Finite Group Databasebeta

← C2×#83566 #11670351

Subgroups — C2×#83566 #11670351

Order 768.

2794 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C22×#942C212yesmaximal, complemented
3C2×#5553C211yesmaximal, characteristic
4C2×#5227C212yesmaximal, complemented
5#83566C2116yesmaximal, direct product factor
6C2×#5090C211yesmaximal, characteristic, complemented
7C2×#5085C211yesmaximal, characteristic, complemented
8C2×#5047C212yesmaximal, complemented
9C2×#5049C212yesmaximal, complemented
10C2×#5027C211yesmaximal, characteristic, complemented
11C2×C4×#216C211yesmaximal, characteristic, complemented
12C2×#5012C211yesmaximal, characteristic, complemented
13C2×#3278—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C22×#235C2212yes—
16C3×D4×Q8—24no—
17C3×D4×Q8C2218yescomplemented
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8—22no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C22×#216C2212yes—
26C22×#216C2211yescharacteristic
27C2×#942—216no—
28C2×#942C22116yescomplemented
29C22×#215C2212yes—
30C22×#215C2212yes—
31C22×#215C2212yes—
32C2×#940—21no—
33C2×#930C2212yes—
34C22×#208C2212yes—
35#5553—24no—
36#5553C2218yes—
37C2×#893—21no—
38C2×#884—22no—
39C2×#874C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41#5227C22116yescomplemented
42C2×#835C2212yes—
43C2×#835C2211yescharacteristic
44C2×#830C2212yes—
45#5090C2218yescomplemented
46#5085C2218yescomplemented
47#5049C22116yescomplemented
48#5047C22116yescomplemented
49C4×#216C2218yescomplemented
50#5027C2218yescomplemented
51#5012C2218yescomplemented
52C2×#810C2211yescharacteristic
53C2×#809C2212yes—
54C2×#807C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57C22×#137—32no—
58C2×#550—31no—
59C2×D4×Q8D311yescharacteristic, complemented
60#3278—316no—
61C2×#492—31no—
62C2×#468—32no—
63C2×#462—31no—
64C2×#460—31no—
65C2×#452—32no—
66C2×#448—32no—
67C2×C4×Q16—31no—
68C2×#436—31no—
69C22×C3×Q8—22no—
70C22×C3×Q8—24no—
71C22×C3×Q8—28no—
72C22×C3×Q8—44no—
73C22×C3×Q8C2312yes—
74C22×C3×Q8C2318yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4D411yescharacteristic
77C23×C3×C4—22no—
78C23×C3×C4C2312yes—
79C22×#67—22no—
80C22×#67—22no—
81C2×#235—24no—
82C2×#235—28no—
83C2×#235—44no—
84C2×#235C2318yes—
85C22×D12—21no—
86C22×D12—21no—
87C22×D12C2312yes—
88C22×C4×D3—22no—
89C22×Dic6C2311yescharacteristic
90C3×C4⋊Q8—28no—
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8C2314yes—
93C3×C22⋊Q8—24no—
94C3×C22⋊Q8—216no—
95C3×C22⋊Q8C2318yes—
96C3×C4×Q8—22no—
97C3×C4×Q8C2314yes—
98C3×C4×D4—28no—
99C3×C4×D4D414yes—
100C3×C4×D4C2314yes—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4D412yes—
109C2×C3×C42—21no—
110C2×C3×C42C2311yescharacteristic
111C2×#216—28no—
112C2×#216D418yescomplemented
113C2×#216C2318yes—
114C2×#216C2314yes—
115#942—264no—
116C2×#215—28no—
117C2×#215—28no—
118C2×#215—44no—
119C2×#215C2318yes—
120C2×#215C2318yes—
121C2×#215C2318yes—
122#940—28no—
123#930—24no—
124#930C2318yes—
125C2×(C2×D3)⋊C4—22no—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×Dic3⋊C4—21no—
128C2×C4×Dic3—21no—
129C2×#208—28no—
130C2×#208C2318yes—
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2312yes—
