Finite Group Databasebeta

← C2×#76555 #10966535

Subgroups — C2×#76555 #10966535

Order 768.

2918 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#947C211yesmaximal, characteristic
2C22×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C2×#5229C211yesmaximal, characteristic, complemented
5C2×#4988C211yesmaximal, characteristic
6C2×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
7C2×#4939C211yesmaximal, characteristic
8#76555C2116yesmaximal, direct product factor
9C2×#4675C212yesmaximal, complemented
10C2×#4673C212yesmaximal, complemented
11C2×#4670C212yesmaximal, complemented
12C2×#4669C212yesmaximal, complemented
13C2×#2592—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C23×Dic6C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×#947—28no—
20C2×#947C2218yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×Dic3⋊C4C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C22×C4×Dic3C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C2×C4×Dic6—28no—
35C2×C4×Dic6C2218yescomplemented
36C3×#391C2218yescomplemented
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39#5229C2218yescomplemented
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4988C2218yes—
44Dic3×C4⋊C4C2218yescomplemented
45#4939C2218yes—
46#4675C22116yescomplemented
47#4673C22116yescomplemented
48#4670C22116yescomplemented
49#4669C22116yescomplemented
50C22×#124—31no—
51C22×C4×Q8—31no—
52#2592—316no—
53C2×#397—32no—
54C2×#393—32no—
55C2×#391—31no—
56C2×#391D311yescharacteristic, complemented
57C2×#390—31no—
58C2×#390—31no—
59C2×#388—32no—
60C2×#388—32no—
61C2×C4×C4⋊C4—31no—
62C24×D3C2312yes—
63C22×C3×Q8—24no—
64C22×C3×Q8—24no—
65C22×C3×Q8—24no—
66C22×C3×Q8C2314yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×Dic3C2312yes—
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2311yescharacteristic
77C22×D12—24no—
78C22×D12—24no—
79C22×D12—24no—
80C22×D12C2314yes—
81C22×Dic6—24no—
82C22×Dic6—24no—
83C22×Dic6—24no—
84C22×Dic6C2314yes—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C42—24no—
88C2×C3×C42C2314yes—
89#947—232no—
90C2×(C2×D3)⋊C4—28no—
91C2×(C2×D3)⋊C4—28no—
92C2×(C2×D3)⋊C4—28no—
93C2×(C2×D3)⋊C4—28no—
94C2×(C2×D3)⋊C4—44no—
95C2×(C2×D3)⋊C4—44no—
96C2×(C2×D3)⋊C4C2318yes—
97C2×(C2×D3)⋊C4C2318yes—
98C2×(C2×D3)⋊C4C2318yes—
99C2×(C2×D3)⋊C4C2318yes—
100C2×(C2×D3)⋊C4C2318yes—
101C2×(C2×D3)⋊C4C2318yes—
102C2×C4⋊Dic3—24no—
103C2×C4⋊Dic3—24no—
104C2×C4⋊Dic3C2314yes—
105C2×C4⋊Dic3C2314yes—
106C2×Dic3⋊C4—24no—
107C2×Dic3⋊C4—24no—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2314yes—
110C2×C4×Dic3—24no—
111C2×C4×Dic3—44no—
112C2×C4×Dic3D418yes—
113C2×C4×Dic3C2314yes—
114C2×C4×Dic3C2318yes—
115C2×C4×Dic3C2314yes—
116C4×Dic6—232no—
117C3×#95C2318yes—
118C3×#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C23×Q8—31no—
123C23×Q8C2×D311yescharacteristic
124C23×D4—31no—
125C2×#124—38no—
126C2×#124—68no—
127C2×C4×Q8—38no—
128C2×C4×Q8—68no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C42—31no—
141C22×C42—31no—
142C22×C42—32no—
143C22×C42C2×D311yescharacteristic
144#397—316no—
145#393—316no—
146#391—38no—
147#391C2×D318yescomplemented
148#390—38no—
149#390—38no—
150#388—316no—
151#388—316no—
152C4×C4⋊C4—38no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95C2×D312yes—
157C2×#95C2×D312yes—
158C24×C3C2411yescharacteristic
159C23×D3—24no—
160C23×D3—24no—
161C23×D3—24no—
162C23×D3—24no—
163C23×D3—24no—
164C23×D3—24no—
165C23×D3—24no—
166C23×D3C2414yes—
167C2×C3×Q8—28no—
168C2×C3×Q8—28no—
169C2×C3×Q8—28no—
170C2×C3×Q8—416no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D414yes—
183C22×C3×C4#4114yes—
184C22×C3×C4#4114yes—
185C22×C3×C4#4114yes—
186C22×C3×C4C2414yes—
