Finite Group Databasebeta

← C2×#78718 #11661851

Subgroups — C2×#78718 #11661851

Order 768.

4742 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5964C211yesmaximal, characteristic
2C2×#5825C211yesmaximal, characteristic, complemented
3C22×#909C212yesmaximal, complemented
4C2×C3×#413C211yesmaximal, characteristic, complemented
5C2×#5139C212yesmaximal, complemented
6#78718C2116yesmaximal, direct product factor
7C2×#4752C214yesmaximal, complemented
8C2×#4746C214yesmaximal, complemented
9C2×#3022—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11#5964—24no—
12#5964—28no—
13#5964C2218yes—
14C2×#1010—22no—
15C2×#1009—22no—
16C2×D3×#41—24no—
17C2×#1007C2212yes—
18C2×#1004—22no—
19C2×#1004C2212yes—
20C22×#235—22no—
21C22×#235C2212yes—
22C23×Dic6C2211yescharacteristic, complemented
23C22×C3×M16C2212yescomplemented
24#5825—24no—
25#5825C2218yescomplemented
26C2×#944C2212yescomplemented
27C2×#936C2212yescomplemented
28C2×#935—22no—
29C2×#926C2211yescharacteristic, complemented
30C2×#924—22no—
31C2×#909—24no—
32C2×#909—28no—
33C2×#909C2218yescomplemented
34C2×#909C22116yescomplemented
35C22×Dic12—22no—
36C22×#204—22no—
37C3×#413C2218yescomplemented
38C2×C3×#101C2212yescomplemented
39C2×C3×#100C2212yescomplemented
40#5139C22116yescomplemented
41C2×#825C2218yescomplemented
42#4752C22132yescomplemented
43#4746C22132yescomplemented
44C2×#585—31no—
45C22×#137—32no—
46C2×#536—31no—
47#3022—316no—
48C2×#456—34no—
49C2×#455—34no—
50C2×#421—32no—
51C2×#413D311yescharacteristic, complemented
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—24no—
55C2×C3×#41D414yes—
56C2×C3×#41C2314yes—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4D412yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4D412yes—
61C23×C3×C4C2311yescharacteristic
62C22×#67—24no—
63C22×#67—22no—
64C23×Dic3—21no—
65#1010—216no—
66#1010—216no—
67#1009—216no—
68#1009—216no—
69D3×#41—232no—
70D3×#41—232no—
71#1007—24no—
72#1007—28no—
73#1007C2318yes—
74C2×#239—24no—
75C2×D3×Q8—24no—
76#1004—216no—
77#1004—216no—
78#1004—24no—
79#1004—28no—
80#1004C2318yes—
81C2×#237—28no—
82C2×#237—24no—
83C2×D3×D4—28no—
84C2×D3×D4—24no—
85C2×#235—24no—
86C2×#235—24no—
87C2×#235—216no—
88C2×#235—22no—
89C2×#235—24no—
90C2×#235—216no—
91C2×#235—22no—
92C2×#235—24no—
93C2×#235—24no—
94C2×#235D414yes—
95C2×#235D418yes—
96C2×#235C2314yes—
97C2×#235C2318yes—
98C22×D12—22no—
99C22×D12D412yes—
100C22×C4×D3—24no—
101C22×C4×D3—22no—
102C22×Dic6—22no—
103C22×Dic6—22no—
104C22×Dic6—22no—
105C22×Dic6—24no—
106C22×Dic6—21no—
107C22×Dic6—22no—
108C22×Dic6D412yes—
109C22×Dic6C2312yes—
110C22×Dic6C2314yescomplemented
111C2×C3×M16—22no—
112C2×C3×M16—24no—
113C2×C3×M16C2314yes—
114C2×C3×M16C2318yescomplemented
115C22×C3×C8—22no—
116C2×#217—24no—
117C2×#217—21no—
118#944—24no—
119#944C2318yescomplemented
120#936—24no—
121#936C2318yescomplemented
