Finite Group Databasebeta

← C2×#19499 #6354571

Subgroups — C2×#19499 #6354571

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic, complemented
2C2×#3116C211yesmaximal, characteristic
3C2×#3076C211yesmaximal, characteristic
4C2×#2991C211yesmaximal, characteristic
5#19499C2116yesmaximal, direct product factor
6C2×#1703C211yesmaximal, characteristic, complemented
7C2×#1699C211yesmaximal, characteristic, complemented
8C2×#1695C211yesmaximal, characteristic, complemented
9C2×#1691C211yesmaximal, characteristic, complemented
10C2×#1684C211yesmaximal, characteristic, complemented
11C2×#1680C211yesmaximal, characteristic, complemented
12C2×#1676C211yesmaximal, characteristic, complemented
13C2×#1672C211yesmaximal, characteristic, complemented
14C2×#1573C211yesmaximal, characteristic, complemented
15C2×#1559C211yesmaximal, characteristic, complemented
16C2×#1550C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20#3231—24no—
21#3231C2218yescomplemented
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#487C2211yescharacteristic
26#3116—24no—
27#3116C2218yes—
28C2×#485—21no—
29C2×#485—21no—
30C2×#485—21no—
31#3076—24no—
32#3076C2218yes—
33C2×#479—21no—
34C2×#479—21no—
35C2×#479—21no—
36C2×#477C2211yescharacteristic
37C2×#477C2211yescharacteristic
38C2×#475C2211yescharacteristic
39C2×#478—21no—
40C2×#476C2211yescharacteristic
41C2×#476C2211yescharacteristic
42C2×#474C2211yescharacteristic
43C2×#462C2211yescharacteristic
44C2×#461C2211yescharacteristic
45C2×#452—21no—
46#2991—24no—
47#2991C2218yes—
48C2×C8×Q8C2211yescharacteristic
49C2×C8×Q8C2211yescharacteristic
50C2×#441—21no—
51C2×#440—21no—
52C2×#436—21no—
53C2×#436—21no—
54C2×#436C2211yescharacteristic
55C2×#436C2211yescharacteristic
56C2×#432C2211yescharacteristic
57C2×#431—21no—
58C2×#431C2211yescharacteristic
59C2×#427—21no—
60C2×#423—21no—
61C2×#423C2211yescharacteristic
62C22×C4⋊C8C2211yescharacteristic
63C2×#416—21no—
64#1703C2218yescomplemented
65#1699C2218yescomplemented
66#1695C2218yescomplemented
67#1691C2218yescomplemented
68#1684C2218yescomplemented
69#1680C2218yescomplemented
70#1676C2218yescomplemented
71#1672C2218yescomplemented
72#1573C2218yescomplemented
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#334C2211yescharacteristic
76C2×#334C2211yescharacteristic
77#1559C2218yescomplemented
78#1550C2218yescomplemented
79C2×#333C2211yescharacteristic
80C2×#333C2211yescharacteristic
81C2×#333C2211yescharacteristic
82C2×#333C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#332C2211yescharacteristic
86C2×#332C2211yescharacteristic
87C22×#41D411yescharacteristic
88C22×Q16—21no—
89C22×SD16—21no—
90C22×M16—21no—
91C22×M16—21no—
92C22×M16—21no—
93C23×C8—21no—
94#533—22no—
95#533—22no—
96#533—22no—
97#533—22no—
98#533—22no—
99#533—22no—
100#533C2314yes—
101#533C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—21no—
107C2×#125—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C4×#41—22no—
113C4×#41—22no—
114C4×#41D414yes—
115C4×#41C2314yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#500—22no—
133#500C2314yes—
134#498—22no—
135#498C2314yes—
136#487—22no—
137#487—22no—
138#487C2314yes—
139#487C2314yes—
140#485—28no—
141#485—28no—
142#485—28no—
143#479—28no—
144#479—28no—
145#479—28no—
146#478—28no—
147#477—22no—
148#477—22no—
149#477C2314yes—
150#477C2314yes—
151#476—22no—
152#476—22no—
153#476C2314yes—
154#476C2314yes—
155#475—22no—
156#475C2314yes—
157#474—22no—
158#474C2314yes—
159#462—22no—
160#462C2314yes—
161#461—22no—
162#461C2314yes—
163#452—28no—
164C8×Q8—22no—
165C8×Q8—22no—
166C8×Q8C2314yes—