187C22×C3×C4C2414yes—
188C22×C3×C4C2412yes—
189C22×C3×C4C2412yes—
190C22×C3×C4C2412yes—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D414yes—
204C22×Dic3C2×D414yes—
205C22×Dic3#4114yes—
206C22×Dic3C2414yes—
207C22×Dic3C2412yes—
208C22×Dic3C2412yes—
209C22×Dic3C2412yes—
210C22×Dic3C2412yes—
211C2×D12—28no—
212C2×D12—28no—
213C2×D12—28no—
214C2×D12—416no—
215C2×Dic6—232no—
216C2×Dic6—28no—
217C2×Dic6—28no—
218C2×Dic6—28no—
219C3×C4⋊C4—28no—
220C3×C42—28no—
221(C2×D3)⋊C4—216no—
222(C2×D3)⋊C4—216no—
223(C2×D3)⋊C4—216no—
224(C2×D3)⋊C4—216no—
225(C2×D3)⋊C4—48no—
226(C2×D3)⋊C4—48no—
227C4⋊Dic3—28no—
228C4⋊Dic3—28no—
229Dic3⋊C4—28no—
230Dic3⋊C4—28no—
231C4×Dic3—28no—
232C4×Dic3—48no—
233C4×Dic3C2×D4116yes—
234C25—32no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—34no—
239C22×Q8—64no—
240C22×Q8—64no—
241C22×Q8—64no—
242C22×Q8C22×D314yes—
243C22×D4—34no—
244C22×D4—64no—
245C22×D4—64no—
246C22×D4—64no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4C22×D311yescharacteristic
253C23×C4C22×D312yes—
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257#124—632no—
258C4×Q8—632no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—34no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4C22×D314yes—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—38no—
271C2×C22⋊C4—38no—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—38no—
275C2×C22⋊C4—68no—
276C2×C22⋊C4—68no—
277C2×C22⋊C4—68no—
278C2×C22⋊C4—68no—
279C2×C22⋊C4—124no—
280C2×C22⋊C4—124no—
281C2×C42—24no—
282C2×C42—38no—
283C2×C42—34no—
284C2×C42—34no—
285C2×C42—38no—
286C2×C42—64no—
287C2×C42—124no—
288C2×C42C22×D314yes—
289#95—38no—
290#95—38no—
291#95—38no—
292#95C22×D318yes—
293#95C22×D318yes—
294C23×C3C22×D412yes—
295C23×C3C2×#4112yes—
296C23×C3C2×#4112yes—
297C23×C3C2×#4112yes—
298C23×C3C2×#4112yes—
299C23×C3H(5,𝔽2)12yes—
300C23×C32−1+412yes—
301C23×C3C2511yescharacteristic
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—24no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—48no—
310C22×D3—48no—
311C22×D3—48no—
312C22×D3—48no—
313C22×D3—48no—
314C22×D3—48no—
315C22×D3—48no—
316C3×Q8—416no—
317C2×C3×C4—28no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
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—22no—
328C2×C3×C4—22no—
329C2×C3×C4—48no—
330C2×C3×C4—44no—
331C2×C3×C4—48no—
332C2×C3×C4#124116yes—
333C2×C3×C4C22×D414yes—
334C2×C3×C4C2×#4114yes—
335C2×C3×C4C2×#4114yes—
336C2×C3×C4C2×#4114yes—
337C2×Dic3—28no—
338C2×Dic3—28no—
339C2×Dic3—28no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—44no—
353C2×Dic3—48no—
354C2×Dic3C22×D414yes—
355C2×Dic3C22×D414yes—
356C2×Dic3C2×#4114yes—
357D12—416no—
358Dic6—232no—
359C24—34no—
360C24—64no—
361C24—64no—
362C24—64no—
363C24—64no—
364C24—64no—
365C24—64no—
366C24—64no—
367C24C23×D311yescharacteristic
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—416no—
372C2×Q8—68no—
373C2×Q8—68no—
374C2×Q8—632no—
375C2×Q8—68no—
376C2×D4—68no—
377C2×D4—68no—
378C2×D4—68no—
379C2×D4—1216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—34no—
394C22×C4—32no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—34no—
398C22×C4—32no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—64no—
406C22×C4—62no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—64no—
410C22×C4—62no—
411C22×C4#23514yes—
412C22×C4D3×D414yes—
413C22×C4#23714yes—
414C22×C4#23914yes—
415C22×C4C23×D314yes—
416C22×C4C23×D314yes—
417C22×C4C23×D312yes—
418C22×C4C23×D312yes—
419C22×C4C23×D312yes—
420C4⋊C4—28no—
421C4⋊C4—68no—
422C4⋊C4—68no—
423C4⋊C4—68no—
424C4⋊C4—68no—
425C22⋊C4—616no—
426C22⋊C4—616no—
427C22⋊C4—616no—
428C22⋊C4—616no—
429C22⋊C4—128no—
430C22⋊C4—128no—
431C42—28no—
432C42—316no—
