Finite Group Databasebeta

← C2×#30252 #6433779

Subgroups — C2×#30252 #6433779

Order 512.

2820 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#470C211yesmaximal, characteristic, complemented
2C22×#468C211yesmaximal, characteristic
3C22×#467C211yesmaximal, characteristic, complemented
4C22×#465C211yesmaximal, characteristic
5C2×#2621C211yesmaximal, characteristic, complemented
6C2×#2620C211yesmaximal, characteristic
7#30252C2116yesmaximal, direct product factor
8C2×#2097C211yesmaximal, characteristic, complemented
9C2×#2096C211yesmaximal, characteristic, complemented
10C2×#2095C211yesmaximal, characteristic, complemented
11C2×#2092C211yesmaximal, characteristic, complemented
12C2×#2016C211yesmaximal, characteristic, complemented
13C2×#2013C211yesmaximal, characteristic, complemented
14C2×#2015C211yesmaximal, characteristic, complemented
15C2×#2015C211yesmaximal, characteristic, complemented
16C2×#1729C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C23×SD16C2211yescharacteristic
20C23×SD16C2211yescharacteristic
21C23×M16C2211yescharacteristic
22C24×C8C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C22×#124C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×#121C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C2×#470—28no—
31C2×#470C2218yescomplemented
32C2×#468—28no—
33C2×#468C2218yes—
34C2×#467—28no—
35C2×#467C2218yescomplemented
36C2×#465—28no—
37C2×#465C2218yes—
38C22×#107C2211yescharacteristic
39C22×#107C2211yescharacteristic
40C22×#106C2211yescharacteristic
41C22×#106C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C22×#103C2211yescharacteristic
47C22×#103C2211yescharacteristic
48C22×#102C2211yescharacteristic
49C22×#102C2211yescharacteristic
50#2621—24no—
51#2621C2218yescomplemented
52#2620—24no—
53#2620C2218yes—
54C2×#398—22no—
55C2×#397—22no—
56C2×#396—22no—
57C2×#394—22no—
58C2×#391C2211yescharacteristic
59C2×#391C2211yescharacteristic
60C2×#391C2211yescharacteristic
61C2×#390C2211yescharacteristic
62C2×#384C2211yescharacteristic
63C2×#382C2211yescharacteristic
64#2097C2218yescomplemented
65#2096C2218yescomplemented
66#2095C2218yescomplemented
67#2092C2218yescomplemented
68#2016C2218yescomplemented
69#2015C2218yescomplemented
70#2015C2218yescomplemented
71#2013C2218yescomplemented
72#1729C2218yescomplemented
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×Q16—24no—
84C22×Q16—24no—
85C22×Q16—42no—
86C22×Q16—42no—
87C22×Q16—42no—
88C22×Q16—42no—
89C22×Q16C2314yes—
90C22×Q16C2314yes—
91C22×SD16—24no—
92C22×SD16—24no—
93C22×SD16—42no—
94C22×SD16—42no—
95C22×SD16—42no—
96C22×SD16—42no—
97C22×SD16C2314yes—
98C22×SD16C2314yes—
99C22×M16—42no—
100C22×M16—42no—
101C22×M16D418yes—
102C22×M16C2314yes—
103C23×C8—42no—
104C23×C8—42no—
105C23×C8D418yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109C23×C8C2311yescharacteristic
110C23×C8C2314yes—
111C2×C4⋊Q8—22no—
112C2×C4⋊Q8—42no—
113C2×C4⋊Q8—42no—
114C2×C4⋊Q8C2314yes—
115C2×#124—22no—
116C2×#124—42no—
117C2×#124—42no—
118C2×#124C2314yes—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—24no—
128C2×C22⋊Q8C2314yes—
129C2×C22⋊Q8C2314yes—
130C2×C22⋊Q8C2314yes—
131C2×#121—22no—
132C2×#121—24no—
133C2×#121—24no—
134C2×#121C2314yes—
135C2×#117—42no—
136C2×#117—42no—
137C2×#117D414yes—
138C2×#117C2314yes—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—22no—
144C22×C4⋊C4D412yes—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C22⋊C4—21no—