167C8×Q8C2314yes—
168#441—28no—
169#440—28no—
170#436—28no—
171#436—28no—
172#436—22no—
173#436—22no—
174#436C2314yes—
175#436C2314yes—
176#432—22no—
177#432C2314yes—
178#431—28no—
179#431D414yes—
180#431C2314yes—
181#427—28no—
182C2×#107—21no—
183C2×#107—21no—
184C2×#107—21no—
185C2×#107—21no—
186C2×#107—21no—
187C2×#106—21no—
188C2×#106—21no—
189C2×#106—21no—
190C2×#106—21no—
191C2×#106—21no—
192#423—22no—
193#423—28no—
194#423C2314yes—
195C2×C4⋊C8—21no—
196C2×C4⋊C8—21no—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—22no—
199C2×C4⋊C8—21no—
200C2×C4⋊C8D411yescharacteristic
201C2×C4⋊C8D411yescharacteristic
202C2×C4⋊C8C2311yescharacteristic
203C2×C4⋊C8C2311yescharacteristic
204C2×C4⋊C8C2311yescharacteristic
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×C4⋊C8C2314yes—
209#416—28no—
210C2×#103—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#103—21no—
217C2×#103—21no—
218C2×#103—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#333C2314yes—
237#333C2314yes—
238#333C2314yes—
239#333C2314yes—
240#332C2314yes—
241#332C2314yes—
242#332C2314yes—
243#332C2314yes—
244C2×#41—22no—
245C2×#41—24no—
246C2×#41—42no—
247C2×#41C2×D414yes—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8C2×D411yescharacteristic
253C22×D4—21no—
254C22×D4C2×D411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4#4111yescharacteristic
261C2×Q16—24no—
262C2×Q16—44no—
263C2×SD16—24no—
264C2×SD16—44no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—82no—
269C2×M16—82no—
270C2×M16—82no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—24no—
274C22×C8—21no—
275C22×C8—21no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—44no—
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8C2×D411yescharacteristic
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
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⋊Q8C2412yes—
296C4⋊Q8C2412yes—
297C4⋊Q8C2412yes—
298C4⋊Q8C2412yes—
299#125—24no—
300#125—24no—
301#125—42no—
302#125—42no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307C22⋊Q8—42no—
308C22⋊Q8—42no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311C4×Q8—22no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8C2×D414yes—
323C4×Q8C2×D412yes—
324C4×Q8C2412yes—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4C2×D414yes—
333C4×D4C2×D412yes—
334C4×D4C2412yes—
335#117—22no—
336#117—24no—
337#117—24no—
338#117—24no—
339#117—24no—
340#117—42no—
341#117—42no—
342#117C2×D414yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4—41no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4—41no—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—41no—
363C2×C22⋊C4—41no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—41no—
370C2×C42—41no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42#4111yescharacteristic
376C2×C42#4112yes—
377C2×C42C2411yescharacteristic
378C2×C42C2412yes—
379#107—24no—
380#107—24no—
381#107—24no—
382#107—24no—
383#107—24no—
384#106—24no—
385#106—24no—
386#106—24no—
387#106—24no—
388#106—24no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391C4⋊C8—21no—
392C4⋊C8—21no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8C2×D414yes—
400C4⋊C8C2×D414yes—
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#103—24no—
416#102—24no—
417#102—24no—
418#102—24no—
419C22⋊C8—24no—
420C22⋊C8—24no—
421C22⋊C8—24no—
422C22⋊C8C2×D414yes—
423C22⋊C8C2×D414yes—
424#97—24no—
425C4×C8—24no—
426C4×C8—24no—
427C4×C8C2×D412yes—
428C4×C8C2×D412yes—
429C4×C8C2412yes—
430C4×C8C2412yes—
431C24—21no—
432C24#12111yescharacteristic
433#41—28no—
434#41—44no—
