Finite Group Databasebeta

← C2×#74689 #10963127

Subgroups — C2×#74689 #10963127

Order 768.

5566 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×Dic6C211yesmaximal, characteristic, complemented
2C22×D4×Dic3C211yesmaximal, characteristic
3C22×#878C212yesmaximal, complemented
4C2×C3×#386C211yesmaximal, characteristic
5C2×#5220C211yesmaximal, characteristic
6C2×#5210C211yesmaximal, characteristic, complemented
7C2×#4947C212yesmaximal, complemented
8C2×#4950C211yesmaximal, characteristic, complemented
9C2×#4942C211yesmaximal, characteristic, complemented
10#74689C2116yesmaximal, direct product factor
11C2×#4656C212yesmaximal, complemented
12C2×#4646C212yesmaximal, complemented
13C2×#2409—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C24×Dic3C2211yescharacteristic
19C23×Dic6—28no—
20C23×Dic6—216no—
21C23×Dic6C2212yes—
22C23×Dic6C2212yes—
23C23×Dic6C2218yescomplemented
24C22×C3×C22⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2212yes—
26C22×#217C2211yescharacteristic
27C22×#217C2211yescharacteristic
28C22×#217C2212yes—
29C22×#217C2211yescharacteristic
30C2×D4×Dic3—28no—
31C2×D4×Dic3C2218yes—
32C22×C4⋊Dic3C2211yescharacteristic
33C22×C4⋊Dic3C2212yes—
34C22×Dic3⋊C4C2212yes—
35C22×Dic3⋊C4C2212yes—
36C22×C4×Dic3C2211yescharacteristic
37C2×#878—216no—
38C2×#878C22116yescomplemented
39C3×#386C2218yes—
40C2×C3×#95C2212yes—
41#5220C2218yes—
42#5210C2218yescomplemented
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45C2×C3⋊#95C2212yes—
46C2×C3⋊#95C2212yes—
47#4950C2218yescomplemented
48#4947C22116yescomplemented
49#4942C2218yescomplemented
50#4656C22116yescomplemented
51#4646C22116yescomplemented
52C24×Q8—31no—
53C22×C22⋊Q8—32no—
54C22×C4×D4—31no—
55#2409—316no—
56C2×#395—32no—
57C2×#393—32no—
58C2×#391—32no—
59C2×#386—31no—
60C2×#386D311yescharacteristic
61C2×#385—31no—
62C2×#385—31no—
63C2×#380—31no—
64C25×C3C2311yescharacteristic
65C25×C3C2311yescharacteristic
66C22×C3×D4—24no—
67C22×C3×D4—24no—
68C22×C3×D4Q818yescomplemented
69C22×C3×D4C2314yes—
70C23×C3×C4—24no—
71C23×C3×C4—28no—
72C23×C3×C4C2314yes—
73C23×C3×C4C2312yes—
74C23×C3×C4C2312yes—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—24no—
77C23×Dic3—28no—
78C23×Dic3—28no—
79C23×Dic3—28no—
80C23×Dic3—24no—
81C23×Dic3D418yes—
82C23×Dic3C2314yes—
83C23×Dic3C2314yes—
84C23×Dic3C2311yescharacteristic
85C23×Dic3C2311yescharacteristic
86C23×Dic3C2312yes—
87C23×Dic3C2311yescharacteristic
88C23×Dic3C2312yes—
89C23×Dic3C2314yes—
90C23×Dic3C2311yescharacteristic
91C22×Dic6—2128no—
92C22×Dic6—28no—
93C22×Dic6—28no—
94C22×Dic6—28no—
95C22×Dic6—28no—
96C22×Dic6—28no—
97C22×Dic6—232no—
98C22×Dic6—264no—
99C22×Dic6D4116yescomplemented
100C22×Dic6C2318yes—
101C22×Dic6C2318yes—
102C2×C3×C22⋊C4—24no—
103C2×C3×C22⋊C4—28no—
104C2×C3×C22⋊C4C2314yes—
105C2×C3×C22⋊C4C2318yes—
106C2×#217—24no—
107C2×#217—24no—
108C2×#217—28no—
109C2×#217—24no—
110C2×#217C2314yes—
111C2×#217C2314yes—
112C2×#217C2318yes—
113C2×#217C2314yes—
114D4×Dic3—232no—
115C2×C4⋊Dic3—28no—
116C2×C4⋊Dic3—24no—
117C2×C4⋊Dic3C2314yes—
118C2×C4⋊Dic3C2318yes—
119C2×Dic3⋊C4—28no—
120C2×Dic3⋊C4—28no—
121C2×Dic3⋊C4C2318yes—
122C2×Dic3⋊C4C2318yes—
123C2×C4×Dic3—24no—
124C2×C4×Dic3C2314yes—
125#878—264no—
126C3×#95C2318yes—
127C3⋊#95C2318yes—
128C3⋊#95C2318yes—
129C3⋊#95C2318yes—
130C3⋊#95C2318yes—
