Finite Group Databasebeta

← C2×#82856 #10964234

Subgroups — C2×#82856 #10964234

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic
2C22×#935C211yesmaximal, characteristic
3C22×D4×Dic3C211yesmaximal, characteristic
4C23×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×#5211C211yesmaximal, characteristic, complemented
6#82856C2116yesmaximal, direct product factor
7C2×#4982C212yesmaximal, complemented
8C2×#4976C212yesmaximal, complemented
9C2×#4952C212yesmaximal, complemented
10C2×#4943C211yesmaximal, characteristic, complemented
11C2×#4942C212yesmaximal, complemented
12C2×#4942C211yesmaximal, characteristic, complemented
13C2×#2486—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C24×Dic3C2211yescharacteristic
19C2×C3×#123—28no—
20C2×C3×#123C2218yes—
21C22×C3×C4⋊C4C2212yes—
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4C2212yes—
24C22×#217C2211yescharacteristic
25C22×#217C2211yescharacteristic
26C22×#217C2212yes—
27C22×#217C2211yescharacteristic
28C2×#935—28no—
29C2×#935C2218yes—
30C2×D4×Dic3—28no—
31C2×D4×Dic3C2218yes—
32C22×C4⋊Dic3C2211yescharacteristic
33C22×Dic3⋊C4—28no—
34C22×Dic3⋊C4C2212yes—
35C22×Dic3⋊C4C2212yes—
36C22×Dic3⋊C4C2212yes—
37C22×Dic3⋊C4C2212yes—
38C22×Dic3⋊C4C2218yescomplemented
39C22×C4×Dic3C2211yescharacteristic
40#5211C2218yescomplemented
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45C2×C3⋊#95C2212yes—
46#4982C22116yescomplemented
47#4976C22116yescomplemented
48#4952C22116yescomplemented
49#4943C2218yescomplemented
50#4942C22116yescomplemented
51#4942C2218yescomplemented
52C22×#123—31no—
53C22×#123D311yescharacteristic
54C22×C4×D4—31no—
55C23×C4⋊C4—31no—
56#2486—316no—
57C2×#398—32no—
58C2×#397—32no—
59C2×#389—32no—
60C2×#381—31no—
61C2×#381—31no—
62C2×#380—32no—
63C2×#380—31no—
64C25×C3C2311yescharacteristic
65C25×C3C2311yescharacteristic
66C22×C3×D4—24no—
67C22×C3×D4—24no—
68C22×C3×D4D418yes—
69C22×C3×D4C2314yes—
70C23×C3×C4—24no—
71C23×C3×C4—24no—
72C23×C3×C4—24no—
73C23×C3×C4C2312yes—
74C23×C3×C4C2314yes—
75C23×C3×C4C2312yes—
76C23×C3×C4C2311yescharacteristic
77C23×Dic3—28no—
78C23×Dic3—24no—
79C23×Dic3—24no—
80C23×Dic3—24no—
81C23×Dic3—24no—
82C23×Dic3—28no—
83C23×Dic3D418yes—
84C23×Dic3C2314yes—
85C23×Dic3C2314yes—
86C23×Dic3C2311yescharacteristic
87C23×Dic3C2312yes—
88C23×Dic3C2311yescharacteristic
89C23×Dic3C2312yes—
90C23×Dic3C2314yes—
91C23×Dic3C2311yescharacteristic
92C23×Dic3C2311yescharacteristic
93C3×#123—232no—
94C2×C3×C4⋊C4—28no—
95C2×C3×C4⋊C4C2318yes—
96C2×C3×C22⋊C4—24no—
97C2×C3×C22⋊C4—28no—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C22⋊C4C2318yes—
100C2×#217—24no—
101C2×#217—24no—
102C2×#217—28no—
103C2×#217—24no—
104C2×#217C2318yes—
105C2×#217C2314yes—
106C2×#217C2314yes—
107C2×#217C2314yes—
108#935—232no—
109D4×Dic3—232no—
110C2×C4⋊Dic3—24no—
111C2×C4⋊Dic3C2314yes—
112C2×Dic3⋊C4—28no—
113C2×Dic3⋊C4—28no—
114C2×Dic3⋊C4—28no—
115C2×Dic3⋊C4—232no—
116C2×Dic3⋊C4—28no—
117C2×Dic3⋊C4C2318yes—
118C2×Dic3⋊C4C2318yes—
