Finite Group Databasebeta

← C2×#81669 #11665113

Subgroups — C2×#81669 #11665113

Order 768.

1998 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×#5728C211yesmaximal, characteristic, complemented
2C2×#5680C211yesmaximal, characteristic
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C2×#5081C211yesmaximal, characteristic, complemented
5C2×#5060C211yesmaximal, characteristic, complemented
6#81669C2116yesmaximal, direct product factor
7C2×#4813C212yesmaximal, complemented
8C2×#4789C212yesmaximal, complemented
9C2×#4778C212yesmaximal
10C2×#4757C212yesmaximal, complemented
11C2×#4703C211yesmaximal, characteristic, complemented
12C2×D3×#97C211yesmaximal, characteristic, complemented
13C2×#3181—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×#946—21no—
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3—21no—
19C2×Q8×Dic3C2211yescharacteristic
20C22×#216C2212yes—
21C2×#938C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C22×#214C2212yes—
24C22×D3×C8C2212yes—
25#5728—24no—
26#5728C2218yescomplemented
27#5680—24no—
28#5680C2218yes—
29C2×#897C2212yes—
30C2×#892—21no—
31C2×#890—21no—
32C2×#887C2212yes—
33C2×#883—21no—
34C2×#67⋊C4—21no—
35C2×#879—21no—
36C2×#877—21no—
37C2×C42×D3C2211yescharacteristic
38C2×#870C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C3×#489C2218yescomplemented
41C2×C3×#103C2212yes—
42C2×C3×#102C2212yes—
43C2×C3×#97C2211yescharacteristic
44C2×#816C2212yes—
45C2×#816C2212yes—
46C2×#815C2212yes—
47#5081C2218yescomplemented
48#5060C2218yescomplemented
49C2×#808C2212yes—
50C2×#807C2212yes—
51C2×#803C2211yescharacteristic
52#4813C22116yescomplemented
53#4789C22116yescomplemented
54#4778C22116yes—
55#4757C22116yescomplemented
56D3×#97C2218yescomplemented
57#4703C2218yescomplemented
58C2×#533—31no—
59C2×#529—31no—
60#3181—316no—
61C2×#501—31no—
62C2×#497—31no—
63C2×#491—31no—
64C2×#489D311yescharacteristic, complemented
65C2×#467—32no—
66C2×#465—32no—
67C2×#441—32no—
68C2×#440—32no—
69C22×#97—31no—
70C22×C3×Q8—21no—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×D4C2311yescharacteristic
73C22×#67—21no—
74C23×Dic3—21no—
75C22×C4×D3D411yescharacteristic
76C22×C4×D3D411yescharacteristic
77C22×C4×D3C2311yescharacteristic
78C22×Dic6C2312yes—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8C2314yes—
82C3×#124—22no—
83C3×#124C2314yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91#946—22no—
92#946—28no—
93#946C2314yes—
94Q8×Dic3—22no—
95Q8×Dic3—28no—
96Q8×Dic3C2314yes—
97C2×#216—28no—
98C2×#216C2318yes—
99#938—22no—
100#938C2314yes—
101D4×Dic3—22no—
102D4×Dic3C2314yes—
103C2×#214—28no—
104C2×#214C2318yes—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—22no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—22no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—22no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2312yes—
123C2×D3×C8—28no—
124C2×D3×C8C2318yes—
125#897—24no—
126#897C2318yes—
127#892—28no—
128#890—28no—
129#887—24no—
130#887C2318yes—
131#883—28no—
132#67⋊C4—28no—
133#879—28no—
134#877—28no—
135C42×D3D414yes—
136C42×D3C2314yes—
137#870—22no—
138#870C2314yes—
139#869—22no—
140#869C2314yes—
141C3×#103C2318yes—
142C3×#102C2318yes—
143C3×#97C2314yes—
144#816C2318yes—
145#816C2318yes—
146#815C2318yes—
147#808C2318yes—
148#807C2318yes—
149#803C2314yes—
150C22×Q16—32no—
151C22×SD16—32no—
152C23×C8—32no—
153#533—38no—
154#533—64no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8C2×D311yescharacteristic
157#529—38no—
158#529—64no—
159C2×#126—61no—
160C2×#125—31no—
161C2×#125—61no—
162C2×#124C2×D311yescharacteristic
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×C42—31no—
176#501—38no—
177#497—38no—
178#491—38no—
