Finite Group Databasebeta

← C2×#19728 #6357365

Subgroups — C2×#19728 #6357365

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic, complemented
2C2×#3226C211yesmaximal, characteristic
3C2×#3210C211yesmaximal, characteristic
4C2×#3209C211yesmaximal, characteristic, complemented
5C2×#3182C211yesmaximal, characteristic
6C2×#3165C211yesmaximal, characteristic, complemented
7#19728C2116yesmaximal, direct product factor
8C2×#1723C211yesmaximal, characteristic, complemented
9C2×#1723C211yesmaximal, characteristic, complemented
10C2×#1722C211yesmaximal, characteristic, complemented
11C2×#1720C211yesmaximal, characteristic, complemented
12C2×#1718C211yesmaximal, characteristic, complemented
13C2×#1714C211yesmaximal, characteristic, complemented
14C2×#1713C211yesmaximal, characteristic, complemented
15C2×#1708C211yesmaximal, characteristic, complemented
16C2×#1453C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21#3231—24no—
22#3231C2218yescomplemented
23#3226—24no—
24#3226C2218yes—
25#3210—24no—
26#3210C2218yes—
27#3209—24no—
28#3209C2218yescomplemented
29#3182—24no—
30#3182C2218yes—
31#3165—24no—
32#3165C2218yescomplemented
33C2×#501—22no—
34C2×#500C2211yescharacteristic
35C2×#500C2211yescharacteristic
36C2×#500C2211yescharacteristic
37C2×#500C2211yescharacteristic
38C2×#500C2211yescharacteristic
39C2×#500C2211yescharacteristic
40C2×#498C2211yescharacteristic
41C2×#498C2211yescharacteristic
42C2×#498C2211yescharacteristic
43C2×#494C2211yescharacteristic
44C2×#494C2211yescharacteristic
45C2×#494C2211yescharacteristic
46C2×#492C2211yescharacteristic
47C2×#492C2211yescharacteristic
48C2×#487C2211yescharacteristic
49C2×#487C2211yescharacteristic
50C2×#487C2211yescharacteristic
51C2×#487C2211yescharacteristic
52C2×#487C2211yescharacteristic
53C2×#487C2211yescharacteristic
54C2×#486C2211yescharacteristic
55C2×#486C2211yescharacteristic
56C2×#486C2211yescharacteristic
57C2×#486C2211yescharacteristic
58C2×#485—22no—
59C2×#480—22no—
60C2×#479—22no—
61C2×#474—22no—
62C2×#468—22no—
63C2×#464—22no—
64C2×#462—22no—
65C2×#452—22no—
66C2×#451—22no—
67C2×#440—22no—
68C2×#431C2211yescharacteristic
69C2×#431C2211yescharacteristic
70C2×#428—22no—
71C2×C4×M16C2211yescharacteristic
72#1723C2218yescomplemented
73#1723C2218yescomplemented
74#1722C2218yescomplemented
75#1720C2218yescomplemented
76#1718C2218yescomplemented
77#1714C2218yescomplemented
78#1713C2218yescomplemented
79#1708C2218yescomplemented
80#1453C2218yescomplemented
81C2×#322C2211yescharacteristic
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×C82C2211yescharacteristic
85C22×#41—21no—
86C23×Q8—21no—
87C22×Q16—22no—
88C22×Q16—22no—
89C22×Q16—22no—
90C22×SD16—22no—
91C22×SD16—22no—
92C22×M16D412yes—
93#533—22no—
94#533—22no—
95#533—22no—
96#533—22no—
97#533—22no—
98#533—22no—
99#533C2314yes—
100#533C2314yes—
101#532—22no—
102#532—22no—
103#532—22no—
104#532C2314yes—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2314yes—
119C2×#127C2311yescharacteristic
120C2×#125—21no—
121C2×#125—21no—
122C2×#124—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×#121—21no—
128C2×#121—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×#117—21no—
136C2×#117—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C42C2311yescharacteristic
140#501—216no—
141#500—22no—
142#500—22no—
143#500—22no—
144#500—22no—
145#500—22no—
146#500—22no—
147#500C2314yes—
148#500C2314yes—
149#500C2314yes—
150#500C2314yes—
151#500C2314yes—
152#500C2314yes—
153#498—22no—
154#498—22no—
155#498—22no—
156#498C2314yes—
157#498C2314yes—
158#498C2314yes—
159#494—22no—
160#494—22no—
161#494—22no—
162#494C2314yes—
