Finite Group Databasebeta

← C2×#81682 #11664467

Subgroups — C2×#81682 #11664467

Order 768.

2246 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×D3×#124C211yesmaximal, characteristic
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic, complemented
5C2×#5055C211yesmaximal, characteristic, complemented
6#81682C2116yesmaximal, direct product factor
7C2×#4802C212yesmaximal, complemented
8C2×#4800C212yesmaximal, complemented
9C2×#4770C212yesmaximal
10C2×#4768C212yesmaximal, complemented
11C2×#4714C211yesmaximal, characteristic, complemented
12C2×#4707C211yesmaximal, characteristic, complemented
13C2×#3140—31nomaximal, Sylow
14C22×D3×Q8C2211yescharacteristic
15C22×D3×D4C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic, complemented
17C2×C3×#124C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#946—21no—
20C2×Q8×Dic3—21no—
21C2×#936C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2212yes—
23#5728—24no—
24#5728C2218yescomplemented
25D3×#124—24no—
26D3×#124C2218yes—
27C2×#897—21no—
28C2×#892C2212yes—
29C2×#890C2212yescomplemented
30C2×#887—21no—
31C2×#885—21no—
32C2×#885—21no—
33C2×D3×C22⋊C4—21no—
34C2×D3×C22⋊C4—21no—
35C2×#875C2211yescharacteristic
36C2×C42×D3C2211yescharacteristic
37C2×#869C2211yescharacteristic, complemented
38C3×#489C2218yescomplemented
39C2×C3×#103C2212yescomplemented
40C2×C3×#102C2212yes—
41C2×C3×#97C2211yescharacteristic, complemented
42C2×#835C2212yescomplemented
43C2×#832C2212yes—
44C2×#821C2212yes—
45C2×#816C2212yescomplemented
46C2×C8×Dic3C2212yescomplemented
47#5089C2218yescomplemented
48#5055C2218yescomplemented
49C2×#810C2212yescomplemented
50C2×#809C2212yes—
51C2×#803C2211yescharacteristic, complemented
52#4802C22116yescomplemented
53#4800C22116yescomplemented
54#4770C22116yes—
55#4768C22116yescomplemented
56#4714C2218yescomplemented
57#4707C2218yescomplemented
58C2×#533—31no—
59C22×#124—31no—
60#3140—316no—
61C2×#489—31no—
62C2×#489—31no—
63C2×#489—31no—
64C2×#489D311yescharacteristic, complemented
65C2×#488—32no—
66C2×#488—32no—
67C2×#419—32no—
68C2×#418—32no—
69C2×C4×M16—31no—
70C24×D3—21no—
71C22×C3×Q8—21no—
72C22×C3×Q8C2311yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C22×#67—21no—
75C2×D3×Q8—22no—
76C2×D3×Q8—28no—
77C2×D3×Q8C2314yes—
78C2×D3×D4—22no—
79C2×D3×D4—28no—
80C2×D3×D4C2314yes—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3D411yescharacteristic
84C22×C4×D3D411yescharacteristic
85C22×C4×D3C2311yescharacteristic
86C22×Dic6—21no—
87C22×Dic6—21no—
88C22×Dic6C2311yescharacteristic
89C22×C3×C8C2312yes—
90C3×C4⋊Q8—22no—
91C3×C4⋊Q8C2314yescomplemented
92C3×#124—22no—
93C3×#124C2314yes—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101#947—22no—
102#947C2314yes—
103#946—28no—
104Q8×Dic3—28no—
105#936—22no—
106#936C2314yes—
107C2×(C2×D3)⋊C4—22no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—22no—
114C2×C4⋊Dic3C2312yes—
115C2×Dic3⋊C4—22no—
116C2×C4×Dic3—22no—
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121C2×ΓL(1,𝔽25)—28no—
122C2×ΓL(1,𝔽25)C2318yes—
123#897—28no—
124#892—24no—
125#892C2318yes—
126#890—24no—
127#890C2318yescomplemented
128#887—28no—
129#885—28no—
130#885—28no—
131D3×C22⋊C4—28no—
132D3×C22⋊C4—28no—
133#875—22no—
134#875C2314yes—
135C42×D3D414yes—
136C42×D3C2314yes—
137#869—22no—
