Finite Group Databasebeta

← C2×#77485 #10964233

Subgroups — C2×#77485 #10964233

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×#883C212yesmaximal
3C22×#67⋊C4C211yesmaximal, characteristic
4C2×C3×#397C211yesmaximal, characteristic, complemented
5C2×#4983C211yesmaximal, characteristic, complemented
6C2×#4953C211yesmaximal, characteristic, complemented
7C2×#4942C212yesmaximal, complemented
8#77485C2116yesmaximal, direct product factor
9C2×#4664C212yesmaximal, complemented
10C2×#4667C211yesmaximal, characteristic, complemented
11C2×#4661C211yesmaximal, characteristic, complemented
12C2×#4647C212yesmaximal, complemented
13C2×#2486—31nomaximal, Sylow
14C23×#67C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C23×C4×D3C2212yes—
17C23×C4×D3C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×#217C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×C4⋊Dic3C2212yes—
25C22×C4⋊Dic3C2211yescharacteristic
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2211yescharacteristic
30C22×C4×Dic3C2211yescharacteristic
31C2×D3×C4⋊C4—28no—
32C2×D3×C4⋊C4C2218yescomplemented
33C2×#883—216no—
34C2×#883C22116yes—
35C2×#67⋊C4—28no—
36C2×#67⋊C4C2218yes—
37C3×#397C2218yescomplemented
38C2×C3×#95C2212yes—
39C2×C3×#95C2211yescharacteristic
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2211yescharacteristic
43C2×C3⋊#95C2212yes—
44#4983C2218yescomplemented
45#4953C2218yescomplemented
46#4942C22116yescomplemented
47#4667C2218yescomplemented
48#4664C22116yescomplemented
49#4661C2218yescomplemented
50#4647C22116yescomplemented
51C22×#123—32no—
52C22×C4×D4—31no—
53C23×C4⋊C4—31no—
54#2486—316no—
55C2×#398—31no—
56C2×#398—31no—
57C2×#397—31no—
58C2×#397D311yescharacteristic, complemented
59C2×#389—32no—
60C2×#381—32no—
61C2×#380—31no—
62C2×#380—32no—
63C25×C3C2311yescharacteristic
64C24×D3C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C22×#67—24no—
70C22×#67—24no—
71C22×#67D418yes—
72C22×#67C2314yes—
73C23×Dic3—24no—
74C23×Dic3—28no—
75C23×Dic3C2312yes—
76C23×Dic3C2314yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2312yes—
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2311yescharacteristic
81C22×C4×D3—28no—
82C22×C4×D3—28no—
83C22×C4×D3—28no—
84C22×C4×D3—28no—
85C22×C4×D3D418yes—
86C22×C4×D3C2318yes—
87C22×C4×D3C2314yes—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C22⋊C4—28no—
91C2×C3×C22⋊C4—24no—
92C2×C3×C22⋊C4C2318yes—
93C2×C3×C22⋊C4C2314yes—
94C2×#217—28no—
95C2×#217C2318yes—
96C2×(C2×D3)⋊C4—24no—
97C2×(C2×D3)⋊C4—28no—
98C2×(C2×D3)⋊C4C2314yes—
99C2×(C2×D3)⋊C4C2318yes—
100C2×C4⋊Dic3—24no—
101C2×C4⋊Dic3—28no—
102C2×C4⋊Dic3C2318yes—
103C2×C4⋊Dic3C2314yes—
104C2×Dic3⋊C4—24no—
105C2×Dic3⋊C4—28no—
106C2×Dic3⋊C4—28no—
107C2×Dic3⋊C4—28no—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111C2×Dic3⋊C4C2314yes—
112C2×C4×Dic3—24no—
113C2×C4×Dic3C2314yes—
114D3×C4⋊C4—232no—
115#883—264no—
116#67⋊C4—232no—
117C3×#95C2318yes—
118C3×#95C2314yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C3⋊#95C2314yes—
123C23×D4—31no—
124C24×C4—31no—
125C24×C4—31no—
126C24×C4—32no—
127C2×#123—316no—
128C2×#123—616no—
129C2×C4×D4—38no—
130C2×C4×D4—68no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—32no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4—38no—
138C22×C4⋊C4—68no—
139C22×C4⋊C4C2×D311yescharacteristic
140C22×C22⋊C4—32no—
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4C2×D311yescharacteristic
144C22×C22⋊C4C2×D312yes—
145C22×C42—31no—
146#398—38no—
147#398—38no—
148#397—38no—
149#397C2×D318yescomplemented
150#389—316no—
151#381—316no—
152#380—316no—
153#380—38no—
154C2×#95—31no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95—32no—
158C2×#95C2×D311yescharacteristic
159C2×#95C2×D312yes—
160C24×C3—22no—
161C24×C3—24no—