435#41—48no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—82no—
451C2×Q8—82no—
452C2×Q8#13714yes—
453C2×Q8C22×D412yes—
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4C22×D412yes—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4#12112yes—
481C22×C4#12112yes—
482C22×C4#12111yescharacteristic
483C22×C4#12111yescharacteristic
484C22×C4#12111yescharacteristic
485C22×C4#12112yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4111yescharacteristic
491C22×C4C2×#4112yes—
492Q16—48no—
493SD16—48no—
494M16—84no—
495M16—84no—
496M16—84no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—48no—
508C2×C8—42no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—42no—
512C2×C8—42no—
513C2×C8—42no—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C2×C8C22×D412yes—
517C2×C8C22×D412yes—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
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—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4C22×D412yes—
544C4⋊C4C22×D412yes—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C42—24no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—44no—
556C42—44no—
557C42#12112yes—
558C42#12112yes—
559C42#12112yes—
560C42C22×D411yescharacteristic
561C42C22×D412yes—
562C42C22×D411yescharacteristic
563C42C22×D412yes—
564C42C2×#4112yes—
565C42C2×#4111yescharacteristic
566C42C2511yescharacteristic
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23C2×#12112yes—
575C23C2×#12111yescharacteristic
576Q8—24no—
577Q8—22no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—82no—
583Q8—82no—
584Q8—82no—
585Q8—82no—
586Q8—82no—
587Q8—82no—
588Q8C2×#13714yes—
589D4—22no—
590D4—22no—
591D4—48no—
592D4—42no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12112yes—
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×#12111yescharacteristic
630C2×C4C2×#13712yes—
631C2×C4#57512yes—
632C2×C4#57712yes—
633C2×C4#57812yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C23×D411yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C8—22no—
638C8—22no—
639C8—22no—
640C8—22no—
641C8—42no—
642C8—42no—
643C8—42no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—84no—
655C22#304612yes—
656C22#305612yes—
657C22#306712yes—
658C22C22×#12111yescharacteristic
659C4—22no—
660C4—22no—
661C4—24no—
662C4—22no—
663C4—21no—
664C4—21no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#304612yes—
672C4#305612yes—
673C4C22×#13711yescharacteristic
674C4C2×#57511yescharacteristic
675C4C2×#57711yescharacteristic
676C4C2×#57811yescharacteristic
677C2—42no—
678C2—82no—
679C2#1949914yesdirect product factor
680C2C2×#304611yescharacteristic
681C2C2×#305611yescharacteristic
682C2C2×#306711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC22×#41D4
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
5. #19499C2direct (N × H)
6. C2×#1703C2semidirect
7. C2×#1699C2semidirect
8. C2×#1695C2semidirect
9. C2×#1691C2semidirect
10. C2×#1684C2semidirect
11. C2×#1680C2semidirect
12. C2×#1676C2semidirect
13. C2×#1672C2semidirect
14. C2×#1573C2semidirect
15. C2×#1559C2semidirect
16. C2×#1550C2semidirect
21. #3231C22semidirect
64. #1703C22semidirect
65. #1699C22semidirect
66. #1695C22semidirect
67. #1691C22semidirect
68. #1684C22semidirect
69. #1680C22semidirect
70. #1676C22semidirect
71. #1672C22semidirect
72. #1573C22semidirect
77. #1559C22semidirect
78. #1550C22semidirect
679. C2#19499direct (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×#31162
C2×#30762
C2×#29912
#194992
C2×#17032
C2×#16992
C2×#16952
C2×#16912
C2×#16842
C2×#16802
C2×#16762
C2×#16722
C2×#15732
C2×#15592
C2×#15502