Finite Group Databasebeta

← C2×#29846 #6435525

Subgroups — C2×#29846 #6435525

Order 512.

2412 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic, complemented
2C22×#431C211yesmaximal, characteristic, complemented
3C2×#2851C211yesmaximal, characteristic, complemented
4C2×#2850C211yesmaximal, characteristic
5#29846C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic, complemented
9C2×#2065C211yesmaximal, characteristic, complemented
10C2×#2036C211yesmaximal, characteristic, complemented
11C2×#2034C211yesmaximal, characteristic
12C2×#2035C211yesmaximal, characteristic, complemented
13C2×#2033C211yesmaximal, characteristic
14C2×#1991C211yesmaximal, characteristic, complemented
15C2×#1958C211yesmaximal, characteristic, complemented
16C2×#1846C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#488—28no—
23C2×#488C2218yescomplemented
24C2×#431—28no—
25C2×#431C2218yescomplemented
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C22×C22⋊C8C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43C22×C4×C8C2211yescharacteristic
44#2851—24no—
45#2851C2218yescomplemented
46#2850—24no—
47#2850C2218yes—
48C2×#391C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#387—21no—
55C2×#387—21no—
56C2×#386—21no—
57C2×#386—21no—
58C2×#385—21no—
59C2×#385—21no—
60C2×#382C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×#380C2211yescharacteristic
63C2×#380C2211yescharacteristic
64#2068C2218yescomplemented
65#2067C2218yescomplemented
66#2066C2218yescomplemented
67#2065C2218yescomplemented
68#2036C2218yescomplemented
69#2035C2218yescomplemented
70#2034C2218yes—
71#2033C2218yes—
72#1991C2218yescomplemented
73#1958C2218yescomplemented
74#1846C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×#125—22no—
90C2×#125—24no—
91C2×#125—24no—
92C2×#125C2314yes—
93C2×#124—22no—
94C2×#124—24no—
95C2×#124—24no—
96C2×#124C2314yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—42no—
99C2×C22⋊Q8—42no—
100C2×C22⋊Q8C2314yes—
101C2×#121—22no—
102C2×#121—42no—
103C2×#121—42no—
104C2×#121C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—24no—
125C22×C42—24no—
126C22×C42D414yes—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#488—232no—
133#431—232no—
134C2×#107—42no—
135C2×#107—42no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106—42no—
139C2×#106—42no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8—24no—
143C2×C4⋊C8—24no—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—42no—
149C2×#103—42no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102—24no—
155C2×#102—24no—
156C2×#102—42no—
157C2×#102—42no—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8—24no—
163C2×C22⋊C8—24no—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C4×C8—24no—
167C2×C4×C8D418yescomplemented
168C2×C4×C8C2314yes—
169C2×C4×C8C2314yes—
170#391—22no—
171#391C2314yes—
172#390—22no—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2314yes—
178#388—28no—
179#388—28no—
180#387—28no—
181#387—28no—
182#386—28no—
183#386—28no—
184#385—28no—
185#385—28no—
186#382—22no—
187#382—22no—
188#382C2314yes—
189#382C2314yes—
190#380—22no—
191#380—22no—
192#380C2314yes—
193#380C2314yes—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208C25—21no—
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4C2412yes—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4C2×D412yes—
250C23×C4C2×D414yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8C2×D414yes—
270C22×C8C2×D414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#125—28no—
276#125—28no—
277#125—48no—
278#124—28no—
279#124—28no—
280#124—48no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284#121—44no—
285#121—44no—
286#121—48no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—216no—
342C2×C42—28no—
343C2×C42—28no—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D414yes—
349C2×C42#4114yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352#107—44no—
