Finite Group Databasebeta

← C2×#78539 #11664964

Subgroups — C2×#78539 #11664964

Order 768.

1986 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic
2C2×#5470C211yesmaximal, characteristic, complemented
3C2×C3×C4×M16C211yesmaximal, characteristic, complemented
4C2×#5137C212yesmaximal, complemented
5C2×#5134C212yesmaximal
6#78539C2116yesmaximal, direct product factor
7C2×#4715C211yesmaximal, characteristic, complemented
8C2×#4703C211yesmaximal, characteristic, complemented
9C2×#4714C211yesmaximal, characteristic, complemented
10C2×#4702C211yesmaximal, characteristic, complemented
11C2×C4×Dic12C212yesmaximal, complemented
12C2×C4×#204C212yesmaximal, complemented
13C2×#3172—31nomaximal, Sylow
14C22×C3×M16C2212yes—
15C22×C3×C42C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#929—21no—
18C2×C4×#67—21no—
19C2×#926—21no—
20C2×#924—21no—
21C2×#924C2212yes—
22C2×#924C2211yescharacteristic
23C22×Dic12C2212yes—
24C22×#204C2212yes—
25#5479—24no—
26#5479C2218yes—
27C2×#876—21no—
28C2×#875C2211yescharacteristic
29C2×C4×D12C2211yescharacteristic
30C2×#872—21no—
31#5470—24no—
32#5470C2218yescomplemented
33C2×#870—21no—
34C2×#870C2211yescharacteristic
35C2×#869C2211yescharacteristic
36C2×#869C2211yescharacteristic
37C2×C4×Dic6—21no—
38C2×C4×Dic6C2212yes—
39C2×C4×Dic6C2211yescharacteristic
40C3×C4×M16C2218yescomplemented
41C2×C3×#97C2211yescharacteristic
42C2×C3×#97C2211yescharacteristic
43C2×C3×C4×C8C2212yes—
44#5137C22116yescomplemented
45#5134C22116yes—
46C2×#821C2212yes—
47C2×#818C2212yes—
48C2×#817C2212yes—
49C2×#816C2212yes—
50C2×#816C2212yes—
51C2×#816C2212yes—
52#4715C2218yescomplemented
53#4714C2218yescomplemented
54#4703C2218yescomplemented
55#4702C2218yescomplemented
56C4×Dic12C22116yescomplemented
57C4×#204C22116yescomplemented
58C2×#533—31no—
59C2×#529—31no—
60#3172—316no—
61C2×#501—31no—
62C2×#497—31no—
63C2×#491—31no—
64C2×#489—31no—
65C2×#470—32no—
66C2×#468—32no—
67C2×C4×Q16—32no—
68C2×C4×SD16—32no—
69C2×C4×M16D311yescharacteristic, complemented
70C23×C3×C4D411yescharacteristic
71C23×C3×C4D411yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C22×#67—21no—
74C22×D12C2311yescharacteristic
75C22×C4×D3—21no—
76C22×Dic6—21no—
77C22×Dic6C2312yes—
78C22×Dic6C2311yescharacteristic
79C2×C3×M16—28no—
80C2×C3×M16C2318yes—
81C22×C3×C8C2312yes—
82C22×C3×C8C2312yes—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D414yes—
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2314yes—
89C2×#217—22no—
90C2×#217—21no—
91#930—22no—
92#930C2314yes—
93#929—28no—
94C4×#67—28no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99#926—28no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106#924—24no—
107#924—28no—
108#924—22no—
109#924C2318yes—
110#924C2314yes—
111C2×Dic3⋊C4—22no—
112C2×Dic3⋊C4—22no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—22no—
118C2×C4×Dic3—21no—
119C2×Dic12—28no—
120C2×Dic12C2318yes—
121C2×#204—28no—
122C2×#204C2318yes—
123#876—28no—
124#875—22no—
125#875C2314yes—
126C4×D12—22no—
127C4×D12C2314yes—
128#872—28no—
129#870—28no—
130#870—22no—
131#870C2314yes—
132#869—22no—
133#869—22no—
134#869C2314yes—
135#869C2314yes—
136C4×Dic6—28no—
137C4×Dic6—24no—
138C4×Dic6—22no—
139C4×Dic6C2318yes—
140C4×Dic6C2314yes—
141C3×#97C2314yes—
142C3×#97C2314yes—
143C3×C4×C8C2318yes—
144#821C2318yes—
145#818C2318yes—
146#817C2318yes—
147#816C2318yes—
148#816C2318yes—
149#816C2318yes—
150C22×Q16—32no—
151C22×SD16—32no—
152C22×M16C2×D312yes—
153#533—38no—
154#533—64no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8—31no—
157#529—38no—
158#529—64no—
159C2×#126—61no—
160C2×#125—31no—
161C2×#125—61no—
162C2×#124—31no—
163C2×#123—61no—
164C2×C22⋊Q8—31no—
165C2×C22⋊Q8—32no—
166C2×C22⋊Q8—61no—
167C2×#121—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—32no—
170C2×C4×Q8—61no—
171C2×C4×D4—31no—
172C2×C4×D4—61no—
173C2×#117—61no—
174C2×#117—61no—
