Finite Group Databasebeta

← C2×#76780 #10963126

Subgroups — C2×#76780 #10963126

Order 768.

5566 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
1C23×D3×Q8C211yesmaximal, characteristic, complemented
2C22×#878C212yesmaximal, complemented
3C22×#67⋊C4C211yesmaximal, characteristic
4C2×C3×#393C211yesmaximal, characteristic, complemented
5C2×#5230C211yesmaximal, characteristic, complemented
6C2×#4956C212yesmaximal
7C2×#4947C211yesmaximal, characteristic, complemented
8#76780C2116yesmaximal, direct product factor
9C2×#4668C212yesmaximal, complemented
10C2×#4667C211yesmaximal, characteristic, complemented
11C2×#4656C211yesmaximal, characteristic, complemented
12C2×#4646C212yesmaximal, complemented
13C2×#2409—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C22×D3×Q8—28no—
18C22×D3×Q8—216no—
19C22×D3×Q8C2218yescomplemented
20C23×C4×D3C2212yes—
21C23×C4×D3C2211yescharacteristic
22C23×Dic6C2211yescharacteristic
23C23×Dic6C2212yes—
24C22×C3×C22⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2212yes—
26C22×#217C2212yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2211yescharacteristic
29C22×C4⋊Dic3C2212yes—
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C2×#67⋊C4—28no—
35C2×#67⋊C4C2218yes—
36C2×#878—216no—
37C2×#878C22116yescomplemented
38C3×#393C2218yescomplemented
39C2×C3×#95C2212yes—
40C2×C3×#95C2211yescharacteristic
41#5230C2218yescomplemented
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2211yescharacteristic
44C2×C3⋊#95C2212yes—
45C2×C3⋊#95C2212yes—
46#4956C22116yes—
47#4947C2218yescomplemented
48#4668C22116yescomplemented
49#4667C2218yescomplemented
50#4656C2218yescomplemented
51#4646C22116yescomplemented
52C24×Q8—31no—
53C22×C22⋊Q8—32no—
54C22×C4×D4—31no—
55#2409—316no—
56C2×#395—31no—
57C2×#395—31no—
58C2×#393—31no—
59C2×#393D311yescharacteristic, complemented
60C2×#391—32no—
61C2×#386—32no—
62C2×#385—32no—
63C2×#380—31no—
64C25×C3C2311yescharacteristic
65C24×D3C2311yescharacteristic
66C22×C3×Q8—24no—
67C22×C3×Q8—28no—
68C22×C3×Q8C2314yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2312yes—
73C22×#67—24no—
74C22×#67—24no—
75C22×#67Q818yescomplemented
76C22×#67C2314yes—
77C23×Dic3—28no—
78C23×Dic3—24no—
79C23×Dic3C2314yes—
80C23×Dic3C2311yescharacteristic
81C23×Dic3C2312yes—
82C23×Dic3C2312yes—
83C23×Dic3C2311yescharacteristic
84C23×Dic3C2311yescharacteristic
85C2×D3×Q8—2128no—
86C2×D3×Q8—232no—
87C2×D3×Q8—264no—
88C22×C4×D3—28no—
89C22×C4×D3—28no—
90C22×C4×D3—28no—
91C22×C4×D3—28no—
92C22×C4×D3D418yes—
93C22×C4×D3C2318yes—
94C22×C4×D3C2314yes—
95C22×Dic6—24no—
96C22×Dic6—28no—
97C22×Dic6—28no—
98C22×Dic6—28no—
99C22×Dic6D4116yescomplemented
100C22×Dic6C2314yes—
101C22×Dic6C2318yes—
102C2×C3×C22⋊C4—28no—
103C2×C3×C22⋊C4—24no—
104C2×C3×C22⋊C4C2314yes—
105C2×C3×C22⋊C4C2318yes—
106C2×#217—28no—
107C2×#217C2318yes—
108C2×(C2×D3)⋊C4—24no—
109C2×(C2×D3)⋊C4—28no—
110C2×(C2×D3)⋊C4C2318yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×C4⋊Dic3—28no—
113C2×C4⋊Dic3C2318yes—
114C2×Dic3⋊C4—28no—
115C2×Dic3⋊C4—24no—
116C2×Dic3⋊C4—28no—
117C2×Dic3⋊C4C2318yes—
118C2×Dic3⋊C4C2314yes—
119C2×Dic3⋊C4C2318yes—
120C2×C4×Dic3—24no—
121C2×C4×Dic3C2314yes—
122#67⋊C4—232no—
123#878—264no—
124C3×#95C2318yes—
125C3×#95C2314yes—
