Finite Group Databasebeta

← C2×#73998 #11001056

Subgroups — C2×#73998 #11001056

Order 768.

3182 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Dic3C211yesmaximal, characteristic
2C22×#890C211yesmaximal, characteristic, complemented
3C2×C3×#380C211yesmaximal, characteristic
4C2×#5212C211yesmaximal, characteristic
5C2×#4984C211yesmaximal, characteristic, complemented
6C2×#4972C211yesmaximal, characteristic, complemented
7C2×#4977C211yesmaximal, characteristic, complemented
8C2×#4950C212yesmaximal
9C2×#4946C212yesmaximal
10#73998C2116yesmaximal, direct product factor
11C2×#4655C212yesmaximal, complemented
12C2×#4654C212yesmaximal, complemented
13C2×#2851—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×#217C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2212yes—
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2212yes—
27C22×C4×Dic3—28no—
28C22×C4×Dic3C2218yes—
29C22×C4×Dic3C2211yescharacteristic
30C22×C4×Dic3C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C22×C4×Dic3C2211yescharacteristic
33C2×#890—28no—
34C2×#890C2218yescomplemented
35C3×#380C2218yes—
36C2×C3×#95C2212yes—
37#5212C2218yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4984C2218yescomplemented
44#4977C2218yescomplemented
45#4972C2218yescomplemented
46#4950C22116yes—
47#4946C22116yes—
48#4655C22116yescomplemented
49#4654C22116yescomplemented
50C22×#125—31no—
51C23×C42—31no—
52#2851—316no—
53C2×#389—31no—
54C2×#389—31no—
55C2×#387—32no—
56C2×#387—32no—
57C2×#385—32no—
58C2×#385—32no—
59C2×#382—31no—
60C2×#380—31no—
61C2×#380D311yescharacteristic
62C25×C3C2311yescharacteristic
63C23×C3×C4—24no—
64C23×C3×C4—24no—
65C23×C3×C4Q818yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2314yes—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3—24no—
72C23×Dic3—24no—
73C23×Dic3—24no—
74C23×Dic3—24no—
75C23×Dic3—24no—
76C23×Dic3—24no—
77C23×Dic3C2314yes—
78C23×Dic3C2314yes—
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2311yescharacteristic
84C23×Dic3C2311yescharacteristic
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C22⋊C4—28no—
90C2×C3×C22⋊C4C2318yes—
91C2×#217—28no—
92C2×#217C2318yes—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—28no—
96C2×C4⋊Dic3—28no—
97C2×C4⋊Dic3C2318yes—
98C2×C4⋊Dic3C2318yes—
99C2×C4⋊Dic3C2314yes—
100C2×C4⋊Dic3C2314yes—
101C2×Dic3⋊C4—28no—
102C2×Dic3⋊C4—28no—
103C2×Dic3⋊C4C2318yes—
104C2×Dic3⋊C4C2318yes—
105C2×C4×Dic3—24no—
106C2×C4×Dic3—24no—
107C2×C4×Dic3—24no—
108C2×C4×Dic3—232no—
109C2×C4×Dic3D418yes—
110C2×C4×Dic3C2314yes—
111C2×C4×Dic3C2314yes—
112C2×C4×Dic3C2314yes—
113C2×C4×Dic3C2314yes—
114#890—232no—
115C3×#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C24×C4—31no—
122C24×C4—31no—
123C24×C4C2×D311yescharacteristic
124C2×#125—38no—
125C2×#125—68no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C4⋊C4C2×D311yescharacteristic
134C22×C22⋊C4—32no—
135C22×C22⋊C4C2×D312yes—
136C22×C42—31no—
137C22×C42—38no—
138C22×C42—31no—
139C22×C42—31no—
140C22×C42—31no—
141C22×C42—68no—
142#389—38no—
143#389—38no—
144#387—316no—
145#387—316no—
146#385—316no—
147#385—316no—
148#382—38no—
149#380—38no—
150#380C2×D318yes—
151C2×#95—32no—
152C2×#95—32no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95C2×D312yes—
157C24×C3—22no—
158C24×C3—22no—
159C24×C3—22no—
160C24×C3—22no—
161C24×C3C2×Q814yes—
162C24×C3#4114yes—
163C24×C3#4114yes—
164C24×C3C2412yes—
165C24×C3C2411yescharacteristic
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—232no—
169C22×C3×C4—28no—
170C22×C3×C4—28no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×Q814yes—
179C22×C3×C4C2×Q8116yes—
180C22×C3×C4C2×Q814yes—
181C22×C3×C4#4114yes—
182C22×C3×C4C2414yes—
183C22×C3×C4C2414yes—
184C22×C3×C4C2412yes—