133C22×C3⋊C8C2311yescharacteristic
134#893—28no—
135#884—216no—
136#874—22no—
137#874C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140#835C2314yes—
141#835C2318yes—
142#830C2318yes—
143#810C2314yes—
144#809C2318yes—
145#807C2314yes—
146C4⋊(C3⋊C8)C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C23×Q8C2×D312yes—
150C2×#137—316no—
151C2×#137—616no—
152C22×Q16—31no—
153C22×Q16—32no—
154C22×SD16—32no—
155C22×SD16—32no—
156C22×SD16—32no—
157C22×M16—32no—
158#550—38no—
159#550—64no—
160D4×Q8—24no—
161D4×Q8C2×D318yescomplemented
162C2×C4⋊Q8—21no—
163C2×C4⋊Q8C2×D311yescharacteristic
164C2×#127—31no—
165C2×#127—61no—
166C2×C22⋊Q8—22no—
167C2×C22⋊Q8C2×D312yes—
168C2×#121—32no—
169C2×#121—62no—
170C2×C4×Q8—31no—
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4—21no—
173C2×C4×D4—61no—
174C2×C4×D4C2×D311yescharacteristic
175#492—38no—
176#468—316no—
177#462—38no—
178#460—38no—
179#452—316no—
180#448—316no—
181C4×Q16—38no—
182#436—38no—
183C2×#107—31no—
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—32no—
187C2×#103—31no—
188C2×#102—32no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3C2×D412yes—
192C23×D3—22no—
193C2×C3×Q8—21no—
194C2×C3×Q8—28no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—216no—
198C2×C3×Q8—24no—
199C2×C3×Q8—24no—
200C2×C3×Q8—48no—
201C2×C3×Q8—48no—
202C2×C3×Q8—416no—
203C2×C3×Q8C2×D414yes—
204C2×C3×Q8C2414yes—
205C2×C3×Q8C2412yes—
206C2×C3×D4—24no—
207C2×C3×D4C2×D414yes—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—28no—
214C22×C3×C4—48no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D414yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2414yes—
220C22×C3×C4C2411yescharacteristic
221C2×#67—28no—
222C2×#67—28no—
223C2×#67—48no—
224C2×#67—84no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227#235—216no—
228#235—48no—
229#235—416no—
230C2×D12—24no—
231C2×D12—24no—
232C2×D12—22no—
233C2×D12—24no—
234C2×D12—24no—
235C2×D12—44no—
236C2×D12—82no—
237C2×D12C2414yes—
238C2×C4×D3—28no—
239C2×C4×D3—48no—
240C2×Dic6—21no—
241C2×Dic6—22no—
242C2×Dic6C2×D414yes—
243C2×Dic6C2412yes—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—28no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D414yes—
250C3×C4⋊C4C2412yes—
251C3×C22⋊C4—28no—
252C3×C22⋊C4C2×D418yes—
253C3×C42—24no—
254C3×C42C2×D412yes—
255C3×C42C2412yes—
256#216—216no—
257#216C2×D4116yescomplemented
258#215—216no—
259#215—216no—
260#215—48no—
261(C2×D3)⋊C4—28no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2412yes—
264Dic3⋊C4—24no—
265C4×Dic3—24no—
266#208—216no—
267C2×C3⋊C8—24no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8C2×D414yes—
270C2×C3⋊C8C2412yes—
271C2×C3⋊C8C2414yes—
272C2×C3⋊C8C2412yes—
273C2×#41—38no—
274C2×#41—68no—
275C2×#41—64no—
276C2×#41—124no—
277C22×Q8—22no—
278C22×Q8—24no—
279C22×Q8—28no—
280C22×Q8—31no—
281C22×Q8—44no—
282C22×Q8C22×D312yes—
283C22×Q8C22×D318yes—
284C22×Q8C22×D311yescharacteristic
285C22×D4—32no—
286C22×D4—62no—
287C22×D4—62no—
288C22×D4—61no—
289C22×D4—61no—
290C22×D4#6711yescharacteristic
291C23×C4—22no—
292C23×C4—62no—
293C23×C4C22×D312yes—
294#137—664no—
295C2×Q16—38no—
296C2×Q16—34no—
297C2×Q16—38no—
298C2×Q16—68no—
