Finite Group Databasebeta

← C2×#19578 #6354680

Subgroups — C2×#19578 #6354680

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3251C211yesmaximal, characteristic, complemented
2C2×#3212C211yesmaximal, characteristic, complemented
3C2×#3207C211yesmaximal, characteristic, complemented
4C2×#2994C211yesmaximal, characteristic, complemented
5#19578C2116yesmaximal, direct product factor
6C2×#1700C211yesmaximal, characteristic, complemented
7C2×#1688C211yesmaximal, characteristic, complemented
8C2×#1701C211yesmaximal, characteristic, complemented
9C2×#1685C211yesmaximal, characteristic, complemented
10C2×#1682C211yesmaximal, characteristic, complemented
11C2×#1666C211yesmaximal, characteristic, complemented
12C2×#1679C211yesmaximal, characteristic, complemented
13C2×#1667C211yesmaximal, characteristic, complemented
14C2×#1574C211yesmaximal, characteristic, complemented
15C2×#1547C211yesmaximal, characteristic, complemented
16C2×#1540C211yesmaximal, characteristic, complemented
17C2×#532C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3251—24no—
21#3251C2218yescomplemented
22#3212—24no—
23#3212C2218yescomplemented
24#3207—24no—
25#3207C2218yescomplemented
26C2×#495—21no—
27C2×#493C2211yescharacteristic
28C2×#492C2211yescharacteristic
29C2×#488—21no—
30C2×#487C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#475C2211yescharacteristic
33C2×#478—21no—
34C2×#474C2211yescharacteristic
35C2×#469—21no—
36C2×#468—21no—
37C2×#462C2211yescharacteristic
38C2×#460C2211yescharacteristic
39C2×#460C2211yescharacteristic
40C2×#463—21no—
41C2×#463—21no—
42C2×#463—21no—
43C2×#461C2211yescharacteristic
44C2×#459C2211yescharacteristic
45C2×#459C2211yescharacteristic
46C2×#452—21no—
47C2×#449—21no—
48C2×#449—21no—
49C2×#449—21no—
50#2994—24no—
51#2994C2218yescomplemented
52C2×#446C2211yescharacteristic
53C2×C8×Q8C2211yescharacteristic
54C2×#436C2211yescharacteristic
55C2×#435—21no—
56C2×C8×D4—21no—
57C2×C8×D4C2211yescharacteristic
58C2×#433—21no—
59C2×#431C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C2×#424—21no—
62C2×#423C2211yescharacteristic
63C22×C4⋊C8C2211yescharacteristic
64#1701C2218yescomplemented
65#1700C2218yescomplemented
66#1688C2218yescomplemented
67#1685C2218yescomplemented
68#1682C2218yescomplemented
69#1679C2218yescomplemented
70#1667C2218yescomplemented
71#1666C2218yescomplemented
72#1574C2218yescomplemented
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#334C2211yescharacteristic
76C2×#334C2211yescharacteristic
77#1547C2218yescomplemented
78#1540C2218yescomplemented
79C2×#332C2211yescharacteristic
80C2×#332C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#331C2211yescharacteristic
84C2×#331C2211yescharacteristic
85C2×#331C2211yescharacteristic
86C2×#331C2211yescharacteristic
87C22×#41—21no—
88C22×#41—21no—
89C22×Q16—21no—
90C22×SD16—21no—
91C22×SD16—21no—
92C22×SD16—21no—
93C22×SD16—21no—
94C22×SD16—21no—
95C22×D8—21no—
96C22×D8—21no—
97C22×D8—21no—
98C22×D8—21no—
99C22×M16—21no—
100C22×M16—21no—
101C23×C8—21no—
102C23×C8—21no—
103#532—22no—
104#532—22no—
105#532—22no—
106#532—22no—
107#532—22no—
108#532—22no—
109#532C2314yes—
110#532C2314yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#127C2311yescharacteristic
115#529—22no—
116#529—22no—
117#529D414yes—
