Finite Group Databasebeta

← C2×#25829 #6445774

Subgroups — C2×#25829 #6445774

Order 512.

3154 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#3165C211yesmaximal, characteristic, complemented
3C2×#3079C211yesmaximal, characteristic
4C2×#2862C211yesmaximal, characteristic, complemented
5#25829C2116yesmaximal, direct product factor
6C2×#2061C214yesmaximal, complemented
7C2×#1997C211yesmaximal, characteristic, complemented
8C2×#1994C211yesmaximal, characteristic, complemented
9C2×#1860C214yesmaximal
10C2×C4×#104C211yesmaximal, characteristic
11#3503—24no—
12#3503—24no—
13#3503—24no—
14#3503C2218yes—
15C2×Q82—21no—
16C2×#557—21no—
17C2×#554—21no—
18C2×#554—22no—
19C2×D4×Q8—22no—
20C2×D4×Q8—21no—
21C2×#544—22no—
22C2×#544C2212yes—
23C2×#533—21no—
24C2×#533C2211yescharacteristic
25C22×C4⋊Q8—21no—
26C22×C4⋊Q8C2211yescharacteristic
27C22×C4⋊Q8C2211yescharacteristic
28C2×#520C2212yes—
29C2×C4×#41C2211yescharacteristic
30#3165—24no—
31#3165C2218yescomplemented
32C2×#501—22no—
33C2×#500—21no—
34C2×#498—21no—
35#3079—24no—
36#3079C2218yes—
37C2×#477—21no—
38C2×#475—21no—
39C2×#476—21no—
40C2×#474—21no—
41C2×#428C2212yes—
42C2×#428C2214yes—
43C2×#423C2211yescharacteristic
44C22×#104C2212yes—
45C2×#417C2212yes—
46C2×C4×M16C2211yescharacteristic
47#2862—24no—
48#2862C2218yescomplemented
49C2×#391—22no—
50C2×#391—22no—
51C2×#382C2214yes—
52C2×C43C2211yescharacteristic
53#2061C22132yescomplemented
54#1997C2218yescomplemented
55#1994C2218yescomplemented
56#1860C22132yes—
57C4×#104C2218yes—
58C2×#343C2214yes—
59C2×#339C2212yes—
60C2×#339C2214yes—
61C22×#41C2311yescharacteristic
62C23×Q8—22no—
63C23×Q8—21no—
64C23×Q8D412yes—
65C22×M16C2312yes—
66C22×M16C2312yes—
67Q82—28no—
68Q82—28no—
69#557—28no—
70#557—28no—
71#554—28no—
72#554—216no—
73#554—28no—
74#554—216no—
75D4×Q8—216no—
76D4×Q8—28no—
77D4×Q8—216no—
78D4×Q8—28no—
79#544—216no—
80#544—216no—
81#544—24no—
82#544—24no—
83#544—24no—
84#544C2318yes—
85#533—28no—
86#533—28no—
87#533—22no—
88#533—22no—
89#533D414yes—
90#533C2314yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8D411yescharacteristic
112C2×C4⋊Q8D411yescharacteristic
113C2×C4⋊Q8D414yes—
114C2×C4⋊Q8C2314yes—
115C2×C4⋊Q8C2314yes—
116C2×#125—24no—
117C2×#125—22no—
118C2×#125—22no—
119C2×#125D412yes—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—24no—
127#520—24no—
128#520—24no—
129#520—24no—
130#520C2318yes—
131C4×#41—22no—
132C4×#41—22no—
133C4×#41D414yes—
134C4×#41C2314yes—
135C2×C4×Q8—22no—
136C2×C4×Q8—22no—
137C2×C4×Q8—22no—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8D411yescharacteristic
141C2×C4×Q8D411yescharacteristic
142C2×C4×D4—22no—
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4—24no—
146C2×C4×D4D411yescharacteristic
147C2×C4×D4D411yescharacteristic
148C2×#117—22no—
149C2×#117—21no—
150C2×#117C2311yescharacteristic
151C2×#117C2312yes—
152C22×C4⋊C4—22no—
153C22×C4⋊C4—22no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D412yes—
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2312yes—
158C22×C42—21no—
159C22×C42—21no—
160C22×C42—21no—
161C22×C42C2312yes—
162C22×C42C2311yescharacteristic
163#501—216no—
164#500—28no—
165#498—28no—
166#477—28no—
167#476—28no—
168#475—28no—
169#474—28no—
170#428—24no—
171#428—28no—
172#428C2318yes—
173#428C23116yes—
174C2×#107—22no—
175C2×#107—24no—
176C2×#106—22no—
177C2×#106—24no—
178#423—22no—
179#423C2314yes—
