Finite Group Databasebeta

← C2×#26414 #6441909

Subgroups — C2×#26414 #6441909

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#26414C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#1987C211yesmaximal, characteristic, complemented
9C2×#1921C211yesmaximal, characteristic, complemented
10C2×#1892C211yesmaximal, characteristic
11C2×#1872C211yesmaximal, characteristic
12C2×#1863C211yesmaximal, characteristic, complemented
13C2×#1852C211yesmaximal, characteristic
14C2×#1850C211yesmaximal, characteristic
15C2×#1803C211yesmaximal, characteristic, complemented
16C22×#340C211yesmaximal, characteristic
17C24×C8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2440—24no—
34#2440C2218yes—
35C2×#398—21no—
36C2×#395—21no—
37#2360—24no—
38#2360C2218yes—
39C2×#391C2211yescharacteristic
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#385—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4C2211yescharacteristic
52#2100C2218yescomplemented
53#2087C2218yescomplemented
54#2068C2218yescomplemented
55#2024C2218yescomplemented
56#1987C2218yescomplemented
57#1921C2218yescomplemented
58#1892C2218yes—
59#1872C2218yes—
60#1863C2218yescomplemented
61#1852C2218yes—
62#1850C2218yes—
63#1803C2218yescomplemented
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2218yes—
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8—24no—
83C23×C8—42no—
84C23×C8—42no—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2314yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8C2314yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—48no—
98C2×C22⋊Q8Q818yescomplemented
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—24no—
102C2×C4×Q8—42no—
103C2×C4×Q8C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—42no—
119C22×C4⋊C4—42no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C2×#107—24no—
138C2×#107—42no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×C4⋊C8—42no—
142C2×C4⋊C8—42no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—42no—
147C2×#103—42no—
148C2×#103—42no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8C2314yes—
155#398—28no—
156#395—28no—
157#391—22no—
158#391C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—28no—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#387—28no—
167#385—28no—
168#385—28no—
169#384—22no—
170#384C2314yes—
171#380—22no—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—28no—
176C4×C22⋊C4D414yes—
177C4×C22⋊C4C2314yes—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95D411yescharacteristic
186C2×#95D411yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25C2×D411yescharacteristic
200C22×Q8—21no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—82no—
210C22×Q8C2×Q814yes—
211C22×Q8C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—82no—
231C23×C4—82no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×Q814yes—
239C23×C4#4114yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C22×C8—28no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8—416no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—28no—
263C4⋊Q8—44no—
264C4⋊Q8—48no—
265C22⋊Q8—216no—
266C22⋊Q8—28no—
267C22⋊Q8—432no—
268C22⋊Q8C2×Q8116yescomplemented
269C4×Q8—28no—
270C4×Q8—44no—
271C4×Q8—48no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×Q814yes—
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4#4114yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4Q1618yes—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—44no—
341C2×C42—82no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—28no—
347#107—44no—
348C4⋊C8—44no—
349C4⋊C8—44no—
350#103—28no—
351#103—44no—
352#103—44no—
353#103—44no—
354C22⋊C8—28no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24C22⋊Q814yes—
369C24C2×Q1614yes—
370C24AGL(1,ℤ/8)14yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—84no—
388C2×Q8—816no—
389C2×Q8C22×Q814yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—816no—
451C22×C4—816no—
452C22×C4C22⋊Q814yes—
453C22×C4C22⋊Q818yes—
454C22×C4C2×Q1614yes—
455C22×C4C2×Q1614yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×Q814yes—
463C22×C4C22×Q812yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4C2511yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—416no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—44no—
487C2×C8—44no—
488C4⋊C4—216no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C22×Q814yes—
517C4⋊C4C2×#4114yes—
518C4⋊C4C2×#4114yes—
519C22⋊C4—216no—
520C22⋊C4—28no—
521C22⋊C4—48no—
522C22⋊C4C2×Q16116yes—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—48no—
530C42—84no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—22no—
540C23—22no—
541C23#477116yes—
542C23C2×C22⋊Q814yes—
543C23C2×C22⋊Q812yes—
544C23#54612yes—
545C23#54612yes—
546C23C22×Q1614yes—
547C23C22×Q1612yes—
548C23C2×AGL(1,ℤ/8)14yes—
549C23C2×AGL(1,ℤ/8)12yes—
550C23#57712yes—
551C23#57712yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—42no—
556Q8—42no—
557Q8—48no—
558Q8—816no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—216no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—816no—
610C2×C4—816no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4C2×C22⋊Q814yes—
616C2×C4C2×C22⋊Q812yes—
617C2×C4#52914yes—
618C2×C4#53814yes—
619C2×C4#54612yes—
620C2×C4#54612yes—
621C2×C4#55414yes—
622C2×C4#55414yes—
623C2×C4C22×Q1614yes—
624C2×C4C22×Q1614yes—
625C2×C4C23×Q811yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C8—42no—
632C8—42no—
633C8—42no—
634C8—42no—
635C22—232no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22#245014yes—
643C22C2×#47718yes—
644C22C2×#47714yes—
645C22#308414yes—
646C22#329914yes—
647C22#329914yes—
648C22#330614yes—
649C22#331014yes—
650C22C22×C22⋊Q811yescharacteristic
651C22C2×#54611yescharacteristic
652C22C2×#54611yescharacteristic
653C22C23×Q1611yescharacteristic
654C22C22×AGL(1,ℤ/8)11yescharacteristic
655C22C2×#57711yescharacteristic
656C22C2×#57711yescharacteristic
657C4—24no—
658C4—24no—
659C4—24no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—44no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#245014yes—
671C4C2×#52911yescharacteristic
672C4C2×#53811yescharacteristic
673C4C2×#55411yescharacteristic
674C4C2×#55411yescharacteristic
675C2—28no—
676C2#2641418yesdirect product factor
677C2C2×#245011yescharacteristic
678C2C22×#47711yescharacteristic
679C2C2×#308411yescharacteristic
680C2C2×#329911yescharacteristic
681C2C2×#329911yescharacteristic
682C2C2×#330611yescharacteristic
683C2C2×#331011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26414C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2087C2semidirect
6. C2×#2068C2semidirect
7. C2×#2024C2semidirect
8. C2×#1987C2semidirect
9. C2×#1921C2semidirect
12. C2×#1863C2semidirect
15. C2×#1803C2semidirect
52. #2100C22semidirect
53. #2087C22semidirect
54. #2068C22semidirect
55. #2024C22semidirect
56. #1987C22semidirect
57. #1921C22semidirect
60. #1863C22semidirect
63. #1803C22semidirect
98. C2×C22⋊Q8Q8semidirect
268. C22⋊Q8C2×Q8semidirect
676. C2#26414direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#23602
#264142
C2×#21002
C2×#20872
C2×#20682
C2×#20242
C2×#19872
C2×#19212
C2×#18922
C2×#18722
C2×#18632
C2×#18522
C2×#18502
C2×#18032
C22×#3402