Finite Group Databasebeta

← C2×#19156 #6353624

Subgroups — C2×#19156 #6353624

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic, complemented
2C2×#3076C211yesmaximal, characteristic, complemented
3C2×#3035C211yesmaximal, characteristic
4C2×#2965C211yesmaximal, characteristic, complemented
5#19156C2116yesmaximal, direct product factor
6C2×#1700C211yesmaximal, characteristic, complemented
7C2×#1698C211yesmaximal, characteristic, complemented
8C2×#1692C211yesmaximal, characteristic, complemented
9C2×#1686C211yesmaximal, characteristic, complemented
10C2×#1699C211yesmaximal, characteristic, complemented
11C2×#1697C211yesmaximal, characteristic, complemented
12C2×#1691C211yesmaximal, characteristic, complemented
13C2×#1685C211yesmaximal, characteristic, complemented
14C22×#335C211yesmaximal, characteristic, complemented
15C2×#1542C211yesmaximal, characteristic, complemented
16C2×#1540C211yesmaximal, characteristic, complemented
17C22×#135—21no—
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3167—24no—
22#3167C2218yescomplemented
23C2×#498C2211yescharacteristic
24C2×#492C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#486C2211yescharacteristic
27#3076—24no—
28#3076C2218yescomplemented
29C2×#479—21no—
30C2×#477C2211yescharacteristic
31C2×#475C2211yescharacteristic
32C2×#478—21no—
33C2×#476C2211yescharacteristic
34C2×#474C2211yescharacteristic
35C2×#465—21no—
36C2×#465—21no—
37#3035—24no—
38#3035C2218yes—
39C2×#464—21no—
40C2×#462C2211yescharacteristic
41C2×#460C2211yescharacteristic
42C2×#463—21no—
43C2×#461C2211yescharacteristic
44C2×#459C2211yescharacteristic
45#2965—24no—
46#2965C2218yescomplemented
47C2×#446C2211yescharacteristic
48C2×#446C2211yescharacteristic
49C2×C4×SD16—21no—
50C2×C4×SD16—21no—
51C2×#436C2211yescharacteristic
52C2×#436C2211yescharacteristic
53C2×#435—21no—
54C2×#435—21no—
55C2×#431—21no—
56C2×#431C2211yescharacteristic
57C2×#427—21no—
58C22×C4⋊C8C2211yescharacteristic
59C22×C4⋊C8C2211yescharacteristic
60C2×#416—21no—
61C2×#416—21no—
62C2×C4×M16—21no—
63C22×C4×C8C2211yescharacteristic
64#1700C2218yescomplemented
65#1699C2218yescomplemented
66#1698C2218yescomplemented
67#1697C2218yescomplemented
68#1692C2218yescomplemented
69#1691C2218yescomplemented
70#1686C2218yescomplemented
71#1685C2218yescomplemented
72C2×#335C2211yescharacteristic
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#335C2211yescharacteristic
76C2×#335C2218yescomplemented
77#1542C2218yescomplemented
78C2×#333C2211yescharacteristic
79C2×#333C2211yescharacteristic
80#1540C2218yescomplemented
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#331C2211yescharacteristic
86C2×#331C2211yescharacteristic
87C22×#41—21no—
88C22×#41D411yescharacteristic
89C2×#135—28no—
90C2×#135—48no—
91C22×Q16—21no—
92C22×SD16—21no—
93C22×SD16—21no—
94C22×SD16—21no—
95C22×SD16—21no—
96C22×D8—21no—
97C22×M16—21no—
98C22×M16—21no—
99C23×C8—21no—
100C23×C8—21no—
101C23×C8D411yescharacteristic
102C23×C8D411yescharacteristic
103#533—22no—
104#533—22no—
105#533—22no—
106#533C2314yes—
107#532—22no—
108#532—22no—
109#532—22no—
110#532C2314yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×#127C2311yescharacteristic
115C2×#125—21no—
116C2×#124—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×#121—21no—
120C2×#121—21no—
121C4×#41—22no—
122C4×#41—22no—
123C4×#41D414yes—
124C4×#41C2314yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4D411yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×#117—21no—