163#494C2314yes—
164#494C2314yes—
165#492—22no—
166#492—22no—
167#492C2314yes—
168#492C2314yes—
169#487—22no—
170#487—22no—
171#487—22no—
172#487—22no—
173#487—22no—
174#487—22no—
175#487C2314yes—
176#487C2314yes—
177#487C2314yes—
178#487C2314yes—
179#487C2314yes—
180#487C2314yes—
181#486—22no—
182#486—22no—
183#486—22no—
184#486—22no—
185#486C2314yes—
186#486C2314yes—
187#486C2314yes—
188#486C2314yes—
189#485—216no—
190#480—216no—
191#479—216no—
192#474—216no—
193#468—216no—
194#464—216no—
195#462—216no—
196#452—216no—
197#451—216no—
198#440—216no—
199#431D414yes—
200#431D414yes—
201#431C2314yes—
202#431C2314yes—
203#428—216no—
204C2×#107—22no—
205C2×#107—22no—
206C2×#107—22no—
207C2×#107—22no—
208C2×#107—22no—
209C2×#107—22no—
210C2×#106—22no—
211C2×#106—22no—
212C2×#106—22no—
213C2×C4⋊C8D412yes—
214C2×C4⋊C8D412yes—
215C2×#103—22no—
216C2×#103—22no—
217C2×#103—22no—
218C2×#103—22no—
219C2×#103—22no—
220C2×#103—22no—
221C2×#102—22no—
222C2×#102—22no—
223C2×#102—22no—
224C2×#102—22no—
225C2×C22⋊C8D412yes—
226C2×C22⋊C8D412yes—
227C4×M16D414yes—
228C4×M16C2314yes—
229C2×#97D412yes—
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232C2×C4×C8C2311yescharacteristic
233C2×C4×C8C2311yescharacteristic
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236#322C2314yes—
237#322C2314yes—
238#322C2314yes—
239C82C2314yes—
240C2×#41—24no—
241C2×#41—42no—
242C2×#41—84no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—42no—
252C22×Q8—41no—
253C22×Q8—84no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—41no—
258C23×C4—41no—
259C23×C4—41no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C2×Q16—28no—
264C2×Q16—28no—
265C2×Q16—28no—
266C2×Q16—84no—
267C2×Q16—84no—
268C2×Q16—84no—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—48no—
272C2×SD16—84no—
273C2×M16—28no—
274C2×M16C2×D418yes—
275C22×C8C2×D412yes—
276C22×C8C2×D412yes—
277C22×C8C2×D412yes—
278C22×C8C2×D412yes—
279C22×C8C2×D412yes—
280C22×C8C2×D412yes—
281C4⋊Q8—24no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—24no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—24no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—42no—
306C4⋊Q8—42no—
307C4⋊Q8—42no—
308C4⋊Q8C2412yes—
309C4⋊Q8C2412yes—
310C4⋊Q8C2412yes—
311C4⋊Q8C2412yes—
312C4⋊Q8C2412yes—
313C4⋊Q8C2412yes—
314C4⋊Q8C2412yes—
315#127—21no—
316#127—21no—
317#127—21no—
318#127C2412yes—
319#125—24no—
320#125—24no—
321#125—42no—
322#125—42no—
323#124—24no—
324#124—42no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328C22⋊Q8—24no—
329C22⋊Q8—42no—
330C22⋊Q8—42no—
331C22⋊Q8—42no—
332C22⋊Q8—42no—
333#121—24no—
334#121—24no—
335#121—42no—
336#121—42no—
337C4×Q8—24no—
338C4×Q8—24no—
339C4×Q8—24no—
340C4×Q8—24no—
341C4×Q8—42no—
342C4×Q8—42no—
343C4×Q8—42no—
344C4×Q8—42no—
345C4×D4—24no—
346C4×D4—24no—
347C4×D4—42no—
348C4×D4—42no—
349#117—24no—
350#117—24no—
351#117—42no—
352#117—42no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—41no—
381C2×C42—41no—
382C2×C42—41no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2×D412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2412yes—
391#107—28no—
392#107—28no—
393#107—28no—
394#107—28no—
395#107—28no—
396#107—28no—
397#106—28no—
398#106—28no—
399#106—28no—
400C4⋊C8C2×D418yes—
401C4⋊C8C2×D418yes—
402#103—28no—
403#103—28no—
404#103—28no—
405#103—28no—
406#103—28no—
407#103—28no—
408#102—28no—
409#102—28no—
410#102—28no—
411#102—28no—
412C22⋊C8C2×D418yes—
413C22⋊C8C2×D418yes—
414#97C2×D418yes—
415C4×C8C2×D412yes—
416C4×C8C2×D412yes—
417C4×C8C2×D412yes—
418C4×C8C2×D412yes—
419C4×C8C2×D412yes—
420C4×C8C2×D412yes—
421C4×C8C2412yes—