175C22×C42C2×D311yescharacteristic
176#501—38no—
177#497—38no—
178#491—38no—
179#489—38no—
180#470—316no—
181#468—316no—
182C4×Q16—316no—
183C4×SD16—316no—
184C2×#107—32no—
185C2×#106—32no—
186C2×#103—32no—
187C2×#103—32no—
188C2×#103—32no—
189C2×#102—32no—
190C4×M16C2×D318yescomplemented
191C2×#97C2×D311yescharacteristic
192C2×#97C2×D311yescharacteristic
193C2×C4×C8C2×D312yes—
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4—42no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D414yes—
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2412yes—
208C22×C3×C4C2411yescharacteristic
209C2×#67—24no—
210C2×#67—44no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C2×D12—24no—
217C2×D12—21no—
218C2×D12C2412yes—
219C2×C4×D3—24no—
220C2×C4×D3—44no—
221C2×Dic6—24no—
222C2×Dic6—24no—
223C2×Dic6—24no—
224C2×Dic6—22no—
225C2×Dic6—24no—
226C2×Dic6—21no—
227C2×Dic6—44no—
228C2×Dic6C2414yes—
229C2×Dic6C2412yes—
230C3×M16—216no—
231C2×C3×C8—24no—
232C2×C3×C8#4118yes—
233C2×C3×C8C2414yes—
234C2×C3×C8C2414yes—
235C3×C42C2×D412yes—
236C3×C42C2×D414yes—
237C3×C42C2×D412yes—
238C3×C42C2×D414yes—
239C3×C42C2412yes—
240C3×C42C2412yes—
241#217—28no—
242#217—24no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—24no—
247C4⋊Dic3—22no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—24no—
251C4⋊Dic3—24no—
252C4⋊Dic3—21no—
253C4⋊Dic3C2412yes—
254C4⋊Dic3C2412yes—
255C4⋊Dic3C2414yes—
256Dic3⋊C4—28no—
257Dic3⋊C4—28no—
258Dic3⋊C4—24no—
259Dic3⋊C4—24no—
260Dic3⋊C4—24no—
261Dic3⋊C4—24no—
262C4×Dic3—28no—
263C4×Dic3—24no—
264Dic12—216no—
265#204—216no—
266C22×Q8—31no—
267C22×Q8—32no—
268C22×Q8—61no—
269C22×D4—31no—
270C22×D4—61no—
271C23×C4—61no—
272C23×C4D1211yescharacteristic
273C23×C4D1211yescharacteristic
274C23×C4C22×D311yescharacteristic
275C2×Q16—38no—
276C2×Q16—68no—
277C2×SD16—38no—
278C2×SD16—68no—
279C2×M16—28no—
280C2×M16C22×D318yes—
281C22×C8C22×D312yes—
282C22×C8C22×D312yes—
283C4⋊Q8—34no—
284C4⋊Q8—34no—
285C4⋊Q8—62no—
286C4⋊Q8—62no—
287#126—68no—
288#125—34no—
289#125—62no—
290#125—68no—
291#124—34no—
292#124—62no—
293#123—68no—
294C22⋊Q8—34no—
295C22⋊Q8—38no—
296C22⋊Q8—62no—
297C22⋊Q8—64no—
298C22⋊Q8—68no—
299#121—34no—
300#121—62no—
301C4×Q8—34no—
302C4×Q8—38no—
303C4×Q8—62no—
304C4×Q8—64no—
305C4×Q8—68no—
306C4×D4—34no—
307C4×D4—68no—
308C4×D4—62no—
309#117—68no—
310#117—68no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—62no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—62no—
322C2×C4⋊C4—61no—
323C2×C22⋊C4—62no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—61no—
330C2×C42—62no—
331C2×C42D1211yescharacteristic
332C2×C42D1214yes—
333C2×C42D1211yescharacteristic
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D314yes—
336C2×C42C22×D311yescharacteristic
337#107—38no—
338#106—38no—
339#103—38no—
340#103—38no—
341#103—38no—
342#102—38no—
343#97C22×D314yes—
344#97C22×D314yes—
345C4×C8C22×D318yes—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3C22×D412yes—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3—42no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—28no—
361C2×C3×C4—24no—
362C2×C3×C4—44no—
363C2×C3×C4#13714yes—
364C2×C3×C4C22×D412yes—
365C2×C3×C4C22×D412yes—
366C2×C3×C4C22×D412yes—
367C2×C3×C4C22×D412yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D412yes—
370C2×C3×C4C2×#4114yes—
371C2×C3×C4C2511yescharacteristic
372#67—48no—
373C2×Dic3—22no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—44no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—44no—
382C2×Dic3—42no—
383D12—24no—
384D12—42no—
385C4×D3—48no—
386Dic6—24no—
387Dic6—24no—
388Dic6—24no—
389Dic6—48no—
390Dic6—44no—
391Dic6—42no—
392C3×C8—24no—
393C3×C8C2×#4118yes—
394C24—61no—
395C24C2×D1211yescharacteristic
396C2×Q8—32no—
397C2×Q8—34no—
398C2×Q8—61no—
399C2×Q8—64no—
400C2×Q8—62no—
401C2×Q8—64no—
402C2×Q8—64no—
403C2×Q8—64no—
