Finite Group Databasebeta

← C2×#19249 #6353754

Subgroups — C2×#19249 #6353754

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3248C211yesmaximal, characteristic
2C2×#3233C211yesmaximal, characteristic
3C2×#3185C211yesmaximal, characteristic, complemented
4C2×#2969C211yesmaximal, characteristic
5#19249C2116yesmaximal, direct product factor
6C2×#1692C211yesmaximal, characteristic, complemented
7C2×#1690C211yesmaximal, characteristic, complemented
8C2×#1703C211yesmaximal, characteristic, complemented
9C2×#1699C211yesmaximal, characteristic, complemented
10C2×#1684C211yesmaximal, characteristic, complemented
11C2×#1680C211yesmaximal, characteristic, complemented
12C2×#1671C211yesmaximal, characteristic, complemented
13C2×#1669C211yesmaximal, characteristic, complemented
14C2×#1565C211yesmaximal, characteristic, complemented
15C2×#1544C211yesmaximal, characteristic, complemented
16C2×#1542C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3248—24no—
21#3248C2218yes—
22#3233—24no—
23#3233C2218yes—
24#3185—24no—
25#3185C2218yescomplemented
26C2×#499—21no—
27C2×#498C2211yescharacteristic
28C2×#494C2211yescharacteristic
29C2×#488—21no—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#485—21no—
33C2×#482—21no—
34C2×#479—21no—
35C2×#477C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#475C2211yescharacteristic
38C2×#476C2211yescharacteristic
39C2×#474C2211yescharacteristic
40C2×#474C2211yescharacteristic
41C2×#470—21no—
42C2×#470—21no—
43C2×#467—21no—
44C2×#465—21no—
45C2×#464—21no—
46C2×#462C2211yescharacteristic
47C2×#461C2211yescharacteristic
48#2969—24no—
49#2969C2218yes—
50C2×#446C2211yescharacteristic
51C2×C8×Q8C2211yescharacteristic
52C2×#441—21no—
53C2×#441—21no—
54C2×#436—21no—
55C2×#436C2211yescharacteristic
56C2×#435—21no—
57C2×C8×D4C2211yescharacteristic
58C2×#433—21no—
59C2×#431C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C22×C4⋊C8C2211yescharacteristic
62C2×#405—21no—
63C2×C4×M16C2211yescharacteristic
64#1703C2218yescomplemented
65#1699C2218yescomplemented
66#1692C2218yescomplemented
67#1690C2218yescomplemented
68#1684C2218yescomplemented
69#1680C2218yescomplemented
70#1671C2218yescomplemented
71#1669C2218yescomplemented
72#1565C2218yescomplemented
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#334C2211yescharacteristic
76C2×#334C2211yescharacteristic
77#1544C2218yescomplemented
78#1542C2218yescomplemented
79C2×#333C2211yescharacteristic
80C2×#333C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#332C2211yescharacteristic
86C2×#332C2211yescharacteristic
87C22×Q16—21no—
88C22×Q16—21no—
89C22×Q16—21no—
90C22×SD16—21no—
91C22×M16—21no—
92C22×M16D411yescharacteristic
93C22×M16D411yescharacteristic
94C23×C8—21no—
95C23×C8—21no—
96#533—22no—
97#533—22no—
98#533—22no—
99#533—22no—
100#533—22no—
101#533—22no—
102#533C2314yes—
103#533C2314yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108#529—22no—
109#529—22no—
110#529D414yes—
111#529C2314yes—
112C2×#126—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125D411yescharacteristic
116C2×#124D411yescharacteristic
117C2×#123—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8D411yescharacteristic
123C2×#121D411yescharacteristic
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C2×#117—21no—
134C22×C42C2311yescharacteristic
135#499—28no—
136#498—22no—
137#498C2314yes—
138#494—22no—
139#494C2314yes—
140#488—28no—
141#487—22no—
142#487—22no—
143#487C2314yes—
144#487C2314yes—
145#485—28no—
146#482—28no—
147#479—28no—
148#477—22no—
149#477C2314yes—