180C2×C4⋊C8—21no—
181C2×C4⋊C8—21no—
182C2×#104Q8116yescomplemented
183C2×#104C2318yes—
184#417—24no—
185#417C2318yes—
186C2×#103—22no—
187C2×#102—22no—
188C4×M16—22no—
189C4×M16C2314yes—
190C2×#97—21no—
191C2×C4×C8—21no—
192#391—216no—
193#391—216no—
194#382—28no—
195#382C23116yes—
196C2×#95—28no—
197C43—22no—
198C43C2314yes—
199#343C23116yes—
200#339C2318yes—
201#339C23116yes—
202C2×#41—21no—
203C2×#41—21no—
204C2×#41—24no—
205C2×#41C2×D412yes—
206C2×#41C2×Q818yes—
207C2×#41C2412yes—
208C22×Q8—24no—
209C22×Q8—216no—
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—48no—
217C22×Q8—48no—
218C22×Q8—42no—
219C22×Q8—41no—
220C22×Q8—41no—
221C22×Q8—88no—
222C22×Q8C2×D411yescharacteristic
223C22×Q8C2×D418yes—
224C22×D4—21no—
225C22×D4C2×D411yescharacteristic
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D414yes—
235C23×C4C2411yescharacteristic
236C2×M16—22no—
237C2×M16—24no—
238C2×M16—22no—
239C2×M16—28no—
240C2×M16C2×Q818yescomplemented
241C2×M16C2414yes—
242C2×M16C2414yes—
243C22×C8—22no—
244C22×C8—22no—
245C4⋊Q8—28no—
246C4⋊Q8—216no—
247C4⋊Q8—216no—
248C4⋊Q8—232no—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—42no—
270C4⋊Q8C2×D414yes—
271C4⋊Q8C2×D414yes—
272#125—28no—
273#125—232no—
274#125—28no—
275#125—28no—
276#125—28no—
277#125—24no—
278#125C2×D418yes—
279C22⋊Q8—216no—
280C22⋊Q8—216no—
281C22⋊Q8—232no—
282C22⋊Q8—232no—
283C22⋊Q8—216no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—216no—
289C4×Q8—28no—
290C4×Q8—216no—
291C4×Q8—216no—
292C4×Q8—28no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8C2×D414yes—
300C4×Q8C2×D414yes—
301C4×D4—216no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—216no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4C2×D414yes—
312C4×D4C2×D414yes—
313#117—28no—
314#117—21no—
315#117—21no—
316#117—22no—
317#117—22no—
318#117—28no—
319#117—24no—
320#117—24no—
321#117C2×D412yes—
322#117C2×D414yes—
323#117C2412yes—
324#117C2414yes—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—216no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4C2×D418yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2414yes—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—44no—
379C2×C42—44no—
380C2×C42C2×D412yes—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385C2×C42C2×D412yes—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×Q818yescomplemented
388C2×C42C2414yes—
389C2×C42C2412yes—
390#107—28no—
391#107—216no—
392#106—28no—
393#106—216no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396#104C2×Q8132yescomplemented
397#103—28no—
398#102—28no—
399#97—24no—
400C4×C8—24no—
401#95—232no—
402C24—21no—
403C24C22×D411yescharacteristic
404#41—24no—
405#41—28no—
406#41—28no—
407#41—24no—
408#41C4⋊Q818yes—
409#41C22×Q818yes—
410C2×Q8—216no—
411C2×Q8—21no—
412C2×Q8—24no—
413C2×Q8—28no—
414C2×Q8—24no—
415C2×Q8—28no—
416C2×Q8—22no—
417C2×Q8—416no—
418C2×Q8—416no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—48no—
423C2×Q8—44no—
424C2×Q8—84no—
425C2×Q8—88no—
426C2×Q8—84no—
427C2×Q8—88no—
428C2×Q8—88no—
429C2×Q8—832no—
430C2×Q8—84no—
431C2×Q8C4⋊Q814yes—
432C2×Q8C22×D412yes—
433C2×D4—22no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4C4⋊Q814yes—
439C2×D4C22×D412yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—216no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—88no—
467C22×C4#12012yes—