138#869C2314yescomplemented
139C3×#103C2318yescomplemented
140C3×#102C2318yes—
141C3×#97C2314yescomplemented
142#835C2318yescomplemented
143#832C2318yes—
144#821C2318yes—
145#816C2318yescomplemented
146C8×Dic3C2318yescomplemented
147#810C2318yescomplemented
148#809C2318yes—
149#803C2314yescomplemented
150C23×Q8—31no—
151C23×D4—31no—
152C22×M16—32no—
153#533—38no—
154#533—64no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8C2×D311yescharacteristic, complemented
157C2×#125—32no—
158C2×#124—31no—
159C2×#124—31no—
160C2×#124—38no—
161C2×#124—31no—
162C2×#124—61no—
163C2×#124—61no—
164C2×#124—64no—
165C2×#124C2×D311yescharacteristic
166C2×C22⋊Q8—61no—
167C2×C22⋊Q8—61no—
168C2×C4×Q8—61no—
169C2×C4×Q8—61no—
170C2×#117—32no—
171C22×C22⋊C4—61no—
172C22×C22⋊C4—61no—
173C22×C42—31no—
174#489—38no—
175#489—38no—
176#489—38no—
177#489C2×D318yescomplemented
178#488—316no—
179#488—316no—
180#419—316no—
181#418—316no—
182C2×#103—32no—
183C2×#103—32no—
184C2×#103—32no—
185C2×#103C2×D312yescomplemented
186C2×#102—32no—
187C2×#102—32no—
188C2×#102—32no—
189C2×#102C2×D312yes—
190C4×M16—38no—
191C2×#97—31no—
192C2×#97C2×D311yescharacteristic, complemented
193C2×C4×C8—32no—
194C24×C3—21no—
195C23×D3—21no—
196C23×D3—24no—
197C23×D3—44no—
198C23×D3—44no—
199C23×D3—44no—
200C23×D3C2×D411yescharacteristic
201C2×C3×Q8—24no—
202C2×C3×Q8—21no—
203C2×C3×Q8—24no—
204C2×C3×Q8—44no—
205C2×C3×Q8C2412yes—
206C2×C3×D4—21no—
207C2×C3×D4—24no—
208C2×C3×D4C2412yes—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—24no—
215C2×#67—44no—
216C22×Dic3—22no—
217C22×Dic3—21no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221D3×Q8—216no—
222D3×Q8—48no—
223D3×D4—216no—
224D3×D4—48no—
225C2×D12—24no—
226C2×D12—21no—
227C2×D12C2412yes—
228C2×C4×D3—24no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—44no—
232C2×C4×D3C2×D414yes—
233C2×C4×D3C2×D414yes—
234C2×C4×D3C2×D412yes—
235C2×C4×D3#4118yes—
236C2×C4×D3C2412yes—
237C2×Dic6—24no—
238C2×Dic6—24no—
239C2×Dic6—21no—
240C2×Dic6—24no—
241C2×Dic6—44no—
242C2×Dic6—44no—
243C2×Dic6C2412yes—
244C2×C3×C8—24no—
245C2×C3×C8C2414yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4C2414yes—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C42C2×D412yes—
252C3×C42C2412yes—
253#217—24no—
254#217—24no—
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—24no—
260C4⋊Dic3—22no—
261C4⋊Dic3—24no—
262C4⋊Dic3—28no—
263C4⋊Dic3C2414yes—
264Dic3⋊C4—28no—
265C4×Dic3—28no—
266C4×Dic3C2×D412yes—
267C4×Dic3C2×D414yes—
268C4×Dic3C2×D414yes—
269C4×Dic3C2412yes—
270C2×C3⋊C8—24no—
271C2×C3⋊C8C2414yes—
272ΓL(1,𝔽25)—216no—
273C25—61no—
274C22×Q8—21no—
275C22×Q8—34no—
276C22×Q8—31no—
277C22×Q8—61no—
278C22×Q8—62no—
279C22×Q8—68no—
280C22×Q8—61no—
281C22×Q8C22×D311yescharacteristic
282C22×D4—34no—
283C22×D4—31no—
284C22×D4—68no—
285C22×D4—62no—
286C22×D4—61no—
287C22×D4C22×D311yescharacteristic
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—31no—
291C23×C4—61no—
292C2×M16—38no—
293C2×M16—68no—
294C22×C8—32no—
295C22×C8C22×D312yes—
296C4⋊Q8—22no—
297C4⋊Q8—34no—
298C4⋊Q8—62no—
299C4⋊Q8C22×D314yescomplemented
300#125—38no—
301#125—64no—
302#124—22no—
303#124—34no—
304#124—34no—
305#124—34no—
306#124—68no—
307#124—68no—
308#124—62no—
309#124—62no—
310#124—62no—
