Finite Group Databasebeta

← C2×#24098 #6452971

Subgroups — C2×#24098 #6452971

Order 512.

3618 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3C23×#107C211yesmaximal, characteristic, complemented
4C2×#2487C211yesmaximal, characteristic
5#24098C2116yesmaximal, direct product factor
6C2×#2057C212yesmaximal, complemented
7C2×#2036C212yesmaximal, complemented
8C2×#1892C211yesmaximal, characteristic, complemented
9C2×#1869C211yesmaximal, characteristic
10C2×#1863C212yesmaximal, complemented
11C2×#1852C212yesmaximal, complemented
12C2×#1824C211yesmaximal, characteristic, complemented
13C24×C8C2211yescharacteristic
14C2×#554—28no—
15C2×#554—24no—
16C2×#554—28no—
17C2×#554C2218yes—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125—21no—
20C22×C22⋊Q8—21no—
21C22×C22⋊Q8—21no—
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4—21no—
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C2×#484—28no—
31C2×#484C2218yescomplemented
32C22×#107—28no—
33C22×#107C2212yes—
34C22×#107C2212yes—
35C22×#107C2212yes—
36C22×#107C2218yescomplemented
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×C22⋊C8C2211yescharacteristic
40C22×C22⋊C8C2212yes—
41#2487—24no—
42#2487C2218yes—
43C2×#400—21no—
44C2×#399—21no—
45C2×#389C2212yes—
46C2×#386—21no—
47C2×#385—21no—
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50#2057C22116yescomplemented
51#2036C22116yescomplemented
52#1892C2218yescomplemented
53#1869C2218yes—
54#1863C22116yescomplemented
55#1852C22116yescomplemented
56#1824C2218yescomplemented
57C2×#340C2212yes—
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—21no—
62C23×Q8C2311yescharacteristic
63C23×D4D411yescharacteristic
64C24×C4—21no—
65C24×C4—21no—
66C24×C4—21no—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C23×C8—24no—
70C23×C8—28no—
71C23×C8C2312yes—
72C23×C8C2314yes—
73C23×C8C2312yes—
74#554—264no—
75#554—232no—
76#554—232no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8C2314yes—
81C2×#125—216no—
82C2×#125—28no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8D418yes—
93C2×C22⋊Q8C2314yes—
94C2×C22⋊Q8C2314yes—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8C2314yes—
99C2×C4×D4—216no—
100C2×C4×D4—28no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4D414yes—
104C2×C4×D4C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D418yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#484—232no—
140C2×#107—28no—
141C2×#107—28no—
142C2×#107—28no—
143C2×#107—232no—
144C2×#107C2318yes—
145C2×#107C2318yes—
146C2×#107C2318yes—
147C2×#103—28no—
148C2×#103—28no—
149C2×#103C2318yes—
150C2×#103C2318yes—
151C2×C22⋊C8—24no—
152C2×C22⋊C8—28no—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2318yes—
155#400—28no—
156#399—28no—
157#389—24no—
158#389C2318yes—
159#386—28no—
160#385—28no—
161#380—24no—
162#380—22no—
163#380C2314yes—
164#380C2318yes—
165C2×#95—21no—
166C2×#95—22no—
167C2×#95—22no—
168#340C2318yes—
169#340C2318yes—
170#340C2318yes—
171#336C2318yes—
172C25C2×D411yescharacteristic
173C25C2×D411yescharacteristic
174C22×Q8—28no—
175C22×Q8—22no—
176C22×Q8—21no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—84no—
181C22×Q8C2×D414yes—
182C22×Q8C2412yes—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4Q1618yes—
186C22×D4C2×D414yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—84no—
206C23×C4—84no—
207C23×C4—84no—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C22×C8—232no—
220C22×C8—28no—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—216no—
224C22×C8—24no—
225C22×C8—24no—
226C22×C8—24no—
227C22×C8#4118yes—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—216no—
231C4⋊Q8—28no—
232C4⋊Q8—28no—
233#125—264no—
234C22⋊Q8—216no—
235C22⋊Q8—264no—
236C22⋊Q8—264no—
237C22⋊Q8—28no—
238C22⋊Q8—216no—
239C22⋊Q8—28no—
240C22⋊Q8—28no—
241C22⋊Q8C2×D4116yes—
242C4×Q8—216no—
243C4×Q8—28no—
244C4×Q8—28no—
245C4×D4—264no—
246C4×D4—216no—
247C4×D4—28no—
