Finite Group Databasebeta

← C2×#8715 #189806

Subgroups — C2×#8715 #189806

Order 512.

2786 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2280C211yesmaximal, characteristic
2C2×#1779C211yesmaximal, characteristic
3C2×C22⋊#95C211yesmaximal, characteristic
4#8715C218yesmaximal, direct product factor
5C2×#1294C212yesmaximal, complemented
6C2×#1294C212yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C22×#117C411yescharacteristic, complemented
9C22×#117C411yescharacteristic, complemented
10C23×C22⋊C4—21no—
11C22×C22⋊C8—21no—
12C22×C22⋊C8—21no—
13C22×C22⋊C8C2212yes—
14#2280—22no—
15#2280—22no—
16#2280C414yescomplemented
17#2280C2214yes—
18C2×#381—21no—
19C2×#380C411yescharacteristic, complemented
20C2×#380C411yescharacteristic, complemented
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#379—21no—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4—21no—
26C22×#95—21no—
27C22×#95C2212yes—
28#1779—22no—
29#1779C2214yes—
30C22⋊#95—22no—
31C22⋊#95C2214yes—
32#1294C2218yescomplemented
33#1294C2218yescomplemented
34C26D411yescharacteristic
35C24×C4—28no—
36C24×C4—22no—
37C24×C4—44no—
38C24×C4—44no—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D412yes—
46C24×C4D412yes—
47C24×C4C2312yes—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—44no—
53C2×#117—44no—
54C2×#117C2×C414yescomplemented
55C2×#117C2×C414yescomplemented
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—28no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—44no—
69C22×C22⋊C4—44no—
70C22×C42—21no—
71C22×C42—21no—
72C2×C22⋊C8—28no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—44no—
77C2×C22⋊C8—44no—
78C2×C22⋊C8C2314yes—
79#381—24no—
80#381—24no—
81#380—22no—
82#380—22no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380C2×C412yes—
88#380C2×C412yes—
89#380C2×C414yescomplemented
90#380C2×C414yescomplemented
91#380C2312yes—
92#380C2312yes—
93#379—24no—
94#379—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—28no—
104C2×#95—22no—
105C2×#95—24no—
106C2×#95—44no—
107C2×#95C2314yes—
108C25—24no—
109C25—24no—
110C25—22no—
111C25—22no—
112C25—24no—
113C25—24no—
114C25—42no—
115C25—42no—
116C25C22⋊C411yescharacteristic
117C25C22⋊C411yescharacteristic
118C25C22⋊C412yes—
119C25C2×D412yes—
120C25C2×D411yescharacteristic
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—28no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—28no—
132C23×C4—28no—
133C23×C4—24no—
134C23×C4—28no—
135C23×C4—28no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4—464no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4C22⋊C412yes—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C412yes—
166C23×C4C22⋊C412yes—
167C23×C4C22⋊C412yes—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C412yes—
170C23×C4C22×C412yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4118yes—
181C23×C4C2411yescharacteristic
182C22×C8—24no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8—44no—
186#117—48no—
187#117—48no—
188#117—48no—
189#117—48no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C22⋊C4—216no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—432no—
213C2×C22⋊C4—48no—
214C2×C22⋊C4—48no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—44no—
224C2×C42—44no—
225C22⋊C8—28no—
226C22⋊C8—48no—
227C22⋊C8—48no—
228C22⋊C8—44no—
229#95—22no—
230#95—22no—
231#95—24no—
232#95—24no—
233#95—22no—
234#95—22no—
235#95—28no—
236#95—48no—
237#95—44no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—216no—
243C24—216no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—24no—
250C24—22no—
251C24—24no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—24no—
256C24—22no—
257C24—22no—
258C24—24no—
259C24—24no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—416no—
272C24—416no—
273C24—416no—
274C24—416no—
275C24—48no—
276C24—48no—
277C24—48no—
278C24—48no—
279C24#9914yes—
280C24#10414yes—
281C24C2×C22⋊C412yes—
