Finite Group Databasebeta

← C2×#19542 #6354628

Subgroups — C2×#19542 #6354628

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3251C211yesmaximal, characteristic, complemented
2C2×#3247C211yesmaximal, characteristic, complemented
3C2×#3207C211yesmaximal, characteristic, complemented
4C2×#2993C211yesmaximal, characteristic, complemented
5#19542C2116yesmaximal, direct product factor
6C2×#1698C211yesmaximal, characteristic, complemented
7C2×#1686C211yesmaximal, characteristic, complemented
8C2×#1697C211yesmaximal, characteristic, complemented
9C2×#1685C211yesmaximal, characteristic, complemented
10C2×#1684C211yesmaximal, characteristic, complemented
11C2×#1668C211yesmaximal, characteristic, complemented
12C2×#1683C211yesmaximal, characteristic, complemented
13C2×#1667C211yesmaximal, characteristic, complemented
14C2×#1571C211yesmaximal, characteristic, complemented
15C2×#1560C211yesmaximal, characteristic, complemented
16C2×#1547C211yesmaximal, characteristic, complemented
17C2×#532C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3251—24no—
21#3251C2218yescomplemented
22#3247—24no—
23#3247C2218yescomplemented
24#3207—24no—
25#3207C2218yescomplemented
26C2×#493C2211yescharacteristic
27C2×#492C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#486C2211yescharacteristic
30C2×#477C2211yescharacteristic
31C2×#475C2211yescharacteristic
32C2×#478—21no—
33C2×#478—21no—
34C2×#476C2211yescharacteristic
35C2×#474C2211yescharacteristic
36C2×#469—21no—
37C2×#468—21no—
38C2×#466—21no—
39C2×#465—21no—
40C2×#460C2211yescharacteristic
41C2×#460C2211yescharacteristic
42C2×#463—21no—
43C2×#463—21no—
44C2×#463—21no—
45C2×#463—21no—
46C2×#459C2211yescharacteristic
47C2×#459C2211yescharacteristic
48C2×#449—21no—
49C2×#449—21no—
50#2993—24no—
51#2993C2218yescomplemented
52C2×#446C2211yescharacteristic
53C2×C8×Q8C2211yescharacteristic
54C2×#436—21no—
55C2×#436C2211yescharacteristic
56C2×#435—21no—
57C2×C8×D4C2211yescharacteristic
58C2×#433—21no—
59C2×#431C2211yescharacteristic
60C2×#424—21no—
61C2×#423C2211yescharacteristic
62C2×#423C2211yescharacteristic
63C22×C4⋊C8C2211yescharacteristic
64#1698C2218yescomplemented
65#1697C2218yescomplemented
66#1686C2218yescomplemented
67#1685C2218yescomplemented
68#1684C2218yescomplemented
69#1683C2218yescomplemented
70#1668C2218yescomplemented
71#1667C2218yescomplemented
72#1571C2218yescomplemented
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#335C2211yescharacteristic
76C2×#335C2211yescharacteristic
77#1560C2218yescomplemented
78#1547C2218yescomplemented
79C2×#333C2211yescharacteristic
80C2×#333C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#331C2211yescharacteristic
84C2×#331C2211yescharacteristic
85C2×#331C2211yescharacteristic
86C2×#331C2211yescharacteristic
87C22×#41—21no—
88C22×#41—21no—
89C22×SD16—21no—
90C22×SD16—21no—
91C22×SD16—21no—
92C22×SD16—21no—
93C22×D8—21no—
94C22×D8—21no—
95C22×D8—21no—
96C22×D8—21no—
97C22×M16—21no—
98C22×M16—21no—
99C22×M16—21no—
100C23×C8—21no—
101C23×C8—21no—
102#532—22no—
103#532—22no—
104#532—22no—
105#532—22no—
106#532—22no—
107#532—22no—
