Finite Group Databasebeta

← C2×#23980 #6451809

Subgroups — C2×#23980 #6451809

Order 512.

4084 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C22×#451C211yesmaximal, characteristic, complemented
3C22×#419C211yesmaximal, characteristic, complemented
4C2×#2409C211yesmaximal, characteristic
5#23980C2116yesmaximal, direct product factor
6C2×#2024C212yesmaximal, complemented
7C2×#2001C212yesmaximal, complemented
8C2×#1894C211yesmaximal, characteristic, complemented
9C2×#1878C212yesmaximal, complemented
10C2×#1873C212yesmaximal, complemented
11C2×#1870C211yesmaximal, characteristic
12C2×#1825C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C23×Q16C2212yes—
15C23×M16C2211yescharacteristic
16C2×#556—28no—
17C2×#556—24no—
18C2×#556—28no—
19C2×#556C2218yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126—21no—
22C22×#126—21no—
23C22×#124—21no—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C4×Q8—21no—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C2×#451—28no—
33C2×#451C2218yescomplemented
34C2×#419—28no—
35C2×#419C2218yescomplemented
36C22×#103C2212yes—
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#103C2212yes—
40C22×C22⋊C8C2211yescharacteristic
41C22×C22⋊C8C2212yes—
42#2409—24no—
43#2409C2218yes—
44C2×#395—21no—
45C2×#393—21no—
46C2×#391C2212yes—
47C2×#386—21no—
48C2×#385—21no—
49C2×#380C2211yescharacteristic
50#2024C22116yescomplemented
51#2001C22116yescomplemented
52#1894C2218yescomplemented
53#1878C22116yescomplemented
54#1873C22116yescomplemented
55#1870C2218yes—
56#1825C2218yescomplemented
57C2×#340C2212yes—
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—21no—
62C23×Q8—21no—
63C23×Q8—22no—
64C23×Q8—28no—
65C23×Q8—24no—
66C23×Q8—24no—
67C23×Q8—44no—
68C23×Q8C2311yescharacteristic
69C23×Q8C2312yes—
70C23×Q8C2314yes—
71C23×D4D411yescharacteristic
72C24×C4—21no—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C22×Q16—28no—
76C22×Q16—28no—
77C22×Q16—44no—
78C22×Q16C2318yes—
79C22×M16—24no—
80C22×M16—44no—
81C22×M16C2314yes—
82C23×C8C2312yes—
83C23×C8C2312yes—
84#556—264no—
85#556—232no—
86#556—232no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8C2314yes—
91C2×#126—216no—
92C2×#126—216no—
93C2×#126—28no—
94C2×#126—28no—
95C2×#124—216no—
96C2×#124—28no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—44no—
105C2×C22⋊Q8D418yes—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2318yes—
108C2×C22⋊Q8C2314yes—
109C2×C4×Q8—216no—
110C2×C4×Q8—24no—
111C2×C4×Q8—28no—
112C2×C4×Q8—24no—
113C2×C4×Q8—22no—
114C2×C4×Q8C2314yes—
115C2×C4×D4—24no—
116C2×C4×D4—24no—
117C2×C4×D4D414yes—
118C2×C4×D4C2314yes—
119C2×#117—24no—
120C2×#117—22no—
121C2×#117—22no—
122C2×#117—24no—
123C2×#117—24no—
124C2×#117D418yes—
125C2×#117C2314yes—
126C2×#117C2314yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—21no—
143C22×C22⋊C4D411yescharacteristic
144C22×C22⋊C4D411yescharacteristic
145C22×C42—21no—
146C22×C42—21no—
147C22×C42—21no—
148C22×C42C2311yescharacteristic
149#451—232no—
150#419—232no—
151C2×#103—28no—
152C2×#103—28no—
153C2×#103—28no—
154C2×#103—44no—
155C2×#103C2318yes—
156C2×#103C2318yes—
157C2×#103C2318yes—
158C2×#103C2318yes—
159C2×C22⋊C8—24no—
160C2×C22⋊C8—44no—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2318yes—
163#395—28no—
164#393—28no—
165#391—24no—
166#391C2318yes—
167#386—28no—
168#385—28no—
169#380—22no—
170#380C2314yes—
171C2×#95—21no—
172C2×#95—22no—
173C2×#95—22no—
174#340C2318yes—
175#340C2318yes—
176#340C2318yes—
177#336C2318yes—
178C25C2×D411yescharacteristic
179C25C2×D411yescharacteristic
180C22×Q8—28no—
181C22×Q8—22no—
182C22×Q8—21no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—232no—
188C22×Q8—216no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—48no—
198C22×Q8—48no—
199C22×Q8—416no—
200C22×Q8—432no—
201C22×Q8—44no—
202C22×Q8—416no—
203C22×Q8—44no—
204C22×Q8—48no—
205C22×Q8—44no—
206C22×Q8C2×D414yes—
207C22×Q8#4118yes—
208C22×Q8C2412yes—
209C22×Q8C2414yes—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—42no—
213C22×D4C2×D414yes—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—48no—
228C23×C4—44no—
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C2×Q16—216no—
241C2×Q16—216no—
242C2×Q16—48no—
243C2×Q16—432no—
244C2×M16—28no—
245C2×M16—48no—
246C2×M16—416no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C4⋊Q8—216no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258#126—264no—