131C23×Q8—32no—
132C23×Q8—32no—
133C23×Q8—38no—
134C23×Q8—68no—
135C23×Q8—616no—
136C23×D4C2×D311yescharacteristic
137C24×C4—31no—
138C24×C4—31no—
139C24×C4—31no—
140C24×C4C2×D311yescharacteristic
141C2×C22⋊Q8—316no—
142C2×C22⋊Q8—616no—
143C2×C4×D4—38no—
144C2×C4×D4—68no—
145C22×C4⋊C4—31no—
146C22×C4⋊C4—32no—
147C22×C4⋊C4—32no—
148C22×C4⋊C4—32no—
149C22×C22⋊C4—31no—
150C22×C22⋊C4—32no—
151C22×C22⋊C4—31no—
152C22×C22⋊C4—31no—
153C22×C22⋊C4C2×D312yes—
154C22×C22⋊C4C2×D311yescharacteristic
155C22×C42—31no—
156#395—316no—
157#393—316no—
158#391—316no—
159#386—38no—
160#386C2×D318yes—
161#385—38no—
162#385—38no—
163#380—38no—
164C2×#95—32no—
165C2×#95—32no—
166C2×#95—32no—
167C2×#95—32no—
168C2×#95C2×D312yes—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3—22no—
173C24×C3—22no—
174C24×C3—24no—
175C24×C3C2×D414yes—
176C24×C3C2×Q814yes—
177C24×C3C2×Q814yescomplemented
178C24×C3#4114yes—
179C24×C3C2412yes—
180C24×C3C2412yes—
181C24×C3C2411yescharacteristic
182C2×C3×D4—232no—
183C2×C3×D4—28no—
184C2×C3×D4—28no—
185C2×C3×D4C2×Q8116yescomplemented
186C22×C3×C4—232no—
187C22×C3×C4—22no—
188C22×C3×C4—28no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—216no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4C2×D418yes—
197C22×C3×C4C2×Q814yescomplemented
198C22×C3×C4C2414yes—
199C22×C3×C4C2414yes—
200C22×C3×C4C2412yes—
201C22×Dic3—232no—
202C22×Dic3—28no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—232no—
207C22×Dic3—216no—
208C22×Dic3—216no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—22no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3—216no—
217C22×Dic3—28no—
218C22×Dic3—416no—
219C22×Dic3C2×D414yes—
220C22×Dic3C2×D418yes—
221C22×Dic3C2×D4116yes—
222C22×Dic3C2×D414yes—
223C22×Dic3C2×D418yes—
224C22×Dic3C2412yes—
225C22×Dic3C2414yes—
226C22×Dic3C2412yes—
227C22×Dic3C2412yes—
228C22×Dic3C2414yes—
229C22×Dic3C2412yes—
230C2×Dic6—264no—
231C2×Dic6—2256no—
232C2×Dic6—216no—
233C2×Dic6—264no—
234C2×Dic6—216no—
235C2×Dic6—216no—
236C2×Dic6—216no—
237C2×Dic6—216no—
238C2×Dic6C2×D4132yescomplemented
239C3×C22⋊C4—28no—
240C3×C22⋊C4—216no—
241#217—28no—
242#217—28no—
243#217—216no—
244#217—28no—
245C4⋊Dic3—216no—
246C4⋊Dic3—28no—
247Dic3⋊C4—216no—
248Dic3⋊C4—216no—
249C4×Dic3—28no—
250C25C22×D311yescharacteristic
251C25C22×D311yescharacteristic
252C22×Q8—316no—
253C22×Q8—38no—
254C22×Q8—38no—
255C22×Q8—68no—
256C22×Q8—68no—
257C22×Q8—68no—
258C22×Q8—68no—
259C22×Q8—632no—
260C22×Q8—664no—
261C22×Q8—6128no—
262C22×Q8—68no—
263C22×D4—24no—
264C22×D4—24no—
265C22×D4Dic618yescomplemented
266C22×D4C22×D314yes—
267C23×C4—24no—
268C23×C4—28no—
269C23×C4—31no—
270C23×C4—38no—
271C23×C4—34no—
272C23×C4—31no—
273C23×C4—32no—
274C23×C4—34no—
275C23×C4—34no—
276C23×C4—32no—
277C23×C4—31no—
278C23×C4—31no—
279C23×C4—68no—
280C23×C4—68no—
281C23×C4—64no—
282C23×C4—68no—
283C23×C4—64no—
284C23×C4C22×D314yes—
285C23×C4C22×D312yes—
286C23×C4C22×D311yescharacteristic
287C23×C4C22×D312yes—
288C22⋊Q8—664no—
289C4×D4—632no—
290C2×C4⋊C4—34no—
291C2×C4⋊C4—38no—
292C2×C4⋊C4—38no—
293C2×C4⋊C4—38no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—68no—
296C2×C4⋊C4—68no—
297C2×C4⋊C4—68no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—34no—
301C2×C22⋊C4—34no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—34no—
