Finite Group Databasebeta

← C2×#24107 #6452980

Subgroups — C2×#24107 #6452980

Order 512.

3504 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3C22×#428C211yesmaximal, characteristic, complemented
4C2×#2486C211yesmaximal, characteristic
5#24107C2116yesmaximal, direct product factor
6C2×#2057C212yesmaximal, complemented
7C2×#2036C212yesmaximal, complemented
8C2×#1894C211yesmaximal, characteristic, complemented
9C2×#1870C211yesmaximal, characteristic
10C2×#1865C212yesmaximal, complemented
11C2×#1854C212yesmaximal, complemented
12C2×#1825C211yesmaximal, characteristic, complemented
13C23×M16C2211yescharacteristic
14C2×#556—28no—
15C2×#556—24no—
16C2×#556—28no—
17C2×#556C2218yes—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126—21no—
20C22×#126—21no—
21C22×#124—21no—
22C22×#123—21no—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8—21no—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C2×#484—28no—
32C2×#484C2218yescomplemented
33C2×#428—28no—
34C2×#428C2218yescomplemented
35C22×#107C2212yes—
36C22×#107C2212yes—
37C22×#106C2212yes—
38C22×#103C2212yes—
39C22×#103C2212yes—
40C22×C22⋊C8C2211yescharacteristic
41C22×C22⋊C8C2212yes—
42#2486—24no—
43#2486C2218yes—
44C2×#398—21no—
45C2×#397—21no—
46C2×#389C2212yes—
47C2×#381—21no—
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50#2057C22116yescomplemented
51#2036C22116yescomplemented
52#1894C2218yescomplemented
53#1870C2218yes—
54#1865C22116yescomplemented
55#1854C22116yescomplemented
56#1825C2218yescomplemented
57C2×#340C2212yes—
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—21no—
62C23×Q8C2311yescharacteristic
63C23×D4D411yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C22×M16—24no—
68C22×M16—28no—
69C22×M16C2314yes—
70C23×C8C2312yes—
71C23×C8C2312yes—
72#556—264no—
73#556—232no—
74#556—232no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8C2314yes—
79C2×#126—216no—
80C2×#126—216no—
81C2×#126—28no—
82C2×#126—28no—
83C2×#124—216no—
84C2×#124—28no—
85C2×#123—28no—
86C2×#123—48no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8D418yes—
93C2×C22⋊Q8C2314yes—
94C2×C22⋊Q8C2314yes—
95C2×C4×Q8—216no—
96C2×C4×Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×Q8—24no—
99C2×C4×Q8—22no—
100C2×C4×Q8C2314yes—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4D414yes—
104C2×C4×D4C2314yes—
105C2×#117—24no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117D418yes—
111C2×#117C2314yes—
112C2×#117C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141#484—232no—
142#428—232no—
143C2×#107—28no—
144C2×#107—28no—
145C2×#107C2318yes—
146C2×#107C2318yes—
147C2×#106—28no—
148C2×#106C2318yes—
149C2×#103—28no—
150C2×#103—28no—
151C2×#103C2318yes—
152C2×#103C2318yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—28no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2318yes—
157#398—28no—
158#397—28no—
159#389—24no—
160#389C2318yes—
161#381—28no—
162#380—24no—
163#380—22no—
164#380C2314yes—
165#380C2318yes—
166C2×#95—21no—
167C2×#95—22no—
168C2×#95—22no—
169#340C2318yes—
170#340C2318yes—
171#340C2318yes—
172#336C2318yes—
173C25C2×D411yescharacteristic
174C25C2×D411yescharacteristic
175C22×Q8—28no—
176C22×Q8—22no—
177C22×Q8—21no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8C2×D414yes—
184C22×Q8C2412yes—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—42no—
188C22×D4C2×D414yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C2×M16—232no—
217C2×M16—28no—
218C2×M16—216no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8#4118yes—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—216no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230#126—264no—
231#126—264no—
232#124—264no—
233#123—432no—
234C22⋊Q8—216no—
235C22⋊Q8—28no—
