Finite Group Databasebeta

← C2×#74738 #10964355

Subgroups — C2×#74738 #10964355

Order 768.

3374 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic
2C23×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#386C211yesmaximal, characteristic
4C2×#5220C211yesmaximal, characteristic
5C2×#5210C211yesmaximal, characteristic, complemented
6C2×#4949C212yesmaximal, complemented
7C2×#4950C211yesmaximal, characteristic, complemented
8C2×#4942C212yesmaximal, complemented
9C2×#4942C211yesmaximal, characteristic, complemented
10#74738C2116yesmaximal, direct product factor
11C2×#4660C212yesmaximal, complemented
12C2×#4647C212yesmaximal, complemented
13C2×#2487—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C24×Dic3C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2211yescharacteristic
22C22×#217C2211yescharacteristic
23C22×#217C2211yescharacteristic
24C22×#217C2212yes—
25C2×D4×Dic3—28no—
26C2×D4×Dic3C2218yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2212yes—
29C22×Dic3⋊C4—28no—
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×Dic3⋊C4C2212yes—
34C22×Dic3⋊C4C2218yescomplemented
35C22×C4×Dic3C2211yescharacteristic
36C3×#386C2218yes—
37C2×C3×#95C2212yes—
38#5220C2218yes—
39#5210C2218yescomplemented
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44#4950C2218yescomplemented
45#4949C22116yescomplemented
46#4942C2218yescomplemented
47#4942C22116yescomplemented
48#4660C22116yescomplemented
49#4647C22116yescomplemented
50C22×C4×D4—31no—
51C23×C4⋊C4—31no—
52#2487—316no—
53C2×#400—32no—
54C2×#399—32no—
55C2×#389—32no—
56C2×#386—31no—
57C2×#386D311yescharacteristic
58C2×#385—31no—
59C2×#385—31no—
60C2×#380—31no—
61C2×#380—32no—
62C25×C3C2311yescharacteristic
63C25×C3C2311yescharacteristic
64C22×C3×D4—24no—
65C22×C3×D4—24no—
66C22×C3×D4Q818yes—
67C22×C3×D4C2314yes—
68C23×C3×C4—24no—
69C23×C3×C4—44no—
70C23×C3×C4C2312yes—
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2314yes—
74C23×Dic3—28no—
75C23×Dic3—24no—
76C23×Dic3—24no—
77C23×Dic3—28no—
78C23×Dic3—44no—
79C23×Dic3D418yes—
80C23×Dic3C2314yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2312yes—
83C23×Dic3C2312yes—
84C23×Dic3C2311yescharacteristic
85C23×Dic3C2314yes—
86C23×Dic3C2311yescharacteristic
87C23×Dic3C2314yes—
88C23×Dic3C2311yescharacteristic
89C2×C3×C22⋊C4—24no—
90C2×C3×C22⋊C4—44no—
91C2×C3×C22⋊C4C2314yes—
92C2×C3×C22⋊C4C2318yes—
93C2×#217—24no—
94C2×#217—24no—
95C2×#217—24no—
96C2×#217—44no—
97C2×#217C2314yes—
98C2×#217C2314yes—
99C2×#217C2314yes—
100C2×#217C2318yes—
101D4×Dic3—232no—
102C2×C4⋊Dic3—24no—
103C2×C4⋊Dic3—44no—
104C2×C4⋊Dic3C2314yes—
105C2×C4⋊Dic3C2318yes—
106C2×Dic3⋊C4—28no—
107C2×Dic3⋊C4—28no—
108C2×Dic3⋊C4—232no—
109C2×Dic3⋊C4—28no—
110C2×Dic3⋊C4—44no—
111C2×Dic3⋊C4C2318yes—
112C2×Dic3⋊C4C2318yes—
113C2×Dic3⋊C4C2318yes—
114C2×Dic3⋊C4C2318yes—
115C2×C4×Dic3—24no—
116C2×C4×Dic3C2314yes—
117C3×#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C23×D4C2×D311yescharacteristic
123C24×C4—31no—
124C24×C4—31no—
125C24×C4—31no—
126C24×C4C2×D311yescharacteristic
127C2×C4×D4—38no—
128C2×C4×D4—68no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—38no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—32no—
136C22×C4⋊C4—68no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—31no—
139C22×C22⋊C4—31no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4C2×D311yescharacteristic
142C22×C22⋊C4C2×D312yes—
143C22×C42—31no—
144#400—316no—
145#399—316no—
146#389—316no—
147#386—38no—
148#386C2×D318yes—
149#385—38no—
150#385—38no—
151#380—316no—
152#380—38no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95C2×D312yes—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—22no—
161C24×C3—24no—
162C24×C3—22no—
163C24×C3—24no—
164C24×C3C2×D414yes—
165C24×C3C2×Q814yes—
