Finite Group Databasebeta

← C2×#83584 #11670355

Subgroups — C2×#83584 #11670355

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×#550C211yesmaximal, characteristic
2C22×#951C212yesmaximal, complemented
3C2×D4×Dic6C211yesmaximal, characteristic, complemented
4C2×#5239C212yesmaximal, complemented
5#83584C2116yesmaximal, direct product factor
6C2×#5075C211yesmaximal, characteristic, complemented
7C2×#5076C211yesmaximal, characteristic, complemented
8C2×#5040C212yesmaximal, complemented
9C2×#5042C212yesmaximal, complemented
10C2×#5031C211yesmaximal, characteristic, complemented
11C2×C4×#216C211yesmaximal, characteristic, complemented
12C2×#5012C211yesmaximal, characteristic, complemented
13C2×#3278—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C23×Dic6C2212yes—
16C3×#550—24no—
17C3×#550C2218yes—
18C2×C3×#127—21no—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#121—22no—
21C2×C3×#121C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×#951—216no—
26C2×#951C22116yescomplemented
27C22×#216C2212yes—
28C22×#216C2211yescharacteristic
29C2×D4×Dic3—21no—
30C22×#214C2212yes—
31C22×#214C2212yes—
32C22×#214C2212yes—
33C2×#924C2212yes—
34C22×#208C2212yes—
35D4×Dic6—24no—
36D4×Dic6C2218yescomplemented
37C2×#888—21no—
38C2×#878—22no—
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41#5239C22116yescomplemented
42C2×#835C2211yescharacteristic
43C2×#832C2212yes—
44C2×#830C2212yes—
45#5076C2218yescomplemented
46#5075C2218yescomplemented
47#5042C22116yescomplemented
48#5040C22116yescomplemented
49#5031C2218yescomplemented
50C4×#216C2218yescomplemented
51#5012C2218yescomplemented
52C2×#810C2212yes—
53C2×#810C2211yescharacteristic
54C2×#807C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57C22×#137—32no—
58C2×#550D311yescharacteristic
59C2×D4×Q8—31no—
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—
69C2×C3×#41—24no—
70C2×C3×#41—28no—
71C2×C3×#41—44no—
72C2×C3×#41C2318yes—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×D4—22no—
75C22×C3×D4—22no—
76C22×C3×D4—21no—
77C22×C3×D4—21no—
78C22×C3×D4D411yescharacteristic
79C22×C3×D4C2312yes—
80C23×C3×C4—22no—
81C23×C3×C4C2312yes—
82C23×Dic3—22no—
83C22×Dic6—28no—
84C22×Dic6—22no—
85C22×Dic6—24no—
86C22×Dic6—44no—
87C22×Dic6C2312yes—
88C22×Dic6C2318yes—
89C22×Dic6C2311yescharacteristic
90C3×#127—28no—
91C3×#127—22no—
92C3×#127C2314yes—
93C3×#121—24no—
94C3×#121—216no—
95C3×#121C2318yes—
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⋊C4D411yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4D412yes—
106C2×C3×C42—21no—
107C2×C3×C42C2311yescharacteristic
108C2×#217—22no—
109#951—264no—
110C2×#216—28no—
111C2×#216D418yescomplemented
112C2×#216C2318yes—
113C2×#216C2314yes—
114D4×Dic3—28no—
115C2×#214—28no—
116C2×#214—28no—
117C2×#214—44no—
118C2×#214C2318yes—
119C2×#214C2318yes—
120C2×#214C2318yes—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2311yescharacteristic
124#924—24no—
125#924C2318yes—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—22no—
128C2×C4×Dic3—21no—
129C2×#208—28no—
130C2×#208C2318yes—
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2312yes—
133C22×C3⋊C8C2311yescharacteristic
134#888—28no—
135#878—216no—
136#869—22no—
137#869C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140#835C2314yes—
141#832C2318yes—
142#830C2318yes—
143#810C2314yes—
144#810C2318yes—
145#807C2314yes—
146C4⋊(C3⋊C8)C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41C2×D312yes—
149C23×Q8—32no—
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—24no—
159#550C2×D318yes—
160D4×Q8—38no—
161D4×Q8—64no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—61no—