136C2×#117—21no—
137C2×#117D411yescharacteristic
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#498—22no—
141#498C2314yes—
142#492—22no—
143#492C2314yes—
144#487—22no—
145#487C2314yes—
146#486—22no—
147#486C2314yes—
148#479—28no—
149#478—28no—
150#477—22no—
151#477C2314yes—
152#476—22no—
153#476C2314yes—
154#475—22no—
155#475C2314yes—
156#474—22no—
157#474C2314yes—
158#465—28no—
159#465—28no—
160#464—28no—
161#463—28no—
162#462—22no—
163#462C2314yes—
164#461—22no—
165#461C2314yes—
166#460—22no—
167#460C2314yes—
168#459—22no—
169#459C2314yes—
170#446—22no—
171#446—22no—
172#446C2314yes—
173#446C2314yes—
174C4×SD16—28no—
175C4×SD16—28no—
176#436—22no—
177#436—22no—
178#436C2314yes—
179#436C2314yes—
180#435—28no—
181#435—28no—
182#431—22no—
183#431—28no—
184#431C2314yes—
185#427—28no—
186C2×#107—21no—
187C2×#107—21no—
188C2×#106—21no—
189C2×#106—21no—
190C2×#106—21no—
191C2×#106—21no—
192C2×C4⋊C8—21no—
193C2×C4⋊C8—21no—
194C2×C4⋊C8—21no—
195C2×C4⋊C8—22no—
196C2×C4⋊C8—21no—
197C2×C4⋊C8—22no—
198C2×C4⋊C8C2311yescharacteristic
199C2×C4⋊C8C2311yescharacteristic
200C2×C4⋊C8C2311yescharacteristic
201C2×C4⋊C8C2311yescharacteristic
202C2×C4⋊C8C2311yescharacteristic
203C2×C4⋊C8C2311yescharacteristic
204C2×C4⋊C8C2314yes—
205C2×C4⋊C8C2314yes—
206#416—28no—
207#416—28no—
208C2×#103—21no—
209C2×#103—21no—
210C2×#103—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#103—21no—
214C2×#102—21no—
215C2×#102—21no—
216C2×#102—21no—
217C2×#102—21no—
218C2×#102—21no—
219C2×#102—21no—
220C2×C22⋊C8—21no—
221C2×C22⋊C8—21no—
222C2×C22⋊C8—21no—
223C4×M16—28no—
224C2×#97—21no—
225C2×#97—21no—
226C2×C4×C8—21no—
227C2×C4×C8D411yescharacteristic
228C2×C4×C8D414yes—
229C2×C4×C8D411yescharacteristic
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232C2×C4×C8C2314yes—
233#335C2314yes—
234#335C2314yes—
235#335C2314yes—
236#335C2314yes—
237#333C2314yes—
238#333C2314yes—
239#332C2314yes—
240#332C2314yes—
241#332C2314yes—
242#332C2314yes—
243#331C2314yes—
244#331C2314yes—
245C2×#41—22no—
246C2×#41—24no—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—42no—
250C2×#41—44no—
251C2×#41—44no—
252C2×#41C2×D414yes—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8C2×D411yescharacteristic
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—41no—
262C22×D4C2×D411yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—41no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4#4111yescharacteristic
270#135—432no—
271C2×Q16—24no—
272C2×Q16—44no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—44no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×SD16—44no—
281C2×D8—24no—
282C2×D8—44no—
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—44no—
286C2×M16—44no—
287C22×C8—24no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—24no—
291C22×C8—21no—
292C22×C8—21no—
293C22×C8—24no—
294C22×C8—21no—
295C22×C8—21no—
296C22×C8—42no—
297C22×C8—44no—
298C22×C8—44no—
299C22×C8C2×D411yescharacteristic
300C22×C8C2×D411yescharacteristic
301C22×C8C2×D411yescharacteristic
302C22×C8C2×D414yes—
303C22×C8C2×D414yes—
304C22×C8C2×D411yescharacteristic
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8C2412yes—
315C4⋊Q8C2412yes—
316C4⋊Q8C2412yes—
317#127—21no—
318#127—21no—
319#127—21no—
320#127C2412yes—
321#125—24no—
322#125—42no—
323#124—24no—
324#124—42no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—42no—