166C24×C3C2×Q814yes—
167C24×C3#4114yes—
168C24×C3C2412yes—
169C24×C3C2412yes—
170C24×C3C2411yescharacteristic
171C2×C3×D4—232no—
172C2×C3×D4—28no—
173C2×C3×D4—28no—
174C2×C3×D4C2×Q8116yes—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—28no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—48no—
185C22×C3×C4—416no—
186C22×C3×C4C2×Q814yes—
187C22×C3×C4C2414yes—
188C22×C3×C4C2414yes—
189C22×C3×C4C2412yes—
190C22×Dic3—24no—
191C22×Dic3—232no—
192C22×Dic3—216no—
193C22×Dic3—22no—
194C22×Dic3—28no—
195C22×Dic3—24no—
196C22×Dic3—22no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—28no—
205C22×Dic3—216no—
206C22×Dic3—48no—
207C22×Dic3—416no—
208C22×Dic3—416no—
209C22×Dic3C2×D414yes—
210C22×Dic3C2×D4116yes—
211C22×Dic3C2×D414yes—
212C22×Dic3#4118yes—
213C22×Dic3C2412yes—
214C22×Dic3C2414yes—
215C22×Dic3C2414yes—
216C22×Dic3C2412yes—
217C22×Dic3C2412yes—
218C22×Dic3C2412yes—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—48no—
221#217—28no—
222#217—28no—
223#217—28no—
224#217—48no—
225C4⋊Dic3—28no—
226C4⋊Dic3—48no—
227Dic3⋊C4—216no—
228Dic3⋊C4—216no—
229Dic3⋊C4—216no—
230Dic3⋊C4—48no—
231C4×Dic3—28no—
232C25C22×D311yescharacteristic
233C25C22×D311yescharacteristic
234C22×D4—24no—
235C22×D4—24no—
236C22×D4Dic618yes—
237C22×D4C22×D314yes—
238C23×C4—24no—
239C23×C4—31no—
240C23×C4—38no—
241C23×C4—34no—
242C23×C4—31no—
243C23×C4—34no—
244C23×C4—31no—
245C23×C4—32no—
246C23×C4—32no—
247C23×C4—31no—
248C23×C4—34no—
249C23×C4—44no—
250C23×C4—68no—
251C23×C4—64no—
252C23×C4—68no—
253C23×C4—64no—
254C23×C4—124no—
255C23×C4C22×D312yes—
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D314yes—
259C4×D4—632no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—38no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—38no—
264C2×C4⋊C4—38no—
265C2×C4⋊C4—38no—
266C2×C4⋊C4—68no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4—632no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—68no—
271C2×C4⋊C4—124no—
272C2×C4⋊C4—124no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—34no—
276C2×C22⋊C4—34no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—124no—
283C2×C22⋊C4C22×D314yes—
284C2×C22⋊C4C22×D318yes—
285C2×C42—34no—
286C2×C42—64no—
287#95—38no—
288#95—38no—
289#95—38no—
290#95—38no—
291#95C22×D318yes—
292C23×C3—28no—
293C23×C3—28no—
294C23×C3—216no—
295C23×C3—216no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—28no—
299C23×C3—216no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—44no—
305C23×C3—44no—
306C23×C3—48no—
307C23×C3C22⋊Q8116yes—
308C23×C3C22×D414yes—
309C23×C3C22×D412yes—
310C23×C3C22×Q814yes—
311C23×C3C22×Q814yes—
312C23×C3C22×Q812yes—
313C23×C3C2×#4114yes—
314C23×C3C2×#4112yes—
315C23×C3C2×#4114yes—
316C23×C3H(5,𝔽2)14yes—
317C23×C3C2511yescharacteristic
318C3×D4—232no—
319C2×C3×C4—28no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—416no—
329C2×C3×C4—48no—
330C2×C3×C4—48no—
331C2×C3×C4C22×Q814yes—
332C2×Dic3—28no—
333C2×Dic3—24no—
334C2×Dic3—232no—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—416no—
346C2×Dic3—44no—
347C2×Dic3—44no—
348C2×Dic3—48no—
349C2×Dic3—48no—
350C2×Dic3—416no—
351C2×Dic3#123116yes—
352C2×Dic3C22×D414yes—
353C2×Dic3C22×D414yes—
354C2×Dic3C2×#4118yes—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24C2×Dic614yes—
362C24C2×Dic614yes—
363C24D3×D414yes—
364C24#23714yes—
365C24C23×D312yes—
366C24C23×D312yes—
367C24C23×D311yescharacteristic
368C2×D4—232no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4C2×Dic6116yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—34no—
382C22×C4—38no—
383C22×C4—32no—
384C22×C4—34no—
385C22×C4—316no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—34no—
389C22×C4—34no—
390C22×C4—32no—
391C22×C4—48no—
392C22×C4—416no—
393C22×C4—632no—
394C22×C4—616no—