299C2×SD16—38no—
300C2×SD16—38no—
301C2×SD16—38no—
302C2×SD16—68no—
303C2×SD16—68no—
304C2×SD16—124no—
305C2×M16—38no—
306C2×M16—68no—
307C22×C8—31no—
308C22×C8—32no—
309C22×C8—31no—
310C4⋊Q8—28no—
311C4⋊Q8—22no—
312C4⋊Q8C22×D314yes—
313#127—34no—
314#127—68no—
315#127—62no—
316C22⋊Q8—24no—
317C22⋊Q8—216no—
318C22⋊Q8C22×D318yes—
319#121—38no—
320#121—64no—
321#121—616no—
322C4×Q8—22no—
323C4×Q8—34no—
324C4×Q8—62no—
325C4×Q8C22×D314yes—
326C4×D4—28no—
327C4×D4—68no—
328C4×D4#6714yes—
329C4×D4C22×D314yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4#6711yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—62no—
340C2×C22⋊C4#6712yes—
341C2×C42—21no—
342C2×C42—61no—
343C2×C42C22×D311yescharacteristic
344#107—34no—
345C4⋊C8—34no—
346#103—34no—
347#103—38no—
348#103—34no—
349#102—38no—
350C22⋊C8—38no—
351C4×C8—34no—
352C23×C3—24no—
353C23×C3—24no—
354C23×C3—24no—
355C23×C3C22×D414yes—
356C23×C3C22×D411yescharacteristic
357C22×D3—24no—
358C22×D3—44no—
359C22×D3—44no—
360C22×D3—44no—
361C3×Q8—24no—
362C3×Q8—22no—
363C3×Q8—24no—
364C3×Q8—24no—
365C3×Q8—416no—
366C3×Q8—44no—
367C3×Q8C22×D414yes—
368C3×D4—28no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—28no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—416no—
379C2×C3×C4—44no—
380C2×C3×C4—42no—
381C2×C3×C4#13714yescomplemented
382C2×C3×C4C22×D412yes—
383C2×C3×C4C22×D412yes—
384C2×C3×C4C22×D414yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D412yes—
387C2×C3×C4H(5,𝔽2)12yes—
388C2×C3×C4C2511yescharacteristic
389#67—416no—
390#67—88no—
391C2×Dic3—24no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—42no—
395D12—24no—
396D12—24no—
397D12—44no—
398D12—48no—
399D12—84no—
400C4×D3—416no—
401Dic6—24no—
402Dic6—22no—
403Dic6C22×D414yes—
404C3⋊C8—24no—
405C3⋊C8—22no—
406C3⋊C8C22×D414yes—
407C24—62no—
408C24C2×#6712yes—
409#41—616no—
410#41—128no—
411#41—1216no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—216no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—34no—
420C2×Q8—32no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—416no—
424C2×Q8—62no—
425C2×Q8—61no—
426C2×Q8D3×D414yes—
427C2×Q8C23×D314yes—
428C2×Q8C23×D312yes—
429C2×D4—24no—
430C2×D4—34no—
431C2×D4—68no—
432C2×D4—68no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4—62no—
437C2×D4—64no—
438C2×D4—128no—
439C2×D4—124no—
440C2×D4—242no—
441C2×D4—244no—
442C2×D4C2×#6714yes—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—48no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—68no—
453C22×C4—128no—
454C22×C4C2×#6714yes—
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6712yes—
457C22×C4C2×#6711yescharacteristic
458C22×C4C23×D314yes—
459C22×C4C23×D311yescharacteristic
460Q16—316no—
461Q16—616no—
462SD16—616no—
463SD16—616no—
464SD16—128no—
465M16—616no—
466C2×C8—34no—
467C2×C8—32no—
468C2×C8—32no—
469C2×C8—34no—
470C2×C8—64no—
471C2×C8—62no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—28no—
476C4⋊C4—21no—
477C4⋊C4—32no—
478C4⋊C4—61no—
479C4⋊C4—64no—
480C4⋊C4C2×#6714yes—
481C4⋊C4C23×D312yes—
482C22⋊C4—28no—
483C22⋊C4—68no—
484C22⋊C4C2×#6718yes—
485C42—24no—
486C42—64no—
487C42C2×#6712yes—