108#532C2314yes—
109#532C2314yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127C2311yescharacteristic
113C2×#127C2311yescharacteristic
114#529—22no—
115#529—22no—
116#529D414yes—
117#529C2314yes—
118C2×#126—21no—
119C2×#125D411yescharacteristic
120C2×#124—21no—
121C2×#124—21no—
122C2×#124D411yescharacteristic
123C2×#123—21no—
124C2×C22⋊Q8D411yescharacteristic
125C2×#121—21no—
126C2×#121—21no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121D411yescharacteristic
130C2×C4×Q8C2311yescharacteristic
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C22×C42C2311yescharacteristic
139#493—22no—
140#493C2314yes—
141#492—22no—
142#492C2314yes—
143#487—22no—
144#487C2314yes—
145#486—22no—
146#486C2314yes—
147#478—28no—
148#478—28no—
149#477—22no—
150#477C2314yes—
151#476—22no—
152#476C2314yes—
153#475—22no—
154#475C2314yes—
155#474—22no—
156#474C2314yes—
157#469—28no—
158#468—28no—
159#466—28no—
160#465—28no—
161#463—28no—
162#463—28no—
163#463—28no—
164#463—28no—
165#460—22no—
166#460—22no—
167#460C2314yes—
168#460C2314yes—
169#459—22no—
170#459—22no—
171#459C2314yes—
172#459C2314yes—
173#449—28no—
174#449—28no—
175#446—22no—
176#446C2314yes—
177C8×Q8—22no—
178C8×Q8C2314yes—
179#436—28no—
180#436—22no—
181#436C2314yes—
182#435—28no—
183C8×D4—22no—
184C8×D4C2314yes—
185#433—28no—
186#431D414yes—
187#431C2314yes—
188C2×#107—21no—
189C2×#107—21no—
190C2×#106—21no—
191C2×#106—21no—
192#424—28no—
193#423—22no—
194#423—22no—
195#423C2314yes—
196#423C2314yes—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—21no—
199C2×C4⋊C8—21no—
200C2×C4⋊C8—21no—
201C2×C4⋊C8—22no—
202C2×C4⋊C8—21no—
203C2×C4⋊C8D411yescharacteristic
204C2×C4⋊C8D411yescharacteristic
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×C4⋊C8C2311yescharacteristic
209C2×C4⋊C8C2311yescharacteristic
210C2×C4⋊C8C2311yescharacteristic
211C2×C4⋊C8C2314yes—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#102—21no—
217C2×#102—21no—
218C2×#102—21no—
219C2×#102—21no—
220C2×#102—21no—
221C2×#102—21no—
222C2×#102—21no—
223C2×#102—21no—
224C2×C22⋊C8—21no—
225C2×C22⋊C8—21no—
226C2×C22⋊C8D411yescharacteristic
227C2×C22⋊C8D411yescharacteristic
228C2×#97—21no—
229C2×C4×C8—21no—
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232#335C2314yes—
233#335C2314yes—
234#335C2314yes—
235#335C2314yes—
236#333C2314yes—
237#333C2314yes—
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—84no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8C2×D411yescharacteristic
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—41no—
262C22×D4—41no—
263C22×D4C2×D411yescharacteristic
264C23×C4—21no—
265C23×C4—41no—
266C23×C4—41no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4#4111yescharacteristic
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—82no—
275C2×SD16—82no—
276C2×SD16—82no—
277C2×SD16—82no—
278C2×D8—24no—
279C2×D8—24no—
280C2×D8—24no—
281C2×D8—24no—
282C2×D8—82no—
283C2×D8—82no—
284C2×D8—82no—
285C2×D8—82no—
286C2×M16—24no—
287C2×M16—24no—
288C2×M16—24no—
289C2×M16—82no—
290C2×M16—82no—
291C2×M16—82no—
292C22×C8—21no—
293C22×C8—21no—