395C22×C4—68no—
396C22×C4—616no—
397C22×C4—64no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—68no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—1216no—
410C22×C4—128no—
411C22×C4—1216no—
412C22×C4C2×Dic614yes—
413C22×C4C23×D314yes—
414C22×C4C23×D314yes—
415C22×C4C23×D312yes—
416C4⋊C4—616no—
417C4⋊C4—616no—
418C4⋊C4—68no—
419C4⋊C4—616no—
420C4⋊C4—128no—
421C4⋊C4—128no—
422C22⋊C4—28no—
423C22⋊C4—48no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C22⋊C4—68no—
427C22⋊C4—128no—
428C42—68no—
429C22×C3—24no—
430C22×C3—24no—
431C22×C3—28no—
432C22×C3—28no—
433C22×C3—232no—
434C22×C3—24no—
435C22×C3—28no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—44no—
439C22×C3—416no—
440C22×C3C2×C22⋊Q818yes—
441C22×C3C2×C22⋊Q814yes—
442C22×C3C2×#12314yes—
443C22×C3#53514yes—
444C22×C3#55318yes—
445C22×C3#55418yes—
446C22×C3C23×D411yescharacteristic
447C22×C3C23×Q811yescharacteristic
448C22×C3C22×#4111yescharacteristic
449C22×C3C22×#4112yes—
450C22×C3C2×H(5,𝔽2)12yes—
451C3×C4—24no—
452C3×C4—44no—
453C3×C4—44no—
454Dic3—24no—
455Dic3—42no—
456Dic3—42no—
457Dic3—44no—
458Dic3—44no—
459Dic3C2×#12318yes—
460C23—28no—
461C23—24no—
462C23—24no—
463C23—216no—
464C23—28no—
465C23—216no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—216no—
472C23—44no—
473C23—44no—
474C23—48no—
475C23#878116yes—
476C23C22×Dic614yes—
477C23C22×Dic614yes—
478C23C22×Dic612yes—
479C23C2×D3×D414yes—
480C23C2×D3×D412yes—
481C23C2×#23714yes—
482C23C2×#23712yes—
483C23#100414yes—
484C23D3×#4114yes—
485C23C24×D311yescharacteristic
486D4—232no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—316no—
497C2×C4—34no—
498C2×C4—38no—
499C2×C4—34no—
500C2×C4—416no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—632no—
504C2×C4—68no—
505C2×C4—64no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—1216no—
517C2×C4—1216no—
518C2×C4—124no—
519C2×C4—128no—
520C2×C4—124no—
521C2×C4—128no—
522C2×C4C22×Dic614yes—
523C2×C3—28no—
524C2×C3—44no—
525C2×C3#248718yescomplemented
526C2×C3C22×C22⋊Q811yescharacteristic
527C2×C3C22×#12311yescharacteristic
528C2×C3C2×#53511yescharacteristic
529C2×C3C2×#55312yes—
530C2×C3C2×#55412yes—
531C22—28no—
532C22—24no—
533C22—28no—
534C22—232no—
535C22—24no—
536C22—28no—
537C22—24no—
538C22—42no—
539C22—44no—
540C22—416no—
541C22—42no—
542C22C2×#87818yes—
543C22C2×#87814yes—
544C22#553918yes—
545C22#559214yes—
546C22#560118yes—
547C22D3×#12314yes—
548C22C23×Dic611yescharacteristic
549C22C22×D3×D411yescharacteristic
550C22C22×#23711yescharacteristic
551C22C2×#100412yes—
552C22C2×D3×#4112yes—
553C4—24no—
554C4—38no—
555C4—44no—
556C4—44no—
557C4—64no—
558C4—124no—
559C4—122no—
560C4—124no—
561C4—122no—
562C3C2×#248711yescharacteristic, complemented, Sylow
563C2—28no—
564C2—44no—
565C2#7473818yesdirect product factor
566C2C22×#87811yescharacteristic
567C2C2×#553912yes—
568C2C2×#559211yescharacteristic
569C2C2×#560112yes—
570C2C2×D3×#12311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#386C2
socleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#386D3
hypocenterC3C2×#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C23×Dic3⋊C4C2semidirect
5. C2×#5210C2semidirect
6. C2×#4949C2semidirect
7. C2×#4950C2semidirect
8. C2×#4942C2semidirect
9. C2×#4942C2semidirect
10. #74738C2direct (N × H)
11. C2×#4660C2semidirect
12. C2×#4647C2semidirect
34. C22×Dic3⋊C4C22semidirect
39. #5210C22semidirect
44. #4950C22semidirect
45. #4949C22semidirect
46. #4942C22semidirect
47. #4942C22semidirect
48. #4660C22semidirect
49. #4647C22semidirect
525. C2×C3#2487semidirect
562. C3C2×#2487semidirect
565. C2#74738direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C23×Dic3⋊C42
C2×C3×#3862
C2×#52202
C2×#52102
C2×#49492
C2×#49502
C2×#49422
#747382
C2×#46602
C2×#46472
C2×#24873