Finite Group Databasebeta

← C2×#80449 #11670426

Subgroups — C2×#80449 #11670426

Order 768.

2806 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#916C212yesmaximal, complemented
2C2×#5726C211yesmaximal, characteristic, complemented
3C2×#5555C211yesmaximal, characteristic, complemented
4C2×C3×#461C211yesmaximal, characteristic, complemented
5C2×#5173C212yesmaximal, complemented
6C2×#5092C211yesmaximal, characteristic, complemented
7C2×C4×#213C211yesmaximal, characteristic, complemented
8#80449C2116yesmaximal, direct product factor
9C2×#4841C211yesmaximal, characteristic, complemented
10C2×#4834C211yesmaximal, characteristic, complemented
11C2×#4813C212yesmaximal, complemented
12C2×#4808C212yesmaximal, complemented
13C2×#3283—31nomaximal, Sylow
14C22×#239C2212yes—
15C22×#237C2212yes—
16C22×C3×SD16C2212yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#946C2212yes—
20C22×#216C2212yes—
21C22×#216C2212yes—
22C22×#213C2211yescharacteristic
23C2×#930—21no—
24C22×C4⋊Dic3—21no—
25C2×#916—216no—
26C2×#916C22116yescomplemented
27C22×#204C2212yes—
28C22×D3×C8C2212yes—
29#5726—24no—
30#5726C2218yescomplemented
31#5555—24no—
32#5555C2218yescomplemented
33C2×#897C2212yes—
34C2×#895—21no—
35C2×#894—22no—
36C2×D3×C4⋊C4—21no—
37C2×#886—22no—
38C2×C4×D12C2211yescharacteristic
39C2×C4×D12C2211yescharacteristic
40C2×#869C2211yescharacteristic
41C3×#461C2218yescomplemented
42C2×C3×C4⋊C8C2211yescharacteristic
43C2×C3×#103C2212yes—
44C2×#832C2211yescharacteristic
45#5173C22116yescomplemented
46C2×(C2×D3)⋊C8C2212yes—
47C2×#816C2212yes—
48#5092C2218yescomplemented
49C4×#213C2218yescomplemented
50C2×#809C2211yescharacteristic
51C2×#807C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4841C2218yescomplemented
54#4834C2218yescomplemented
55#4813C22116yescomplemented
56#4808C22116yescomplemented
57C22×#135—32no—
58C2×#547—31no—
59C2×#547—31no—
60#3283—316no—
61C2×#494—31no—
62C2×#467—32no—
63C2×#461—31no—
64C2×#461D311yescharacteristic, complemented
65C2×#452—32no—
66C2×#452—32no—
67C2×C4×D8—31no—
68C2×C8×D4—31no—
69C22×C3×Q8C2312yes—
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C22×#67—22no—
73C23×Dic3—22no—
74C2×#239—24no—
75C2×#239—28no—
76C2×#239—44no—
77C2×#239C2318yes—
78C2×#237—28no—
79C2×#237—24no—
80C2×#237—44no—
81C2×#237C2318yes—
82C22×D12—22no—
83C22×D12D411yescharacteristic
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C22×C4×D3—22no—
87C22×C4×D3C2312yes—
88C22×Dic6C2312yes—
89C2×C3×SD16—28no—
90C2×C3×SD16C2318yes—
91C22×C3×C8C2312yes—
92C3×C4⋊Q8—22no—
93C3×C4⋊Q8C2314yes—
94C3×C4×D4—22no—
95C3×C4×D4C2314yes—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C42C2311yescharacteristic
101#946—24no—
102#946C2318yes—
103C2×#216—28no—
104C2×#216—44no—
105C2×#216C2318yes—
106C2×#216C2318yes—
107C2×#213D418yescomplemented
108C2×#213C2314yes—
109#930—28no—
110C2×(C2×D3)⋊C4—22no—
111C2×(C2×D3)⋊C4—22no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4D412yes—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—22no—
117C2×C4⋊Dic3—28no—
118C2×C4⋊Dic3D411yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—22no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#916—264no—
124C2×#204—28no—
125C2×#204C2318yes—
126C2×D3×C8—28no—
