Finite Group Databasebeta

← C2×#8448 #186604

Subgroups — C2×#8448 #186604

Order 512.

2418 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C22×#332C211yesmaximal, characteristic
6C22×Q8⋊C8C211yesmaximal, characteristic
7C22×Q8⋊C8C211yesmaximal, characteristic
8#8448C218yesmaximal, direct product factor
9C2×#557—24no—
10C2×#557—22no—
11C2×#557—24no—
12C2×#557—22no—
13C2×#557—24no—
14C2×#557—22no—
15C2×#557C418yescomplemented
16C2×#557C2214yes—
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125—21no—
20C22×#125—22no—
21C22×C4×Q8—21no—
22C22×C4×Q8—21no—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103—21no—
28C22×#103—21no—
29C22×C4×C8C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#1990—22no—
32#1990C2214yes—
33#1803—22no—
34#1803—22no—
35#1803C2214yes—
36#1803C2214yes—
37C2×#340—22no—
38C2×#340—21no—
39C2×#340—21no—
40C2×#332—24no—
41C2×#332C2214yes—
42C2×Q8⋊C8—24no—
43C2×Q8⋊C8—24no—
44C2×Q8⋊C8C2214yes—
45C2×Q8⋊C8C2214yes—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C23×Q8D411yescharacteristic
49C23×C8—21no—
50C23×C8—21no—
51C23×C8—21no—
52#557—216no—
53#557—216no—
54#557—216no—
55#557—216no—
56#557—28no—
57#557—216no—
58#557—216no—
59#557—28no—
60#557—28no—
61#557C2×C4116yescomplemented
62C2×C4⋊Q8—28no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8C2×C414yes—
71C2×C4⋊Q8D414yes—
72C2×C4⋊Q8D412yes—
73C2×C4⋊Q8C2312yes—
74C2×#125—28no—
75C2×#125—216no—
76C2×#125—28no—
77C2×#125—24no—
78C2×#125—216no—
79C2×#125—28no—
80C2×#125—28no—
81C2×#125—24no—
82C2×C4×Q8—24no—
83C2×C4×Q8—28no—
84C2×C4×Q8—24no—
85C2×C4×Q8—28no—
86C2×C4×Q8—28no—
87C2×C4×Q8—24no—
88C2×C4×Q8—28no—
89C2×C4×Q8—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—21no—
95C2×C4×Q8—22no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8C2×C414yes—
99C2×C4×Q8C2×C414yes—
100C2×C4×Q8D414yes—
101C2×C4×Q8D414yes—
102C2×C4×Q8D412yes—
103C2×C4×Q8D412yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D412yes—
118C22×C42—22no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C2×C4⋊C8—22no—
123C2×C4⋊C8—22no—
124C2×C4⋊C8—22no—
125C2×C4⋊C8—22no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#103—24no—
131C2×#103—24no—
132C2×#103—44no—
133C2×#103—44no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136C2×C4×C8—21no—
137C2×C4×C8C2312yes—
138#384—21no—
139#384—21no—
140#384—21no—
141#384C2312yes—
142#340—24no—
143#340—24no—
144#340—28no—
145#332—28no—
146Q8⋊C8—28no—
147Q8⋊C8—28no—
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—24no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8C22⋊C414yes—
167C22×Q8C22⋊C414yes—
168C22×Q8SD1614yes—
169C22×Q8Q1614yes—
170C22×Q8C2×D412yes—
171C22×Q8C2×D412yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—41no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C22×C8—24no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C4⋊Q8—216no—
194C4⋊Q8—232no—
195C4⋊Q8—216no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8C22⋊C418yes—
204C4⋊Q8C22×C414yes—
205C4⋊Q8C2×D414yes—
206#125—232no—
207#125—264no—
208#125—216no—
209#125—232no—
210#125—232no—
211#125—216no—
212C4×Q8—216no—
213C4×Q8—216no—
214C4×Q8—232no—
215C4×Q8—232no—
216C4×Q8—216no—
217C4×Q8—216no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8C22⋊C418yes—
233C4×Q8C22⋊C418yes—
234C4×Q8C22×C414yes—
235C4×Q8C22×C414yes—
236C4×Q8C2×D414yes—
237C4×Q8C2×D414yes—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4C22⋊C414yes—
274C2×C4⋊C4C22⋊C414yes—
275C2×C4⋊C4C22⋊C418yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D414yes—
279C2×C42—28no—
280C2×C42—216no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42C22×C412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2411yescharacteristic
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303#103—48no—
304#103—48no—
305C4×C8—22no—
306C4×C8—22no—
307C4×C8—42no—
308C24#12011yescharacteristic
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—28no—
318C2×Q8—28no—
319C2×Q8—24no—
320C2×Q8—28no—
321C2×Q8—24no—
322C2×Q8—28no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—28no—