119C2×Dic3⋊C4C2318yes—
120C2×Dic3⋊C4C2318yes—
121C2×C4×Dic3—24no—
122C2×C4×Dic3C2314yes—
123C3⋊#95C2318yes—
124C3⋊#95C2318yes—
125C3⋊#95C2318yes—
126C3⋊#95C2318yes—
127C3⋊#95C2318yes—
128C23×D4C2×D311yescharacteristic
129C24×C4—31no—
130C24×C4—31no—
131C24×C4—31no—
132C24×C4C2×D311yescharacteristic
133C2×#123—28no—
134C2×#123—38no—
135C2×#123—68no—
136C2×#123C2×D318yes—
137C2×C4×D4—38no—
138C2×C4×D4—68no—
139C22×C4⋊C4—32no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4—32no—
142C22×C4⋊C4—32no—
143C22×C4⋊C4—38no—
144C22×C4⋊C4—32no—
145C22×C4⋊C4—68no—
146C22×C4⋊C4C2×D312yes—
147C22×C22⋊C4—31no—
148C22×C22⋊C4—32no—
149C22×C22⋊C4—31no—
150C22×C22⋊C4—31no—
151C22×C22⋊C4C2×D312yes—
152C22×C22⋊C4C2×D311yescharacteristic
153C22×C42—31no—
154#398—316no—
155#397—316no—
156#389—316no—
157#381—38no—
158#381—38no—
159#380—316no—
160#380—38no—
161C2×#95—32no—
162C2×#95—32no—
163C2×#95—32no—
164C2×#95—32no—
165C2×#95—32no—
166C24×C3—24no—
167C24×C3—24no—
168C24×C3—22no—
169C24×C3—24no—
170C24×C3—22no—
171C24×C3—24no—
172C24×C3C2×D414yes—
173C24×C3C2×D414yes—
174C24×C3C2×D414yes—
175C24×C3#4114yes—
176C24×C3C2412yes—
177C24×C3C2412yes—
178C24×C3C2411yescharacteristic
179C2×C3×D4—232no—
180C2×C3×D4—28no—
181C2×C3×D4—28no—
182C2×C3×D4C2×D4116yes—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—28no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—28no—
190C22×C3×C4—24no—
191C22×C3×C4—28no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—416no—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2412yes—
199C22×Dic3—24no—
200C22×Dic3—232no—
201C22×Dic3—216no—
202C22×Dic3—22no—
203C22×Dic3—28no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—28no—
209C22×Dic3—24no—
210C22×Dic3—28no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—28no—
216C22×Dic3—216no—
217C22×Dic3—416no—
218C22×Dic3—416no—
219C22×Dic3C2×D414yes—
220C22×Dic3C2×D4116yes—
221C22×Dic3C2×D414yes—
222C22×Dic3#4118yes—
223C22×Dic3C2412yes—
224C22×Dic3C2412yes—
225C22×Dic3C2414yes—
226C22×Dic3C2414yes—
227C22×Dic3C2412yes—
228C22×Dic3C2412yes—
229C3×C4⋊C4—216no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—216no—
232#217—28no—
233#217—28no—
234#217—216no—
235#217—28no—
236C4⋊Dic3—28no—
237Dic3⋊C4—216no—
238Dic3⋊C4—216no—
239Dic3⋊C4—216no—
240Dic3⋊C4—216no—
241C4×Dic3—28no—
242C25C22×D311yescharacteristic
243C25C22×D311yescharacteristic
244C22×D4—24no—
245C22×D4—24no—
246C22×D4#6718yes—
247C22×D4C22×D314yes—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—31no—
252C23×C4—38no—
253C23×C4—34no—
254C23×C4—31no—
255C23×C4—34no—
256C23×C4—34no—
257C23×C4—32no—
258C23×C4—31no—
259C23×C4—32no—
260C23×C4—31no—
261C23×C4—68no—
262C23×C4—64no—
263C23×C4—68no—
264C23×C4—64no—
265C23×C4—64no—
266C23×C4—64no—
267C23×C4C22×D314yes—
268C23×C4C22×D312yes—
269C23×C4C22×D312yes—
270C23×C4C22×D311yescharacteristic
271#123—232no—
272#123—632no—
273C4×D4—632no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—38no—
276C2×C4⋊C4—38no—