127C2×D3×C8C2318yes—
128#897—24no—
129#897C2318yes—
130#895—28no—
131#894—216no—
132D3×C4⋊C4—28no—
133#886—216no—
134C4×D12—22no—
135C4×D12D414yes—
136C4×D12C2314yes—
137C4×D12C2314yes—
138#869—22no—
139#869C2314yes—
140C3×C4⋊C8C2314yes—
141C3×#103C2318yes—
142#832C2314yes—
143(C2×D3)⋊C8C2318yes—
144#816C2318yes—
145#809C2314yes—
146#807C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C22×#41—32no—
150C2×#135—316no—
151C2×#135—616no—
152C22×Q16—32no—
153C22×Q16—32no—
154C22×SD16—32no—
155C22×SD16C2×D312yes—
156C22×D8—31no—
157C23×C8—32no—
158#547—38no—
159#547—38no—
160#547—64no—
161#547—64no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8C2×D311yescharacteristic
164C2×#123—62no—
165C2×#123—62no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—32no—
168C2×#121—61no—
169C2×#121—61no—
170C2×C4×D4—31no—
171C2×C4×D4—31no—
172C2×C4×D4C2×D311yescharacteristic
173C22×C4⋊C4—61no—
174C22×C4⋊C4—61no—
175#494—38no—
176#467—316no—
177#461—38no—
178#461C2×D318yescomplemented
179#452—316no—
180#452—316no—
181C4×D8—38no—
182C8×D4—38no—
183C2×#107—31no—
184C2×C4⋊C8C2×D311yescharacteristic
185C2×#103—32no—
186C2×#103C2×D312yes—
187C2×#102—31no—
188C2×#102—31no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3—21no—
192C23×D3—21no—
193C23×D3C2×D412yes—
194C2×C3×Q8—22no—
195C2×C3×Q8—24no—
196C2×C3×Q8—24no—
197C2×C3×Q8C2414yes—
198C2×C3×D4—21no—
199C2×C3×D4—22no—
200C2×C3×D4C2×D414yes—
201C2×C3×D4C2412yes—
202C22×C3×C4—22no—
203C22×C3×C4—21no—
204C22×C3×C4—24no—
205C22×C3×C4—44no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—28no—
210C2×#67—48no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3—28no—
214C22×Dic3—48no—
215C22×Dic3C2×D412yes—
216#239—216no—
217#239—48no—
218#239—416no—
219#237—216no—
220#237—48no—
221#237—416no—
222C2×D12—24no—
223C2×D12—21no—
224C2×D12—22no—
225C2×D12—28no—
226C2×D12—48no—
227C2×D12C2×D414yes—
228C2×D12C2×D414yes—
229C2×D12C2412yes—
230C2×C4×D3—24no—
231C2×C4×D3—24no—
232C2×C4×D3—24no—
233C2×C4×D3—28no—
234C2×C4×D3—48no—
235C2×C4×D3—44no—
236C2×C4×D3C2×D414yes—
237C2×C4×D3C2414yes—
238C2×Dic6—24no—
239C2×Dic6—24no—
240C2×Dic6—22no—
241C2×Dic6C2414yes—
242C3×SD16—216no—
243C2×C3×C8—24no—
244C2×C3×C8C2414yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2412yes—
249C3×C22⋊C4—24no—
250C3×C42C2×D412yes—
251C3×C42C2412yes—
252#216—216no—
253#216—48no—
254#213C2×D4116yescomplemented
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—28no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4C2×D418yes—
259C4⋊Dic3—24no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—28no—
263C4⋊Dic3C2×D414yes—
264C4⋊Dic3C2412yes—
265Dic3⋊C4—28no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8C2×D414yes—
268C2×C3⋊C8C2412yes—
269C2×C3⋊C8C2412yes—
270#204—216no—
271D3×C8—216no—
272C2×#41—38no—
273C2×#41—38no—
274C2×#41—64no—
275C2×#41—64no—
276C2×#41—68no—
277C2×#41—68no—
278C2×#41—124no—
279C2×#41—124no—
280C22×Q8—32no—
281C22×Q8C22×D312yes—
282C22×D4—31no—
283C22×D4—31no—
284C22×D4—62no—
285C22×D4—62no—
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4—32no—
289C23×C4—62no—
290C23×C4—62no—
291C23×C4—61no—
292#135—664no—
293C2×Q16—38no—
294C2×Q16—38no—
295C2×Q16—68no—
296C2×Q16—124no—
297C2×SD16—28no—