126C3⋊#95C2318yes—
127C3⋊#95C2318yes—
128C3⋊#95C2318yes—
129C3⋊#95C2314yes—
130C23×Q8—38no—
131C23×Q8—32no—
132C23×Q8—31no—
133C23×Q8—68no—
134C23×Q8—616no—
135C23×Q8C2×D311yescharacteristic
136C23×D4—31no—
137C24×C4—31no—
138C24×C4—31no—
139C24×C4—32no—
140C2×C22⋊Q8—316no—
141C2×C22⋊Q8—616no—
142C2×C4×D4—38no—
143C2×C4×D4—68no—
144C22×C4⋊C4—31no—
145C22×C4⋊C4—32no—
146C22×C4⋊C4—32no—
147C22×C4⋊C4—32no—
148C22×C22⋊C4—32no—
149C22×C22⋊C4—31no—
150C22×C22⋊C4—32no—
151C22×C22⋊C4C2×D311yescharacteristic
152C22×C22⋊C4C2×D312yes—
153C22×C42—31no—
154#395—38no—
155#395—38no—
156#393—38no—
157#393C2×D318yescomplemented
158#391—316no—
159#386—316no—
160#385—316no—
161#380—38no—
162C2×#95—31no—
163C2×#95—32no—
164C2×#95—32no—
165C2×#95—32no—
166C2×#95C2×D311yescharacteristic
167C2×#95C2×D312yes—
168C24×C3—24no—
169C24×C3—22no—
170C24×C3—22no—
171C24×C3—24no—
172C24×C3C2×Q814yescomplemented
173C24×C3C2412yes—
174C24×C3C2411yescharacteristic
175C23×D3—24no—
176C23×D3—24no—
177C23×D3C2×D414yes—
178C23×D3C2×Q814yes—
179C23×D3#4114yes—
180C23×D3C2412yes—
181C2×C3×Q8—232no—
182C2×C3×Q8—28no—
183C2×C3×Q8—216no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4C2×D414yes—
193C22×C3×C4C2×D418yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2414yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2414yes—
198C2×#67—232no—
199C2×#67—28no—
200C2×#67—28no—
201C2×#67C2×Q8116yescomplemented
202C22×Dic3—22no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—22no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—216no—
212C22×Dic3—28no—
213C22×Dic3—416no—
214C22×Dic3C2×D418yes—
215C22×Dic3C2×D414yes—
216C22×Dic3C2×D418yes—
217C22×Dic3C2×Q814yescomplemented
218C22×Dic3C2414yes—
219C22×Dic3C2412yes—
220C22×Dic3C2414yes—
221C22×Dic3C2412yes—
222C22×Dic3C2412yes—
223D3×Q8—2256no—
224C2×C4×D3—264no—
225C2×C4×D3—216no—
226C2×C4×D3—216no—
227C2×C4×D3—232no—
228C2×C4×D3—216no—
229C2×C4×D3—216no—
230C2×C4×D3C2×D4116yes—
231C2×Dic6—232no—
232C2×Dic6—28no—
233C2×Dic6—264no—
234C2×Dic6—216no—
235C2×Dic6—216no—
236C2×Dic6—216no—
237C2×Dic6C2×D4132yescomplemented
238C3×C22⋊C4—216no—
239C3×C22⋊C4—28no—
240#217—216no—
241(C2×D3)⋊C4—28no—
242(C2×D3)⋊C4—216no—
243C4⋊Dic3—216no—
244Dic3⋊C4—216no—
245Dic3⋊C4—28no—
246Dic3⋊C4—216no—
247C4×Dic3—28no—
248C25—31no—
249C25C22×D311yescharacteristic
250C22×Q8—24no—
251C22×Q8—28no—
252C22×Q8—316no—
253C22×Q8—38no—
254C22×Q8—34no—
255C22×Q8—68no—
256C22×Q8—68no—
257C22×Q8—68no—
258C22×Q8—632no—
259C22×Q8—664no—
260C22×Q8—6128no—
261C22×Q8—64no—
262C22×Q8C22×D314yes—
263C22×D4—38no—
264C22×D4—34no—
265C22×D4—64no—
266C22×D4—64no—
267C23×C4—31no—
268C23×C4—32no—
269C23×C4—31no—
270C23×C4—38no—
271C23×C4—34no—
272C23×C4—34no—
273C23×C4—32no—
274C23×C4—31no—
275C23×C4—38no—
276C23×C4—68no—
277C23×C4—68no—
278C23×C4—68no—
279C23×C4—64no—
280C23×C4—68no—
281C23×C4—68no—
282C23×C4C22×D312yes—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D311yescharacteristic
285C23×C4C22×D312yes—
286C22⋊Q8—664no—
287C4×D4—632no—
288C2×C4⋊C4—34no—
289C2×C4⋊C4—38no—
290C2×C4⋊C4—38no—
291C2×C4⋊C4—38no—
292C2×C4⋊C4—68no—
293C2×C4⋊C4—68no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—68no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—38no—
