Finite Group Databasebeta

← C2×#81683 #11664475

Subgroups — C2×#81683 #11664475

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#5668C211yesmaximal, characteristic
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic, complemented
5C2×#5055C211yesmaximal, characteristic, complemented
6#81683C2116yesmaximal, direct product factor
7C2×#4803C212yesmaximal, complemented
8C2×#4792C212yesmaximal, complemented
9C2×#4771C212yesmaximal, complemented
10C2×#4760C212yesmaximal, complemented
11C2×#4714C211yesmaximal, characteristic, complemented
12C2×#4707C211yesmaximal, characteristic, complemented
13C2×#3141—31nomaximal, Sylow
14C22×#239C2211yescharacteristic
15C22×D3×Q8—21no—
16C22×#237C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×#944—21no—
20C2×#944C2211yescharacteristic
21C2×#940C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2212yes—
23D3×C4⋊Q8—24no—
24D3×C4⋊Q8C2218yescomplemented
25#5668—24no—
26#5668C2218yes—
27C2×D3×C4⋊C4—21no—
28C2×D3×C4⋊C4C2212yes—
29C2×#888—21no—
30C2×#888C2212yes—
31C2×#884—21no—
32C2×#884—21no—
33C2×#67⋊C4—21no—
34C2×#67⋊C4—21no—
35C2×#875C2211yescharacteristic
36C2×C42×D3C2211yescharacteristic
37C2×#869C2211yescharacteristic
38C3×#489C2218yescomplemented
39C2×C3×#103C2212yes—
40C2×C3×#102C2212yes—
41C2×C3×#97C2211yescharacteristic
42C2×#835C2212yes—
43C2×#832C2212yes—
44C2×#821C2212yes—
45C2×#816C2212yes—
46C2×C8×Dic3C2212yes—
47#5089C2218yescomplemented
48#5055C2218yescomplemented
49C2×#810C2212yes—
50C2×#809C2212yes—
51C2×#803C2211yescharacteristic
52#4803C22116yescomplemented
53#4792C22116yescomplemented
54#4771C22116yescomplemented
55#4760C22116yescomplemented
56#4714C2218yescomplemented
57#4707C2218yescomplemented
58C2×#532—31no—
59C22×C4⋊Q8—31no—
60#3141—316no—
61C2×#489—31no—
62C2×#489—31no—
63C2×#489—31no—
64C2×#489D311yescharacteristic, complemented
65C2×#487—32no—
66C2×#486—32no—
67C2×#417—32no—
68C2×#417—32no—
69C2×C4×M16—31no—
70C22×C3×Q8—21no—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×D4C2311yescharacteristic
73C22×#67—21no—
74C22×#67—21no—
75C22×#67—21no—
76C23×Dic3—21no—
77C2×#239—22no—
78C2×#239—28no—
79C2×#239C2314yes—
80C2×D3×Q8—28no—
81C2×D3×Q8—48no—
82C2×#237—22no—
83C2×#237—28no—
84C2×#237C2314yes—
85C22×D12—21no—
86C22×D12C2311yescharacteristic
87C22×C4×D3—22no—
88C22×C4×D3—21no—
89C22×C4×D3D411yescharacteristic
90C22×C4×D3D411yescharacteristic
91C22×C4×D3C2311yescharacteristic
92C22×Dic6—21no—
93C22×Dic6—22no—
94C22×Dic6C2311yescharacteristic
95C22×C3×C8C2312yes—
96C3×C4⋊Q8—22no—
97C3×C4⋊Q8C2314yes—
98C3×#124—22no—
99C3×#124C2314yes—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42C2311yescharacteristic
105#944—22no—
106#944—28no—
107#944C2314yes—
108#940—22no—
109#940C2314yes—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2312yes—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—22no—
117C2×Dic3⋊C4—22no—
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2312yes—
122C2×ΓL(1,𝔽25)—28no—
123C2×ΓL(1,𝔽25)C2318yes—
124D3×C4⋊C4—24no—
125D3×C4⋊C4—28no—
126D3×C4⋊C4C2318yes—
127#888—24no—
128#888—28no—
129#888C2318yes—
130#884—28no—
131#884—28no—
132#67⋊C4—28no—
133#67⋊C4—28no—
134#875—22no—
135#875C2314yes—