304C2×C22⋊C4—64no—
305C2×C22⋊C4—68no—
306C2×C22⋊C4—64no—
307C2×C22⋊C4—64no—
308C2×C22⋊C4C22×D318yes—
309C2×C22⋊C4C22×D314yes—
310C2×C42—34no—
311C2×C42—64no—
312#95—38no—
313#95—38no—
314#95—38no—
315#95—38no—
316#95C22×D318yes—
317C23×C3—28no—
318C23×C3—28no—
319C23×C3—216no—
320C23×C3—216no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—28no—
324C23×C3—216no—
325C23×C3—24no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—24no—
329C23×C3—44no—
330C23×C3—44no—
331C23×C3—48no—
332C23×C3C22⋊Q8116yescomplemented
333C23×C3C22×D414yes—
334C23×C3C22×D412yes—
335C23×C3C22×D414yes—
336C23×C3C22×Q814yes—
337C23×C3C22×Q812yes—
338C23×C3C22×Q814yescomplemented
339C23×C3C2×#4114yes—
340C23×C3C2×#4112yes—
341C23×C32−1+414yes—
342C23×C3C2511yescharacteristic
343C3×D4—232no—
344C2×C3×C4—232no—
345C2×C3×C4—216no—
346C2×C3×C4—28no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—48no—
355C2×C3×C4C22×D418yes—
356C2×C3×C4C22×Q814yescomplemented
357C2×Dic3—232no—
358C2×Dic3—28no—
359C2×Dic3—216no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—232no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—44no—
372C2×Dic3—44no—
373C2×Dic3—48no—
374C2×Dic3—416no—
375C2×Dic3#120116yes—
376C2×Dic3C22×D414yes—
377C2×Dic3C22×D418yes—
378C2×Dic3C22×D414yes—
379C2×Dic3C22×D418yes—
380Dic6—264no—
381Dic6—264no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—22no—
387C24—24no—
388C24C2×Dic614yes—
389C24C2×Dic614yescomplemented
390C24D3×D414yes—
391C24#23714yes—
392C24C23×D312yes—
393C24C23×D312yes—
394C24C23×D311yescharacteristic
395C2×Q8—332no—
396C2×Q8—616no—
397C2×Q8—616no—
398C2×Q8—616no—
399C2×Q8—616no—
400C2×Q8—664no—
401C2×Q8—6256no—
402C2×Q8—664no—
403C2×Q8—616no—
404C2×D4—232no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4C2×Dic6116yescomplemented
408C22×C4—232no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—34no—
419C22×C4—38no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—38no—
423C22×C4—316no—
424C22×C4—32no—
425C22×C4—34no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—616no—
430C22×C4—632no—
431C22×C4—616no—
432C22×C4—68no—
433C22×C4—616no—
434C22×C4—632no—
435C22×C4—68no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—64no—
441C22×C4—62no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—1216no—
447C22×C4C2×Dic614yescomplemented
448C22×C4D3×D418yes—
449C22×C4C23×D314yes—
450C22×C4C23×D312yes—
451C22×C4C23×D314yes—
452C4⋊C4—68no—
453C4⋊C4—616no—
454C4⋊C4—616no—
455C4⋊C4—616no—
456C22⋊C4—28no—
457C22⋊C4—216no—
458C22⋊C4—68no—
459C22⋊C4—616no—
460C22⋊C4—68no—
461C22⋊C4—68no—
462C42—68no—
463C22×C3—24no—
464C22×C3—232no—
465C22×C3—24no—
466C22×C3—28no—
467C22×C3—28no—
468C22×C3—24no—
469C22×C3—28no—
470C22×C3—42no—
471C22×C3—416no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3C2×#12014yes—
475C22×C3C2×C22⋊Q814yes—
476C22×C3C2×C22⋊Q818yescomplemented
477C22×C3#53614yes—
478C22×C3#54818yes—
479C22×C3D4×Q818yes—
480C22×C3C23×D411yescharacteristic
481C22×C3C23×D412yes—
482C22×C3C23×Q811yescharacteristic
483C22×C3C22×#4111yescharacteristic
484C22×C3C2×2−1+412yes—
485C3×C4—28no—
486C3×C4—24no—
487C3×C4—44no—
488Dic3—24no—
489Dic3—28no—
490Dic3—42no—
491Dic3—42no—
492Dic3—44no—
493Dic3C2×#12018yes—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—216no—
498C23—28no—