299C2×C22⋊C4—34no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—64no—
302C2×C22⋊C4—68no—
303C2×C22⋊C4—68no—
304C2×C22⋊C4C22×D314yes—
305C2×C22⋊C4C22×D318yes—
306C2×C42—34no—
307C2×C42—64no—
308#95—34no—
309#95—38no—
310#95—38no—
311#95—38no—
312#95C22×D314yes—
313#95C22×D318yes—
314C23×C3—28no—
315C23×C3—216no—
316C23×C3—24no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—24no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—48no—
323C23×C3C22×D412yes—
324C23×C3C22×D414yes—
325C23×C3C22×Q812yes—
326C23×C3C22×Q814yescomplemented
327C23×C3C2×#4112yes—
328C23×C32−1+414yes—
329C23×C3C2511yescharacteristic
330C22×D3—28no—
331C22×D3—28no—
332C22×D3—216no—
333C22×D3—216no—
334C22×D3—24no—
335C22×D3—24no—
336C22×D3C22⋊Q8116yescomplemented
337C22×D3C22×D414yes—
338C22×D3C22×Q814yes—
339C22×D3C2×#4114yes—
340C3×Q8—232no—
341C2×C3×C4—28no—
342C2×C3×C4—216no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—44no—
352C2×C3×C4—48no—
353C2×C3×C4C22×D414yes—
354C2×C3×C4C22×D418yes—
355#67—232no—
356C2×Dic3—216no—
357C2×Dic3—28no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—22no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—44no—
368C2×Dic3—48no—
369C2×Dic3—416no—
370C2×Dic3#120116yes—
371C2×Dic3C22×D418yes—
372C2×Dic3C22×D414yes—
373C2×Dic3C22×D418yes—
374C2×Dic3C22×Q814yescomplemented
375C4×D3—264no—
376C4×D3—232no—
377Dic6—232no—
378Dic6—264no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—34no—
384C24—34no—
385C24—34no—
386C24—32no—
387C24—64no—
388C24—64no—
389C24D3×Q814yescomplemented
390C24C23×D312yes—
391C24C23×D311yescharacteristic
392C2×Q8—232no—
393C2×Q8—28no—
394C2×Q8—216no—
395C2×Q8—332no—
396C2×Q8—616no—
397C2×Q8—616no—
398C2×Q8—616no—
399C2×Q8—664no—
400C2×Q8—632no—
401C2×Q8—6256no—
402C2×Q8—68no—
403C2×D4—316no—
404C2×D4—68no—
405C2×D4—68no—
406C2×D4—632no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—34no—
416C22×C4—38no—
417C22×C4—32no—
418C22×C4—38no—
419C22×C4—34no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—316no—
423C22×C4—34no—
424C22×C4—32no—
425C22×C4—616no—
426C22×C4—616no—
427C22×C4—632no—
428C22×C4—616no—
429C22×C4—664no—
430C22×C4—68no—
431C22×C4—616no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4—64no—
435C22×C4—62no—
436C22×C4—616no—
437C22×C4—64no—
438C22×C4—62no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—1216no—
443C22×C4D3×D414yes—
444C22×C4D3×D418yes—
445C22×C4C23×D312yes—
446C22×C4C23×D314yes—
447C22×C4C23×D312yes—
448C22×C4C23×D314yes—
449C4⋊C4—616no—
450C4⋊C4—616no—
451C4⋊C4—68no—
452C4⋊C4—616no—
453C22⋊C4—28no—
454C22⋊C4—216no—
455C22⋊C4—68no—
456C22⋊C4—616no—
457C22⋊C4—616no—
458C42—68no—
459C22×C3—24no—
460C22×C3—28no—
461C22×C3—42no—
462C22×C3—416no—
463C22×C3—42no—
464C22×C3—44no—
465C22×C3C2×#12014yes—
466C22×C3C2×C22⋊Q814yes—
467C22×C3#53614yes—
468C22×C3#54818yes—
469C22×C3D4×Q818yes—
470C22×C3C23×D411yescharacteristic
471C22×C3C23×D412yes—
472C22×C3C23×Q811yescharacteristic
473C22×C3C22×#4111yescharacteristic
474C22×C3C2×2−1+412yes—
475C2×D3—24no—
476C2×D3—232no—
477C2×D3—24no—
478C2×D3—28no—
479C2×D3—28no—
480C2×D3C2×C22⋊Q818yescomplemented
481C3×C4—24no—
482C3×C4—28no—
483C3×C4—42no—
484C3×C4—44no—
485Dic3—28no—
486Dic3—24no—
487Dic3—42no—
488Dic3—44no—