136C42×D3D414yes—
137C42×D3C2314yes—
138#869—22no—
139#869C2314yes—
140C3×#103C2318yes—
141C3×#102C2318yes—
142C3×#97C2314yes—
143#835C2318yes—
144#832C2318yes—
145#821C2318yes—
146#816C2318yes—
147C8×Dic3C2318yes—
148#810C2318yes—
149#809C2318yes—
150#803C2314yes—
151C22×#41—31no—
152C22×#41—31no—
153C23×Q8—61no—
154C22×M16—32no—
155#532—38no—
156#532—64no—
157C2×C4⋊Q8—38no—
158C2×C4⋊Q8—32no—
159C2×C4⋊Q8—31no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—61no—
162C2×C4⋊Q8—64no—
163C2×C4⋊Q8—61no—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×#127—31no—
166C2×#124—31no—
167C2×#124C2×D311yescharacteristic
168C2×#121—61no—
169C2×#121—61no—
170C2×C4×D4—61no—
171C2×C4×D4—61no—
172C22×C4⋊C4—32no—
173C22×C4⋊C4—61no—
174C22×C42—31no—
175#489—38no—
176#489—38no—
177#489—38no—
178#489C2×D318yescomplemented
179#487—316no—
180#486—316no—
181#417—316no—
182#417—316no—
183C2×#103—32no—
184C2×#103—32no—
185C2×#103—32no—
186C2×#103C2×D312yes—
187C2×#102—32no—
188C2×#102—32no—
189C2×#102—32no—
190C2×#102C2×D312yes—
191C4×M16—38no—
192C2×#97—31no—
193C2×#97C2×D311yescharacteristic
194C2×C4×C8—32no—
195C24×C3—21no—
196C23×D3—21no—
197C23×D3C2×D411yescharacteristic
198C2×C3×Q8—21no—
199C2×C3×Q8—24no—
200C2×C3×Q8—24no—
201C2×C3×Q8—44no—
202C2×C3×Q8C2412yes—
203C2×C3×D4—21no—
204C2×C3×D4—24no—
205C2×C3×D4C2412yes—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C2×#67—24no—
212C2×#67—24no—
213C2×#67—24no—
214C2×#67—44no—
215C2×#67—44no—
216C2×#67—44no—
217C22×Dic3—22no—
218C22×Dic3—21no—
219C22×Dic3—24no—
220C22×Dic3—44no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224#239—216no—
225#239—48no—
226D3×Q8—432no—
227#237—216no—
228#237—48no—
229C2×D12—24no—
230C2×D12—24no—
231C2×D12—21no—
232C2×D12—44no—
233C2×D12C2412yes—
234C2×C4×D3—24no—
235C2×C4×D3—24no—
236C2×C4×D3—28no—
237C2×C4×D3—24no—
238C2×C4×D3—48no—
239C2×C4×D3—44no—
240C2×C4×D3C2×D414yes—
241C2×C4×D3C2×D414yes—
242C2×C4×D3C2×D412yes—
243C2×C4×D3#4118yes—
244C2×C4×D3C2412yes—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—21no—
248C2×Dic6—28no—
249C2×Dic6—44no—
250C2×Dic6—48no—
251C2×Dic6C2412yes—
252C2×C3×C8—24no—
253C2×C3×C8C2414yes—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4C2414yes—
257C3×C22⋊C4—24no—
258C3×C22⋊C4—24no—
259C3×C42C2×D412yes—
260C3×C42C2412yes—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263C4⋊Dic3—22no—
264C4⋊Dic3—24no—
265C4⋊Dic3C2414yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268Dic3⋊C4—28no—
269Dic3⋊C4—28no—
270C4×Dic3C2×D412yes—
271C4×Dic3C2×D414yes—
272C4×Dic3C2×D414yes—
273C4×Dic3C2412yes—
274C2×C3⋊C8—24no—
275C2×C3⋊C8C2414yes—
276ΓL(1,𝔽25)—216no—
277C2×#41—34no—
278C2×#41—34no—
279C2×#41—62no—
280C2×#41—62no—
281C2×#41—68no—
282C2×#41—68no—
283C22×Q8—21no—
284C22×Q8—31no—
285C22×Q8—62no—
286C22×Q8—68no—
287C22×Q8—61no—
288C22×Q8—128no—
289C22×Q8C22×D311yescharacteristic
290C22×D4—31no—
291C22×D4—61no—
292C22×D4—61no—
293C22×D4—61no—
294C22×D4—61no—
295C22×D4C22×D311yescharacteristic
296C23×C4—31no—
297C23×C4—31no—
298C23×C4—31no—
299C23×C4—61no—
300C23×C4—62no—
301C23×C4—61no—
302C2×M16—38no—
303C2×M16—68no—
304C22×C8—32no—
305C22×C8C22×D312yes—
306C4⋊Q8—22no—