164C2×#127—21no—
165C2×#127C2×D311yescharacteristic
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—62no—
168C2×#121—22no—
169C2×#121C2×D312yes—
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—31no—
187C2×#103—32no—
188C2×#102—32no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3—22no—
192C24×C3C2×D412yes—
193C3×#41—216no—
194C3×#41—48no—
195C3×#41—416no—
196C2×C3×Q8—21no—
197C2×C3×Q8—22no—
198C2×C3×Q8C2×D414yes—
199C2×C3×Q8C2412yes—
200C2×C3×D4—28no—
201C2×C3×D4—28no—
202C2×C3×D4—24no—
203C2×C3×D4—24no—
204C2×C3×D4—24no—
205C2×C3×D4—22no—
206C2×C3×D4—24no—
207C2×C3×D4—24no—
208C2×C3×D4—48no—
209C2×C3×D4—44no—
210C2×C3×D4—82no—
211C2×C3×D4—84no—
212C2×C3×D4C2×D414yes—
213C2×C3×D4C2414yes—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—21no—
218C22×C3×C4—28no—
219C22×C3×C4—48no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D414yes—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2414yes—
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—28no—
228C22×Dic3—21no—
229C22×Dic3—22no—
230C22×Dic3—48no—
231C2×Dic6—21no—
232C2×Dic6—22no—
233C2×Dic6—24no—
234C2×Dic6—24no—
235C2×Dic6—216no—
236C2×Dic6—22no—
237C2×Dic6—28no—
238C2×Dic6—48no—
239C2×Dic6—416no—
240C2×Dic6—48no—
241C2×Dic6C2×D414yes—
242C2×Dic6C2414yes—
243C2×Dic6C2412yes—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D414yes—
247C3×C4⋊C4C2412yes—
248C3×C22⋊C4—28no—
249C3×C22⋊C4C2×D418yes—
250C3×C42—24no—
251C3×C42C2×D412yes—
252C3×C42C2412yes—
253#217—28no—
254#216—216no—
255#216C2×D4116yescomplemented
256#214—216no—
257#214—216no—
258#214—48no—
259C4⋊Dic3—21no—
260C4⋊Dic3—24no—
261C4⋊Dic3—24no—
262C4⋊Dic3C2412yes—
263Dic3⋊C4—24no—
264Dic3⋊C4—28no—
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—24no—
274C2×#41—28no—
275C2×#41—44no—
276C2×#41C22×D318yes—
277C22×Q8—31no—
278C22×Q8—32no—
279C22×Q8—38no—
280C22×Q8—68no—
281C22×Q8—62no—
282C22×Q8—64no—
283C22×Q8—124no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—22no—
286C22×D4—22no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4#6711yescharacteristic
290C22×D4C22×D312yes—
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—34no—
311C4⋊Q8—68no—
312C4⋊Q8—62no—
313#127—28no—
314#127—22no—
315#127C22×D314yes—
316C22⋊Q8—38no—
317C22⋊Q8—64no—
318C22⋊Q8—616no—
319#121—24no—
320#121—216no—
321#121C22×D318yes—
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—31no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—62no—
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—34no—
348#103—38no—
349#102—38no—
350C22⋊C8—38no—
351C4×C8—34no—
352C23×C3—24no—
353C23×C3—24no—
354C23×C3—24no—
355C23×C3—24no—
356C23×C3—44no—
357C23×C3—44no—
358C23×C3—44no—
359C23×C3C22×D414yes—
360C23×C3C22×D411yescharacteristic
361C3×Q8—24no—
362C3×Q8—22no—
363C3×Q8C22×D414yes—
364C3×D4—28no—
365C3×D4—24no—
366C3×D4—24no—
367C3×D4—416no—
368C3×D4—44no—
369C3×D4—48no—
370C3×D4—84no—
371C3×D4—88no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—28no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—416no—
381C2×C3×C4—42no—
382C2×C3×C4#13714yescomplemented
383C2×C3×C4C22×D412yes—
384C2×C3×C4C22×D412yes—
385C2×C3×C4C22×D414yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×D412yes—
388C2×C3×C4H(5,𝔽2)12yes—
389C2×C3×C4C2511yescharacteristic
390C2×Dic3—24no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—24no—
394C2×Dic3—416no—
395C2×Dic3—44no—
396C2×Dic3—42no—
397Dic6—24no—
398Dic6—22no—
399Dic6—24no—
400Dic6—24no—
401Dic6—416no—
402Dic6—44no—
403Dic6C22×D414yes—
404C3⋊C8—24no—
405C3⋊C8—22no—
406C3⋊C8C22×D414yes—
407C24—22no—