150C22×C22⋊C4—21no—
151C22×C22⋊C4—21no—
152C22×C22⋊C4—22no—
153C22×C22⋊C4—21no—
154C22×C22⋊C4—22no—
155C22×C22⋊C4D412yes—
156C22×C42C2311yescharacteristic
157C22×C42C2311yescharacteristic
158#470—232no—
159#468—232no—
160#467—232no—
161#465—232no—
162C2×#107—24no—
163C2×#107—24no—
164C2×#107C2314yes—
165C2×#107C2314yes—
166C2×#106—24no—
167C2×#106—24no—
168C2×#106C2314yes—
169C2×#106C2314yes—
170C2×#103—24no—
171C2×#103—24no—
172C2×#103—24no—
173C2×#103—24no—
174C2×#103—24no—
175C2×#103—24no—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#103C2314yes—
180C2×#103C2314yes—
181C2×#103C2314yes—
182C2×#102—24no—
183C2×#102—24no—
184C2×#102C2314yes—
185C2×#102C2314yes—
186#398—216no—
187#397—216no—
188#396—216no—
189#394—216no—
190#391—22no—
191#391—22no—
192#391—22no—
193#391C2314yes—
194#391C2314yes—
195#391C2314yes—
196#390—22no—
197#390C2314yes—
198#384—22no—
199#384C2314yes—
200#382—22no—
201#382C2314yes—
202C2×#95—22no—
203C2×#95—22no—
204C2×#95—22no—
205C2×#95—22no—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210C25—21no—
211C25C2×D411yescharacteristic
212C22×Q8—21no—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×D4—24no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—44no—
245C22×D4—84no—
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D414yes—
263C23×C4C2×D414yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×Q16—416no—
273C2×Q16—416no—
274C2×SD16—28no—
275C2×SD16—28no—
276C2×SD16—44no—
277C2×SD16—44no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×SD16—416no—
281C2×SD16—416no—
282C2×M16—416no—
283C2×M16—44no—
284C2×M16—44no—
285C2×M16C2×D4116yes—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—416no—
295C22×C8—44no—
296C22×C8—44no—
297C22×C8C2×D414yes—
298C22×C8C2×D414yes—
299C22×C8C2×D414yes—
300C22×C8C2×D414yes—
301C22×C8C2×D4116yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308C4⋊Q8—48no—
309#124—44no—
310#124—44no—
311#124—48no—
312C22⋊Q8—28no—
313C22⋊Q8—28no—
314C22⋊Q8—28no—
315C22⋊Q8—28no—
316C22⋊Q8—28no—
317C22⋊Q8—28no—
318C22⋊Q8—48no—
319C22⋊Q8—48no—
320C22⋊Q8—48no—
321#121—28no—
322#121—28no—
323#121—48no—
324#117—48no—
325#117—44no—
326#117—44no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—28no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4—48no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—84no—
350C2×C4⋊C4C2×D418yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—44no—
362C2×C22⋊C4—48no—
363C2×C22⋊C4—44no—
364C2×C22⋊C4—44no—
365C2×C22⋊C4—44no—
366C2×C22⋊C4—44no—
367C2×C22⋊C4—84no—
368C2×C22⋊C4C2×D418yes—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377#107—28no—
378#107—28no—
379#106—28no—
380#106—28no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—28no—
385#103—28no—
386#103—28no—
387#102—28no—
388#102—28no—
389#95—28no—
390#95—28no—
391#95—28no—
392#95—28no—
393C24—22no—
394C24—22no—
395C24—24no—
396C24—24no—
397C24—22no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—44no—
402C24—44no—
403C24#12714yes—
404C24C22×D412yes—
405C24C22×D411yescharacteristic
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—48no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—88no—
460C2×D4—816no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4#12714yes—
504C22×C4#12718yes—
505C22×C4C22×D414yes—
506C22×C4C22×D414yes—
507C22×C4C22×D412yes—
508C22×C4C22×D414yes—