298C2×SD16—38no—
299C2×SD16—68no—
300C2×SD16C22×D318yes—
301C2×D8—38no—
302C2×D8—34no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—38no—
306C22×C8—68no—
307C22×C8C22×D312yes—
308C4⋊Q8—22no—
309C4⋊Q8—34no—
310C4⋊Q8—62no—
311C4⋊Q8C22×D314yes—
312#123—616no—
313#123—616no—
314C22⋊Q8—38no—
315C22⋊Q8—38no—
316C22⋊Q8—64no—
317C22⋊Q8—64no—
318#121—68no—
319#121—68no—
320C4×D4—22no—
321C4×D4—34no—
322C4×D4—34no—
323C4×D4—34no—
324C4×D4—62no—
325C4×D4C22×D314yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—68no—
331C2×C4⋊C4—68no—
332C2×C4⋊C4—62no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—62no—
340C2×C22⋊C4—62no—
341C2×C22⋊C4—61no—
342C2×C42C22×D311yescharacteristic
343#107—34no—
344C4⋊C8C22×D314yes—
345#103—38no—
346#103C22×D318yes—
347#102—34no—
348#102—34no—
349C22⋊C8—38no—
350C4×C8—34no—
351C23×C3—24no—
352C23×C3—22no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3C22×D411yescharacteristic
356C22×D3—24no—
357C22×D3—24no—
358C22×D3—24no—
359C22×D3—24no—
360C22×D3—22no—
361C22×D3—42no—
362C22×D3—42no—
363C22×D3C22×D414yes—
364C3×Q8—24no—
365C3×Q8—24no—
366C3×Q8—44no—
367C3×D4—24no—
368C3×D4—22no—
369C3×D4C22×D414yes—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—44no—
374C2×C3×C4—42no—
375C2×C3×C4—48no—
376C2×C3×C4#13514yescomplemented
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4H(5,𝔽2)12yes—
382C2×C3×C4C2511yescharacteristic
383#67—416no—
384C2×Dic3—24no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—416no—
388C2×Dic3—44no—
389C2×Dic3—42no—
390C2×Dic3C22×D414yes—
391D12—24no—
392D12—28no—
393D12—22no—
394D12—416no—
395D12C22×D414yes—
396C4×D3—28no—
397C4×D3—24no—
398C4×D3—24no—
399C4×D3—416no—
400C4×D3—48no—
401Dic6—24no—
402Dic6—24no—
403Dic6—44no—
404C3×C8—24no—
405C3⋊C8—22no—
406C3⋊C8C22×D414yes—
407C24—21no—
408C24—32no—
409C24—61no—
410#41—616no—
411#41—616no—
412#41—128no—
413#41—1216no—
414#41—128no—
415#41—1216no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—34no—
420C2×Q8—62no—
421C2×Q8—64no—
422C2×Q8—64no—
423C2×Q8C23×D314yes—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—34no—
427C2×D4—34no—
428C2×D4—32no—
429C2×D4—68no—
430C2×D4—68no—
431C2×D4—64no—
432C2×D4—61no—
433C2×D4—62no—
434C2×D4—128no—
435C2×D4—128no—
436C2×D4D3×D414yes—
437C2×D4C23×D312yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—32no—
442C22×C4—34no—
443C22×C4—34no—
444C22×C4—44no—
445C22×C4—68no—
446C22×C4—68no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—61no—
450C22×C4—62no—
451C22×C4—64no—
452C22×C4—128no—
453C22×C4—128no—
454C22×C4—124no—
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—616no—
459Q16—128no—
460SD16—216no—
461SD16—616no—
462D8—316no—
463C2×C8—24no—
464C2×C8—34no—
465C2×C8—32no—
466C2×C8—32no—
467C2×C8—616no—
468C2×C8—62no—
469C2×C8C23×D314yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—34no—
474C4⋊C4—32no—
475C4⋊C4—68no—
476C4⋊C4—68no—
477C4⋊C4—61no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4C23×D312yes—
481C22⋊C4—24no—
482C22⋊C4—38no—
483C22⋊C4—68no—
484C22⋊C4—68no—
485C22⋊C4—64no—
486C42D3×D412yes—
487C42C23×D312yes—
488C22×C3—24no—
489C22×C3—48no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3D4212yes—
493C22×C3C2×#13512yes—
494C22×C3#57812yes—