408C24C2×#6712yes—
409#41—216no—
410#41—48no—
411#41—416no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—34no—
415C2×Q8—32no—
416C2×Q8—34no—
417C2×Q8—616no—
418C2×Q8—62no—
419C2×Q8—61no—
420C2×Q8—68no—
421C2×Q8—62no—
422C2×Q8—64no—
423C2×Q8—64no—
424C2×Q8—128no—
425C2×Q8—128no—
426C2×Q8—1216no—
427C2×Q8D3×D414yes—
428C2×Q8C23×D312yes—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—82no—
440C2×D4—84no—
441C2×D4C2×#6714yes—
442C2×D4C23×D314yes—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—28no—
448C22×C4—48no—
449C22×C4—61no—
450C22×C4—68no—
451C22×C4—61no—
452C22×C4—62no—
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—21no—
474C4⋊C4—32no—
475C4⋊C4—61no—
476C4⋊C4—64no—
477C4⋊C4—64no—
478C4⋊C4—68no—
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×C3—44no—
494C22×C3—44no—
495C22×C3—44no—
496C22×C3—88no—
497C22×C3D4212yes—
498C22×C3C2×#13712yes—
499C22×C3#57712yes—
500C22×C3C23×D411yescharacteristic
501C3×C4—24no—
502C3×C4—24no—
503C3×C4—22no—
504C3×C4—42no—
505C3×C4D4212yes—
506C3×C4C2×#13714yescomplemented
507C3×C4C23×D411yescharacteristic
508C3×C4C2×H(5,𝔽2)11yescharacteristic
509Dic3—24no—
510Dic3—44no—
511Dic3—42no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23C22×#6714yes—
520C23C22×#6711yescharacteristic
521Q8—22no—
522Q8—24no—
523Q8—34no—
524Q8—62no—
525Q8—64no—
526Q8—64no—
527Q8—64no—
528Q8—1216no—
529Q8—124no—
530Q8C2×D3×D414yes—
531D4—24no—
532D4—24no—
533D4—28no—
534D4—416no—
535D4—44no—
536D4—48no—
537D4—84no—
538D4—88no—
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—416no—
548C2×C4—42no—
549C2×C4—62no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—64no—
553C2×C4—1216no—
554C2×C4—124no—
555C2×C4—122no—
556C2×C4#95114yescomplemented
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—84no—
569C2×C3#327814yescomplemented
570C2×C3C2×D4211yescharacteristic
571C2×C3C22×#13711yescharacteristic
572C2×C3C2×#57711yescharacteristic
573C22—24no—
574C22—24no—
575C22—24no—
576C22—48no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—88no—
581C22D4×#6712yes—
582C22C2×#95112yes—
583C22#583612yes—
584C22C23×#6711yescharacteristic
585C4—24no—
586C4—24no—
587C4—22no—
588C4—42no—
589C4—64no—
590C4—124no—
591C4—122no—
592C4D4×#6712yes—
593C4C2×#95114yescomplemented
594C4C22×D3×D411yescharacteristic
595C4C2×#100411yescharacteristic
596C3C2×#327811yescharacteristic, complemented, Sylow
597C2—44no—
598C2—84no—
599C2#8358414yesdirect product factor
600C2C2×D4×#6711yescharacteristic
601C2C22×#95111yescharacteristic
602C2C2×#583611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC2×C3×#550C2
socleC23×C3C22×D4
hypercenterC2×#550D3
hypocenterC3C2×#3278
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C2×D4×Dic6C2semidirect
4. C2×#5239C2semidirect
5. #83584C2direct (N × H)
6. C2×#5075C2semidirect
7. C2×#5076C2semidirect
8. C2×#5040C2semidirect
9. C2×#5042C2semidirect
10. C2×#5031C2semidirect
11. C2×C4×#216C2semidirect
12. C2×#5012C2semidirect
26. C2×#951C22semidirect
36. D4×Dic6C22semidirect
41. #5239C22semidirect
45. #5076C22semidirect
46. #5075C22semidirect
47. #5042C22semidirect
48. #5040C22semidirect
49. #5031C22semidirect
50. C4×#216C22semidirect
51. #5012C22semidirect
111. C2×#216D4semidirect
255. #216C2×D4semidirect
382. C2×C3×C4#137semidirect
506. C3×C4C2×#137semidirect
556. C2×C4#951semidirect
569. C2×C3#3278semidirect
593. C4C2×#951semidirect
596. C3C2×#3278semidirect
599. C2#83584direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C22×#9512
C2×D4×Dic62
C2×#52392
#835842
C2×#50752
C2×#50762
C2×#50402
C2×#50422
C2×#50312
C2×C4×#2162
C2×#50122
C2×#32783