4×Q8—62no—
340C4×Q8—62no—
341C4×D4—22no—
342C4×D4—34no—
343C4×D4—62no—
344C4×D4—68no—
345C4×D4—68no—
346C4×D4—68no—
347C4×D4—68no—
348C4×D4C22×D314yes—
349#117—34no—
350#117—68no—
351#117—62no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4C22×D311yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—61no—
360C2×C22⋊C4—61no—
361C2×C22⋊C4—62no—
362C2×C22⋊C4—62no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—31no—
366C2×C42—34no—
367C2×C42—34no—
368C2×C42—68no—
369C2×C42—61no—
370C2×C42—61no—
371C2×C42—61no—
372C2×C42C22×D311yescharacteristic
373C4⋊C8—38no—
374C4⋊C8—34no—
375C4⋊C8C22×D314yes—
376#103—38no—
377#103—38no—
378#102—38no—
379#102C22×D318yes—
380C23×C3—24no—
381C23×C3—22no—
382C23×C3—24no—
383C23×C3—42no—
384C23×C3—42no—
385C23×C3—44no—
386C23×C3—44no—
387C23×C3—44no—
388C23×C3C22×D411yescharacteristic
389C22×D3—22no—
390C22×D3—22no—
391C22×D3#12114yes—
392C22×D3C22×D412yes—
393C3×D4—24no—
394C3×D4—24no—
395C3×D4—22no—
396C3×D4—24no—
397C3×D4—48no—
398C3×D4—44no—
399C3×D4—48no—
400C3×D4C22×D414yes—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—42no—
404C2×C3×C4—48no—
405C2×C3×C4#13514yescomplemented
406C2×C3×C4C22×D412yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D412yes—
409C2×C3×C4C22×D412yes—
410C2×C3×C4C2×#4112yes—
411C2×C3×C4C2511yescharacteristic
412#67—216no—
413#67—416no—
414C2×Dic3—24no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—216no—
419C2×Dic3—22no—
420C2×Dic3—24no—
421C2×Dic3—48no—
422C2×Dic3—44no—
423C2×Dic3—42no—
424C2×Dic3—416no—
425C2×Dic3#12114yes—
426C2×Dic3C22×D412yes—
427C2×Dic3C22×D412yes—
428C2×Dic3C22×D412yes—
429C4×D3—28no—
430C4×D3—44no—
431C4×D3#12118yes—
432C4×D3C22×D414yes—
433C4×D3C2×#4114yes—
434Dic6—24no—
435Dic6—22no—
436Dic6—24no—
437Dic6—24no—
438Dic6—44no—
439Dic6C22×D414yes—
440C3×C8—24no—
441C3⋊C8—24no—
442C24—21no—
443C24—22no—
444C24—31no—
445#41—316no—
446#41—68no—
447#41—616no—
448#41—616no—
449#41—616no—
450#41—1216no—
451C2×Q8—34no—
452C2×Q8—32no—
453C2×Q8—34no—
454C2×Q8—61no—
455C2×Q8—62no—
456C2×Q8—62no—
457C2×Q8—64no—
458C2×Q8—64no—
459C2×D4—21no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—64no—
469C2×D4—68no—
470C2×D4—68no—
471C2×D4—128no—
472C2×D4D3×D414yes—
473C2×D4C23×D314yes—
474C2×D4C23×D312yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—31no—
478C22×C4—32no—
479C22×C4—34no—
480C22×C4—34no—
481C22×C4—32no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—34no—
485C22×C4—34no—
486C22×C4—44no—
487C22×C4—64no—
488C22×C4—68no—
489C22×C4—64no—
490C22×C4—68no—
491C22×C4—61no—
492C22×C4—64no—
493C22×C4—61no—
494C22×C4—62no—
495C22×C4—122no—
496C22×C4—128no—
497C22×C4—124no—
498C22×C4D3×D411yescharacteristic
499C22×C4D3×D411yescharacteristic
500C22×C4C23×D311yescharacteristic
501Q16—616no—
502SD16—616no—
503SD16—616no—
504D8—216no—
505C2×C8—24no—
506C2×C8—34no—
507C2×C8—616no—
508C2×C8—64no—
509C2×C8C23×D314yes—
510C4⋊C4—21no—
511C4⋊C4—32no—
512C4⋊C4—64no—
513C4⋊C4—61no—
514C4⋊C4—64no—
515C4⋊C4—64no—
516C4⋊C4—64no—
517C4⋊C4C23×D312yes—
518C22⋊C4—24no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—68no—
522C22⋊C4—68no—
523C42—34no—
524C42—34no—
525C42—32no—
526C42—32no—
527C42—64no—
528C42—64no—
529C42—64no—
530C42D3×D412yes—
531C42C23×D312yes—
532C22×C3—24no—
533C22×C3—48no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—44no—
537C22×C3—44no—
538C22×C3—44no—
539C22×C3—88no—
540C22×C3C2×#12112yes—