282C24C2×C22⋊C412yes—
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285C24#12314yes—
286C24#13514yes—
287C24AGL(1,ℤ/8)12yes—
288C24#13712yes—
289C24C22×D411yescharacteristic
290C22×C4—232no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—416no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—416no—
347C22×C4—432no—
348C22×C4—432no—
349C22×C4—432no—
350C22×C4—432no—
351C22×C4—416no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—44no—
357C22×C4—416no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4#9914yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C412yes—
382C22×C4#11714yes—
383C22×C4#11714yes—
384C22×C4#12314yes—
385C22×C4#12314yes—
386C22×C4#12314yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C2×#4114yes—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—88no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C22⋊C4—48no—
403C22⋊C4—432no—
404C22⋊C4—48no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—48no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C42—48no—
413C42—48no—
414C23—28no—
415C23—28no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—28no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—28no—
430C23—28no—
431C23—28no—
432C23—216no—
433C23—28no—
434C23—28no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—28no—
446C23—28no—
447C23—28no—
448C23—22no—
449C23—22no—
450C23—28no—
451C23—48no—
452C23—48no—
453C23—48no—
454C23—48no—
455C23—48no—
456C23—48no—
457C23—48no—
458C23—48no—
459C23—464no—
460C23—464no—
461C23—44no—
462C23—44no—
463C23—48no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—48no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23#38112yes—
476C23#38112yes—
477C23C2×#9914yes—
478C23C2×#9912yes—
479C23C2×#9912yes—
480C23#41612yes—
481C23#41612yes—
482C23#41712yes—
483C23#41712yes—
484C23C2×#10414yes—
485C23C2×#10412yes—
486C23#42112yes—
487C23#48218yes—
488C23#48518yes—
489C23C22×C22⋊C411yescharacteristic
490C23C2×#12312yes—
491C23C2×#13512yes—
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494C2×C4—216no—
495C2×C4—216no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—832no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4#38112yes—
550C2×C4#38112yes—
551C2×C4C2×#9914yes—
552C2×C4C2×#11712yes—
553C2×C4C2×#12312yes—
554C8—84no—
555C22—216no—
556C22—216no—
557C22—22no—
558C22—22no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—416no—
578C22—416no—
579C22—432no—
580C22—416no—
581C22—416no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22#150114yes—
591C22#150414yes—
592C22#151314yes—
593C22#151614yes—
594C22#181914yes—
595C22#182514yes—
596C22#182514yes—
597C22#182614yes—
598C22C2×#38111yescharacteristic
599C22C22×#9911yescharacteristic
600C22C22×#9911yescharacteristic
601C22C2×#41611yescharacteristic
602C22C2×#41711yescharacteristic
603C22C22×#10411yescharacteristic
604C22C2×#42111yescharacteristic
605C22C2×#48212yes—
606C22C2×#48512yes—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—24no—
611C4—22no—
612C4—42no—
613C4—88no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C2—22no—
619C2—22no—
620C2—24no—
621C2—44no—
622C2—44no—
623C2#871518yesdirect product factor
624C2C2×#150111yescharacteristic
625C2C2×#150411yescharacteristic
626C2C2×#151311yescharacteristic
627C2C2×#151611yescharacteristic
628C2C2×#181911yescharacteristic
629C2C2×#182511yescharacteristic
630C2C2×#182611yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8715C2direct (N × H)
5. C2×#1294C2semidirect
6. C2×#1294C2semidirect
8. C22×#117C4semidirect
9. C22×#117C4semidirect
16. #2280C4semidirect
19. C2×#380C4semidirect
20. C2×#380C4semidirect
32. #1294C22semidirect
33. #1294C22semidirect
54. C2×#117C2×C4semidirect
55. C2×#117C2×C4semidirect
89. #380C2×C4semidirect
90. #380C2×C4semidirect
623. C2#8715direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#22802
C2×#17792
C2×C22⋊#952
#87152
C2×#12942