307C4⋊Q8—38no—
308C4⋊Q8—34no—
309C4⋊Q8—34no—
310C4⋊Q8—68no—
311C4⋊Q8—64no—
312C4⋊Q8—68no—
313C4⋊Q8—62no—
314C4⋊Q8—62no—
315C4⋊Q8C22×D314yes—
316#127—34no—
317#127—62no—
318#124—22no—
319#124—34no—
320#124—62no—
321#124C22×D314yes—
322#121—68no—
323#121—68no—
324C4×D4—68no—
325C4×D4—68no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—38no—
329C2×C4⋊C4—64no—
330C2×C4⋊C4—68no—
331C2×C4⋊C4—62no—
332C2×C4⋊C4—62no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4C22×D312yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42—34no—
345C2×C42—34no—
346C2×C42C22×D311yescharacteristic
347#103—38no—
348#103—38no—
349#103—38no—
350#103C22×D318yes—
351#102—38no—
352#102—38no—
353#102—38no—
354#102C22×D318yes—
355#97—34no—
356#97C22×D314yes—
357C4×C8—38no—
358C23×C3—22no—
359C23×C3—42no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3C22×D411yescharacteristic
363C22×D3—22no—
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—42no—
367C22×D3—42no—
368C22×D3—42no—
369C22×D3#12414yes—
370C22×D3C22×D412yes—
371C3×Q8—24no—
372C3×Q8—48no—
373C3×Q8—42no—
374C3×D4—24no—
375C3×D4—42no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—24no—
380C2×C3×C4—42no—
381C2×C3×C4—44no—
382C2×C3×C4C22×D412yes—
383C2×C3×C4C22×D412yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4114yes—
386C2×C3×C4C2511yescharacteristic
387#67—48no—
388#67—48no—
389#67—48no—
390C2×Dic3—24no—
391C2×Dic3—22no—
392C2×Dic3—42no—
393C2×Dic3—44no—
394C2×Dic3—48no—
395C2×Dic3#12414yes—
396C2×Dic3AGL(1,ℤ/8)14yes—
397C2×Dic3#13714yes—
398C2×Dic3C22×D412yes—
399C2×Dic3C22×D412yes—
400C2×Dic3C22×D412yes—
401D12—24no—
402D12—48no—
403D12—42no—
404C4×D3—28no—
405C4×D3—28no—
406C4×D3—416no—
407C4×D3—48no—
408C4×D3#12418yes—
409C4×D3C2×#4118yes—
410Dic6—24no—
411Dic6—48no—
412Dic6—42no—
413Dic6—416no—
414C3×C8—24no—
415C3⋊C8—24no—
416C24—21no—
417C24—31no—
418C24—61no—
419#41—616no—
420#41—616no—
421#41—128no—
422#41—128no—
423C2×Q8—21no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—32no—
427C2×Q8—44no—
428C2×Q8—68no—
429C2×Q8—61no—
430C2×Q8—64no—
431C2×Q8—64no—
432C2×Q8—124no—
433C2×Q8—128no—
434C2×Q8—1232no—
435C2×Q8C23×D312yes—
436C2×D4—21no—
437C2×D4—24no—
438C2×D4—32no—
439C2×D4—64no—
440C2×D4—64no—
441C2×D4—64no—
442C2×D4—64no—
443C2×D4—61no—
444C2×D4—64no—
445C2×D4—124no—
446C2×D4—124no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4C23×D312yes—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—32no—
456C22×C4—38no—
457C22×C4—32no—
458C22×C4—34no—
459C22×C4—34no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—68no—
464C22×C4—64no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—124no—
468C22×C4—124no—
469C22×C4—128no—
470C22×C4D3×D411yescharacteristic
471C22×C4D3×D411yescharacteristic
472C22×C4C23×D311yescharacteristic
473M16—616no—
474C2×C8—24no—
475C2×C8—34no—
476C2×C8—64no—
477C2×C8C23×D314yes—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—34no—
481C4⋊C4—68no—
482C4⋊C4—68no—
483C4⋊C4—64no—
484C4⋊C4—64no—
485C4⋊C4—62no—
486C4⋊C4—64no—
487C4⋊C4C23×D314yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—64no—
491C22⋊C4—64no—
492C42—34no—
493C42—34no—
494C42—32no—
495C42—32no—
496C42D3×D412yes—