404C2×Q8—124no—
405C2×D4—32no—
406C2×D4—64no—
407C2×D4—61no—
408C2×D4—64no—
409C2×D4—124no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—64no—
419C22×C4—124no—
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1212yes—
422C22×C4C2×D1211yescharacteristic
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C2×D1214yes—
426C22×C4C2×D1214yes—
427C22×C4C2×D1211yescharacteristic
428C22×C4C23×D312yes—
429C22×C4C23×D311yescharacteristic
430Q16—616no—
431SD16—616no—
432M16—216no—
433C2×C8—24no—
434C2×C8#23518yes—
435C2×C8C23×D314yes—
436C2×C8C23×D314yes—
437C4⋊C4—34no—
438C4⋊C4—32no—
439C4⋊C4—32no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—62no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—68no—
449C4⋊C4—64no—
450C4⋊C4—68no—
451C4⋊C4—64no—
452C22⋊C4—68no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C42—64no—
459C42—68no—
460C42C2×D1214yes—
461C42C2×D1212yes—
462C42C2×D1212yes—
463C42C2×D1214yes—
464C42C23×D312yes—
465C42C23×D312yes—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—44no—
470C22×C3#53212yes—
471C22×C3C2×#13712yes—
472C22×C3#57512yes—
473C22×C3C23×D411yescharacteristic
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—84no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4#53212yes—
483C3×C4C2×#13714yes—
484C3×C4C22×#4112yes—
485Dic3—44no—
486Dic3—42no—
487Dic3—42no—
488Dic3—44no—
489Dic3—42no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—62no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23C22×D1212yes—
498C23C22×D1211yescharacteristic
499Q8—64no—
500Q8—64no—
501Q8—64no—
502Q8—128no—
503Q8—122no—
504Q8—124no—
505D4—64no—
506D4—128no—
507D4—122no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—44no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—122no—
522C2×C4—124no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—124no—
526C2×C4—128no—
527C2×C4#90914yes—
528C2×C4C22×D1212yes—
529C2×C4C22×D1212yes—
530C2×C4C22×D1212yes—
531C2×C4C22×D1212yes—
532C2×C4C22×D1211yescharacteristic
533C2×C4C22×D1212yes—
534C2×C4C2×#23514yes—
535C2×C4C24×D311yescharacteristic
536C8—24no—
537C8C2×#23518yes—
538C2×C3—42no—
539C2×C3#317214yescomplemented
540C2×C3C2×#53211yescharacteristic
541C2×C3C22×#13711yescharacteristic
542C2×C3C2×#57511yescharacteristic
543D3—82no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—44no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—244no—
552C22#547712yes—
553C22C2×#90912yes—
554C22#574812yes—
555C22C23×D1211yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—122no—
561C4—124no—
562C4—122no—
563C4—122no—
564C4—124no—
565C4#547712yes—
566C4C2×#90914yes—
567C4C22×#23512yes—
568C3C2×#317211yescharacteristic, complemented, Sylow
569C2—42no—
570C2—242no—
571C2#7853914yesdirect product factor
572C2C2×#547711yescharacteristic
573C2C2×#574811yescharacteristic
574C2C22×#90911yescharacteristic
subgroupG/H
centerC22×C4C2×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×M16C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×C4×M16D3
hypocenterC3C2×#3172
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×#5470C2semidirect
3. C2×C3×C4×M16C2semidirect
4. C2×#5137C2semidirect
6. #78539C2direct (N × H)
7. C2×#4715C2semidirect
8. C2×#4703C2semidirect
9. C2×#4714C2semidirect
10. C2×#4702C2semidirect
11. C2×C4×Dic12C2semidirect
12. C2×C4×#204C2semidirect
32. #5470C22semidirect
40. C3×C4×M16C22semidirect
44. #5137C22semidirect
52. #4715C22semidirect
53. #4714C22semidirect
54. #4703C22semidirect
55. #4702C22semidirect
56. C4×Dic12C22semidirect
57. C4×#204C22semidirect
69. C2×C4×M16D3semidirect
190. C4×M16C2×D3semidirect
539. C2×C3#3172semidirect
568. C3C2×#3172semidirect
571. C2#78539direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54792
C2×#54702
C2×C3×C4×M162
C2×#51372
C2×#51342
#785392
C2×#47152
C2×#47032
C2×#47142
C2×#47022
C2×C4×Dic122
C2×C4×#2042
C2×#31723