328C22⋊Q8—42no—
329#121—24no—
330#121—24no—
331#121—42no—
332#121—42no—
333C4×Q8—22no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8—24no—
337C4×Q8—24no—
338C4×Q8—42no—
339C4×Q8—42no—
340C4×Q8C2×D414yes—
341C4×Q8C2×D412yes—
342C4×Q8C2412yes—
343C4×D4—24no—
344C4×D4—24no—
345C4×D4—24no—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4—21no—
349C4×D4—24no—
350C4×D4—21no—
351C4×D4—24no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4C2×D414yes—
355C4×D4C2×D412yes—
356C4×D4C2412yes—
357#117—22no—
358#117—24no—
359#117—24no—
360#117—24no—
361#117—42no—
362#117C2×D414yes—
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—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4C2×D411yescharacteristic
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—41no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—24no—
385C2×C42—24no—
386C2×C42—41no—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D412yes—
389C2×C42C2×D412yes—
390C2×C42C2×D411yescharacteristic
391C2×C42#4112yes—
392C2×C42#4111yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2412yes—
395#107—24no—
396#107—24no—
397#106—24no—
398#106—24no—
399#106—24no—
400#106—24no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—21no—
404C4⋊C8—21no—
405C4⋊C8—24no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8—24no—
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8C2412yes—
412C4⋊C8C2412yes—
413C4⋊C8C2412yes—
414C4⋊C8C2412yes—
415C4⋊C8C2412yes—
416C4⋊C8C2412yes—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—24no—
421#103—24no—
422#103—24no—
423#102—24no—
424#102—24no—
425#102—24no—
426#102—24no—
427#102—24no—
428#102—24no—
429C22⋊C8—24no—
430C22⋊C8—24no—
431C22⋊C8—24no—
432#97—24no—
433#97—24no—
434C4×C8—24no—
435C4×C8C2×D414yes—
436C4×C8C2×D414yes—
437C4×C8C2×D412yes—
438C4×C8C2×D412yes—
439C4×C8C2412yes—
440C4×C8C2412yes—
441C24—21no—
442C24—41no—
443C24#12111yescharacteristic
444#41—28no—
445#41—28no—
446#41—48no—
447#41—48no—
448#41—48no—
449#41—88no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—41no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—82no—
464C2×Q8C22×D412yes—
465C2×D4—24no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—82no—
482C2×D4—84no—
483C2×D4C22×D412yes—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—44no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—44no—
502C22×C4—84no—
503C22×C4#12111yescharacteristic
504C22×C4#12111yescharacteristic
505C22×C4#12112yes—
506C22×C4#12112yes—
507C22×C4#12112yes—
508C22×C4#12111yescharacteristic
509C22×C4AGL(1,ℤ/8)14yes—
510C22×C4#13714yes—
511C22×C4C22×D412yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D412yes—
515C22×C4C2×#4111yescharacteristic
516C22×C4C2×#4112yes—
517Q16—48no—
518SD16—48no—
519SD16—48no—
520SD16—48no—
521SD16—48no—
522D8—48no—
523M16—48no—
524M16—48no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—28no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—44no—
536C2×C8—42no—
537C2×C8—42no—
538C2×C8—42no—
539C2×C8—48no—
540C2×C8—42no—
541C2×C8—42no—
542C2×C8—48no—
543C2×C8—42no—
544C2×C8#13514yescomplemented
545C2×C8C22×D412yes—
546C2×C8C22×D412yes—
547C2×C8C22×D412yes—
548C2×C8C22×D412yes—
549C4⋊C4—22no—
550C4⋊C4—24no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4—41no—
563C4⋊C4—41no—
564C4⋊C4—41no—
565C4⋊C4—41no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4C22×D412yes—