433C42—68no—
434C42—128no—
435C22×C3#52514yes—
436C22×C3C2×#12414yes—
437C22×C3#53714yes—
438C22×C3#55614yes—
439C22×C3#55614yes—
440C22×C3#55914yes—
441C22×C3#55914yes—
442C22×C3C23×D411yescharacteristic
443C22×C3C22×#4111yescharacteristic
444C22×C3C22×#4111yescharacteristic
445C22×C3C22×#4111yescharacteristic
446C22×C3C22×#4111yescharacteristic
447C22×C3C2×H(5,𝔽2)11yescharacteristic
448C22×C3C2×2−1+411yescharacteristic
449C2×D3—44no—
450C2×D3—44no—
451C2×D3—44no—
452C2×D3—44no—
453C2×D3—44no—
454C2×D3—44no—
455C2×D3—44no—
456C2×D3—816no—
457C3×C4—24no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—44no—
461C3×C4C2×#12418yes—
462Dic3—24no—
463Dic3—24no—
464Dic3—24no—
465Dic3—42no—
466Dic3—44no—
467C23—64no—
468C23—64no—
469C23—64no—
470C23—64no—
471C23—64no—
472C23—64no—
473C23—64no—
474C23—128no—
475C23—128no—
476C23—128no—
477C23—128no—
478C23—128no—
479C23—128no—
480C23—128no—
481C23C2×#23512yes—
482C23C2×D3×D412yes—
483C23C2×#23712yes—
484C23C2×#23912yes—
485C23#100712yes—
486C23D3×#4112yes—
487C23#100912yes—
488C23C24×D311yescharacteristic
489Q8—416no—
490Q8—632no—
491D4—1216no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—34no—
505C2×C4—34no—
506C2×C4—34no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—68no—
517C2×C4—68no—
518C2×C4—68no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—62no—
523C2×C4—64no—
524C2×C4—62no—
525C2×C4—128no—
526C2×C4—124no—
527C2×C4#885116yes—
528C2×C4C2×#23514yes—
529C2×C4C2×D3×D414yes—
530C2×C4C2×#23714yes—
531C2×C4C2×#23914yes—
532C2×C3#259218yescomplemented
533C2×C3C2×#52511yescharacteristic
534C2×C3C22×#12411yescharacteristic
535C2×C3C2×#53711yescharacteristic
536C2×C3C2×#55611yescharacteristic
537C2×C3C2×#55611yescharacteristic
538C2×C3C2×#55911yescharacteristic
539C2×C3C2×#55911yescharacteristic
540D3—84no—
541C22—124no—
542C22—124no—
543C22—124no—
544C22—124no—
545C22—124no—
546C22—124no—
547C22—124no—
548C22—2416no—
549C22C2×#88514yes—
550C22#556814yes—
551C22#558514yes—
552C22#562714yes—
553C22#562914yes—
554C22#568114yes—
555C22#573214yes—
556C22C22×#23511yescharacteristic
557C22C22×D3×D411yescharacteristic
558C22C22×#23711yescharacteristic
559C22C22×#23911yescharacteristic
560C22C2×#100711yescharacteristic
561C22C2×D3×#4111yescharacteristic
562C22C2×#100911yescharacteristic
563C4—24no—
564C4—44no—
565C4—44no—
566C4—42no—
567C4—64no—
568C4—64no—
569C4—64no—
570C4—124no—
571C4—122no—
572C4C2×#88518yes—
573C3C2×#259211yescharacteristic, complemented, Sylow
574C2—244no—
575C2#7655518yesdirect product factor
576C2C22×#88511yescharacteristic
577C2C2×#556811yescharacteristic
578C2C2×#558511yescharacteristic
579C2C2×#562711yescharacteristic
580C2C2×#562911yescharacteristic
581C2C2×#568111yescharacteristic
582C2C2×#573211yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#391C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#391D3
hypocenterC3C2×#2592
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic6C2semidirect
3. C2×C3×#391C2semidirect
4. C2×#5229C2semidirect
6. C2×Dic3×C4⋊C4C2semidirect
8. #76555C2direct (N × H)
9. C2×#4675C2semidirect
10. C2×#4673C2semidirect
11. C2×#4670C2semidirect
12. C2×#4669C2semidirect
35. C2×C4×Dic6C22semidirect
36. C3×#391C22semidirect
39. #5229C22semidirect
44. Dic3×C4⋊C4C22semidirect
46. #4675C22semidirect
47. #4673C22semidirect
48. #4670C22semidirect
49. #4669C22semidirect
56. C2×#391D3semidirect
147. #391C2×D3semidirect
532. C2×C3#2592semidirect
573. C3C2×#2592semidirect
575. C2#76555direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9472
C22×C4×Dic62
C2×C3×#3912
C2×#52292
C2×#49882
C2×Dic3×C4⋊C42
C2×#49392
#765552
C2×#46752
C2×#46732
C2×#46702
C2×#46692
C2×#25923