118#529C2314yes—
119C2×#126—21no—
120C2×#125D411yescharacteristic
121C2×#124—21no—
122C2×#124—21no—
123C2×#124D411yescharacteristic
124C2×#123—21no—
125C2×C22⋊Q8D411yescharacteristic
126C2×#121—21no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121—21no—
130C2×#121D411yescharacteristic
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4C2311yescharacteristic
138C2×#117—21no—
139C22×C42C2311yescharacteristic
140#495—28no—
141#493—22no—
142#493C2314yes—
143#492—22no—
144#492C2314yes—
145#488—28no—
146#487—22no—
147#487C2314yes—
148#486—22no—
149#486C2314yes—
150#478—28no—
151#475—22no—
152#475C2314yes—
153#474—22no—
154#474C2314yes—
155#469—28no—
156#468—28no—
157#463—28no—
158#463—28no—
159#463—28no—
160#462—22no—
161#462C2314yes—
162#461—22no—
163#461C2314yes—
164#460—22no—
165#460—22no—
166#460C2314yes—
167#460C2314yes—
168#459—22no—
169#459—22no—
170#459C2314yes—
171#459C2314yes—
172#452—28no—
173#449—28no—
174#449—28no—
175#449—28no—
176#446—22no—
177#446C2314yes—
178C8×Q8—22no—
179C8×Q8C2314yes—
180#436—22no—
181#436C2314yes—
182#435—28no—
183C8×D4—28no—
184C8×D4—22no—
185C8×D4C2314yes—
186#433—28no—
187#431—22no—
188#431D414yes—
189#431C2314yes—
190#431C2314yes—
191C2×#107—21no—
192C2×#106—21no—
193#424—28no—
194#423—22no—
195#423C2314yes—
196C2×C4⋊C8—21no—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—21no—
199C2×C4⋊C8—21no—
200C2×C4⋊C8—22no—
201C2×C4⋊C8D411yescharacteristic
202C2×C4⋊C8D411yescharacteristic
203C2×C4⋊C8C2311yescharacteristic
204C2×C4⋊C8C2311yescharacteristic
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×C4⋊C8C2311yescharacteristic
209C2×C4⋊C8C2314yes—
210C2×#103—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#102—21no—
216C2×#102—21no—
217C2×#102—21no—
218C2×#102—21no—
219C2×#102—21no—
220C2×#102—21no—
221C2×#102—21no—
222C2×C22⋊C8—21no—
223C2×C22⋊C8—21no—
224C2×C22⋊C8—21no—
225C2×C22⋊C8D411yescharacteristic
226C2×C22⋊C8D411yescharacteristic
227C2×#97—21no—
228C2×C4×C8—21no—
229C2×C4×C8—21no—
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232#335C2314yes—
233#335C2314yes—
234#334C2314yes—
235#334C2314yes—
236#332C2314yes—
237#332C2314yes—
238#332C2314yes—
239#332C2314yes—
240#331C2314yes—
241#331C2314yes—
242#331C2314yes—
243#331C2314yes—
244C2×#41—24no—
245C2×#41—24no—
246C2×#41—42no—
247C2×#41—42no—
248C2×#41—82no—
249C2×#41—82no—
250C2×#41—82no—
251C2×#41—82no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8C2×D411yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4—41no—
263C22×D4—41no—
264C22×D4C2×D411yescharacteristic
265C23×C4—21no—
266C23×C4—41no—
267C23×C4—41no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4#4111yescharacteristic
271C2×Q16—24no—
272C2×Q16—82no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—82no—
279C2×SD16—82no—
280C2×SD16—82no—
281C2×SD16—82no—
282C2×SD16—82no—
283C2×D8—24no—
284C2×D8—24no—
285C2×D8—24no—
286C2×D8—24no—
287C2×D8—82no—
288C2×D8—82no—
289C2×D8—82no—
290C2×D8—82no—
291C2×M16—24no—
292C2×M16—24no—
293C2×M16—82no—
294C2×M16—82no—
295C22×C8—21no—
296C22×C8—21no—
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—21no—
300C22×C8—21no—
301C22×C8—24no—
302C22×C8—24no—
303C22×C8—82no—
304C22×C8—82no—
305C22×C8C2×D411yescharacteristic
306C22×C8C2×D411yescharacteristic