122#935—216no—
123#926—22no—
124#926C2314yescomplemented
125C2×C4⋊Dic3—21no—
126#924—216no—
127C2×Dic3⋊C4—24no—
128C2×C4×Dic3C2312yes—
129#909—216no—
130#909—232no—
131#909C23132yescomplemented
132C2×Dic12—28no—
133C2×Dic12—216no—
134C2×#204—28no—
135C2×#204—216no—
136C3×#101C2318yescomplemented
137C3×#100C2318yescomplemented
138#825C23132yescomplemented
139#585—38no—
140#585—68no—
141#585—64no—
142C2×2−1+4—32no—
143C2×2−1+4—62no—
144C2×H(5,𝔽2)—32no—
145C2×H(5,𝔽2)—62no—
146C2×H(5,𝔽2)—62no—
147C22×#41—32no—
148C22×#41—64no—
149C22×#41—62no—
150C22×#41C2×D311yescharacteristic
151C23×Q8—31no—
152C2×#137—38no—
153C2×#137—316no—
154C2×#137—64no—
155C2×#137—68no—
156C22×Q16—62no—
157C22×SD16—62no—
158C22×M16C2×D312yescomplemented
159#536—38no—
160#536—64no—
161C2×C4⋊Q8—32no—
162C2×#124—32no—
163C2×#123—62no—
164C2×C22⋊Q8—62no—
165C2×#117—31no—
166#456—332no—
167#455—332no—
168#421—316no—
169C2×#104—38no—
170#413C2×D318yescomplemented
171C2×#101C2×D312yescomplemented
172C2×#100C2×D312yescomplemented
173C24×C3C2×D412yes—
174C24×C3C2×D411yescharacteristic
175C23×D3—24no—
176C23×D3—22no—
177C3×#41—28no—
178C3×#41—28no—
179C3×#41—28no—
180C3×#41—28no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8C2×D412yes—
184C2×C3×Q8C2412yes—
185C2×C3×D4—24no—
186C2×C3×D4—24no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4C2×D418yes—
190C2×C3×D4C2×D412yes—
191C2×C3×D4C2412yes—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—24no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D418yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D414yes—
202C22×C3×C4C2411yescharacteristic
203C22×C3×C4C2412yes—
204C2×#67—28no—
205C2×#67—216no—
206C2×#67—24no—
207C2×#67—28no—
208C2×#67—24no—
209C2×#67—216no—
210C2×#67—28no—
211C22×Dic3—24no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—24no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218#239—216no—
219#239—216no—
220D3×Q8—216no—
221D3×Q8—216no—
222#237—264no—
223#237—216no—
224#237—216no—
225D3×D4—264no—
226D3×D4—216no—
227D3×D4—216no—
228#235—216no—
229#235—232no—
230#235—232no—
231#235—28no—
232#235—28no—
233#235—28no—
234#235—28no—
235#235—216no—
236#235—24no—
237#235—24no—
238#235—28no—
239#235C2×D418yes—
240#235C2×D4116yes—
241#235C2418yes—
242C2×D12—24no—
243C2×D12—216no—
244C2×D12—22no—
245C2×D12—24no—
246C2×D12C2×D414yes—
247C2×D12C2×D418yes—
248C2×C4×D3—28no—
249C2×C4×D3—232no—
250C2×C4×D3—24no—
251C2×C4×D3—28no—
252C2×C4×D3—24no—
253C2×C4×D3—28no—
254C2×Dic6—24no—
255C2×Dic6—216no—
256C2×Dic6—22no—
257C2×Dic6—24no—
258C2×Dic6—24no—
259C2×Dic6—28no—
260C2×Dic6—24no—
261C2×Dic6—28no—
262C2×Dic6—24no—
263C2×Dic6—22no—
264C2×Dic6—41no—
265C2×Dic6—44no—
266C2×Dic6—41no—
267C2×Dic6—44no—
268C2×Dic6—42no—
269C2×Dic6C2×D414yes—
270C2×Dic6C2×D418yes—
271C2×Dic6C2414yes—
272C3×M16—24no—
273C3×M16—28no—