569C4⋊C4C22×D412yes—
570C22⋊C4—24no—
571C22⋊C4—22no—
572C22⋊C4—24no—
573C22⋊C4—22no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C42—24no—
578C42—24no—
579C42—22no—
580C42—22no—
581C42—44no—
582C42#12112yes—
583C42#12112yes—
584C42#12112yes—
585C42C22×D412yes—
586C42C22×D411yescharacteristic
587C42C22×D412yes—
588C42C22×D411yescharacteristic
589C42C2×#4112yes—
590C42C2×#4111yescharacteristic
591C42C2511yescharacteristic
592C23—22no—
593C23—22no—
594C23—22no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—82no—
600C23—82no—
601C23—82no—
602C23#46814yes—
603C23C2×#12112yes—
604C23C2×#12111yescharacteristic
605Q8—22no—
606Q8—22no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—82no—
613Q8—82no—
614Q8—82no—
615Q8—82no—
616D4—22no—
617D4—22no—
618D4—48no—
619D4—42no—
620D4—42no—
621D4—42no—
622D4—42no—
623D4—42no—
624D4—82no—
625D4—82no—
626D4—88no—
627D4—82no—
628D4—82no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—48no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—88no—
655C2×C4#46814yes—
656C2×C4#46818yes—
657C2×C4C2×#12112yes—
658C2×C4C2×#12111yescharacteristic
659C2×C4C2×#12112yes—
660C2×C4C2×#12111yescharacteristic
661C2×C4C2×#12112yes—
662C2×C4C2×#12111yescharacteristic
663C2×C4C2×#13512yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)14yes—
666C2×C4C2×#13712yes—
667C2×C4C2×#13714yes—
668C2×C4#57512yes—
669C2×C4C23×D411yescharacteristic
670C2×C4C23×D411yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C8—22no—
673C8—22no—
674C8—22no—
675C8—42no—
676C8—42no—
677C8—42no—
678C8—42no—
679C8—42no—
680C8—42no—
681C8C2×#13514yescomplemented
682C22—22no—
683C22—28no—
684C22—22no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—164no—
693C22#303512yes—
694C22C2×#46814yes—
695C22C2×#46812yes—
696C22#306712yes—
697C22C22×#12111yescharacteristic
698C4—22no—
699C4—21no—
700C4—21no—
701C4—22no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#303512yes—
709C4C2×#46814yes—
710C4C2×#46812yes—
711C4C22×#13511yescharacteristic
712C4C22×AGL(1,ℤ/8)11yescharacteristic
713C4C22×#13711yescharacteristic
714C4C2×#57511yescharacteristic
715C2—24no—
716C2—82no—
717C2—162no—
718C2#1915614yesdirect product factor
719C2C2×#303511yescharacteristic
720C2C22×#46811yescharacteristic
721C2C2×#306711yescharacteristic
subgroupG/H
centerC22×C4#121
commutator, Frattini, cosocleC42C25
socleC23C2×#121
prime radicalC2×#3035C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3167C2semidirect
2. C2×#3076C2semidirect
4. C2×#2965C2semidirect
5. #19156C2direct (N × H)
6. C2×#1700C2semidirect
7. C2×#1698C2semidirect
8. C2×#1692C2semidirect
9. C2×#1686C2semidirect
10. C2×#1699C2semidirect
11. C2×#1697C2semidirect
12. C2×#1691C2semidirect
13. C2×#1685C2semidirect
14. C22×#335C2semidirect
15. C2×#1542C2semidirect
16. C2×#1540C2semidirect
22. #3167C22semidirect
28. #3076C22semidirect
46. #2965C22semidirect
64. #1700C22semidirect
65. #1699C22semidirect
66. #1698C22semidirect
67. #1697C22semidirect
68. #1692C22semidirect
69. #1691C22semidirect
70. #1686C22semidirect
71. #1685C22semidirect
76. C2×#335C22semidirect
77. #1542C22semidirect
80. #1540C22semidirect
544. C2×C8#135semidirect
681. C8C2×#135semidirect
718. C2#19156direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31672
C2×#30762
C2×#30352
C2×#29652
#191562
C2×#17002
C2×#16982
C2×#16922
C2×#16862
C2×#16992
C2×#16972
C2×#16912
C2×#16852
C22×#3352
C2×#15422
C2×#15402