307C22×C8C2×D411yescharacteristic
308C22×C8C2×D411yescharacteristic
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8C2412yes—
316C4⋊Q8C2412yes—
317#127—21no—
318#127—21no—
319#127—21no—
320#127—21no—
321#127—21no—
322#127—21no—
323#127C2412yes—
324#127C2412yes—
325#126—24no—
326#126—24no—
327#125—22no—
328#125C2×D414yes—
329#124—22no—
330#124—24no—
331#124—24no—
332#124—42no—
333#124—42no—
334#124C2×D414yes—
335#123—24no—
336#123—24no—
337C22⋊Q8—22no—
338C22⋊Q8C2×D414yes—
339#121—22no—
340#121—24no—
341#121—24no—
342#121—24no—
343#121—24no—
344#121—42no—
345#121—42no—
346#121—42no—
347#121—42no—
348#121C2×D414yes—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8C2×D412yes—
352C4×Q8C2412yes—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—21no—
356C4×D4—24no—
357C4×D4—24no—
358C4×D4—21no—
359C4×D4—24no—
360C4×D4—24no—
361C4×D4—42no—
362C4×D4—42no—
363C4×D4—42no—
364C4×D4—42no—
365C4×D4C2×D412yes—
366C4×D4C2412yes—
367#117—24no—
368#117—24no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4C2×D411yescharacteristic
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4—41no—
386C2×C22⋊C4—41no—
387C2×C22⋊C4—41no—
388C2×C42—21no—
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42#4111yescharacteristic
394C2×C42#4112yes—
395C2×C42C2411yescharacteristic
396C2×C42C2412yes—
397#107—24no—
398#106—24no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401C4⋊C8—24no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—24no—
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8C2×D414yes—
410C4⋊C8C2×D414yes—
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#102—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—
432C22⋊C8C2×D414yes—
433C22⋊C8C2×D414yes—
434#97—24no—
435C4×C8—24no—
436C4×C8—24no—
437C4×C8C2×D412yes—
438C4×C8C2×D412yes—
439C4×C8C2412yes—
440C4×C8C2412yes—
441C24—21no—
442C24—41no—
443C24—41no—
444C24#12111yescharacteristic
445#41—84no—
446#41—84no—
447#41—84no—
448#41—84no—
449#41—88no—
450#41—88no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—41no—
457C2×Q8—41no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8C22×D412yes—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—82no—
491C2×D4—82no—
492C2×D4—84no—
493C2×D4—84no—
494C2×D4C22×D412yes—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—84no—
512C22×C4—84no—
513C22×C4#12112yes—
514C22×C4#12112yes—
515C22×C4#12111yescharacteristic
516C22×C4#12111yescharacteristic
517C22×C4#12111yescharacteristic
518C22×C4#12112yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C2×#4111yescharacteristic
524C22×C4C2×#4112yes—
525Q16—84no—
526SD16—84no—
527SD16—84no—
528SD16—84no—
529SD16—84no—
530SD16—84no—
531D8—84no—
532D8—84no—
533D8—84no—
534D8—84no—
535M16—84no—
536M16—84no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—42no—
548C2×C8—42no—
549C2×C8—42no—
550C2×C8—42no—
551C2×C8—42no—
552C2×C8—42no—
553C2×C8—84no—
554C2×C8—84no—
555C2×C8C22×D412yes—
556C2×C8C22×D412yes—
557C2×C8C22×D412yes—
558C2×C8C22×D412yes—
559C4⋊C4—24no—
560C4⋊C4—24no—
561C4⋊C4—24no—
562C4⋊C4—22no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—41no—
571C4⋊C4—41no—
572C4⋊C4—41no—
573C4⋊C4—41no—
574C4⋊C4C22×D412yes—
575C4⋊C4C22×D412yes—
576C22⋊C4—24no—
577C22⋊C4—24no—
578C22⋊C4—24no—
579C22⋊C4—24no—
580C22⋊C4—22no—