353#107—44no—
354#106—44no—
355#106—44no—
356C4⋊C8—28no—
357C4⋊C8—28no—
358#103—28no—
359#103—28no—
360#103—44no—
361#103—44no—
362#102—28no—
363#102—28no—
364#102—44no—
365#102—44no—
366C22⋊C8—28no—
367C22⋊C8—28no—
368C4×C8—28no—
369C4×C8C2×D4116yescomplemented
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24#13514yes—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—816no—
427C22×C4—232no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—416no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4#12114yes—
489C22×C4#12114yes—
490C22×C4#13514yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C22×C4C2511yescharacteristic
504C2×C8—28no—
505C2×C8—28no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—44no—
517C2×C8—44no—
518C2×C8C22×D414yes—
519C2×C8C22×D414yes—
520C4⋊C4—22no—
521C4⋊C4—22no—
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—22no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C42—28no—
552C42—24no—
553C42—28no—
554C42—22no—
555C42—22no—
556C42#12118yes—
557C42C22×D414yes—
558C42C2×#4114yes—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23C2×#12112yes—
587C23#53512yes—
588C23#53612yes—
589C23C2×#13514yes—
590C23C2×#13512yes—
591C23C2×#13512yes—
592C23#57512yes—
593C23#57512yes—
594C23C23×D411yescharacteristic
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—84no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—816no—
605D4—84no—
606C2×C4—232no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—416no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4C2×#12114yes—
659C2×C4C2×#12114yes—
660C2×C4C2×#12112yes—
661C2×C4#53512yes—
662C2×C4#53612yes—
663C2×C4#53714yes—
664C2×C4#53714yes—
665C2×C4#54214yes—
666C2×C4#54214yes—
667C2×C4C2×#13514yes—
668C2×C4C23×D411yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C8—24no—
675C8—24no—
676C8—42no—
677C8—42no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—416no—
685C22—42no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—168no—
694C22#262014yes—
695C22#305014yes—
696C22#305414yes—
697C22#321114yes—
698C22#321214yes—
699C22#323214yes—
700C22#323314yes—
701C22C22×#12111yescharacteristic
702C22C2×#53511yescharacteristic
703C22C2×#53611yescharacteristic
704C22C22×#13511yescharacteristic
705C22C22×#13511yescharacteristic
706C22C2×#57511yescharacteristic
707C22C2×#57511yescharacteristic
708C4—24no—
709C4—24no—
710C4—24no—
711C4—42no—
712C4—42no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4#262014yes—
721C4C2×#53711yescharacteristic
722C4C2×#53711yescharacteristic
723C4C2×#54211yescharacteristic
724C4C2×#54211yescharacteristic
725C2—44no—
726C2—162no—
727C2#2984618yesdirect product factor
728C2C2×#262011yescharacteristic
729C2C2×#305011yescharacteristic
730C2C2×#305411yescharacteristic
731C2C2×#321111yescharacteristic
732C2C2×#321211yescharacteristic
733C2C2×#323211yescharacteristic
734C2C2×#323311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#488C2semidirect
2. C22×#431C2semidirect
3. C2×#2851C2semidirect
5. #29846C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2066C2semidirect
8. C2×#2067C2semidirect
9. C2×#2065C2semidirect
10. C2×#2036C2semidirect
12. C2×#2035C2semidirect
14. C2×#1991C2semidirect
15. C2×#1958C2semidirect
16. C2×#1846C2semidirect
23. C2×#488C22semidirect
25. C2×#431C22semidirect
45. #2851C22semidirect
64. #2068C22semidirect
65. #2067C22semidirect
66. #2066C22semidirect
67. #2065C22semidirect
68. #2036C22semidirect
69. #2035C22semidirect
72. #1991C22semidirect
73. #1958C22semidirect
74. #1846C22semidirect
167. C2×C4×C8D4semidirect
369. C4×C8C2×D4semidirect
727. C2#29846direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4882
C22×#4312
C2×#28512
C2×#28502
#298462
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#20362
C2×#20342
C2×#20352
C2×#20332
C2×#19912
C2×#19582
C2×#18462