499C23—216no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—28no—
505C23—216no—
506C23—44no—
507C23—44no—
508C23—48no—
509C23#878116yescomplemented
510C23C22×Dic614yes—
511C23C22×Dic612yes—
512C23C22×Dic614yescomplemented
513C23C2×D3×D414yes—
514C23C2×D3×D412yes—
515C23C2×D3×D414yes—
516C23C2×#23714yes—
517C23C2×#23712yes—
518C23#101014yes—
519C23C24×D311yescharacteristic
520Q8—664no—
521Q8—664no—
522D4—232no—
523C2×C4—232no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—216no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—34no—
534C2×C4—38no—
535C2×C4—316no—
536C2×C4—34no—
537C2×C4—38no—
538C2×C4—48no—
539C2×C4—632no—
540C2×C4—632no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—68no—
545C2×C4—616no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—1216no—
554C2×C4—124no—
555C2×C4—124no—
556C2×C4—128no—
557C2×C4C22×Dic614yescomplemented
558C2×C4C2×D3×D418yes—
559C2×C3—28no—
560C2×C3—44no—
561C2×C3#240918yescomplemented
562C2×C3C22×#12011yescharacteristic
563C2×C3C22×C22⋊Q811yescharacteristic
564C2×C3C2×#53611yescharacteristic
565C2×C3C2×#54812yes—
566C2×C3C2×D4×Q812yes—
567C22—24no—
568C22—28no—
569C22—232no—
570C22—24no—
571C22—28no—
572C22—24no—
573C22—28no—
574C22—42no—
575C22—44no—
576C22—416no—
577C22—42no—
578C22C2×#87814yes—
579C22C2×#87818yescomplemented
580C22D4×Dic618yes—
581C22D3×#12014yes—
582C22#559818yes—
583C22#564814yes—
584C22C23×Dic611yescharacteristic
585C22C22×D3×D412yes—
586C22C22×D3×D411yescharacteristic
587C22C22×#23711yescharacteristic
588C22C2×#101012yes—
589C4—28no—
590C4—24no—
591C4—38no—
592C4—44no—
593C4—64no—
594C4—68no—
595C4—122no—
596C4—124no—
597C4—122no—
598C3C2×#240911yescharacteristic, complemented, Sylow
599C2—28no—
600C2—44no—
601C2#7468918yesdirect product factor
602C2C22×#87811yescharacteristic
603C2C2×D4×Dic612yes—
604C2C2×D3×#12011yescharacteristic
605C2C2×#559812yes—
606C2C2×#564811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#386C2
socleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#386D3
hypocenterC3C2×#2409
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. C24×Dic6C2semidirect
3. C22×#878C2semidirect
6. C2×#5210C2semidirect
7. C2×#4947C2semidirect
8. C2×#4950C2semidirect
9. C2×#4942C2semidirect
10. #74689C2direct (N × H)
11. C2×#4656C2semidirect
12. C2×#4646C2semidirect
23. C23×Dic6C22semidirect
38. C2×#878C22semidirect
42. #5210C22semidirect
47. #4950C22semidirect
48. #4947C22semidirect
49. #4942C22semidirect
50. #4656C22semidirect
51. #4646C22semidirect
68. C22×C3×D4Q8semidirect
99. C22×Dic6D4semidirect
177. C24×C3C2×Q8semidirect
185. C2×C3×D4C2×Q8semidirect
197. C22×C3×C4C2×Q8semidirect
238. C2×Dic6C2×D4semidirect
265. C22×D4Dic6semidirect
332. C23×C3C22⋊Q8semidirect
338. C23×C3C22×Q8semidirect
356. C2×C3×C4C22×Q8semidirect
389. C24C2×Dic6semidirect
407. C2×D4C2×Dic6semidirect
447. C22×C4C2×Dic6semidirect
476. C22×C3C2×C22⋊Q8semidirect
509. C23#878semidirect
512. C23C22×Dic6semidirect
557. C2×C4C22×Dic6semidirect
561. C2×C3#2409semidirect
579. C22C2×#878semidirect
598. C3C2×#2409semidirect
601. C2#74689direct (N × H)

Maximal subgroups

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

subgroupindex
C24×Dic62
C22×D4×Dic32
C22×#8782
C2×C3×#3862
C2×#52202
C2×#52102
C2×#49472
C2×#49502
C2×#49422
#746892
C2×#46562
C2×#46462
C2×#24093