Finite Group Databasebeta

← C2×#71619 #10794582

Subgroups — C2×#71619 #10794582

Order 768.

2066 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
1C22×C3×C4×D4C211yesmaximal, characteristic
2C22×#834C212yesmaximal
3C2×#5212C212yesmaximal, complemented
4#71619C218yesmaximal, direct product factor
5C2×#4526C212yesmaximal, complemented
6C2×#1799—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C24×C3×C4C411yescharacteristic, complemented
9C24×C3×C4C2211yescharacteristic
10C2×C3×C4×D4—22no—
11C2×C3×C4×D4—22no—
12C2×C3×C4×D4—24no—
13C2×C3×C4×D4C414yescomplemented
14C2×C3×C4×D4C2214yes—
15C22×C3×C4⋊C4C411yescharacteristic, complemented
16C22×C3×C22⋊C4C411yescharacteristic, complemented
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C42C411yescharacteristic, complemented
19C22×#217C2214yes—
20C22×C4⋊Dic3—22no—
21C2×#834—28no—
22C2×#834C2218yes—
23#5212—24no—
24#5212C2218yescomplemented
25C2×C3⋊#95—22no—
26#4526C2218yescomplemented
27C22×C4×D4D311yescharacteristic
28C22×#99—32no—
29C2×#380—32no—
30#1799—38no—
31C2×#342—32no—
32C25×C3C2×C411yescharacteristic
33C25×C3C2311yescharacteristic
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4—24no—
37C22×C3×D4C2×C412yes—
38C22×C3×D4D414yes—
39C22×C3×D4Q814yes—
40C22×C3×D4C2312yes—
41C23×C3×C4—22no—
42C23×C3×C4—21no—
43C23×C3×C4—21no—
44C23×C3×C4—44no—
45C23×C3×C4—44no—
46C23×C3×C4C2×C411yescharacteristic
47C23×C3×C4C2×C411yescharacteristic
48C23×C3×C4C2×C411yescharacteristic
49C23×C3×C4C2×C411yescharacteristic
50C23×C3×C4C2×C411yescharacteristic
51C23×C3×C4C2×C412yes—
52C23×C3×C4C2×C414yescomplemented
53C23×C3×C4D412yes—
54C23×C3×C4D412yes—
55C23×C3×C4D414yes—
56C23×C3×C4C2312yes—
57C23×Dic3—24no—
58C3×C4×D4—28no—
59C3×C4×D4—28no—
60C3×C4×D4—28no—
61C3×C4×D4—28no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4C2×C414yescomplemented
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4C2×C412yes—
70C2×C3×C22⋊C4C2×C414yescomplemented
71C2×C3×C22⋊C4C2312yes—
72C2×C3×C42—22no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42C2×C414yescomplemented
76C2×#217—28no—
77C2×#217—28no—
78C2×#217—24no—
79C2×#217C2318yes—
80C2×C4⋊Dic3—28no—
81C2×C4⋊Dic3—48no—
82#834—216no—
83C3⋊#95—28no—
84C23×D4C2×D311yescharacteristic
85C24×C4Dic311yescharacteristic, complemented
86C24×C4C2×D311yescharacteristic
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4—24no—
90C2×C4×D4Dic314yescomplemented
91C2×C4×D4C2×D314yes—
92C22×C4⋊C4—62no—
93C22×C4⋊C4Dic311yescharacteristic, complemented
94C22×C22⋊C4—34no—
95C22×C22⋊C4Dic311yescharacteristic, complemented
96C22×C22⋊C4C2×D311yescharacteristic
97C22×C42Dic311yescharacteristic, complemented
98C2×#99—38no—
99C2×#99—68no—
100#380—38no—
101#380—64no—
102C2×#95—62no—
103#342—38no—
104C24×C3—21no—
105C24×C3—21no—
106C24×C3—22no—
107C24×C3—42no—
108C24×C3—42no—
109C24×C3—42no—
110C24×C3—42no—
111C24×C3C4⋊C414yescomplemented
112C24×C3C22×C411yescharacteristic
113C24×C3C22×C412yes—
114C24×C3C22×C411yescharacteristic
115C24×C3C2×D412yes—
116C24×C3C2×D412yes—
117C24×C3C2×D412yes—
118C24×C3C2×Q812yes—
119C24×C3#4114yes—
120C24×C3C2411yescharacteristic
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—28no—
124C2×C3×D4—28no—
125C2×C3×D4—28no—
126C2×C3×D4—28no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4C4⋊C418yescomplemented
130C2×C3×D4C2×D414yes—
131C2×C3×D4C2×Q814yes—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—28no—
136C22×C3×C4—24no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—44no—
144C22×C3×C4—48no—
145C22×C3×C4—44no—
146C22×C3×C4—42no—
147C22×C3×C4—44no—
148C22×C3×C4—44no—
149C22×C3×C4—48no—
150C22×C3×C4—48no—
151C22×C3×C4—44no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C22⋊C412yes—
155C22×C3×C4C22⋊C411yescharacteristic
156C22×C3×C4C22⋊C412yes—
157C22×C3×C4C22⋊C414yes—
158C22×C3×C4C22⋊C412yes—
159C22×C3×C4C22⋊C414yes—
160C22×C3×C4C22⋊C411yescharacteristic
161C22×C3×C4C4⋊C414yescomplemented
162C22×C3×C4C22×C412yes—
163C22×C3×C4C22×C412yes—
164C22×C3×C4C22×C412yes—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D414yes—
169C22×Dic3—28no—