274C3×M16C2418yes—
275C2×C3×C8—24no—
276C2×C3×C8—28no—
277#217—216no—
278#217—24no—
279C4⋊Dic3—24no—
280Dic3⋊C4—216no—
281C4×Dic3—22no—
282C4×Dic3C2414yes—
283Dic12—232no—
284#204—232no—
2852−1+4—38no—
2862−1+4—68no—
2872−1+4—616no—
2882−1+4—616no—
2892−1+4—64no—
290H(5,𝔽2)—38no—
291H(5,𝔽2)—68no—
292H(5,𝔽2)—616no—
293H(5,𝔽2)—616no—
294H(5,𝔽2)—616no—
295H(5,𝔽2)—64no—
296H(5,𝔽2)—616no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—24no—
300C2×#41—34no—
301C2×#41—38no—
302C2×#41—34no—
303C2×#41—38no—
304C2×#41—64no—
305C2×#41—64no—
306C2×#41—616no—
307C2×#41—64no—
308C2×#41—68no—
309C2×#41—64no—
310C2×#41—632no—
311C2×#41—632no—
312C2×#41—62no—
313C2×#41—64no—
314C2×#41—64no—
315C2×#41—616no—
316C2×#41—62no—
317C2×#41—64no—
318C2×#41D1214yes—
319C2×#41C22×D314yes—
320C22×Q8—32no—
321C22×Q8—32no—
322C22×Q8—34no—
323C22×Q8—62no—
324C22×Q8—64no—
325C22×Q8—61no—
326C22×Q8—62no—
327C22×Q8—62no—
328C22×Q8—62no—
329C22×Q8—64no—
330C22×Q8C22×D311yescharacteristic
331C22×D4—32no—
332C22×D4—62no—
333C22×D4—64no—
334C22×D4—68no—
335C22×D4—62no—
336C22×D4—64no—
337C22×D4D1212yes—
338C22×D4C22×D311yescharacteristic
339C23×C4—64no—
340C23×C4—62no—
341C23×C4—61no—
342C23×C4D1212yes—
343C23×C4C22×D311yescharacteristic
344#137—332no—
345#137—616no—
346#137—632no—
347C2×Q16—68no—
348C2×Q16—616no—
349C2×SD16—68no—
350C2×SD16—616no—
351C2×M16—22no—
352C2×M16—24no—
353C2×M16C22×D314yes—
354C2×M16C22×D318yescomplemented
355C22×C8—22no—
356C4⋊Q8—38no—
357C4⋊Q8—64no—
358#124—38no—
359#124—64no—
360#123—616no—
361C22⋊Q8—616no—
362#117—34no—
363#117—62no—
364C2×C4⋊C4—61no—
365C2×C4⋊C4—64no—
366C2×C22⋊C4—61no—
367C2×C22⋊C4—64no—
368C2×C42—32no—
369#104—332no—
370#101C22×D318yescomplemented
371#100C22×D318yescomplemented
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—42no—
375C23×C3—44no—
376C23×C3#12012yes—
377C23×C3#12012yes—
378C23×C3C22×D411yescharacteristic
379C23×C3C22×D414yes—
380C23×C3C22×D412yes—
381C22×D3—24no—
382C22×D3—216no—
383C22×D3—28no—
384C22×D3—84no—
385C22×D3—88no—
386C3×Q8—24no—
387C3×Q8—24no—
388C3×D4—216no—
389C3×D4—24no—
390C3×D4—24no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—28no—
394C2×C3×C4—28no—
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4#12014yes—
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4#12014yes—
399C2×C3×C4#12012yes—
400C2×C3×C4C22×D414yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D412yes—
403C2×C3×C4C22×D412yes—
404C2×C3×C4C22×D414yes—
405C2×C3×C4C22×D414yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C2511yescharacteristic
408#67—232no—
409#67—28no—
410#67—216no—
411#67—28no—
412C2×Dic3—216no—
413C2×Dic3—24no—
414C2×Dic3—28no—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—44no—
418C2×Dic3—48no—
419C2×Dic3—42no—
420C2×Dic3—42no—
421D12—216no—
422D12—28no—
423D12—24no—
424D12C22×D418yes—