311#124C22×D314yes—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314C4×Q8—68no—
315C4×Q8—68no—
316#117—38no—
317#117—64no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—62no—
321C2×C4⋊C4—62no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4C22×D312yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—68no—
327C2×C22⋊C4—68no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—62no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—34no—
339C2×C42—34no—
340C2×C42—62no—
341C2×C42C22×D311yescharacteristic
342#103—38no—
343#103—38no—
344#103—38no—
345#103C22×D318yescomplemented
346#102—38no—
347#102—38no—
348#102—38no—
349#102C22×D318yes—
350#97—34no—
351#97C22×D314yescomplemented
352C4×C8—38no—
353C23×C3—22no—
354C23×C3—42no—
355C23×C3—42no—
356C23×C3—42no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—48no—
362C22×D3—48no—
363C22×D3—42no—
364C22×D3—42no—
365C22×D3—42no—
366C22×D3—48no—
367C22×D3—816no—
368C22×D3#12414yes—
369C22×D3C22×D412yes—
370C3×Q8—24no—
371C3×Q8—48no—
372C3×Q8—42no—
373C3×D4—24no—
374C3×D4—42no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—42no—
380C2×C3×C4—44no—
381C2×C3×C4C22×D412yes—
382C2×C3×C4C22×D412yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4114yes—
385C2×C3×C4C2511yescharacteristic
386#67—48no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—22no—
391C2×Dic3—42no—
392C2×Dic3—44no—
393C2×Dic3#12414yes—
394C2×Dic3C22×D412yes—
395C2×Dic3C22×D412yes—
396C2×Dic3C22×D412yes—
397D12—24no—
398D12—42no—
399C4×D3—44no—
400C4×D3—44no—
401C4×D3—48no—
402C4×D3#12418yes—
403C4×D3C2×#4118yes—
404Dic6—24no—
405Dic6—48no—
406Dic6—42no—
407Dic6—48no—
408C3×C8—24no—
409C3⋊C8—24no—
410C24—21no—
411C24—31no—
412C24—64no—
413C24—61no—
414C24—124no—
415C24—124no—
416C24—124no—
417C2×Q8—24no—
418C2×Q8—21no—
419C2×Q8—24no—
420C2×Q8—32no—
421C2×Q8—44no—
422C2×Q8—64no—
423C2×Q8—616no—
424C2×Q8—64no—
425C2×Q8—61no—
426C2×Q8—64no—
427C2×Q8—124no—
428C2×Q8—124no—
429C2×Q8—128no—
430C2×Q8C23×D312yes—
431C2×D4—21no—
432C2×D4—24no—
433C2×D4—32no—
434C2×D4—616no—
435C2×D4—64no—
436C2×D4—61no—
437C2×D4—64no—
438C2×D4—128no—
439C2×D4—124no—
440C2×D4C23×D312yes—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—32no—
445C22×C4—38no—
446C22×C4—32no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—34no—
450C22×C4—34no—
451C22×C4—64no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—124no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460M16—616no—
461C2×C8—24no—
462C2×C8—34no—
463C2×C8—64no—
464C2×C8C23×D314yes—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—34no—
468C4⋊C4—68no—
469C4⋊C4—68no—
470C4⋊C4—62no—
471C4⋊C4—64no—
472C4⋊C4C23×D314yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C22⋊C4—68no—
478C22⋊C4—64no—
479C22⋊C4—64no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C42—34no—
483C42—34no—
484C42—32no—
485C42—32no—
486C42—68no—
487C42D3×D412yes—
488C42C23×D312yes—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—84no—
493C22×C3C2×#12412yes—
494C22×C3#57512yes—
495C22×C3#57512yes—
496C22×C3C23×D411yescharacteristic
497C2×D3—28no—
498C2×D3—22no—
499C2×D3—22no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—816no—
504C2×D3—84no—
505C2×D3C2×#12414yes—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4—44no—