170C22×Dic3—48no—
171C22×Dic3—48no—
172C22×Dic3—48no—
173C3×C4⋊C4—28no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—28no—
177C3×C42—24no—
178C3×C42—24no—
179#217—28no—
180#217—28no—
181#217—48no—
182C4⋊Dic3—416no—
183C25C2×Dic311yescharacteristic
184C25C22×D311yescharacteristic
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4Dic614yes—
189C22×D4D1214yes—
190C22×D4C2×Dic312yes—
191C22×D4C22×D312yes—
192C23×C4—22no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—64no—
198C23×C4C2×Dic311yescharacteristic
199C23×C4C2×Dic311yescharacteristic
200C23×C4C2×Dic311yescharacteristic
201C23×C4C2×Dic311yescharacteristic
202C23×C4C2×Dic311yescharacteristic
203C23×C4C2×Dic312yes—
204C23×C4C2×Dic314yescomplemented
205C23×C4#6712yes—
206C23×C4#6712yes—
207C23×C4#6714yes—
208C23×C4C22×D312yes—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—68no—
216C2×C4⋊C4—128no—
217C2×C4⋊C4C2×Dic314yescomplemented
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—38no—
223C2×C22⋊C4—68no—
224C2×C22⋊C4—68no—
225C2×C22⋊C4—64no—
226C2×C22⋊C4C2×Dic312yes—
227C2×C22⋊C4C2×Dic314yescomplemented
228C2×C22⋊C4C22×D312yes—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C2×Dic314yescomplemented
233#99—616no—
234#95—68no—
235C23×C3—24no—
236C23×C3—22no—
237C23×C3—24no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—24no—
243C23×C3—41no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—44no—
248C23×C3—42no—
249C23×C3—44no—
250C23×C3—48no—
251C23×C3—41no—
252C23×C3—42no—
253C23×C3—48no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—48no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3C2×C22⋊C412yes—
261C23×C3C2×C22⋊C411yescharacteristic
262C23×C3C2×C22⋊C411yescharacteristic
263C23×C3C2×C4⋊C412yes—
264C23×C3C2×C4⋊C412yes—
265C23×C3C2×C4⋊C414yescomplemented
266C23×C3#11712yes—
267C23×C3#11712yes—
268C23×C3#12114yes—
269C23×C3#12114yes—
270C23×C3C22⋊Q814yes—
271C23×C3C22⋊Q814yes—
272C23×C3C23×C411yescharacteristic
273C23×C3C22×D411yescharacteristic
274C23×C3C22×D412yes—
275C23×C3C22×Q811yescharacteristic
276C23×C3C2×#4112yes—
277C3×D4—28no—
278C3×D4—28no—
279C3×D4—216no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—28no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—28no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—44no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—44no—
296C2×C3×C4—48no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—48no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#9914yescomplemented
305C2×C3×C4C2×C22⋊C412yes—
306C2×C3×C4C2×C22⋊C414yes—
307C2×C3×C4C2×C22⋊C412yes—
308C2×C3×C4C2×C4⋊C414yescomplemented
309C2×Dic3—48no—
310C2×Dic3—48no—
311C2×Dic3—48no—
312C2×Dic3—816no—
313C24—22no—
314C24—21no—
315C24—21no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24C4⋊Dic314yescomplemented
321C24C2×Dic612yes—
322C24C2×D1212yes—
323C24#23514yes—
324C24C22×Dic311yescharacteristic
325C24C22×Dic312yes—
326C24C22×Dic311yescharacteristic
327C24C2×#6712yes—
328C24C2×#6712yes—
329C24C23×D311yescharacteristic
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—24no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4C4⋊Dic318yescomplemented
339C2×D4C2×Dic614yes—
340C2×D4C2×D1214yes—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—68no—
364C22×C4—128no—
365C22×C4—128no—
366C22×C4—128no—
367C22×C4C4⋊Dic314yescomplemented
368C22×C4#21712yes—
369C22×C4#21714yes—
370C22×C4#21711yescharacteristic
371C22×C4#21712yes—
372C22×C4#21712yes—
373C22×C4#21714yes—
374C22×C4#21711yescharacteristic
375C22×C4C22×Dic312yes—
376C22×C4C22×Dic312yes—
377C22×C4C22×Dic312yes—
378C22×C4C22×Dic312yes—
379C22×C4C22×Dic312yes—
380C22×C4C2×#6714yes—
381C22×C4C2×#6714yes—
382C4⋊C4—28no—
383C4⋊C4—1216no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—28no—
387C22⋊C4—68no—
388C22⋊C4—68no—
389C22⋊C4—128no—
390C42—24no—
391C42—24no—
392C22×C3—28no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—28no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—22no—
404C22×C3—24no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—42no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—48no—