425C4×D3—232no—
426C4×D3—216no—
427C4×D3—28no—
428C4×D3—28no—
429Dic6—216no—
430Dic6—28no—
431Dic6—24no—
432Dic6—28no—
433Dic6—44no—
434Dic6—44no—
435Dic6—44no—
436Dic6C22×D418yes—
437C3×C8—28no—
438C24—64no—
439C24—62no—
440C24C2×D1212yes—
441C24C2×D1211yescharacteristic
442#41—28no—
443#41—28no—
444#41—28no—
445#41—28no—
446#41—316no—
447#41—38no—
448#41—38no—
449#41—632no—
450#41—616no—
451#41—632no—
452#41—616no—
453#41—664no—
454#41—616no—
455#41—616no—
456#41—616no—
457#41—68no—
458#41—68no—
459#41—68no—
460#41—68no—
461#41—64no—
462#41—616no—
463#41—68no—
464#41—64no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—34no—
468C2×Q8—38no—
469C2×Q8—34no—
470C2×Q8—64no—
471C2×Q8—616no—
472C2×Q8—616no—
473C2×Q8—616no—
474C2×Q8—62no—
475C2×Q8—64no—
476C2×Q8—64no—
477C2×Q8—62no—
478C2×Q8—68no—
479C2×Q8—64no—
480C2×Q8—68no—
481C2×Q8—64no—
482C2×Q8—122no—
483C2×Q8—124no—
484C2×Q8—124no—
485C2×Q8—121no—
486C2×Q8—121no—
487C2×Q8C2×D1212yes—
488C2×Q8C23×D312yes—
489C2×D4—24no—
490C2×D4—24no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—34no—
494C2×D4—38no—
495C2×D4—64no—
496C2×D4—616no—
497C2×D4—616no—
498C2×D4—664no—
499C2×D4—616no—
500C2×D4—68no—
501C2×D4—616no—
502C2×D4—64no—
503C2×D4—68no—
504C2×D4—64no—
505C2×D4—68no—
506C2×D4—616no—
507C2×D4—62no—
508C2×D4—64no—
509C2×D4C2×D1218yes—
510C2×D4C2×D1212yes—
511C2×D4C23×D312yes—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—24no—
515C22×C4—68no—
516C22×C4—632no—
517C22×C4—64no—
518C22×C4—62no—
519C22×C4—64no—
520C22×C4—68no—
521C22×C4—64no—
522C22×C4—68no—
523C22×C4—62no—
524C22×C4—62no—
525C22×C4—64no—
526C22×C4—122no—
527C22×C4—122no—
528C22×C4C2×D1212yes—
529C22×C4C2×D1218yes—
530C22×C4C2×D1211yescharacteristic
531C22×C4C2×D1212yes—
532C22×C4D3×D412yes—
533C22×C4D3×D412yes—
534C22×C4D3×D414yes—
535C22×C4C23×D311yescharacteristic
536C22×C4C23×D312yes—
537Q16—632no—
538SD16—632no—
539M16—24no—
540M16—28no—
541M16C23×D318yes—
542C2×C8—24no—
543C2×C8—28no—
544C4⋊C4—64no—
545C4⋊C4—616no—
546C22⋊C4—64no—
547C22⋊C4—616no—
548C42—34no—
549C42—62no—
550C22×C3—22no—
551C22×C3—24no—
552C22×C3—24no—
553C22×C3—42no—
554C22×C3—42no—
555C22×C3—44no—
556C22×C3—44no—
557C22×C3—82no—
558C22×C3C2×#12014yes—
559C22×C3C2×#12011yescharacteristic
560C22×C3C2×#12011yescharacteristic
561C22×C3C23×D411yescharacteristic
562C2×D3—28no—
563C2×D3—48no—
564C2×D3—88no—
565C2×D3—816no—
566C3×C4—24no—
567C3×C4C2×#12012yes—
568C3×C4C2×#12012yes—
569C3×C4C2×#12012yes—
570C3×C4C23×D412yes—
571Dic3—28no—
572Dic3—44no—
573Dic3—44no—
574C23—22no—
575C23—22no—
576C23—42no—
577C23—44no—
578C23—616no—
579C23—64no—
580C23—68no—
581C23—244no—
582C23—248no—
583C23#88212yes—
584C23#88212yes—
585C23C22×D1211yescharacteristic
586C23C22×D1214yes—
587C23C22×D1212yes—
588Q8—24no—
589Q8—24no—
590Q8—38no—
591Q8—616no—
592Q8—68no—
593Q8—64no—
594Q8—68no—