328C2×Q8—28no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—42no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8#103116yescomplemented
344C2×Q8C2×C22⋊C414yes—
345C2×Q8C2×C22⋊C414yes—
346C2×Q8C2×SD1614yes—
347C2×Q8C2×Q1614yes—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—28no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4#12012yes—
378C22×C4#12011yescharacteristic
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#12011yescharacteristic
382C22×C4#12011yescharacteristic
383C22×C4C2×SD1612yes—
384C22×C4C2×Q1612yes—
385C22×C4#13512yes—
386C22×C4#13512yes—
387C22×C4AGL(1,ℤ/8)12yes—
388C22×C4#13712yes—
389C22×C4#13712yes—
390C22×C4#13712yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C2×C8—24no—
395C2×C8—48no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—84no—
405C2×C8—84no—
406C4⋊C4—232no—
407C4⋊C4—232no—
408C4⋊C4—232no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4C2×C22⋊C414yes—
433C4⋊C4C2×C22⋊C414yes—
434C4⋊C4C2×C22⋊C418yes—
435C42—232no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—28no—
441C42—28no—
442C42C2×C22⋊C412yes—
443C42C2×C22⋊C412yes—
444C42C2×C22⋊C412yes—
445C42#12012yes—
446C42#12012yes—
447C42#12012yes—
448C42C23×C411yescharacteristic
449C42C22×D411yescharacteristic
450C42C22×D411yescharacteristic
451C42C22×D411yescharacteristic
452C23#37912yes—
453C23#44812yes—
454C23#44912yes—
455C23#45112yes—
456C23#45212yes—
457C23#45412yes—
458C23#45612yes—
459C23C2×#12011yescharacteristic
460Q8—24no—
461Q8—24no—
462Q8—24no—
463Q8—216no—
464Q8—24no—
465Q8—24no—
466Q8—44no—
467Q8—44no—
468Q8—42no—
469Q8—44no—
470Q8—42no—
471Q8—44no—
472Q8—48no—
473Q8—416no—
474Q8C2×#10318yescomplemented
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—216no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—416no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—164no—
505C2×C4#37912yes—
506C2×C4#37912yes—
507C2×C4#37912yes—
508C2×C4C2×#10314yes—
509C2×C4#41614yes—
510C2×C4#41714yes—
511C2×C4#41914yes—
512C2×C4#44814yes—
513C2×C4#44812yes—
514C2×C4#44914yes—
515C2×C4#44912yes—
516C2×C4#45114yes—
517C2×C4#45112yes—
518C2×C4#45214yes—
519C2×C4#45212yes—
520C2×C4C22×C22⋊C411yescharacteristic
521C2×C4C22×C22⋊C411yescharacteristic
522C2×C4C22×C22⋊C411yescharacteristic
523C2×C4C2×#12011yescharacteristic
524C2×C4C2×#12011yescharacteristic
525C2×C4C2×#12011yescharacteristic
526C2×C4C22×SD1611yescharacteristic
527C2×C4C22×Q1611yescharacteristic
528C2×C4C2×#13511yescharacteristic
529C2×C4C2×#13511yescharacteristic
530C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
531C2×C4C2×#13711yescharacteristic
532C2×C4C2×#13711yescharacteristic
533C2×C4C2×#13711yescharacteristic
534C8—44no—
535C8—82no—
536C8—82no—
537C22#162614yes—
538C22#163614yes—
539C22#164814yes—
540C22#165314yes—
541C22#178714yes—
542C22#178814yes—
543C22#179314yes—
544C22C2×#37911yescharacteristic
545C22C2×#44811yescharacteristic
546C22C2×#45111yescharacteristic
547C22C2×#44911yescharacteristic
548C22C2×#45211yescharacteristic
549C22C2×#45411yescharacteristic
550C22C2×#45611yescharacteristic
551C4—28no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—44no—
556C4—48no—
557C4—44no—
558C4—44no—
559C4—162no—
560C4#178714yes—
561C4#178814yes—
562C4C22×#10311yescharacteristic
563C4C2×#41611yescharacteristic
564C4C2×#41911yescharacteristic
565C4C2×#41711yescharacteristic
566C4C2×#44811yescharacteristic
567C4C2×#45111yescharacteristic
568C4C2×#44911yescharacteristic
569C4C2×#45211yescharacteristic
570C2#844818yesdirect product factor
571C2C2×#162611yescharacteristic
572C2C2×#163611yescharacteristic
573C2C2×#164811yescharacteristic
574C2C2×#165311yescharacteristic
575C2C2×#178711yescharacteristic
576C2C2×#178811yescharacteristic
577C2C2×#179311yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #8448C2direct (N × H)
15. C2×#557C4semidirect
61. #557C2×C4semidirect
343. C2×Q8#103semidirect
474. Q8C2×#103semidirect
570. C2#8448direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#19902
C2×#18032
C22×#3322
C22×Q8⋊C82
#84482