468C22×C4#12012yes—
469C22×C4#12011yescharacteristic
470C22×C4#12011yescharacteristic
471C22×C4#12011yescharacteristic
472C22×C4#12011yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D418yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×Q814yes—
482C22×C42−1+414yes—
483C22×C4C2511yescharacteristic
484M16—28no—
485M16—24no—
486M16—28no—
487M16—44no—
488M16C22×Q818yescomplemented
489C2×C8—28no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—44no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—28no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4C22×D414yes—
520C4⋊C4C22×D412yes—
521C4⋊C4C22×D414yes—
522C4⋊C4C22×D412yes—
523C22⋊C4—216no—
524C22⋊C4—24no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C42—216no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—24no—
535C42—28no—
536C42—22no—
537C42—24no—
538C42—24no—
539C42—48no—
540C42—48no—
541C42#12012yes—
542C42#12012yes—
543C42#12012yes—
544C42#12012yes—
545C42C22×D412yes—
546C42C22×D412yes—
547C42C22×D412yes—
548C42C22×D412yes—
549C42C22×Q818yescomplemented
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—42no—
555C23—42no—
556C23C2×#12011yescharacteristic
557C23C2×#12011yescharacteristic
558C23D4×Q814yes—
559C23C23×D411yescharacteristic
560Q8—22no—
561Q8—24no—
562Q8—416no—
563Q8—48no—
564Q8—48no—
565Q8—88no—
566Q8—88no—
567Q8—816no—
568Q8—88no—
569Q8C2×C4⋊Q814yes—
570D4—28no—
571D4—28no—
572D4—22no—
573D4—24no—
574D4C2×C4⋊Q814yes—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—84no—
605C2×C4—816no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4#45414yes—
609C2×C4#45514yes—
610C2×C4C2×#12012yes—
611C2×C4C2×#12012yes—
612C2×C4C2×#12011yescharacteristic
613C2×C4C2×#12011yescharacteristic
614C2×C4C2×#12012yes—
615C2×C4C2×#12012yes—
616C2×C4C2×C4⋊Q812yes—
617C2×C4C2×C4⋊Q812yes—
618C2×C4#53612yes—
619C2×C4#53612yes—
620C2×C4D4×Q814yes—
621C2×C4D4×Q814yes—
622C2×C4D4×Q814yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C23×D411yescharacteristic
625C2×C4C23×Q812yes—
626C2×C4C2×2−1+412yes—
627C8—28no—
628C8—44no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—28no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—48no—
638C22#243412yes—
639C22#243412yes—
640C22C2×#45512yes—
641C22C2×#45412yes—
642C22C22×#12011yescharacteristic
643C22C2×D4×Q812yes—
644C4—24no—
645C4—22no—
646C4—22no—
647C4—48no—
648C4—42no—
649C4—48no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—84no—
654C4#243412yes—
655C4#243412yes—
656C4C2×#45514yes—
657C4C2×#45414yes—
658C4C22×C4⋊Q811yescharacteristic
659C4C2×#53611yescharacteristic
660C4C2×D4×Q812yes—
661C2—22no—
662C2—22no—
663C2—44no—
664C2#2582914yesdirect product factor
665C2C2×#243411yescharacteristic
666C2C22×#45511yescharacteristic
667C2C22×#45411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3165C2semidirect
4. C2×#2862C2semidirect
5. #25829C2direct (N × H)
6. C2×#2061C2semidirect
7. C2×#1997C2semidirect
8. C2×#1994C2semidirect
31. #3165C22semidirect
48. #2862C22semidirect
53. #2061C22semidirect
54. #1997C22semidirect
55. #1994C22semidirect
182. C2×#104Q8semidirect
240. C2×M16C2×Q8semidirect
387. C2×C42C2×Q8semidirect
396. #104C2×Q8semidirect
488. M16C22×Q8semidirect
549. C42C22×Q8semidirect
664. C2#25829direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#31652
C2×#30792
C2×#28622
#258292
C2×#20612
C2×#19972
C2×#19942
C2×#18602
C2×C4×#1042