277C2×C4⋊C4—34no—
278C2×C4⋊C4—38no—
279C2×C4⋊C4—38no—
280C2×C4⋊C4—68no—
281C2×C4⋊C4—68no—
282C2×C4⋊C4—632no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—68no—
286C2×C4⋊C4C22×D318yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—34no—
290C2×C22⋊C4—34no—
291C2×C22⋊C4—34no—
292C2×C22⋊C4—38no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—68no—
297C2×C22⋊C4C22×D314yes—
298C2×C22⋊C4C22×D318yes—
299C2×C42—34no—
300C2×C42—64no—
301#95—38no—
302#95—38no—
303#95—38no—
304#95—38no—
305#95—38no—
306C23×C3—28no—
307C23×C3—28no—
308C23×C3—216no—
309C23×C3—216no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—28no—
313C23×C3—216no—
314C23×C3—22no—
315C23×C3—24no—
316C23×C3—22no—
317C23×C3—24no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3—48no—
321C23×C3#121116yes—
322C23×C3C22×D414yes—
323C23×C3C22×D412yes—
324C23×C3C22×D414yes—
325C23×C3C22×D414yes—
326C23×C3C22×D412yes—
327C23×C3C2×#4114yes—
328C23×C3C2×#4112yes—
329C23×C3C2×#4114yes—
330C23×C32−1+414yes—
331C23×C3C2511yescharacteristic
332C3×D4—232no—
333C2×C3×C4—28no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—416no—
343C2×C3×C4—48no—
344C2×C3×C4—48no—
345C2×C3×C4C22×D414yes—
346C2×Dic3—28no—
347C2×Dic3—24no—
348C2×Dic3—232no—
349C2×Dic3—22no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—22no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—416no—
360C2×Dic3—48no—
361C2×Dic3—44no—
362C2×Dic3—44no—
363C2×Dic3—48no—
364C2×Dic3—416no—
365C2×Dic3#123116yescomplemented
366C2×Dic3C22×D414yes—
367C2×Dic3C22×D414yes—
368C2×Dic3C2×#4118yes—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24D3×D414yes—
376C24#23714yes—
377C24C2×#6714yes—
378C24C2×#6714yes—
379C24C23×D312yes—
380C24C23×D312yes—
381C24C23×D311yescharacteristic
382C2×D4—232no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4C2×#67116yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—34no—
398C22×C4—38no—
399C22×C4—32no—
400C22×C4—34no—
401C22×C4—316no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—34no—
406C22×C4—34no—
407C22×C4—416no—
408C22×C4—632no—
409C22×C4—616no—
410C22×C4—68no—
411C22×C4—616no—
412C22×C4—64no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—68no—
416C22×C4—64no—
417C22×C4—68no—
418C22×C4—64no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—68no—
426C22×C4—1216no—
427C22×C4—1216no—
428C22×C4C2×#6714yes—
429C22×C4C23×D314yes—
430C22×C4C23×D314yes—
431C22×C4C23×D312yes—
432C4⋊C4—216no—
433C4⋊C4—616no—
434C4⋊C4—616no—
435C4⋊C4—68no—
436C4⋊C4—616no—
437C4⋊C4—616no—
438C22⋊C4—28no—
439C22⋊C4—216no—
440C22⋊C4—68no—
441C22⋊C4—68no—
442C22⋊C4—68no—
443C22⋊C4—616no—
444C42—68no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—28no—
448C22×C3—28no—
449C22×C3—232no—
450C22×C3—24no—
451C22×C3—28no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3—416no—
456C22×C3C2×#12118yes—
457C22×C3C2×#12114yes—
458C22×C3C2×#12314yes—
459C22×C3#53614yes—
460C22×C3#54718yes—
461C22×C3#55218yes—
462C22×C3C23×D411yescharacteristic