497C42C23×D312yes—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—84no—
502C22×C3C2×#12412yes—
503C22×C3C2×AGL(1,ℤ/8)12yes—
504C22×C3C2×#13712yes—
505C22×C3C23×D411yescharacteristic
506C2×D3—28no—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—84no—
513C2×D3C2×#12414yes—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—44no—
518C3×C4C2×#12412yes—
519C3×C4C22×#4112yes—
520Dic3—42no—
521Dic3—44no—
522Dic3C2×#12414yes—
523Dic3C2×AGL(1,ℤ/8)14yes—
524Dic3C2×#13714yes—
525C23—22no—
526C23—34no—
527C23—32no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—122no—
535C23—122no—
536C23—122no—
537C23C2×D3×D411yescharacteristic
538Q8—24no—
539Q8—48no—
540Q8—42no—
541Q8—64no—
542Q8—128no—
543Q8—1216no—
544Q8—122no—
545D4—24no—
546D4—42no—
547D4—64no—
548D4—128no—
549D4—128no—
550D4—128no—
551D4—128no—
552D4—122no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—34no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—32no—
561C2×C4—34no—
562C2×C4—34no—
563C2×C4—38no—
564C2×C4—38no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—68no—
568C2×C4—68no—
569C2×C4—62no—
570C2×C4—64no—
571C2×C4—128no—
572C2×C4—128no—
573C2×C4—1216no—
574C2×C4—122no—
575C2×C4—124no—
576C2×C4C2×D3×D412yes—
577C2×C4C2×D3×D412yes—
578C2×C4C2×D3×D411yescharacteristic
579C2×C4D3×#4114yes—
580C2×C4C24×D311yescharacteristic
581C8—24no—
582C8—64no—
583C2×C3—82no—
584C2×C3#314114yescomplemented
585C2×C3C22×#12411yescharacteristic
586C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
587C2×C3C22×#13711yescharacteristic
588D3—24no—
589D3—82no—
590C22—34no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—68no—
595C22—62no—
596C22—62no—
597C22—84no—
598C22—122no—
599C22—122no—
600C22—122no—
601C22—244no—
602C22D3×#12412yes—
603C22D3×AGL(1,ℤ/8)12yes—
604C22D3×#13712yes—
605C22C22×D3×D411yescharacteristic
606C4—22no—
607C4—22no—
608C4—34no—
609C4—34no—
610C4—34no—
611C4—42no—
612C4—44no—
613C4—122no—
614C4—124no—
615C4D3×#12412yes—
616C4C2×D3×#4112yes—
617C3C2×#314111yescharacteristic, complemented, Sylow
618C2—64no—
619C2—82no—
620C2—242no—
621C2#8168314yesdirect product factor
622C2C2×D3×#12411yescharacteristic
623C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
624C2C2×D3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC2×#5668C2
hypercenterC2×#489D3
hypocenterC3C2×#3141
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊Q8C2semidirect
3. C2×C3×#489C2semidirect
4. C2×#5089C2semidirect
5. C2×#5055C2semidirect
6. #81683C2direct (N × H)
7. C2×#4803C2semidirect
8. C2×#4792C2semidirect
9. C2×#4771C2semidirect
10. C2×#4760C2semidirect
11. C2×#4714C2semidirect
12. C2×#4707C2semidirect
24. D3×C4⋊Q8C22semidirect
38. C3×#489C22semidirect
47. #5089C22semidirect
48. #5055C22semidirect
52. #4803C22semidirect
53. #4792C22semidirect
54. #4771C22semidirect
55. #4760C22semidirect
56. #4714C22semidirect
57. #4707C22semidirect
64. C2×#489D3semidirect
178. #489C2×D3semidirect
584. C2×C3#3141semidirect
617. C3C2×#3141semidirect
621. C2#81683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#56682
C2×C3×#4892
C2×#50892
C2×#50552
#816832
C2×#48032
C2×#47922
C2×#47712
C2×#47602
C2×#47142
C2×#47072
C2×#31413