185C22×C3×C4C2412yes—
186C22×Dic3—232no—
187C22×Dic3—232no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—28no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—28no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—28no—
199C22×Dic3—28no—
200C22×Dic3—28no—
201C22×Dic3—28no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×D414yes—
210C22×Dic3#4114yes—
211C22×Dic3#4114yes—
212C22×Dic3C2414yes—
213C22×Dic3C2414yes—
214C22×Dic3C2412yes—
215C22×Dic3C2412yes—
216C22×Dic3C2412yes—
217C22×Dic3C2412yes—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—28no—
220C3×C22⋊C4—216no—
221#217—216no—
222C4⋊Dic3—28no—
223C4⋊Dic3—28no—
224C4⋊Dic3—216no—
225C4⋊Dic3—216no—
226Dic3⋊C4—216no—
227Dic3⋊C4—216no—
228C4×Dic3—28no—
229C4×Dic3—28no—
230C4×Dic3—28no—
231C4×Dic3C2×D4116yes—
232C25C22×D311yescharacteristic
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—34no—
240C23×C4—34no—
241C23×C4—32no—
242C23×C4—32no—
243C23×C4—64no—
244C23×C4—64no—
245C23×C4—64no—
246C23×C4—64no—
247C23×C4—64no—
248C23×C4—64no—
249C23×C4Dic618yes—
250C23×C4C22×D312yes—
251C23×C4C22×D312yes—
252C23×C4C22×D314yes—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255#125—632no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—38no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—38no—
263C2×C4⋊C4—38no—
264C2×C4⋊C4—68no—
265C2×C4⋊C4—68no—
266C2×C4⋊C4—68no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4C22×D314yes—
271C2×C4⋊C4C22×D314yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—68no—
275C2×C22⋊C4C22×D318yes—
276C2×C42—38no—
277C2×C42—34no—
278C2×C42—34no—
279C2×C42—34no—
280C2×C42—34no—
281C2×C42—632no—
282C2×C42—64no—
283C2×C42—64no—
284C2×C42—64no—
285#95—38no—
286#95—38no—
287#95—38no—
288#95—38no—
289#95—38no—
290#95C22×D318yes—
291C23×C3—28no—
292C23×C3—28no—
293C23×C3—28no—
294C23×C3—28no—
295C23×C3—28no—
296C23×C3—28no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3#125116yescomplemented
302C23×C3C22×D412yes—
303C23×C3C22×Q814yes—
304C23×C3C22×Q812yes—
305C23×C3C2×#4114yes—
306C23×C3C2×#4112yes—
307C23×C3C2×#4112yes—
308C23×C3C2×#4114yes—
309C23×C3C2×#4112yes—
310C23×C3C2×#4112yes—
311C23×C3C2×#4112yes—
312C23×C3C2511yescharacteristic
313C2×C3×C4—28no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—232no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—48no—
324C2×C3×C4—48no—
325C2×C3×C4C22⋊Q8116yes—
326C2×C3×C4C22×D414yes—
327C2×C3×C4C22×Q814yes—
328C2×C3×C4C22×Q814yes—
329C2×C3×C4C2×#4114yes—
330C2×Dic3—232no—
331C2×Dic3—232no—
332C2×Dic3—28no—
333C2×Dic3—28no—
334C2×Dic3—28no—
335C2×Dic3—28no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—48no—
351C2×Dic3—48no—
352C2×Dic3C22×D414yes—
353C2×Dic3C22×D414yes—
354C2×Dic3C2×#4114yes—
355C2×Dic3C2×#4114yes—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C2×Dic614yes—
361C24#23714yes—
362C24#23914yes—
363C24C23×D312yes—
364C24C23×D311yescharacteristic
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—232no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—34no—
377C22×C4—34no—
378C22×C4—34no—
379C22×C4—34no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—34no—
385C22×C4—34no—
386C22×C4—68no—
387C22×C4—68no—
388C22×C4—68no—
389C22×C4—68no—
390C22×C4—632no—
391C22×C4—632no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—68no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—68no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4C2×Dic614yes—
409C22×C4C2×Dic6116yes—
410C22×C4C2×Dic614yes—
411C22×C4D3×D414yes—
412C22×C4#23714yes—
413C22×C4C23×D314yes—
414C22×C4C23×D314yes—
415C22×C4C23×D312yes—
416C22×C4C23×D312yes—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—616no—
420C4⋊C4—616no—
421C4⋊C4—616no—
422C4⋊C4—616no—
