Finite Group Databasebeta

← C2×#19366 #6354039

Subgroups — C2×#19366 #6354039

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3252C211yesmaximal, characteristic, complemented
2C2×#3251C211yesmaximal, characteristic
3C2×#3131C211yesmaximal, characteristic, complemented
4C2×#2966C211yesmaximal, characteristic, complemented
5#19366C2116yesmaximal, direct product factor
6C2×#1704C211yesmaximal, characteristic, complemented
7C2×#1702C211yesmaximal, characteristic, complemented
8C2×#1703C211yesmaximal, characteristic, complemented
9C2×#1701C211yesmaximal, characteristic, complemented
10C2×#1680C211yesmaximal, characteristic, complemented
11C2×#1678C211yesmaximal, characteristic, complemented
12C2×#1679C211yesmaximal, characteristic, complemented
13C2×#1677C211yesmaximal, characteristic, complemented
14C2×#1566C211yesmaximal, characteristic, complemented
15C2×#1559C211yesmaximal, characteristic, complemented
16C2×#1557C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3252—24no—
21#3252C2218yescomplemented
22#3251—24no—
23#3251C2218yes—
24#3131—24no—
25#3131C2218yescomplemented
26C2×#488—21no—
27C2×#488—21no—
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479—21no—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#478—21no—
36C2×#476C2211yescharacteristic
37C2×#474C2211yescharacteristic
38C2×#470—21no—
39C2×#469—21no—
40C2×#468—21no—
41C2×#468—21no—
42C2×#464—21no—
43C2×#462C2211yescharacteristic
44C2×#460C2211yescharacteristic
45C2×#463—21no—
46C2×#461C2211yescharacteristic
47C2×#459C2211yescharacteristic
48#2966—24no—
49#2966C2218yescomplemented
50C2×#446C2211yescharacteristic
51C2×C8×Q8C2211yescharacteristic
52C2×#436C2211yescharacteristic
53C2×#435—21no—
54C2×C8×D4—21no—
55C2×C8×D4C2211yescharacteristic
56C2×#433—21no—
57C2×#432C2211yescharacteristic
58C2×#423C2211yescharacteristic
59C2×#423C2211yescharacteristic
60C2×#416—21no—
61C2×#416—21no—
62C2×#405—21no—
63C22×C4×C8C2211yescharacteristic
64#1704C2218yescomplemented
65#1703C2218yescomplemented
66#1702C2218yescomplemented
67#1701C2218yescomplemented
68#1680C2218yescomplemented
69#1679C2218yescomplemented
70#1678C2218yescomplemented
71#1677C2218yescomplemented
72#1566C2218yescomplemented
73C2×#334C2211yescharacteristic
74C2×#334C2211yescharacteristic
75C2×#334C2211yescharacteristic
76C2×#334C2211yescharacteristic
77#1559C2218yescomplemented
78#1557C2218yescomplemented
79C2×#333C2211yescharacteristic
80C2×#333C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#331C2211yescharacteristic
86C2×#331C2211yescharacteristic
87C22×#41—21no—
88C22×Q16—21no—
89C22×SD16—21no—
90C22×SD16—21no—
91C22×D8—21no—
92C22×M16—21no—
93C22×M16—21no—
94C22×M16—21no—
95C23×C8—21no—
96C23×C8D411yescharacteristic
97C23×C8D411yescharacteristic
98#533—22no—
99#533—22no—
100#533—22no—
101#533C2314yes—
102#532—22no—
103#532—22no—
104#532—22no—
105#532C2314yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110#529—22no—
111#529—22no—
112#529D414yes—
113#529C2314yes—
114C2×#126—21no—
115C2×#125—21no—
116C2×#125D411yescharacteristic
117C2×#124—21no—
118C2×#124D411yescharacteristic
119C2×#123—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8D411yescharacteristic
123C2×#121—21no—
124C2×#121—21no—
125C2×#121D411yescharacteristic
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C2×#117—21no—
135C22×C42C2311yescharacteristic
136#488—28no—
137#488—28no—
138#487—22no—
139#487—22no—
140#487C2314yes—
141#487C2314yes—
142#486—22no—
143#486—22no—
144#486C2314yes—
145#486C2314yes—
146#479—28no—
147#478—28no—
148#477—22no—
149#477C2314yes—
150#476—22no—
151#476C2314yes—
152#475—22no—
153#475C2314yes—
154#474—22no—
155#474C2314yes—
156#470—28no—
157#469—28no—
158#468—28no—
159#468—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#446C2314yes—
172C8×Q8—22no—
173C8×Q8C2314yes—
174#436—22no—
175#436C2314yes—
176#435—28no—
177C8×D4—28no—
178C8×D4—22no—
179C8×D4C2314yes—
180#433—28no—
181#432—22no—
182#432C2314yes—