236C22⋊Q8—216no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8C2×D4116yes—
240C4×Q8—264no—
241C4×Q8—216no—
242C4×Q8—28no—
243C4×Q8—28no—
244C4×D4—216no—
245C4×D4—28no—
246C4×D4—28no—
247#117—216no—
248#117—216no—
249#117—28no—
250#117—28no—
251#117—28no—
252#117C2×D4116yes—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42C2×D412yes—
315C2×C42C2412yes—
316#107—216no—
317#107—216no—
318#106—216no—
319#103—216no—
320#103—216no—
321C22⋊C8—28no—
322C22⋊C8—216no—
323#95—28no—
324#95—24no—
325#95—28no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24#12014yes—
336C24C22×D412yes—
337C24C22×D412yes—
338C24C22×D411yescharacteristic
339C2×Q8—216no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—216no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—432no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8C22×D414yes—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—44no—
355C2×D4—416no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—816no—
404C22×C4#12014yes—
405C22×C4#12018yes—
406C22×C4AGL(1,ℤ/8)14yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C42−1+414yes—
418C22×C4C2511yescharacteristic
419M16—232no—
420C2×C8—216no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—48no—
427C2×C8C2×#4118yes—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—416no—
447C4⋊C4C22×D414yes—
448C4⋊C4C22×D414yes—
449C4⋊C4C22×D414yes—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C42—216no—
459C42—216no—
460C42—216no—
461C42—24no—
462C42—22no—
463C42—22no—
464C23—28no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—48no—
477C23—48no—
478C23—48no—
479C23—48no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23C2×#12014yes—
484C23C2×#12012yes—
485C23#54714yes—
486C23C2×AGL(1,ℤ/8)12yes—
487C23#57512yes—
488C23#57814yes—
489C23C23×D411yescharacteristic
490Q8—28no—
491Q8—432no—
492Q8—48no—
493Q8—42no—
494Q8—42no—
495D4—416no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—216no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—816no—
536C2×C4—88no—
537C2×C4—88no—
538C2×C4C2×#12014yes—
539C2×C4C2×#12012yes—
540C2×C4#52514yes—
541C2×C4#53614yes—
542C2×C4#54714yes—
543C2×C4#54718yes—
544C2×C4C2×AGL(1,ℤ/8)14yes—
545C2×C4C23×D411yescharacteristic
546C2×C4C23×D411yescharacteristic
547C2×C4C22×#4112yes—
548C2×C4C2×2−1+412yes—
549C8—28no—
550C8—44no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—416no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—88no—
563C22#240614yes—
564C22#300614yes—
565C22#301114yes—
566C22#334918yes—
567C22#335318yes—
568C22C22×#12011yescharacteristic
569C22C2×#54712yes—
570C22C22×AGL(1,ℤ/8)11yescharacteristic
571C22C2×#57511yescharacteristic
572C22C2×#57812yes—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—44no—
577C4—44no—
578C4—44no—
579C4—42no—
580C4—44no—
581C4—84no—
582C4—84no—
583C4#240614yes—
584C4C2×#52511yescharacteristic
585C4C2×#53611yescharacteristic
586C4C2×#54712yes—
587C2—44no—
588C2—82no—
589C2#2410718yesdirect product factor
590C2C2×#240611yescharacteristic
591C2C2×#300611yescharacteristic
592C2C2×#301111yescharacteristic
593C2C2×#334912yes—
594C2C2×#335312yes—
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×#484C2semidirect
3. C22×#428C2semidirect
5. #24107C2direct (N × H)
6. C2×#2057C2semidirect
7. C2×#2036C2semidirect
8. C2×#1894C2semidirect
10. C2×#1865C2semidirect
11. C2×#1854C2semidirect
12. C2×#1825C2semidirect
32. C2×#484C22semidirect
34. C2×#428C22semidirect
50. #2057C22semidirect
51. #2036C22semidirect
52. #1894C22semidirect
54. #1865C22semidirect
55. #1854C22semidirect
56. #1825C22semidirect
589. C2#24107direct (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×#4842
C22×#4282
C2×#24862
#241072
C2×#20572
C2×#20362
C2×#18942
C2×#18702
C2×#18652
C2×#18542
C2×#18252