294C22×C8—21no—
295C22×C8—21no—
296C22×C8—21no—
297C22×C8—21no—
298C22×C8—24no—
299C22×C8—24no—
300C22×C8—82no—
301C22×C8—82no—
302C22×C8C2×D411yescharacteristic
303C22×C8C2×D411yescharacteristic
304C22×C8C2×D411yescharacteristic
305C22×C8C2×D411yescharacteristic
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8C2412yes—
313C4⋊Q8C2412yes—
314#127—21no—
315#127—21no—
316#127—21no—
317#127—21no—
318#127—21no—
319#127—21no—
320#127C2412yes—
321#127C2412yes—
322#126—24no—
323#126—24no—
324#125—22no—
325#125C2×D414yes—
326#124—22no—
327#124—24no—
328#124—24no—
329#124—42no—
330#124—42no—
331#124C2×D414yes—
332#123—24no—
333#123—24no—
334C22⋊Q8—22no—
335C22⋊Q8C2×D414yes—
336#121—22no—
337#121—24no—
338#121—24no—
339#121—24no—
340#121—24no—
341#121—42no—
342#121—42no—
343#121—42no—
344#121—42no—
345#121C2×D414yes—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×Q8C2×D412yes—
349C4×Q8C2412yes—
350C4×D4—24no—
351C4×D4—24no—
352C4×D4—21no—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—24no—
356C4×D4—21no—
357C4×D4—24no—
358C4×D4—42no—
359C4×D4—42no—
360C4×D4—42no—
361C4×D4—42no—
362C4×D4C2×D412yes—
363C4×D4C2412yes—
364#117—24no—
365#117—24no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—41no—
382C2×C22⋊C4—41no—
383C2×C22⋊C4—41no—
384C2×C22⋊C4—41no—
385C2×C42—21no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D412yes—
389C2×C42C2×D412yes—
390C2×C42#4111yescharacteristic
391C2×C42#4112yes—
392C2×C42C2411yescharacteristic
393C2×C42C2412yes—
394#107—24no—
395#107—24no—
396#106—24no—
397#106—24no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—24no—
403C4⋊C8—24no—
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#102—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⋊C8C2×D414yes—
432C22⋊C8C2×D414yes—
433#97—24no—
434C4×C8—24no—
435C4×C8C2×D412yes—
436C4×C8C2×D412yes—
437C4×C8C2412yes—
438C4×C8C2412yes—
439C24—21no—
440C24—41no—
441C24—41no—
442C24#12111yescharacteristic
443#41—84no—
444#41—84no—
445#41—88no—
446#41—88no—
447#41—88no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—82no—
458C2×Q8C22×D412yes—
459C2×D4—24no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—82no—
486C2×D4—82no—
487C2×D4—82no—
488C2×D4—84no—
489C2×D4—84no—
490C2×D4C22×D412yes—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—84no—
508C22×C4—84no—
509C22×C4#12112yes—
510C22×C4#12112yes—
511C22×C4#12111yescharacteristic
512C22×C4#12111yescharacteristic
513C22×C4#12111yescharacteristic
514C22×C4#12112yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C2×#4111yescharacteristic
520C22×C4C2×#4112yes—
521SD16—84no—
522SD16—84no—
523SD16—84no—
524SD16—84no—
525D8—84no—
526D8—84no—
527D8—84no—
528D8—84no—
529M16—84no—
530M16—84no—
531M16—84no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—42no—
543C2×C8—42no—
544C2×C8—42no—
545C2×C8—42no—
546C2×C8—42no—
547C2×C8—42no—
548C2×C8—84no—
549C2×C8—84no—
550C2×C8C22×D412yes—
551C2×C8C22×D412yes—
552C2×C8C22×D412yes—
553C2×C8C22×D412yes—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—41no—
566C4⋊C4—41no—
567C4⋊C4—41no—
568C4⋊C4—41no—
569C4⋊C4C22×D412yes—