183C2×#107—21no—
184C2×#107—21no—
185C2×#106—21no—
186C2×#106—21no—
187#423—22no—
188#423—22no—
189#423C2314yes—
190#423C2314yes—
191C2×C4⋊C8—21no—
192C2×C4⋊C8—21no—
193C2×C4⋊C8—21no—
194C2×C4⋊C8—21no—
195C2×C4⋊C8C2311yescharacteristic
196C2×C4⋊C8C2311yescharacteristic
197C2×C4⋊C8C2311yescharacteristic
198C2×C4⋊C8C2311yescharacteristic
199C2×C4⋊C8C2311yescharacteristic
200C2×C4⋊C8C2311yescharacteristic
201#416—28no—
202#416—28no—
203C2×#103—21no—
204C2×#103—21no—
205C2×#103—21no—
206C2×#103—21no—
207C2×#103—21no—
208C2×#103—21no—
209C2×#103—21no—
210C2×#103—21no—
211C2×#102—21no—
212C2×#102—21no—
213C2×#102—21no—
214C2×#102—21no—
215C2×#102—21no—
216C2×#102—21no—
217C2×#102—21no—
218C2×#102—21no—
219C2×C22⋊C8—21no—
220C2×C22⋊C8—21no—
221C2×C22⋊C8—21no—
222#405—28no—
223C2×#97—21no—
224C2×#97—21no—
225C2×C4×C8—21no—
226C2×C4×C8D411yescharacteristic
227C2×C4×C8D411yescharacteristic
228C2×C4×C8D414yes—
229C2×C4×C8C2311yescharacteristic
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2314yes—
232#334C2314yes—
233#334C2314yes—
234#334C2314yes—
235#334C2314yes—
236#333C2314yes—
237#333C2314yes—
238#332C2314yes—
239#332C2314yes—
240#332C2314yes—
241#332C2314yes—
242#331C2314yes—
243#331C2314yes—
244C2×#41—24no—
245C2×#41—44no—
246C2×#41—44no—
247C2×#41—42no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8C2×D411yescharacteristic
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—41no—
257C22×D4C2×D411yescharacteristic
258C23×C4—21no—
259C23×C4—41no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4#4111yescharacteristic
263C2×Q16—24no—
264C2×Q16—82no—
265C2×SD16—24no—
266C2×SD16—24no—
267C2×SD16—82no—
268C2×SD16—82no—
269C2×D8—24no—
270C2×D8—82no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—44no—
275C2×M16—82no—
276C2×M16—82no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—21no—
282C22×C8—21no—
283C22×C8—24no—
284C22×C8—42no—
285C22×C8—42no—
286C22×C8—44no—
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D414yes—
292C22×C8C2×D414yes—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8C2412yes—
303C4⋊Q8C2412yes—
304C4⋊Q8C2412yes—
305#127—21no—
306#127—21no—
307#127—21no—
308#127C2412yes—
309#126—24no—
310#126—24no—
311#125—22no—
312#125—24no—
313#125—42no—
314#125C2×D414yes—
315#124—22no—
316#124—24no—
317#124—42no—
318#124C2×D414yes—
319#123—24no—
320#123—24no—
321C22⋊Q8—22no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8—42no—
325C22⋊Q8—42no—
326C22⋊Q8C2×D414yes—
327#121—22no—
328#121—24no—
329#121—24no—
330#121—42no—
331#121—42no—
332#121C2×D414yes—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—24no—
336C4×Q8—24no—
337C4×Q8—42no—
338C4×Q8—42no—
339C4×Q8C2×D412yes—
340C4×Q8C2412yes—
341C4×D4—24no—
342C4×D4—24no—
343C4×D4—21no—
344C4×D4—24no—
345C4×D4—21no—
346C4×D4—24no—
347C4×D4—42no—
348C4×D4—42no—
349C4×D4C2×D412yes—
350C4×D4C2412yes—
351#117—24no—
352#117—24no—
353#117—24no—
354#117—42no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
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—41no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—41no—
377C2×C42—21no—
378C2×C42—41no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D412yes—
382C2×C42C2×D412yes—
383C2×C42#4111yescharacteristic
384C2×C42#4112yes—
385C2×C42C2411yescharacteristic
386C2×C42C2412yes—
387#107—24no—
388#107—24no—
389#106—24no—
390#106—24no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8C2412yes—
402C4⋊C8C2412yes—
403C4⋊C8C2412yes—
404C4⋊C8C2412yes—
405C4⋊C8C2412yes—
406C4⋊C8C2412yes—
407#103—24no—
408#103—24no—
409#103—24no—
410#103—24no—
411#103—24no—
412#103—24no—
413#103—24no—
414#103—24no—
415#102—24no—
416#102—24no—
417#102—24no—
418#102—24no—
419#102—24no—
420#102—24no—
421#102—24no—
422#102—24no—
423C22⋊C8—24no—
424C22⋊C8—24no—
425C22⋊C8—24no—
426#97—24no—
427#97—24no—
428C4×C8—24no—
429C4×C8C2×D414yes—
430C4×C8C2×D414yes—
431C4×C8C2×D412yes—
432C4×C8C2×D412yes—
433C4×C8C2412yes—
434C4×C8C2412yes—