463C22×C3C23×D411yescharacteristic
464C22×C3C22×#4111yescharacteristic
465C22×C3C22×#4112yes—
466C22×C3C2×2−1+412yes—
467C3×C4—24no—
468C3×C4—44no—
469C3×C4—44no—
470Dic3—24no—
471Dic3—44no—
472Dic3—42no—
473Dic3—42no—
474Dic3—44no—
475Dic3C2×#12318yescomplemented
476C23—28no—
477C23—216no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—216no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—216no—
488C23—44no—
489C23—48no—
490C23—44no—
491C23#939116yes—
492C23C2×D3×D414yes—
493C23C2×D3×D412yes—
494C23C2×#23714yes—
495C23C2×#23712yes—
496C23D3×#4114yes—
497C23#101014yes—
498C23C22×#6714yes—
499C23C22×#6714yes—
500C23C22×#6712yes—
501C23C24×D311yescharacteristic
502D4—232no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—316no—
513C2×C4—34no—
514C2×C4—38no—
515C2×C4—34no—
516C2×C4—416no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—632no—
520C2×C4—68no—
521C2×C4—64no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—1216no—
533C2×C4—124no—
534C2×C4—124no—
535C2×C4—1216no—
536C2×C4—128no—
537C2×C4—128no—
538C2×C4C22×#6714yes—
539C2×C3—28no—
540C2×C3—44no—
541C2×C3#248618yescomplemented
542C2×C3C22×#12111yescharacteristic
543C2×C3C22×#12311yescharacteristic
544C2×C3C2×#53611yescharacteristic
545C2×C3C2×#54712yes—
546C2×C3C2×#55212yes—
547C22—232no—
548C22—28no—
549C22—24no—
550C22—28no—
551C22—24no—
552C22—24no—
553C22—28no—
554C22—416no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22#564814yes—
559C22#565018yes—
560C22D3×#12314yes—
561C22C2×#93918yes—
562C22C2×#93914yes—
563C22#582418yes—
564C22C22×D3×D411yescharacteristic
565C22C22×#23711yescharacteristic
566C22C2×D3×#4112yes—
567C22C2×#101012yes—
568C22C23×#6711yescharacteristic
569C4—24no—
570C4—38no—
571C4—44no—
572C4—44no—
573C4—64no—
574C4—124no—
575C4—124no—
576C4—122no—
577C4—122no—
578C3C2×#248611yescharacteristic, complemented, Sylow
579C2—28no—
580C2—44no—
581C2#8285618yesdirect product factor
582C2C2×#564811yescharacteristic
583C2C2×#565012yes—
584C2C2×D3×#12311yescharacteristic
585C2C22×#93911yescharacteristic
586C2C2×#582412yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×#123C2
socleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC22×#123D3
hypocenterC3C2×#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
4. C23×Dic3⋊C4C2semidirect
5. C2×#5211C2semidirect
6. #82856C2direct (N × H)
7. C2×#4982C2semidirect
8. C2×#4976C2semidirect
9. C2×#4952C2semidirect
10. C2×#4943C2semidirect
11. C2×#4942C2semidirect
12. C2×#4942C2semidirect
38. C22×Dic3⋊C4C22semidirect
40. #5211C22semidirect
46. #4982C22semidirect
47. #4976C22semidirect
48. #4952C22semidirect
49. #4943C22semidirect
50. #4942C22semidirect
51. #4942C22semidirect
365. C2×Dic3#123semidirect
475. Dic3C2×#123semidirect
541. C2×C3#2486semidirect
578. C3C2×#2486semidirect
581. C2#82856direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C22×#9352
C22×D4×Dic32
C23×Dic3⋊C42
C2×#52112
#828562
C2×#49822
C2×#49762
C2×#49522
C2×#49432
C2×#49422
C2×#24863