150#476—22no—
151#476C2314yes—
152#475—22no—
153#475—22no—
154#475C2314yes—
155#475C2314yes—
156#474—22no—
157#474—22no—
158#474C2314yes—
159#474C2314yes—
160#470—28no—
161#470—28no—
162#467—28no—
163#465—28no—
164#464—28no—
165#462—22no—
166#462C2314yes—
167#461—22no—
168#461C2314yes—
169#446—22no—
170#446C2314yes—
171C8×Q8—22no—
172C8×Q8C2314yes—
173#441—28no—
174#441—28no—
175#436—28no—
176#436—22no—
177#436C2314yes—
178#435—28no—
179C8×D4—22no—
180C8×D4C2314yes—
181#433—28no—
182#431—22no—
183#431—22no—
184#431C2314yes—
185#431C2314yes—
186C2×#107—21no—
187C2×#107—21no—
188C2×#107—21no—
189C2×#106—21no—
190C2×#106—21no—
191C2×#106—21no—
192C2×#106—21no—
193C2×#106—21no—
194C2×C4⋊C8—21no—
195C2×C4⋊C8—21no—
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×#103—21no—
206C2×#103—21no—
207C2×#103—21no—
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×C22⋊C8—21no—
218C2×C22⋊C8—21no—
219C2×C22⋊C8—21no—
220C2×C22⋊C8—21no—
221#405—28no—
222C4×M16D414yes—
223C4×M16C2314yes—
224C2×#97—21no—
225C2×#97D411yescharacteristic
226C2×#97D411yescharacteristic
227C2×C4×C8—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#332C2314yes—
239#332C2314yes—
240#332C2314yes—
241#332C2314yes—
242#332C2314yes—
243#332C2314yes—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8C2×D411yescharacteristic
249C22×D4—21no—
250C22×D4C2×D411yescharacteristic
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4#4111yescharacteristic
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16—82no—
261C2×SD16—24no—
262C2×SD16—82no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—42no—
266C2×M16—44no—
267C2×M16C2×D414yes—
268C2×M16C2×D414yes—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—44no—
278C22×C8—82no—
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#126—24no—
300#126—24no—
301#125—22no—
302#125—24no—
303#125—24no—
304#125—42no—
305#125—42no—
306#125C2×D414yes—
307#124—22no—
308#124C2×D414yes—
309#123—24no—
310#123—24no—
311C22⋊Q8—22no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—42no—
317C22⋊Q8—42no—
318C22⋊Q8—42no—
319C22⋊Q8—42no—
320C22⋊Q8C2×D414yes—
321#121—22no—
322#121C2×D414yes—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8—42no—
330C4×Q8—42no—
331C4×Q8—42no—
332C4×Q8—42no—
333C4×Q8C2×D412yes—
334C4×Q8C2412yes—
335C4×D4—24no—
336C4×D4—24no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4C2×D412yes—
340C4×D4C2412yes—
341#117—24no—
342#117—24no—
343#117—24no—
344#117—24no—
345#117—42no—
346#117—42no—
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—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—41no—
362C2×C4⋊C4—41no—
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—41no—
372C2×C42—21no—
373C2×C42—41no—
374C2×C42—41no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42#4111yescharacteristic
380C2×C42#4112yes—
381C2×C42C2411yescharacteristic
382C2×C42C2412yes—
383#107—24no—
384#107—24no—
385#107—24no—
386#106—24no—
387#106—24no—
388#106—24no—
389#106—24no—
390#106—24no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—24no—
396C4⋊C8—21no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8C2412yes—
401C4⋊C8C2412yes—
402C4⋊C8C2412yes—
403C4⋊C8C2412yes—
404C4⋊C8C2412yes—
405C4⋊C8C2412yes—
406#103—24no—
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—
418C22⋊C8—24no—
419C22⋊C8—24no—
420C22⋊C8—24no—
421C22⋊C8—24no—
422#97—24no—
423#97C2×D414yes—
424#97C2×D414yes—
425C4×C8—24no—
426C4×C8—24no—
427C4×C8—24no—
428C4×C8C2×D412yes—
429C4×C8C2×D412yes—
430C4×C8C2412yes—
431C4×C8C2412yes—
432C24—21no—