435C24—21no—
436C24—41no—
437C24#12111yescharacteristic
438#41—48no—
439#41—48no—
440#41—88no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—82no—
455C2×Q8C22×D412yes—
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—82no—
473C2×D4—84no—
474C2×D4C22×D412yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—44no—
491C22×C4—84no—
492C22×C4#12112yes—
493C22×C4#12112yes—
494C22×C4#12111yescharacteristic
495C22×C4#12111yescharacteristic
496C22×C4#12111yescharacteristic
497C22×C4#12112yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C2×#4111yescharacteristic
503C22×C4C2×#4112yes—
504Q16—84no—
505SD16—84no—
506SD16—84no—
507D8—84no—
508M16—48no—
509M16—84no—
510M16—84no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—44no—
522C2×C8—44no—
523C2×C8—48no—
524C2×C8—42no—
525C2×C8—42no—
526C2×C8—42no—
527C2×C8—42no—
528C2×C8—42no—
529C2×C8—42no—
530C2×C8C22×D412yes—
531C2×C8C22×D412yes—
532C2×C8C22×D412yes—
533C2×C8C22×D412yes—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4C22×D412yes—
556C4⋊C4C22×D412yes—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C42—22no—
567C42—22no—
568C42—44no—
569C42#12112yes—
570C42#12112yes—
571C42#12112yes—
572C42C22×D412yes—
573C42C22×D411yescharacteristic
574C42C22×D412yes—
575C42C22×D411yescharacteristic
576C42C2×#4112yes—
577C42C2×#4111yescharacteristic
578C42C2511yescharacteristic
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—82no—
588C23—82no—
589C23—82no—
590C23C2×#12112yes—
591C23C2×#12111yescharacteristic
592Q8—22no—
593Q8—22no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—82no—
600Q8—82no—
601Q8—82no—
602Q8—82no—
603D4—22no—
604D4—22no—
605D4—48no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—82no—
612D4—82no—
613D4—88no—
614D4—82no—
615D4—82no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—88no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4C2×#12112yes—
645C2×C4C2×#12112yes—
646C2×C4C2×#12111yescharacteristic
647C2×C4C2×#12112yes—
648C2×C4C2×#12111yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4#57512yes—
651C2×C4#57512yes—
652C2×C4#57712yes—
653C2×C4#57712yes—
654C2×C4C23×D411yescharacteristic
655C2×C4C23×D411yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C8—22no—
658C8—22no—
659C8—22no—
660C8—22no—
661C8—42no—
662C8—42no—
663C8—42no—
664C8—42no—
665C8—42no—
666C8—42no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—164no—
679C22#304012yes—
680C22#306012yes—
681C22#306712yes—
682C22C22×#12111yescharacteristic
683C4—22no—
684C4—22no—
685C4—21no—
686C4—21no—
687C4—22no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#304012yes—
695C4#306012yes—
696C4C2×#57511yescharacteristic
697C4C2×#57511yescharacteristic
698C4C2×#57711yescharacteristic
699C4C2×#57711yescharacteristic
700C2—42no—
701C2—82no—
702C2—162no—
703C2#1936614yesdirect product factor
704C2C2×#304011yescharacteristic
705C2C2×#306011yescharacteristic
706C2C2×#306711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#3251C2
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×#3252C2semidirect
3. C2×#3131C2semidirect
4. C2×#2966C2semidirect
5. #19366C2direct (N × H)
6. C2×#1704C2semidirect
7. C2×#1702C2semidirect
8. C2×#1703C2semidirect
9. C2×#1701C2semidirect
10. C2×#1680C2semidirect
11. C2×#1678C2semidirect
12. C2×#1679C2semidirect
13. C2×#1677C2semidirect
14. C2×#1566C2semidirect
15. C2×#1559C2semidirect
16. C2×#1557C2semidirect
21. #3252C22semidirect
25. #3131C22semidirect
49. #2966C22semidirect
64. #1704C22semidirect
65. #1703C22semidirect
66. #1702C22semidirect
67. #1701C22semidirect
68. #1680C22semidirect
69. #1679C22semidirect
70. #1678C22semidirect
71. #1677C22semidirect
72. #1566C22semidirect
77. #1559C22semidirect
78. #1557C22semidirect
703. C2#19366direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32522
C2×#32512
C2×#31312
C2×#29662
#193662
C2×#17042
C2×#17022
C2×#17032
C2×#17012
C2×#16802
C2×#16782
C2×#16792
C2×#16772
C2×#15662
C2×#15592
C2×#15572