581C22⋊C4—22no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C42—22no—
587C42—22no—
588C42#12112yes—
589C42#12112yes—
590C42#12112yes—
591C42C22×D412yes—
592C42C22×D411yescharacteristic
593C42C22×D411yescharacteristic
594C42C22×D412yes—
595C42C2×#4112yes—
596C42C2×#4111yescharacteristic
597C42C2511yescharacteristic
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—82no—
608C23—82no—
609C23—82no—
610C23—82no—
611C23—82no—
612C23—82no—
613C23C2×#12112yes—
614C23C2×#12111yescharacteristic
615Q8—22no—
616Q8—22no—
617Q8—42no—
618Q8—42no—
619Q8—42no—
620Q8—42no—
621Q8—42no—
622Q8—82no—
623Q8—82no—
624D4—22no—
625D4—22no—
626D4—48no—
627D4—42no—
628D4—42no—
629D4—42no—
630D4—42no—
631D4—42no—
632D4—42no—
633D4—42no—
634D4—42no—
635D4—42no—
636D4—82no—
637D4—82no—
638D4—82no—
639D4—82no—
640D4—88no—
641D4—88no—
642D4—82no—
643D4—82no—
644D4—82no—
645D4—82no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—88no—
670C2×C4—88no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4C2×#12112yes—
674C2×C4C2×#12111yescharacteristic
675C2×C4C2×#12112yes—
676C2×C4C2×#12112yes—
677C2×C4C2×#12111yescharacteristic
678C2×C4C2×#12111yescharacteristic
679C2×C4#57612yes—
680C2×C4#57612yes—
681C2×C4#57712yes—
682C2×C4#57712yes—
683C2×C4C23×D411yescharacteristic
684C2×C4C23×D411yescharacteristic
685C2×C4C22×#4111yescharacteristic
686C8—22no—
687C8—22no—
688C8—22no—
689C8—22no—
690C8—42no—
691C8—42no—
692C8—42no—
693C8—42no—
694C8—42no—
695C8—42no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—44no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—84no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—164no—
711C22—164no—
712C22#304512yes—
713C22#306212yes—
714C22#306712yes—
715C22C22×#12111yescharacteristic
716C4—22no—
717C4—22no—
718C4—22no—
719C4—21no—
720C4—21no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—82no—
725C4—82no—
726C4#304512yes—
727C4#306212yes—
728C4C2×#57611yescharacteristic
729C4C2×#57611yescharacteristic
730C4C2×#57711yescharacteristic
731C4C2×#57711yescharacteristic
732C2—42no—
733C2—82no—
734C2—162no—
735C2—162no—
736C2#1957814yesdirect product factor
737C2C2×#304511yescharacteristic
738C2C2×#306211yescharacteristic
739C2C2×#306711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
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×#3251C2semidirect
2. C2×#3212C2semidirect
3. C2×#3207C2semidirect
4. C2×#2994C2semidirect
5. #19578C2direct (N × H)
6. C2×#1700C2semidirect
7. C2×#1688C2semidirect
8. C2×#1701C2semidirect
9. C2×#1685C2semidirect
10. C2×#1682C2semidirect
11. C2×#1666C2semidirect
12. C2×#1679C2semidirect
13. C2×#1667C2semidirect
14. C2×#1574C2semidirect
15. C2×#1547C2semidirect
16. C2×#1540C2semidirect
21. #3251C22semidirect
23. #3212C22semidirect
25. #3207C22semidirect
51. #2994C22semidirect
64. #1701C22semidirect
65. #1700C22semidirect
66. #1688C22semidirect
67. #1685C22semidirect
68. #1682C22semidirect
69. #1679C22semidirect
70. #1667C22semidirect
71. #1666C22semidirect
72. #1574C22semidirect
77. #1547C22semidirect
78. #1540C22semidirect
736. C2#19578direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32512
C2×#32122
C2×#32072
C2×#29942
#195782
C2×#17002
C2×#16882
C2×#17012
C2×#16852
C2×#16822
C2×#16662
C2×#16792
C2×#16672
C2×#15742
C2×#15472
C2×#15402