259#126—264no—
260#124—264no—
261C22⋊Q8—216no—
262C22⋊Q8—28no—
263C22⋊Q8—216no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—216no—
267C22⋊Q8—48no—
268C22⋊Q8—416no—
269C22⋊Q8C2×D4116yes—
270C4×Q8—264no—
271C4×Q8—216no—
272C4×Q8—28no—
273C4×Q8—28no—
274C4×D4—216no—
275C4×D4—28no—
276C4×D4—28no—
277#117—216no—
278#117—216no—
279#117—28no—
280#117—28no—
281#117—28no—
282#117C2×D4116yes—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—28no—
328C2×C42—28no—
329C2×C42—28no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337#103—216no—
338#103—216no—
339#103—216no—
340#103—48no—
341C22⋊C8—28no—
342C22⋊C8—48no—
343#95—24no—
344#95—28no—
345#95—28no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24#12014yes—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C22×D411yescharacteristic
359C2×Q8—216no—
360C2×Q8—24no—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—232no—
365C2×Q8—216no—
366C2×Q8—216no—
367C2×Q8—216no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—416no—
375C2×Q8—432no—
376C2×Q8—432no—
377C2×Q8—48no—
378C2×Q8—432no—
379C2×Q8—464no—
380C2×Q8—416no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8C22×D414yes—
391C2×Q8C2×#4118yes—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—44no—
395C2×D4—416no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—416no—
427C22×C4—432no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4#12014yes—
444C22×C4#12018yes—
445C22×C4#13714yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4114yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4C2511yescharacteristic
458Q16—432no—
459M16—416no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—48no—
467C2×C8—48no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—28no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—416no—
482C4⋊C4—44no—
483C4⋊C4C22×D414yes—
484C4⋊C4C22×D414yes—
485C4⋊C4C22×D414yes—
486C22⋊C4—216no—
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C42—216no—
495C42—216no—
496C42—216no—
497C42—24no—
498C42—22no—
499C42—22no—
500C23—28no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23C2×#12014yes—
520C23C2×#12012yes—
521C23#54614yes—
522C23C2×#13712yes—
523C23#57512yes—
524C23#57814yes—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—28no—
528Q8—28no—
529Q8—432no—
530Q8—432no—
531Q8—48no—
532Q8—416no—
533Q8—42no—
534Q8—42no—
535Q8—44no—
536Q8—44no—
537D4—416no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—216no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—432no—
557C2×C4—44no—
558C2×C4—416no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—88no—
580C2×C4C2×#12014yes—
581C2×C4C2×#12012yes—
582C2×C4#52514yes—
583C2×C4#53514yes—
584C2×C4#54614yes—
585C2×C4#54618yes—
586C2×C4C2×#13714yes—
587C2×C4C23×D411yescharacteristic
588C2×C4C23×D411yescharacteristic
589C2×C4C22×#4112yes—
590C2×C4C2×H(5,𝔽2)12yes—
591C8—44no—
592C8—44no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—416no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—88no—
605C22#240414yes—
606C22#300714yes—
607C22#301114yes—
608C22#331318yes—
609C22#332818yes—
610C22C22×#12011yescharacteristic
611C22C2×#54612yes—
612C22C22×#13711yescharacteristic
613C22C2×#57511yescharacteristic
614C22C2×#57812yes—
615C4—24no—
616C4—24no—
617C4—42no—
618C4—48no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—84no—
625C4#240414yes—
626C4C2×#52511yescharacteristic
627C4C2×#53511yescharacteristic
628C4C2×#54612yes—
629C2—44no—
630C2—82no—
631C2#2398018yesdirect product factor
632C2C2×#240411yescharacteristic
633C2C2×#300711yescharacteristic
634C2C2×#301111yescharacteristic
635C2C2×#331312yes—
636C2C2×#332812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#451C2semidirect
3. C22×#419C2semidirect
5. #23980C2direct (N × H)
6. C2×#2024C2semidirect
7. C2×#2001C2semidirect
8. C2×#1894C2semidirect
9. C2×#1878C2semidirect
10. C2×#1873C2semidirect
12. C2×#1825C2semidirect
33. C2×#451C22semidirect
35. C2×#419C22semidirect
50. #2024C22semidirect
51. #2001C22semidirect
52. #1894C22semidirect
53. #1878C22semidirect
54. #1873C22semidirect
56. #1825C22semidirect
631. C2#23980direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5562
C22×#4512
C22×#4192
C2×#24092
#239802
C2×#20242
C2×#20012
C2×#18942
C2×#18782
C2×#18732
C2×#18702
C2×#18252