423C4⋊C4—68no—
424C4⋊C4—68no—
425C22⋊C4—216no—
426C22⋊C4—616no—
427C42—316no—
428C42—68no—
429C42—68no—
430C42—68no—
431C22×C3—232no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—24no—
438C22×C3C2×C22⋊Q814yes—
439C22×C3#52514yes—
440C22×C3#52514yes—
441C22×C3C2×#12514yes—
442C22×C3C2×#12518yescomplemented
443C22×C3#52914yes—
444C22×C3#52914yes—
445C22×C3#53314yes—
446C22×C3C23×D411yescharacteristic
447C22×C3C23×Q811yescharacteristic
448C22×C3C22×#4111yescharacteristic
449C22×C3C22×#4111yescharacteristic
450C22×C3C22×#4111yescharacteristic
451C22×C3C22×#4111yescharacteristic
452C22×C3C22×#4111yescharacteristic
453C3×C4—24no—
454C3×C4—44no—
455C3×C4—44no—
456C3×C4C2×C22⋊Q818yes—
457Dic3—24no—
458Dic3—24no—
459Dic3—24no—
460Dic3—24no—
461Dic3—44no—
462Dic3—44no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23#890116yescomplemented
474C23C22×Dic614yes—
475C23C22×Dic612yes—
476C23C2×D3×D412yes—
477C23C2×#23714yes—
478C23C2×#23712yes—
479C23C2×#23712yes—
480C23C2×#23914yes—
481C23C2×#23912yes—
482C23D3×#4112yes—
483C23D3×#4112yes—
484C23C24×D311yescharacteristic
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—28no—
488C2×C4—232no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—34no—
496C2×C4—34no—
497C2×C4—34no—
498C2×C4—34no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—632no—
502C2×C4—632no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—68no—
512C2×C4—68no—
513C2×C4—68no—
514C2×C4—68no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—128no—
522C2×C4—128no—
523C2×C4#878116yes—
524C2×C4C22×Dic614yes—
525C2×C4C22×Dic614yes—
526C2×C4C2×D3×D414yes—
527C2×C4C2×#23714yes—
528C2×C3—28no—
529C2×C3#285118yescomplemented
530C2×C3C22×C22⋊Q811yescharacteristic
531C2×C3C2×#52511yescharacteristic
532C2×C3C2×#52511yescharacteristic
533C2×C3C22×#12511yescharacteristic
534C2×C3C2×#52911yescharacteristic
535C2×C3C2×#52911yescharacteristic
536C2×C3C2×#53311yescharacteristic
537C22—232no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22C2×#87814yes—
545C22C2×#89014yes—
546C22C2×#89018yescomplemented
547C22#551714yes—
548C22#560014yes—
549C22#560614yes—
550C22#562414yes—
551C22#562714yes—
552C22C23×Dic611yescharacteristic
553C22C22×D3×D411yescharacteristic
554C22C22×#23711yescharacteristic
555C22C22×#23711yescharacteristic
556C22C22×#23911yescharacteristic
557C22C2×D3×#4111yescharacteristic
558C22C2×D3×#4111yescharacteristic
559C4—24no—
560C4—44no—
561C4—44no—
562C4—64no—
563C4—64no—
564C4—64no—
565C4—64no—
566C4—124no—
567C4—124no—
568C4C2×#87818yes—
569C3C2×#285111yescharacteristic, complemented, Sylow
570C2—28no—
571C2#7399818yesdirect product factor
572C2C22×#87811yescharacteristic
573C2C22×#89011yescharacteristic
574C2C2×#551711yescharacteristic
575C2C2×#560611yescharacteristic
576C2C2×#560011yescharacteristic
577C2C2×#562711yescharacteristic
578C2C2×#562411yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#380C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#380D3
hypocenterC3C2×#2851
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#890C2semidirect
5. C2×#4984C2semidirect
6. C2×#4972C2semidirect
7. C2×#4977C2semidirect
10. #73998C2direct (N × H)
11. C2×#4655C2semidirect
12. C2×#4654C2semidirect
34. C2×#890C22semidirect
43. #4984C22semidirect
44. #4977C22semidirect
45. #4972C22semidirect
48. #4655C22semidirect
49. #4654C22semidirect
301. C23×C3#125semidirect
442. C22×C3C2×#125semidirect
473. C23#890semidirect
529. C2×C3#2851semidirect
546. C22C2×#890semidirect
569. C3C2×#2851semidirect
571. C2#73998direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×Dic32
C22×#8902
C2×C3×#3802
C2×#52122
C2×#49842
C2×#49722
C2×#49772
C2×#49502
C2×#49462
#739982
C2×#46552
C2×#46542
C2×#28513