179#489C2×D318yescomplemented
180#467—316no—
181#465—316no—
182#441—316no—
183#440—316no—
184C2×#107—32no—
185C2×#106—32no—
186C2×#103—32no—
187C2×#103—32no—
188C2×#103C2×D312yes—
189C2×#102C2×D312yes—
190C2×#97—32no—
191C2×#97—31no—
192C2×#97—38no—
193C2×#97C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3C2×D411yescharacteristic
196C2×C3×Q8—21no—
197C2×C3×Q8—24no—
198C2×C3×Q8—24no—
199C2×C3×Q8—44no—
200C2×C3×Q8C2412yes—
201C2×C3×D4—21no—
202C2×C3×D4—24no—
203C2×C3×D4C2412yes—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—24no—
210C2×#67—44no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3—21no—
214C22×Dic3—24no—
215C22×Dic3—44no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C2×C4×D3—24no—
220C2×C4×D3—24no—
221C2×C4×D3—42no—
222C2×C4×D3C2×D412yes—
223C2×C4×D3C2×D414yes—
224C2×C4×D3C2×D414yes—
225C2×C4×D3C2412yes—
226C2×Dic6—24no—
227C2×Dic6—24no—
228C2×Dic6—22no—
229C2×Dic6C2414yes—
230C2×C3×C8—24no—
231C2×C3×C8C2414yes—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4C2414yes—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C42C2×D412yes—
238C3×C42C2412yes—
239#217—24no—
240#217—24no—
241#216—216no—
242#214—216no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—28no—
245C4⋊Dic3—24no—
246C4⋊Dic3—24no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—24no—
250C4⋊Dic3—28no—
251C4⋊Dic3C2412yes—
252C4⋊Dic3C2412yes—
253Dic3⋊C4—24no—
254Dic3⋊C4—24no—
255Dic3⋊C4—24no—
256Dic3⋊C4—28no—
257C4×Dic3—24no—
258C4×Dic3—28no—
259C4×Dic3C2×D412yes—
260C4×Dic3C2×D414yes—
261C4×Dic3C2×D414yes—
262C4×Dic3C2412yes—
263C2×C3⋊C8#4118yes—
264C2×C3⋊C8C2414yes—
265D3×C8—216no—
266C22×Q8—21no—
267C22×Q8—32no—
268C22×Q8C22×D311yescharacteristic
269C22×D4—61no—
270C22×D4C22×D311yescharacteristic
271C23×C4—31no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—61no—
275C2×Q16—38no—
276C2×Q16—68no—
277C2×SD16—38no—
278C2×SD16—68no—
279C22×C8—32no—
280C22×C8—38no—
281C22×C8—68no—
282C22×C8C22×D312yes—
283C4⋊Q8—22no—
284C4⋊Q8—34no—
285C4⋊Q8—62no—
286C4⋊Q8C22×D314yes—
287#126—68no—
288#125—34no—
289#125—68no—
290#125—62no—
291#124—22no—
292#124C22×D314yes—
293#123—68no—
294C22⋊Q8—34no—
295C22⋊Q8—38no—
296C22⋊Q8—64no—
297C22⋊Q8—62no—
298C22⋊Q8—68no—
299#121—34no—
300#121—62no—
301C4×Q8—34no—
302C4×Q8—38no—
303C4×Q8—64no—
304C4×Q8—62no—
305C4×Q8—68no—
306C4×D4—34no—
307C4×D4—62no—
308C4×D4—68no—
309#117—68no—
310#117—68no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—62no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D312yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—34no—
333C2×C42—34no—
334C2×C42—61no—
335C2×C42—62no—
336C2×C42C22×D311yescharacteristic
337#107—38no—
338#106—38no—
339#103—38no—
340#103—38no—
341#103C22×D318yes—
342#102C22×D318yes—
343#97—38no—
344#97—34no—
345#97C22×D314yes—
346C23×C3—22no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—22no—
353C22×D3#13714yes—
354C22×D3C22×D412yes—
355C3×Q8—24no—
356C3×Q8—48no—
357C3×Q8—42no—
358C3×D4—24no—
359C3×D4—42no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—42no—
365C2×C3×C4—44no—
366C2×C3×C4C22×D412yes—
367C2×C3×C4C22×D412yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4114yes—
370C2×C3×C4C2511yescharacteristic
371#67—48no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—22no—
378C2×Dic3—48no—
379C2×Dic3—44no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3C22×D412yes—
383C2×Dic3C22×D412yes—
384C2×Dic3C22×D412yes—
385C4×D3—28no—
386C4×D3—24no—
387C4×D3—24no—
388C4×D3—44no—
389Dic6—24no—
390Dic6—24no—
391Dic6—44no—
392C3×C8—24no—
393C3⋊C8C2×#4118yes—
394C24—21no—
395C24—31no—
396C2×Q8—21no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—34no—
400C2×Q8—44no—
401C2×Q8—62no—