495C22×C3C23×D411yescharacteristic
496C2×D3—24no—
497C2×D3—24no—
498C2×D3—24no—
499C2×D3—24no—
500C2×D3—48no—
501C2×D3—48no—
502C2×D3—42no—
503C2×D3—42no—
504C3×C4—22no—
505C3×C4—44no—
506C3×C4—42no—
507C3×C4D4212yes—
508C3×C4C2×#13514yescomplemented
509C3×C4C23×D411yescharacteristic
510C3×C4C2×H(5,𝔽2)11yescharacteristic
511Dic3—24no—
512Dic3—44no—
513Dic3—42no—
514C23—22no—
515C23—24no—
516C23—34no—
517C23—42no—
518C23—42no—
519C23—62no—
520C23—64no—
521C23—64no—
522C23—64no—
523C23—64no—
524C23—122no—
525C23—122no—
526C23C2×D3×D411yescharacteristic
527Q8—24no—
528Q8—24no—
529Q8—44no—
530Q8—64no—
531Q8—64no—
532Q8—124no—
533D4—22no—
534D4—24no—
535D4—34no—
536D4—62no—
537D4—68no—
538D4—64no—
539D4—1216no—
540D4—1216no—
541D4C2×D3×D414yes—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—34no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—68no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—1216no—
556C2×C4—1216no—
557C2×C4—122no—
558C2×C4—124no—
559C2×C4—128no—
560C2×C4#91614yescomplemented
561C2×C4C2×D3×D412yes—
562C2×C4C2×D3×D412yes—
563C2×C4C2×D3×D412yes—
564C2×C4C2×D3×D411yescharacteristic
565C2×C4#100912yes—
566C2×C4C24×D311yescharacteristic
567C8—24no—
568C8—34no—
569C8—62no—
570C2×C3—44no—
571C2×C3#328314yescomplemented
572C2×C3C2×D4211yescharacteristic
573C2×C3C22×#13511yescharacteristic
574C2×C3C2×#57811yescharacteristic
575D3—44no—
576D3—44no—
577C22—24no—
578C22—48no—
579C22—42no—
580C22—42no—
581C22—64no—
582C22—64no—
583C22—64no—
584C22—64no—
585C22—128no—
586C22—128no—
587C22—122no—
588C22—122no—
589C22#560912yes—
590C22C2×#91612yes—
591C22#577912yes—
592C22C22×D3×D411yescharacteristic
593C4—22no—
594C4—42no—
595C4—44no—
596C4—64no—
597C4—122no—
598C4—124no—
599C4#560912yes—
600C4C2×#91614yescomplemented
601C4C22×D3×D411yescharacteristic
602C4C2×#100911yescharacteristic
603C3C2×#328311yescharacteristic, complemented, Sylow
604C2—44no—
605C2—124no—
606C2—124no—
607C2#8044914yesdirect product factor
608C2C2×#560911yescharacteristic
609C2C22×#91611yescharacteristic
610C2C2×#577911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
hypercenterC2×#461D3
hypocenterC3C2×#3283
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C22×#916C2semidirect
2. C2×#5726C2semidirect
3. C2×#5555C2semidirect
4. C2×C3×#461C2semidirect
5. C2×#5173C2semidirect
6. C2×#5092C2semidirect
7. C2×C4×#213C2semidirect
8. #80449C2direct (N × H)
9. C2×#4841C2semidirect
10. C2×#4834C2semidirect
11. C2×#4813C2semidirect
12. C2×#4808C2semidirect
26. C2×#916C22semidirect
30. #5726C22semidirect
32. #5555C22semidirect
41. C3×#461C22semidirect
45. #5173C22semidirect
48. #5092C22semidirect
49. C4×#213C22semidirect
53. #4841C22semidirect
54. #4834C22semidirect
55. #4813C22semidirect
56. #4808C22semidirect
64. C2×#461D3semidirect
107. C2×#213D4semidirect
178. #461C2×D3semidirect
254. #213C2×D4semidirect
376. C2×C3×C4#135semidirect
508. C3×C4C2×#135semidirect
560. C2×C4#916semidirect
571. C2×C3#3283semidirect
600. C4C2×#916semidirect
603. C3C2×#3283semidirect
607. C2#80449direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9162
C2×#57262
C2×#55552
C2×C3×#4612
C2×#51732
C2×#50922
C2×C4×#2132
#804492
C2×#48412
C2×#48342
C2×#48132
C2×#48082
C2×#32833