489Dic3C2×#12018yes—
490C23—216no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—28no—
496C23—34no—
497C23—34no—
498C23—34no—
499C23—316no—
500C23—44no—
501C23—48no—
502C23—44no—
503C23—616no—
504C23—68no—
505C23—616no—
506C23—64no—
507C23—64no—
508C23—68no—
509C23C2×D3×D414yes—
510C23C2×D3×D412yes—
511C23C2×D3×Q812yes—
512C23C2×D3×Q814yescomplemented
513C23D3×#4112yes—
514C23#101014yes—
515C23C24×D311yescharacteristic
516Q8—232no—
517Q8—664no—
518Q8—632no—
519D4—632no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—216no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—34no—
531C2×C4—38no—
532C2×C4—316no—
533C2×C4—38no—
534C2×C4—34no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—632no—
538C2×C4—664no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—616no—
544C2×C4—68no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—1216no—
551C2×C4—128no—
552C2×C4—124no—
553C2×C4C2×D3×D414yes—
554C2×C4C2×D3×D418yes—
555C2×C3—44no—
556C2×C3#240918yescomplemented
557C2×C3C22×#12011yescharacteristic
558C2×C3C22×C22⋊Q811yescharacteristic
559C2×C3C2×#53611yescharacteristic
560C2×C3C2×#54812yes—
561C2×C3C2×D4×Q812yes—
562D3—28no—
563C22—28no—
564C22—24no—
565C22—38no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—416no—
570C22—64no—
571C22—68no—
572C22—64no—
573C22—68no—
574C22—632no—
575C22D3×#12014yes—
576C22D3×C22⋊Q814yes—
577C22#563218yes—
578C22#567514yes—
579C22#567718yes—
580C22C22×D3×D411yescharacteristic
581C22C22×D3×D412yes—
582C22C22×D3×Q811yescharacteristic
583C22C2×D3×#4111yescharacteristic
584C22C2×#101012yes—
585C4—24no—
586C4—28no—
587C4—38no—
588C4—42no—
589C4—44no—
590C4—68no—
591C4—64no—
592C4—124no—
593C4—122no—
594C3C2×#240911yescharacteristic, complemented, Sylow
595C2—44no—
596C2—68no—
597C2#7678018yesdirect product factor
598C2C2×D3×#12011yescharacteristic
599C2C2×#563212yes—
600C2C2×D3×C22⋊Q811yescharacteristic
601C2C2×#567511yescharacteristic
602C2C2×#567712yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#393C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#393D3
hypocenterC3C2×#2409
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C23×D3×Q8C2semidirect
2. C22×#878C2semidirect
4. C2×C3×#393C2semidirect
5. C2×#5230C2semidirect
7. C2×#4947C2semidirect
8. #76780C2direct (N × H)
9. C2×#4668C2semidirect
10. C2×#4667C2semidirect
11. C2×#4656C2semidirect
12. C2×#4646C2semidirect
19. C22×D3×Q8C22semidirect
37. C2×#878C22semidirect
38. C3×#393C22semidirect
41. #5230C22semidirect
47. #4947C22semidirect
48. #4668C22semidirect
49. #4667C22semidirect
50. #4656C22semidirect
51. #4646C22semidirect
59. C2×#393D3semidirect
75. C22×#67Q8semidirect
99. C22×Dic6D4semidirect
157. #393C2×D3semidirect
172. C24×C3C2×Q8semidirect
201. C2×#67C2×Q8semidirect
217. C22×Dic3C2×Q8semidirect
237. C2×Dic6C2×D4semidirect
326. C23×C3C22×Q8semidirect
336. C22×D3C22⋊Q8semidirect
374. C2×Dic3C22×Q8semidirect
389. C24D3×Q8semidirect
480. C2×D3C2×C22⋊Q8semidirect
512. C23C2×D3×Q8semidirect
556. C2×C3#2409semidirect
594. C3C2×#2409semidirect
597. C2#76780direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3×Q82
C22×#8782
C22×#67⋊C42
C2×C3×#3932
C2×#52302
C2×#49562
C2×#49472
#767802
C2×#46682
C2×#46672
C2×#46562
C2×#46462
C2×#24093