509C22×C4C22×D414yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518Q16—416no—
519Q16—416no—
520SD16—416no—
521SD16—416no—
522M16—416no—
523C2×C8—28no—
524C2×C8—28no—
525C2×C8—28no—
526C2×C8—28no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—416no—
536C2×C8C22×D414yes—
537C2×C8C22×D414yes—
538C2×C8C22×D414yes—
539C2×C8C22×D414yes—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—48no—
549C4⋊C4—416no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—48no—
555C4⋊C4—48no—
556C4⋊C4—48no—
557C4⋊C4—88no—
558C22⋊C4—48no—
559C22⋊C4—416no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—48no—
564C22⋊C4—88no—
565C42—24no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—24no—
578C23—48no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—44no—
584C23—44no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—88no—
589C23—88no—
590C23C2×#12714yes—
591C23C2×#12712yes—
592C23#53412yes—
593C23#53712yes—
594C23#57714yes—
595C23#57814yes—
596C23C23×D411yescharacteristic
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—84no—
610Q8—84no—
611Q8—84no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—816no—
617D4—84no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—216no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—48no—
633C2×C4—48no—
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—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4C2×#12714yes—
659C2×C4C2×#12712yes—
660C2×C4#53412yes—
661C2×C4#53414yes—
662C2×C4#53414yes—
663C2×C4#53712yes—
664C2×C4#53714yes—
665C2×C4#53714yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C23×D411yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—24no—
673C8—24no—
674C8—24no—
675C8—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—416no—
680C22—44no—
681C22—44no—
682C22—82no—
683C22—84no—
684C22—82no—
685C22—82no—
686C22—84no—
687C22—168no—
688C22#265614yes—
689C22#315414yes—
690C22#315614yes—
691C22#320118yes—
692C22#325818yes—
693C22C22×#12711yescharacteristic
694C22C2×#53411yescharacteristic
695C22C2×#53711yescharacteristic
696C22C2×#57712yes—
697C22C2×#57812yes—
698C4—24no—
699C4—44no—
700C4—44no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#265614yes—
709C4C2×#53411yescharacteristic
710C4C2×#53411yescharacteristic
711C4C2×#53711yescharacteristic
712C4C2×#53711yescharacteristic
713C2—44no—
714C2—162no—
715C2#3025218yesdirect product factor
716C2C2×#265611yescharacteristic
717C2C2×#315411yescharacteristic
718C2C2×#315611yescharacteristic
719C2C2×#320112yes—
720C2C2×#325812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#470C2semidirect
3. C22×#467C2semidirect
5. C2×#2621C2semidirect
7. #30252C2direct (N × H)
8. C2×#2097C2semidirect
9. C2×#2096C2semidirect
10. C2×#2095C2semidirect
11. C2×#2092C2semidirect
12. C2×#2016C2semidirect
13. C2×#2013C2semidirect
14. C2×#2015C2semidirect
15. C2×#2015C2semidirect
16. C2×#1729C2semidirect
31. C2×#470C22semidirect
35. C2×#467C22semidirect
51. #2621C22semidirect
64. #2097C22semidirect
65. #2096C22semidirect
66. #2095C22semidirect
67. #2092C22semidirect
68. #2016C22semidirect
69. #2015C22semidirect
70. #2015C22semidirect
71. #2013C22semidirect
72. #1729C22semidirect
715. C2#30252direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4702
C22×#4682
C22×#4672
C22×#4652
C2×#26212
C2×#26202
#302522
C2×#20972
C2×#20962
C2×#20952
C2×#20922
C2×#20162
C2×#20132
C2×#20152
C2×#17292