510C3×C4C2×#12412yes—
511C3×C4C22×#4112yes—
512Dic3—22no—
513Dic3—22no—
514Dic3—42no—
515Dic3—44no—
516Dic3C2×#12414yes—
517C23—22no—
518C23—34no—
519C23—32no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—62no—
524C23—62no—
525C23—62no—
526C23—128no—
527C23—128no—
528C23—128no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—2416no—
533C23C2×D3×D411yescharacteristic
534Q8—24no—
535Q8—48no—
536Q8—42no—
537Q8—64no—
538Q8—128no—
539Q8—128no—
540Q8—122no—
541D4—24no—
542D4—42no—
543D4—64no—
544D4—128no—
545D4—122no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—34no—
551C2×C4—32no—
552C2×C4—38no—
553C2×C4—38no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4—128no—
565C2×C4—122no—
566C2×C4—124no—
567C2×C4C2×D3×D412yes—
568C2×C4C2×D3×D412yes—
569C2×C4C2×D3×D411yescharacteristic
570C2×C4D3×#4114yes—
571C2×C4C24×D311yescharacteristic
572C8—24no—
573C8—64no—
574C2×C3—82no—
575C2×C3#314014yescomplemented
576C2×C3C22×#12411yescharacteristic
577C2×C3C2×#57511yescharacteristic
578C2×C3C2×#57511yescharacteristic
579D3—24no—
580D3—82no—
581C22—34no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—68no—
586C22—62no—
587C22—62no—
588C22—84no—
589C22—122no—
590C22—122no—
591C22—122no—
592C22—2416no—
593C22—244no—
594C22D3×#12412yes—
595C22#577312yes—
596C22#577712yes—
597C22C22×D3×D411yescharacteristic
598C4—22no—
599C4—22no—
600C4—34no—
601C4—42no—
602C4—44no—
603C4—62no—
604C4—62no—
605C4—122no—
606C4—124no—
607C4D3×#12412yes—
608C4C2×D3×#4112yes—
609C3C2×#314011yescharacteristic, complemented, Sylow
610C2—64no—
611C2—82no—
612C2—242no—
613C2#8168214yesdirect product factor
614C2C2×D3×#12411yescharacteristic
615C2C2×#577311yescharacteristic
616C2C2×#577711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenterC2×#489D3
hypocenterC3C2×#3140
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×#5728C2semidirect
3. C2×C3×#489C2semidirect
4. C2×#5089C2semidirect
5. C2×#5055C2semidirect
6. #81682C2direct (N × H)
7. C2×#4802C2semidirect
8. C2×#4800C2semidirect
10. C2×#4768C2semidirect
11. C2×#4714C2semidirect
12. C2×#4707C2semidirect
16. C2×C3×C4⋊Q8C22semidirect
24. #5728C22semidirect
29. C2×#890C22semidirect
37. C2×#869C22semidirect
38. C3×#489C22semidirect
39. C2×C3×#103C22semidirect
41. C2×C3×#97C22semidirect
42. C2×#835C22semidirect
45. C2×#816C22semidirect
46. C2×C8×Dic3C22semidirect
47. #5089C22semidirect
48. #5055C22semidirect
49. C2×#810C22semidirect
51. C2×#803C22semidirect
52. #4802C22semidirect
53. #4800C22semidirect
55. #4768C22semidirect
56. #4714C22semidirect
57. #4707C22semidirect
64. C2×#489D3semidirect
91. C3×C4⋊Q8C23semidirect
127. #890C23semidirect
138. #869C23semidirect
139. C3×#103C23semidirect
141. C3×#97C23semidirect
142. #835C23semidirect
145. #816C23semidirect
146. C8×Dic3C23semidirect
147. #810C23semidirect
149. #803C23semidirect
156. C2×C4⋊Q8C2×D3semidirect
177. #489C2×D3semidirect
185. C2×#103C2×D3semidirect
192. C2×#97C2×D3semidirect
299. C4⋊Q8C22×D3semidirect
345. #103C22×D3semidirect
351. #97C22×D3semidirect
575. C2×C3#3140semidirect
609. C3C2×#3140semidirect
613. C2#81682direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57282
C2×D3×#1242
C2×C3×#4892
C2×#50892
C2×#50552
#816822
C2×#48022
C2×#48002
C2×#47702
C2×#47682
C2×#47142
C2×#47072
C2×#31403