541C22×C3C2×#13512yes—
542C22×C3#57512yes—
543C22×C3C23×D411yescharacteristic
544C2×D3—28no—
545C2×D3—22no—
546C2×D3—22no—
547C2×D3C2×#12114yes—
548C3×C4—22no—
549C3×C4—42no—
550C3×C4C2×#12112yes—
551C3×C4C2×#13514yescomplemented
552C3×C4C23×D411yescharacteristic
553C3×C4C22×#4111yescharacteristic
554Dic3—24no—
555Dic3—22no—
556Dic3—22no—
557Dic3—44no—
558Dic3—42no—
559Dic3C2×#12114yes—
560C23—22no—
561C23—24no—
562C23—24no—
563C23—32no—
564C23—34no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—62no—
571C23—62no—
572C23C2×D3×D411yescharacteristic
573Q8—34no—
574Q8—62no—
575Q8—64no—
576Q8—64no—
577Q8—64no—
578Q8—124no—
579D4—24no—
580D4—22no—
581D4—24no—
582D4—24no—
583D4—48no—
584D4—48no—
585D4—44no—
586D4—616no—
587D4—1216no—
588D4C2×D3×D414yes—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—32no—
592C2×C4—34no—
593C2×C4—34no—
594C2×C4—34no—
595C2×C4—38no—
596C2×C4—32no—
597C2×C4—32no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—616no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—64no—
605C2×C4—68no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—124no—
609C2×C4—1216no—
610C2×C4—128no—
611C2×C4—122no—
612C2×C4—124no—
613C2×C4#91214yescomplemented
614C2×C4C2×D3×D412yes—
615C2×C4C2×D3×D412yes—
616C2×C4C2×D3×D412yes—
617C2×C4C2×D3×D411yescharacteristic
618C2×C4D3×#4112yes—
619C2×C4C24×D311yescharacteristic
620C8—24no—
621C8—64no—
622C2×C3—44no—
623C2×C3—84no—
624C2×C3#303514yescomplemented
625C2×C3C22×#12111yescharacteristic
626C2×C3C22×#13511yescharacteristic
627C2×C3C2×#57511yescharacteristic
628D3—24no—
629C22—24no—
630C22—34no—
631C22—42no—
632C22—42no—
633C22—48no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—62no—
638C22—62no—
639C22—68no—
640C22—88no—
641C22D3×#12112yes—
642C22C2×#91212yes—
643C22#577312yes—
644C22C22×D3×D411yescharacteristic
645C4—22no—
646C4—34no—
647C4—42no—
648C4—62no—
649C4—62no—
650C4—64no—
651C4—122no—
652C4—124no—
653C4D3×#12112yes—
654C4C2×#91214yescomplemented
655C4C22×D3×D411yescharacteristic
656C4C2×D3×#4111yescharacteristic
657C3C2×#303511yescharacteristic, complemented, Sylow
658C2—44no—
659C2—64no—
660C2—84no—
661C2#8033114yesdirect product factor
662C2C2×D3×#12111yescharacteristic
663C2C22×#91211yescharacteristic
664C2C2×#577311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3C2×#3035
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C22×#912C2semidirect
2. C2×#5716C2semidirect
3. C2×C4×#237C2semidirect
4. C2×C3×#459C2semidirect
5. C2×#5153C2semidirect
6. C2×#5075C2semidirect
7. C2×#5019C2semidirect
8. #80331C2direct (N × H)
9. C2×#4849C2semidirect
10. C2×D3×C4⋊C8C2semidirect
11. C2×#4767C2semidirect
12. C2×#4772C2semidirect
30. C2×#912C22semidirect
34. #5716C22semidirect
36. C4×#237C22semidirect
44. C3×#459C22semidirect
48. #5153C22semidirect
51. #5075C22semidirect
52. #5019C22semidirect
55. #4849C22semidirect
56. D3×C4⋊C8C22semidirect
57. #4772C22semidirect
58. #4767C22semidirect
68. C2×#459D3semidirect
86. C2×#237D4semidirect
193. #459C2×D3semidirect
245. #237C2×D4semidirect
405. C2×C3×C4#135semidirect
551. C3×C4C2×#135semidirect
613. C2×C4#912semidirect
624. C2×C3#3035semidirect
654. C4C2×#912semidirect
657. C3C2×#3035semidirect
661. C2#80331direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9122
C2×#57162
C2×C4×#2372
C2×C3×#4592
C2×#51532
C2×#50752
C2×#50192
#803312
C2×#48492
C2×D3×C4⋊C82
C2×#47672
C2×#47722
C2×#30353