422C4×C8C2412yes—
423C4×C8C2412yes—
424C4×C8C2412yes—
425C4×C8C2412yes—
426C4×C8C2412yes—
427C24—41no—
428C24#12711yescharacteristic
429#41—88no—
430#41—88no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—44no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—88no—
454C2×Q8—84no—
455C2×Q8—88no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—44no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—84no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—44no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—44no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4#12712yes—
482C22×C4#12712yes—
483C22×C4#12712yes—
484C22×C4#12711yescharacteristic
485C22×C4#12711yescharacteristic
486C22×C4#12711yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493Q16—88no—
494Q16—88no—
495Q16—88no—
496SD16—416no—
497SD16—88no—
498M16—216no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8#13718yes—
505C2×C8C22×D414yes—
506C2×C8C22×D414yes—
507C2×C8C22×D414yes—
508C2×C8C22×D414yes—
509C2×C8C22×D414yes—
510C2×C8C22×D414yes—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C42—44no—
550C42—44no—
551C42#12712yes—
552C42#12712yes—
553C42#12712yes—
554C42C22×D411yescharacteristic
555C42C22×D411yescharacteristic
556C42C22×D411yescharacteristic
557C42C22×D412yes—
558C42C22×D412yes—
559C42C22×D412yes—
560C42C2511yescharacteristic
561C23—22no—
562C23—22no—
563C23—22no—
564C23—42no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23C2×#12712yes—
569C23C2×#12711yescharacteristic
570Q8—44no—
571Q8—44no—
572Q8—44no—
573Q8—44no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—88no—
578Q8—82no—
579Q8—82no—
580Q8—82no—
581Q8—82no—
582Q8—82no—
583Q8—82no—
584Q8—82no—
585D4—44no—
586D4—44no—
587D4—44no—
588D4—88no—
589D4—82no—
590D4—82no—
591D4—82no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4C2×#12712yes—
618C2×C4C2×#12712yes—
619C2×C4C2×#12712yes—
620C2×C4C2×#12711yescharacteristic
621C2×C4C2×#12711yescharacteristic
622C2×C4C2×#12711yescharacteristic
623C2×C4C2×#13714yes—
624C2×C4#57614yes—
625C2×C4#57814yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C23×D411yescharacteristic
628C2×C4C23×D411yescharacteristic
629C8—24no—
630C8—24no—
631C8—24no—
632C8—24no—
633C8—24no—
634C8C2×#13718yes—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—44no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—164no—
643C22#315212yes—
644C22#315512yes—
645C22#315612yes—
646C22C22×#12711yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#315212yes—
658C4#315512yes—
659C4#315612yes—
660C4C22×#13712yes—
661C4C2×#57612yes—
662C4C2×#57812yes—
663C2—42no—
664C2—162no—
665C2#1972814yesdirect product factor
666C2C2×#315211yescharacteristic
667C2C2×#315511yescharacteristic
668C2C2×#315611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3231C2semidirect
4. C2×#3209C2semidirect
6. C2×#3165C2semidirect
7. #19728C2direct (N × H)
8. C2×#1723C2semidirect
9. C2×#1723C2semidirect
10. C2×#1722C2semidirect
11. C2×#1720C2semidirect
12. C2×#1718C2semidirect
13. C2×#1714C2semidirect
14. C2×#1713C2semidirect
15. C2×#1708C2semidirect
16. C2×#1453C2semidirect
22. #3231C22semidirect
28. #3209C22semidirect
32. #3165C22semidirect
72. #1723C22semidirect
73. #1723C22semidirect
74. #1722C22semidirect
75. #1720C22semidirect
76. #1718C22semidirect
77. #1714C22semidirect
78. #1713C22semidirect
79. #1708C22semidirect
80. #1453C22semidirect
665. C2#19728direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32312
C2×#32262
C2×#32102
C2×#32092
C2×#31822
C2×#31652
#197282
C2×#17232
C2×#17222
C2×#17202
C2×#17182
C2×#17142
C2×#17132
C2×#17082
C2×#14532