488C42C23×D312yes—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—24no—
492C22×C3—48no—
493C22×C3D4212yes—
494C22×C3C2×#13712yes—
495C22×C3#57712yes—
496C22×C3C23×D411yescharacteristic
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—88no—
501C3×C4—24no—
502C3×C4—24no—
503C3×C4—22no—
504C3×C4—44no—
505C3×C4—42no—
506C3×C4D4212yes—
507C3×C4C2×#13714yescomplemented
508C3×C4C23×D411yescharacteristic
509C3×C4C2×H(5,𝔽2)11yescharacteristic
510Dic3—24no—
511Dic3—42no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—64no—
516C23—124no—
517C23—124no—
518C23—124no—
519C23C22×#6714yes—
520C23C22×#6711yescharacteristic
521Q8—22no—
522Q8—24no—
523Q8—24no—
524Q8—24no—
525Q8—34no—
526Q8—416no—
527Q8—44no—
528Q8—62no—
529Q8—64no—
530Q8C2×D3×D414yes—
531D4—28no—
532D4—64no—
533D4—64no—
534D4—1216no—
535D4—124no—
536D4—128no—
537D4—244no—
538D4—248no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—416no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—62no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—1216no—
555C2×C4—122no—
556C2×C4#94214yescomplemented
557C2×C4C2×D3×D412yes—
558C2×C4#100412yes—
559C2×C4C22×#6712yes—
560C2×C4C22×#6714yes—
561C2×C4C22×#6711yescharacteristic
562C2×C4C22×#6712yes—
563C2×C4C24×D311yescharacteristic
564C8—34no—
565C8—64no—
566C8—62no—
567C2×C3—44no—
568C2×C3#327814yescomplemented
569C2×C3C2×D4211yescharacteristic
570C2×C3C22×#13711yescharacteristic
571C2×C3C2×#57711yescharacteristic
572D3—84no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—48no—
577C22—124no—
578C22—124no—
579C22—124no—
580C22—248no—
581C22D4×#6712yes—
582C22C2×#94212yes—
583C22#583712yes—
584C22C23×#6711yescharacteristic
585C4—24no—
586C4—24no—
587C4—22no—
588C4—44no—
589C4—42no—
590C4—64no—
591C4—122no—
592C4D4×#6712yes—
593C4C2×#94214yescomplemented
594C4C22×D3×D411yescharacteristic
595C4C2×#100411yescharacteristic
596C3C2×#327811yescharacteristic, complemented, Sylow
597C2—44no—
598C2—244no—
599C2#8356614yesdirect product factor
600C2C2×D4×#6711yescharacteristic
601C2C22×#94211yescharacteristic
602C2C2×#583711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C22×D4
prime radicalC2×#5553C2
hypercenterC2×D4×Q8D3
hypocenterC3C2×#3278
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C22×#942C2semidirect
4. C2×#5227C2semidirect
5. #83566C2direct (N × H)
6. C2×#5090C2semidirect
7. C2×#5085C2semidirect
8. C2×#5047C2semidirect
9. C2×#5049C2semidirect
10. C2×#5027C2semidirect
11. C2×C4×#216C2semidirect
12. C2×#5012C2semidirect
17. C3×D4×Q8C22semidirect
28. C2×#942C22semidirect
41. #5227C22semidirect
45. #5090C22semidirect
46. #5085C22semidirect
47. #5049C22semidirect
48. #5047C22semidirect
49. C4×#216C22semidirect
50. #5027C22semidirect
51. #5012C22semidirect
59. C2×D4×Q8D3semidirect
112. C2×#216D4semidirect
161. D4×Q8C2×D3semidirect
257. #216C2×D4semidirect
381. C2×C3×C4#137semidirect
507. C3×C4C2×#137semidirect
556. C2×C4#942semidirect
568. C2×C3#3278semidirect
593. C4C2×#942semidirect
596. C3C2×#3278semidirect
599. C2#83566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C22×#9422
C2×#55532
C2×#52272
#835662
C2×#50902
C2×#50852
C2×#50472
C2×#50492
C2×#50272
C2×C4×#2162
C2×#50122
C2×#32783