162C24×C3—22no—
163C24×C3—24no—
164C24×C3C2×D414yes—
165C24×C3C2412yes—
166C24×C3C2411yescharacteristic
167C23×D3—24no—
168C23×D3—24no—
169C23×D3C2×D414yes—
170C23×D3C2×D414yes—
171C23×D3#4114yes—
172C23×D3C2412yes—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2412yes—
184C22×C3×C4C2414yes—
185C22×C3×C4C2414yes—
186C22×C3×C4C2412yes—
187C2×#67—232no—
188C2×#67—28no—
189C2×#67—28no—
190C2×#67C2×D4116yes—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—22no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—28no—
202C22×Dic3—216no—
203C22×Dic3—416no—
204C22×Dic3C2×D414yes—
205C22×Dic3C2×D414yes—
206C22×Dic3#4118yes—
207C22×Dic3C2412yes—
208C22×Dic3C2414yes—
209C22×Dic3C2414yes—
210C22×Dic3C2412yes—
211C22×Dic3C2412yes—
212C2×C4×D3—232no—
213C2×C4×D3—216no—
214C2×C4×D3—216no—
215C2×C4×D3—216no—
216C2×C4×D3—216no—
217C2×C4×D3—432no—
218C2×C4×D3C2×D4116yes—
219C3×C4⋊C4—28no—
220C3×C22⋊C4—216no—
221C3×C22⋊C4—28no—
222#217—216no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—216no—
225C4⋊Dic3—28no—
226C4⋊Dic3—216no—
227Dic3⋊C4—28no—
228Dic3⋊C4—216no—
229Dic3⋊C4—216no—
230Dic3⋊C4—216no—
231C4×Dic3—28no—
232C25—31no—
233C25C22×D311yescharacteristic
234C22×D4—38no—
235C22×D4—34no—
236C22×D4—64no—
237C22×D4—64no—
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—38no—
241C23×C4—34no—
242C23×C4—34no—
243C23×C4—32no—
244C23×C4—32no—
245C23×C4—31no—
246C23×C4—38no—
247C23×C4—68no—
248C23×C4—68no—
249C23×C4—68no—
250C23×C4—68no—
251C23×C4—64no—
252C23×C4—68no—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D312yes—
256C23×C4C22×D311yescharacteristic
257#123—664no—
258C4×D4—632no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—38no—
264C2×C4⋊C4—38no—
265C2×C4⋊C4—38no—
266C2×C4⋊C4—632no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4—68no—
269C2×C4⋊C4—68no—
270C2×C4⋊C4—68no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4C22×D314yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—68no—
281C2×C22⋊C4—68no—
282C2×C22⋊C4C22×D314yes—
283C2×C22⋊C4C22×D318yes—
284C2×C42—34no—
285C2×C42—64no—
286#95—38no—
287#95—38no—
288#95—34no—
289#95—38no—
290#95C22×D314yes—
291#95C22×D318yes—
292C23×C3—28no—
293C23×C3—216no—
294C23×C3—22no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—48no—
301C23×C3C22×D414yes—
302C23×C3C22×D412yes—
303C23×C3C22×D412yes—
304C23×C3C2×#4112yes—
305C23×C3C2×#4114yes—
306C23×C32−1+414yes—
307C23×C3C2511yescharacteristic
308C22×D3—28no—
309C22×D3—28no—
310C22×D3—216no—
311C22×D3—216no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3#121116yes—
315C22×D3C22×D414yes—
316C22×D3C22×D414yes—
317C22×D3C2×#4114yes—
318C2×C3×C4—28no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—48no—
329C2×C3×C4—44no—
330C2×C3×C4—48no—
331C2×C3×C4C22×D414yes—
332#67—232no—
333C2×Dic3—28no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—48no—
345C2×Dic3—44no—
346C2×Dic3—48no—
347C2×Dic3—416no—
348C2×Dic3#123116yescomplemented
349C2×Dic3C22×D414yes—
350C2×Dic3C22×D414yes—
351C2×Dic3C2×#4118yes—
352C4×D3—232no—
353C4×D3—432no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—34no—
359C24—34no—
360C24—34no—
361C24—32no—
362C24—64no—
363C24—64no—
364C24D3×D414yes—
365C24C23×D312yes—
366C24C23×D311yescharacteristic
367C2×D4—316no—
368C2×D4—68no—
369C2×D4—68no—
370C2×D4—632no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—34no—
381C22×C4—38no—
382C22×C4—32no—
383C22×C4—34no—
384C22×C4—32no—
385C22×C4—316no—
386C22×C4—32no—
387C22×C4—34no—
388C22×C4—34no—
389C22×C4—632no—
390C22×C4—616no—
391C22×C4—616no—
392C22×C4—616no—
393C22×C4—616no—
394C22×C4—68no—
395C22×C4—616no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—64no—
399C22×C4—62no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—1216no—