402C2×Q8—64no—
403C2×Q8—64no—
404C2×Q8C23×D312yes—
405C2×D4—21no—
406C2×D4—24no—
407C2×D4—64no—
408C2×D4—124no—
409C2×D4C23×D312yes—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—34no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—34no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—122no—
426C22×C4—124no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430Q16—616no—
431SD16—616no—
432C2×C8—24no—
433C2×C8—38no—
434C2×C8—34no—
435C2×C8—616no—
436C2×C8C23×D314yes—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—32no—
440C4⋊C4—32no—
441C4⋊C4—64no—
442C4⋊C4—68no—
443C4⋊C4—64no—
444C4⋊C4—68no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—64no—
451C4⋊C4C23×D314yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—68no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C42—34no—
459C42—32no—
460C42—34no—
461C42—32no—
462C42—64no—
463C42—68no—
464C42D3×D412yes—
465C42C23×D312yes—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—84no—
470C22×C3#53212yes—
471C22×C3C2×#13712yes—
472C22×C3#57512yes—
473C22×C3C23×D411yescharacteristic
474C2×D3—28no—
475C2×D3—22no—
476C2×D3—22no—
477C2×D3C2×#13714yes—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—42no—
481C3×C4—44no—
482C3×C4#53212yes—
483C3×C4C22×#4112yes—
484Dic3—22no—
485Dic3—22no—
486Dic3—22no—
487Dic3—44no—
488Dic3—42no—
489Dic3—42no—
490C23—22no—
491C23—32no—
492C23—34no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—62no—
497C23—62no—
498C23C2×D3×D411yescharacteristic
499Q8—24no—
500Q8—48no—
501Q8—42no—
502Q8—64no—
503Q8—64no—
504Q8—124no—
505D4—24no—
506D4—42no—
507D4—128no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—32no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—68no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—62no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—124no—
527C2×C4—128no—
528C2×C4—124no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4C2×D3×D412yes—
532C2×C4C2×D3×D412yes—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4D3×#4114yes—
535C2×C4C24×D311yescharacteristic
536C8—24no—
537C8—38no—
538C2×C3—82no—
539C2×C3#318114yescomplemented
540C2×C3C2×#53211yescharacteristic
541C2×C3C22×#13711yescharacteristic
542C2×C3C2×#57511yescharacteristic
543D3—24no—
544C22—34no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—68no—
549C22—62no—
550C22—62no—
551C22—84no—
552C22#566812yes—
553C22#577312yes—
554C22D3×#13712yes—
555C22C22×D3×D411yescharacteristic
556C4—22no—
557C4—22no—
558C4—42no—
559C4—44no—
560C4—62no—
561C4—62no—
562C4—62no—
563C4—124no—
564C4—122no—
565C4—122no—
566C4#566812yes—
567C4C2×D3×#4112yes—
568C3C2×#318111yescharacteristic, complemented, Sylow
569C2—64no—
570C2—82no—
571C2#8166914yesdirect product factor
572C2C2×#566811yescharacteristic
573C2C2×#577311yescharacteristic
574C2C2×D3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenterC2×#489D3
hypocenterC3C2×#3181
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
3. C2×C3×#489C2semidirect
4. C2×#5081C2semidirect
5. C2×#5060C2semidirect
6. #81669C2direct (N × H)
7. C2×#4813C2semidirect
8. C2×#4789C2semidirect
10. C2×#4757C2semidirect
11. C2×#4703C2semidirect
12. C2×D3×#97C2semidirect
26. #5728C22semidirect
40. C3×#489C22semidirect
47. #5081C22semidirect
48. #5060C22semidirect
52. #4813C22semidirect
53. #4789C22semidirect
55. #4757C22semidirect
56. D3×#97C22semidirect
57. #4703C22semidirect
64. C2×#489D3semidirect
179. #489C2×D3semidirect
539. C2×C3#3181semidirect
568. C3C2×#3181semidirect
571. C2#81669direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#56802
C2×C3×#4892
C2×#50812
C2×#50602
#816692
C2×#48132
C2×#47892
C2×#47782
C2×#47572
C2×#47032
C2×D3×#972
C2×#31813