570C4⋊C4C22×D412yes—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C42—22no—
582C42—22no—
583C42#12112yes—
584C42#12112yes—
585C42#12112yes—
586C42C22×D411yescharacteristic
587C42C22×D412yes—
588C42C22×D411yescharacteristic
589C42C22×D412yes—
590C42C2×#4112yes—
591C42C2×#4111yescharacteristic
592C42C2511yescharacteristic
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23—82no—
607C23—82no—
608C23C2×#12112yes—
609C23C2×#12111yescharacteristic
610Q8—22no—
611Q8—22no—
612Q8—42no—
613Q8—42no—
614Q8—42no—
615Q8—82no—
616Q8—82no—
617Q8—82no—
618D4—22no—
619D4—22no—
620D4—42no—
621D4—42no—
622D4—42no—
623D4—42no—
624D4—42no—
625D4—42no—
626D4—42no—
627D4—42no—
628D4—42no—
629D4—82no—
630D4—82no—
631D4—82no—
632D4—82no—
633D4—84no—
634D4—88no—
635D4—88no—
636D4—82no—
637D4—82no—
638D4—82no—
639D4—82no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—88no—
664C2×C4—88no—
665C2×C4—82no—
666C2×C4—82no—
667C2×C4C2×#12112yes—
668C2×C4C2×#12111yescharacteristic
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4C2×#12111yescharacteristic
672C2×C4C2×#12111yescharacteristic
673C2×C4#57612yes—
674C2×C4#57612yes—
675C2×C4#57712yes—
676C2×C4#57812yes—
677C2×C4C23×D411yescharacteristic
678C2×C4C23×D411yescharacteristic
679C2×C4C22×#4111yescharacteristic
680C8—22no—
681C8—22no—
682C8—22no—
683C8—22no—
684C8—42no—
685C8—42no—
686C8—42no—
687C8—42no—
688C8—42no—
689C8—42no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—84no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—164no—
705C22—164no—
706C22#304512yes—
707C22#306112yes—
708C22#306812yes—
709C22C22×#12111yescharacteristic
710C4—22no—
711C4—22no—
712C4—22no—
713C4—21no—
714C4—21no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—82no—
719C4—82no—
720C4#304512yes—
721C4#306112yes—
722C4C2×#57611yescharacteristic
723C4C2×#57611yescharacteristic
724C4C2×#57711yescharacteristic
725C4C2×#57811yescharacteristic
726C2—42no—
727C2—82no—
728C2—162no—
729C2—162no—
730C2#1954214yesdirect product factor
731C2C2×#304511yescharacteristic
732C2C2×#306111yescharacteristic
733C2C2×#306811yescharacteristic
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×#3247C2semidirect
3. C2×#3207C2semidirect
4. C2×#2993C2semidirect
5. #19542C2direct (N × H)
6. C2×#1698C2semidirect
7. C2×#1686C2semidirect
8. C2×#1697C2semidirect
9. C2×#1685C2semidirect
10. C2×#1684C2semidirect
11. C2×#1668C2semidirect
12. C2×#1683C2semidirect
13. C2×#1667C2semidirect
14. C2×#1571C2semidirect
15. C2×#1560C2semidirect
16. C2×#1547C2semidirect
21. #3251C22semidirect
23. #3247C22semidirect
25. #3207C22semidirect
51. #2993C22semidirect
64. #1698C22semidirect
65. #1697C22semidirect
66. #1686C22semidirect
67. #1685C22semidirect
68. #1684C22semidirect
69. #1683C22semidirect
70. #1668C22semidirect
71. #1667C22semidirect
72. #1571C22semidirect
77. #1560C22semidirect
78. #1547C22semidirect
730. C2#19542direct (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×#32472
C2×#32072
C2×#29932
#195422
C2×#16982
C2×#16862
C2×#16972
C2×#16852
C2×#16842
C2×#16682
C2×#16832
C2×#16672
C2×#15712
C2×#15602
C2×#15472