595Q8—124no—
596Q8—124no—
597Q8—124no—
598D4—216no—
599D4—24no—
600D4—24no—
601D4—38no—
602D4—616no—
603D4—632no—
604D4—68no—
605D4—68no—
606D4—616no—
607D4—68no—
608D4—64no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—632no—
614C2×C4—616no—
615C2×C4—68no—
616C2×C4—64no—
617C2×C4—68no—
618C2×C4—616no—
619C2×C4—68no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—124no—
623C2×C4—122no—
624C2×C4—128no—
625C2×C4—122no—
626C2×C4#88212yes—
627C2×C4#88214yes—
628C2×C4#88214yes—
629C2×C4#88211yescharacteristic
630C2×C4#88211yescharacteristic
631C2×C4C22×D1214yes—
632C2×C4C22×D1211yescharacteristic
633C2×C4C22×D1212yes—
634C2×C4C2×D3×D414yes—
635C2×C4C2×D3×D414yes—
636C2×C4C2×D3×D412yes—
637C2×C4C2×D3×D412yes—
638C2×C4C24×D311yescharacteristic
639C8—28no—
640C2×C3—21no—
641C2×C3—24no—
642C2×C3—21no—
643C2×C3—44no—
644C2×C3#302212yescomplemented
645C2×C3C22×#12011yescharacteristic
646D3—48no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—44no—
654C22—68no—
655C22—82no—
656C22—128no—
657C22—248no—
658C22—2416no—
659C22C2×#88211yescharacteristic
660C22C2×#88214yes—
661C22C2×#88211yescharacteristic
662C22C23×D1211yescharacteristic
663C4—24no—
664C4—68no—
665C4—124no—
666C4—124no—
667C4C2×#88212yes—
668C4C2×#88212yes—
669C4C2×#88212yes—
670C4C22×D3×D412yes—
671C3C2×#302211yescharacteristic, complemented, Sylow
672C2—21no—
673C2—21no—
674C2—24no—
675C2—44no—
676C2—128no—
677C2#7871812yesdirect product factor
678C2C22×#88211yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#413C2
socleC22×C3C2×#120
prime radicalC2×#5964C2
hypercenterC2×#413D3
hypocenterC3C2×#3022
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C22×#909C2semidirect
4. C2×C3×#413C2semidirect
5. C2×#5139C2semidirect
6. #78718C2direct (N × H)
7. C2×#4752C2semidirect
8. C2×#4746C2semidirect
22. C23×Dic6C22semidirect
23. C22×C3×M16C22semidirect
25. #5825C22semidirect
26. C2×#944C22semidirect
27. C2×#936C22semidirect
29. C2×#926C22semidirect
33. C2×#909C22semidirect
34. C2×#909C22semidirect
37. C3×#413C22semidirect
38. C2×C3×#101C22semidirect
39. C2×C3×#100C22semidirect
40. #5139C22semidirect
41. C2×#825C22semidirect
42. #4752C22semidirect
43. #4746C22semidirect
51. C2×#413D3semidirect
110. C22×Dic6C23semidirect
114. C2×C3×M16C23semidirect
119. #944C23semidirect
121. #936C23semidirect
124. #926C23semidirect
131. #909C23semidirect
136. C3×#101C23semidirect
137. C3×#100C23semidirect
138. #825C23semidirect
158. C22×M16C2×D3semidirect
170. #413C2×D3semidirect
171. C2×#101C2×D3semidirect
172. C2×#100C2×D3semidirect
354. C2×M16C22×D3semidirect
370. #101C22×D3semidirect
371. #100C22×D3semidirect
644. C2×C3#3022semidirect
671. C3C2×#3022semidirect
677. C2#78718direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#59642
C2×#58252
C22×#9092
C2×C3×#4132
C2×#51392
#787182
C2×#47522
C2×#47462
C2×#30223