407C22×C4—1232no—
408C22×C4D3×D414yes—
409C22×C4C23×D312yes—
410C22×C4C23×D314yes—
411C22×C4C23×D314yes—
412C22×C4C23×D312yes—
413C4⋊C4—28no—
414C4⋊C4—616no—
415C4⋊C4—616no—
416C4⋊C4—616no—
417C4⋊C4—616no—
418C4⋊C4—68no—
419C4⋊C4—68no—
420C22⋊C4—216no—
421C22⋊C4—28no—
422C22⋊C4—68no—
423C22⋊C4—616no—
424C22⋊C4—616no—
425C42—68no—
426C22×C3—24no—
427C22×C3—28no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—44no—
431C22×C3—416no—
432C22×C3C2×#12114yes—
433C22×C3C2×#12314yes—
434C22×C3#53614yes—
435C22×C3#54718yes—
436C22×C3#55218yes—
437C22×C3C23×D411yescharacteristic
438C22×C3C23×D411yescharacteristic
439C22×C3C22×#4111yescharacteristic
440C22×C3C22×#4112yes—
441C22×C3C2×2−1+412yes—
442C2×D3—24no—
443C2×D3—24no—
444C2×D3—28no—
445C2×D3—28no—
446C2×D3—232no—
447C2×D3C2×#12118yes—
448C3×C4—24no—
449C3×C4—44no—
450C3×C4—42no—
451C3×C4—44no—
452Dic3—24no—
453Dic3—44no—
454Dic3—42no—
455Dic3—44no—
456Dic3C2×#12318yescomplemented
457C23—216no—
458C23—22no—
459C23—24no—
460C23—22no—
461C23—24no—
462C23—28no—
463C23—34no—
464C23—34no—
465C23—34no—
466C23—316no—
467C23—44no—
468C23—44no—
469C23—48no—
470C23—68no—
471C23—616no—
472C23—68no—
473C23—64no—
474C23—64no—
475C23—616no—
476C23C2×D3×D414yes—
477C23C2×D3×D412yes—
478C23C2×D3×D412yes—
479C23D3×#4114yes—
480C23D3×#4112yes—
481C23#101014yes—
482C23C24×D311yescharacteristic
483D4—632no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—316no—
495C2×C4—34no—
496C2×C4—38no—
497C2×C4—34no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—632no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—68no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—1216no—
514C2×C4—1232no—
515C2×C4—124no—
516C2×C4—128no—
517C2×C4—128no—
518C2×C4C2×D3×D414yes—
519C2×C3—44no—
520C2×C3#248618yescomplemented
521C2×C3C22×#12111yescharacteristic
522C2×C3C22×#12311yescharacteristic
523C2×C3C2×#53611yescharacteristic
524C2×C3C2×#54712yes—
525C2×C3C2×#55212yes—
526D3—28no—
527C22—24no—
528C22—28no—
529C22—38no—
530C22—42no—
531C22—44no—
532C22—416no—
533C22—42no—
534C22—64no—
535C22—68no—
536C22—632no—
537C22—64no—
538C22—68no—
539C22D3×#12114yes—
540C22D3×#12314yes—
541C22#565518yes—
542C22#567514yes—
543C22#570818yes—
544C22C22×D3×D411yescharacteristic
545C22C22×D3×D411yescharacteristic
546C22C2×D3×#4111yescharacteristic
547C22C2×D3×#4112yes—
548C22C2×#101012yes—
549C4—24no—
550C4—38no—
551C4—42no—
552C4—44no—
553C4—44no—
554C4—64no—
555C4—124no—
556C4—122no—
557C4—124no—
558C3C2×#248611yescharacteristic, complemented, Sylow
559C2—44no—
560C2—68no—
561C2#7748518yesdirect product factor
562C2C2×D3×#12111yescharacteristic
563C2C2×D3×#12311yescharacteristic
564C2C2×#565512yes—
565C2C2×#567511yescharacteristic
566C2C2×#570812yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#397C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#397D3
hypocenterC3C2×#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×D3×C4⋊C4C2semidirect
4. C2×C3×#397C2semidirect
5. C2×#4983C2semidirect
6. C2×#4953C2semidirect
7. C2×#4942C2semidirect
8. #77485C2direct (N × H)
9. C2×#4664C2semidirect
10. C2×#4667C2semidirect
11. C2×#4661C2semidirect
12. C2×#4647C2semidirect
32. C2×D3×C4⋊C4C22semidirect
37. C3×#397C22semidirect
44. #4983C22semidirect
45. #4953C22semidirect
46. #4942C22semidirect
47. #4667C22semidirect
48. #4664C22semidirect
49. #4661C22semidirect
50. #4647C22semidirect
58. C2×#397D3semidirect
149. #397C2×D3semidirect
348. C2×Dic3#123semidirect
456. Dic3C2×#123semidirect
520. C2×C3#2486semidirect
558. C3C2×#2486semidirect
561. C2#77485direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C4⋊C42
C22×#8832
C22×#67⋊C42
C2×C3×#3972
C2×#49832
C2×#49532
C2×#49422
#774852
C2×#46642
C2×#46672
C2×#46612
C2×#46472
C2×#24863