248C4×D4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—84no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D4116yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—84no—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C42—28no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42C2×D412yes—
313C2×C42C2412yes—
314#107—216no—
315#107—216no—
316#107—216no—
317#103—216no—
318#103—216no—
319C22⋊C8—28no—
320C22⋊C8—216no—
321#95—28no—
322#95—24no—
323#95—28no—
324C24—24no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24#12014yes—
331C24C2×Q1614yes—
332C24C2×Q1614yes—
333C24#13714yes—
334C24C22×D412yes—
335C24C22×D412yes—
336C24C22×D411yescharacteristic
337C2×Q8—216no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—216no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—88no—
347C2×Q8—816no—
348C2×Q8C22×D414yes—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—232no—
352C2×D4C2×Q16116yes—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—28no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—816no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—816no—
403C22×C4—816no—
404C22×C4#12014yes—
405C22×C4#12018yes—
406C22×C4C2×Q1614yes—
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
419C2×C8—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—216no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—216no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—88no—
449C4⋊C4C22×D414yes—
450C4⋊C4C22×D414yes—
451C4⋊C4C22×D414yes—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—48no—
457C22⋊C4—88no—
458C42—216no—
459C42—24no—
460C42—22no—
461C42—22no—
462C23—216no—
463C23—216no—
464C23—28no—
465C23—24no—
466C23—28no—
467C23—24no—
468C23—216no—
469C23—22no—
470C23—24no—
471C23—28no—
472C23—22no—
473C23—24no—
474C23—48no—
475C23—44no—
476C23—44no—
477C23#451116yes—
478C23C2×#12014yes—
479C23C2×#12012yes—
480C23#54714yes—
481C23C22×Q1614yes—
482C23C22×Q1614yes—
483C23C22×Q1612yes—
484C23C2×#13714yes—
485C23C2×#13712yes—
486C23#57614yes—
487C23C23×D411yescharacteristic
488Q8—28no—
489Q8—48no—
490Q8—42no—
491Q8—42no—
492Q8—816no—
493D4—232no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—22no—
499C2×C4—216no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—816no—
532C2×C4—816no—
533C2×C4—816no—
534C2×C4—88no—
535C2×C4—88no—
536C2×C4C2×#12014yes—
537C2×C4C2×#12012yes—
538C2×C4#52514yes—
539C2×C4#53614yes—
540C2×C4#54714yes—
541C2×C4#54718yes—
542C2×C4C22×Q1614yes—
543C2×C4C23×D411yescharacteristic
544C2×C4C23×D411yescharacteristic
545C2×C4C22×#4112yes—
546C2×C4C2×2−1+412yes—
547C8—28no—
548C8—44no—
549C22—232no—
550C22—28no—
551C22—28no—
552C22—24no—
553C22—24no—
554C22—28no—
555C22—24no—
556C22—416no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22#240614yes—
561C22C2×#45118yes—
562C22C2×#45114yes—
563C22#300814yes—
564C22#333818yes—
565C22#334618yes—
566C22C22×#12011yescharacteristic
567C22C2×#54712yes—
568C22C23×Q1611yescharacteristic
569C22C22×#13711yescharacteristic
570C22C2×#57612yes—
571C4—24no—
572C4—24no—
573C4—42no—
574C4—44no—
575C4—42no—
576C4—44no—
577C4—44no—
578C4—44no—
579C4—84no—
580C4—84no—
581C4#240614yes—
582C4C2×#52511yescharacteristic
583C4C2×#53611yescharacteristic
584C4C2×#54712yes—
585C2—28no—
586C2—44no—
587C2#2409818yesdirect product factor
588C2C2×#240611yescharacteristic
589C2C22×#45111yescharacteristic
590C2C2×#300811yescharacteristic
591C2C2×#333812yes—
592C2C2×#334612yes—
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. C23×#107C2semidirect
5. #24098C2direct (N × H)
6. C2×#2057C2semidirect
7. C2×#2036C2semidirect
8. C2×#1892C2semidirect
10. C2×#1863C2semidirect
11. C2×#1852C2semidirect
12. C2×#1824C2semidirect
31. C2×#484C22semidirect
36. C22×#107C22semidirect
50. #2057C22semidirect
51. #2036C22semidirect
52. #1892C22semidirect
54. #1863C22semidirect
55. #1852C22semidirect
56. #1824C22semidirect
587. C2#24098direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5542
C22×#4842
C23×#1072
C2×#24872
#240982
C2×#20572
C2×#20362
C2×#18922
C2×#18692
C2×#18632
C2×#18522
C2×#18242