433C24#12111yescharacteristic
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8C22×D412yes—
453C2×D4—24no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—82no—
458C2×D4C22×D412yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4#12112yes—
476C22×C4#12112yes—
477C22×C4#12111yescharacteristic
478C22×C4#12111yescharacteristic
479C22×C4#12111yescharacteristic
480C22×C4#12112yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4112yes—
487Q16—48no—
488Q16—48no—
489Q16—84no—
490SD16—84no—
491M16—28no—
492M16—44no—
493M16—48no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—42no—
504C2×C8—42no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—42no—
508C2×C8—42no—
509C2×C8—48no—
510C2×C8—84no—
511C2×C8#13514yes—
512C2×C8C22×D412yes—
513C2×C8C22×D412yes—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
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—24no—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C42—22no—
554C42—22no—
555C42—44no—
556C42—44no—
557C42#12112yes—
558C42#12112yes—
559C42#12112yes—
560C42C22×D412yes—
561C42C22×D412yes—
562C42C22×D411yescharacteristic
563C42C22×D411yescharacteristic
564C42C2×#4111yescharacteristic
565C42C2×#4112yes—
566C42C2511yescharacteristic
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23C2×#12112yes—
575C23C2×#12111yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—82no—
586Q8—82no—
587Q8—82no—
588Q8—82no—
589Q8—82no—
590D4—22no—
591D4—22no—
592D4—42no—
593D4—84no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
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—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4C2×#12112yes—
624C2×C4C2×#12112yes—
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×#12112yes—
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×#13512yes—
630C2×C4#57512yes—
631C2×C4#57712yes—
632C2×C4#57812yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C8—22no—
637C8—22no—
638C8—22no—
639C8—42no—
640C8—42no—
641C8—42no—
642C8—42no—
643C8—42no—
644C8—42no—
645C8C2×#13514yes—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—84no—
654C22#303512yes—
655C22#306312yes—
656C22#306912yes—
657C22C22×#12111yescharacteristic
658C4—22no—
659C4—22no—
660C4—21no—
661C4—21no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#303512yes—
671C4#306312yes—
672C4C22×#13511yescharacteristic
673C4C2×#57511yescharacteristic
674C4C2×#57711yescharacteristic
675C4C2×#57811yescharacteristic
676C2—42no—
677C2—82no—
678C2#1924914yesdirect product factor
679C2C2×#303511yescharacteristic
680C2C2×#306311yescharacteristic
681C2C2×#306911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#121D4
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
3. C2×#3185C2semidirect
5. #19249C2direct (N × H)
6. C2×#1692C2semidirect
7. C2×#1690C2semidirect
8. C2×#1703C2semidirect
9. C2×#1699C2semidirect
10. C2×#1684C2semidirect
11. C2×#1680C2semidirect
12. C2×#1671C2semidirect
13. C2×#1669C2semidirect
14. C2×#1565C2semidirect
15. C2×#1544C2semidirect
16. C2×#1542C2semidirect
25. #3185C22semidirect
64. #1703C22semidirect
65. #1699C22semidirect
66. #1692C22semidirect
67. #1690C22semidirect
68. #1684C22semidirect
69. #1680C22semidirect
70. #1671C22semidirect
71. #1669C22semidirect
72. #1565C22semidirect
77. #1544C22semidirect
78. #1542C22semidirect
678. C2#19249direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32482
C2×#32332
C2×#31852
C2×#29692
#192492
C2×#16922
C2×#16902
C2×#17032
C2×#16992
C2×#16842
C2×#16802
C2×#16712
C2×#16692
C2×#15652
C2×#15442
C2×#15422