415C22×C3—48no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3#38012yes—
419C22×C3#38014yes—
420C22×C3#38012yes—
421C22×C3C2×#9912yes—
422C22×C3#41012yes—
423C22×C3C22×C22⋊C411yescharacteristic
424C22×C3C22×C4⋊C411yescharacteristic
425C22×C3C2×#11711yescharacteristic
426C22×C3C2×#12112yes—
427C22×C3C2×C22⋊Q812yes—
428C3×C4—22no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4C2×#9914yescomplemented
434Dic3—88no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23—44no—
447C23—44no—
448C23—48no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—42no—
453C23—48no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—48no—
458C23—42no—
459C23—42no—
460C23#92414yes—
461C23#92414yes—
462C23C2×C4⋊Dic312yes—
463C23C2×C4⋊Dic312yes—
464C23C2×C4⋊Dic314yescomplemented
465C23#92612yes—
466C23#92612yes—
467C23#93014yes—
468C23#93014yes—
469C23C2×#21712yes—
470C23C2×#21711yescharacteristic
471C23C2×#21711yescharacteristic
472C23C22×Dic611yescharacteristic
473C23C22×D1211yescharacteristic
474C23C2×#23512yes—
475C23C23×Dic311yescharacteristic
476C23C22×#6712yes—
477D4—216no—
478D4—28no—
479D4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—128no—
505C2×C4—128no—
506C2×C4—128no—
507C2×C4—2416no—
508C2×C4#83414yescomplemented
509C2×C4C2×C4⋊Dic314yescomplemented
510C2×C4C2×#21712yes—
511C2×C4C2×#21714yes—
512C2×C4C2×#21712yes—
513C2×C3—22no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—22no—
517C2×C3—22no—
518C2×C3—24no—
519C2×C3—44no—
520C2×C3#179914yescomplemented
521C2×C3C2×#38011yescharacteristic
522C2×C3C22×#9911yescharacteristic
523C2×C3C2×#41011yescharacteristic
524C22—21no—
525C22—21no—
526C22—22no—
527C22—22no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—28no—
533C22—24no—
534C22—24no—
535C22—28no—
536C22—24no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—44no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—44no—
545C22—48no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22—48no—
550C22#521212yes—
551C22#521214yes—
552C22#521212yes—
553C22C2×#83412yes—
554C22#524012yes—
555C22C2×#92412yes—
556C22C22×C4⋊Dic311yescharacteristic
557C22C2×#92611yescharacteristic
558C22C2×#93012yes—
559C22C22×#21711yescharacteristic
560C4—24no—
561C4—22no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—248no—
566C4C2×#83414yescomplemented
567C3C2×#179911yescharacteristic, complemented, Sylow
568C2—22no—
569C2—22no—
570C2—21no—
571C2—21no—
572C2—24no—
573C2—22no—
574C2—44no—
575C2#7161914yesdirect product factor
576C2C2×#521211yescharacteristic
577C2C22×#83411yescharacteristic
578C2C2×#524011yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×C4×D4C2
socleC23×C3C2×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC24×C3C24
hypercenterC22×C4×D4D3
hypocenterC3C2×#1799
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
3. C2×#5212C2semidirect
4. #71619C2direct (N × H)
5. C2×#4526C2semidirect
8. C24×C3×C4C4semidirect
13. C2×C3×C4×D4C4semidirect
15. C22×C3×C4⋊C4C4semidirect
16. C22×C3×C22⋊C4C4semidirect
18. C22×C3×C42C4semidirect
24. #5212C22semidirect
26. #4526C22semidirect
52. C23×C3×C4C2×C4semidirect
64. C2×C3×C4⋊C4C2×C4semidirect
70. C2×C3×C22⋊C4C2×C4semidirect
75. C2×C3×C42C2×C4semidirect
85. C24×C4Dic3semidirect
90. C2×C4×D4Dic3semidirect
93. C22×C4⋊C4Dic3semidirect
95. C22×C22⋊C4Dic3semidirect
97. C22×C42Dic3semidirect
111. C24×C3C4⋊C4semidirect
129. C2×C3×D4C4⋊C4semidirect
161. C22×C3×C4C4⋊C4semidirect
204. C23×C4C2×Dic3semidirect
217. C2×C4⋊C4C2×Dic3semidirect
227. C2×C22⋊C4C2×Dic3semidirect
232. C2×C42C2×Dic3semidirect
265. C23×C3C2×C4⋊C4semidirect
304. C2×C3×C4#99semidirect
308. C2×C3×C4C2×C4⋊C4semidirect
320. C24C4⋊Dic3semidirect
338. C2×D4C4⋊Dic3semidirect
367. C22×C4C4⋊Dic3semidirect
433. C3×C4C2×#99semidirect
464. C23C2×C4⋊Dic3semidirect
508. C2×C4#834semidirect
509. C2×C4C2×C4⋊Dic3semidirect
520. C2×C3#1799semidirect
566. C4C2×#834semidirect
567. C3C2×#1799semidirect
575. C2#71619direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C22×#8342
C2×#52122
#716192
C2×#45262
C2×#17993