Finite Group Databasebeta

← C2×#26541 #6447274

Subgroups — C2×#26541 #6447274

Order 512.

2342 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#26541C2116yesmaximal, direct product factor
4C2×#2093C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#1982C211yesmaximal, characteristic
9C2×#1961C211yesmaximal, characteristic
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1940C211yesmaximal, characteristic, complemented
12C2×#1864C211yesmaximal, characteristic, complemented
13C2×#1864C211yesmaximal, characteristic, complemented
14C2×#1804C211yesmaximal, characteristic, complemented
15C2×#1802C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×C4×D4—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2570—24no—
38#2570C2218yescomplemented
39C2×#400—21no—
40C2×#398—21no—
41#2366—24no—
42#2366C2218yes—
43C2×#390—21no—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#387—21no—
50C2×#386—21no—
51C2×#386—21no—
52C2×#384C2211yescharacteristic
53C2×#382C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2093C2218yescomplemented
59#2092C2218yescomplemented
60#2067C2218yescomplemented
61#2065C2218yescomplemented
62#1982C2218yes—
63#1961C2218yes—
64#1949C2218yescomplemented
65#1940C2218yescomplemented
66#1864C2218yescomplemented
67#1864C2218yescomplemented
68#1804C2218yescomplemented
69#1802C2218yescomplemented
70#1776C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4—21no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#127—22no—
88C2×#127—24no—
89C2×#127Q818yes—
90C2×#127C2314yes—
91C2×C4×D4—28no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4—48no—
99C2×C4×D4C2314yes—
100C2×C4×D4C2314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C2×#107—24no—
125C2×#107—42no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106—24no—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8—42no—
134C2×C4⋊C8—42no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—42no—
141C2×#102—42no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97—24no—
155C2×#97C2314yes—
156#400—28no—
157#398—28no—
158#390—28no—
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#387—28no—
168#387—28no—
169#386—28no—
170#386—28no—
171#384—22no—
172#384C2314yes—
173#382—22no—
174#382—22no—
175#382C2314yes—
176#382C2314yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4D414yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95D411yescharacteristic
192C2×#95D411yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C25—21no—
201C22×D4—28no—
202C22×D4—28no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—82no—
220C22×D4—82no—
221C22×D4C2×Q814yes—
222C22×D4C2×Q814yes—
223C22×D4C2412yes—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—82no—
238C23×C4—82no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
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—22no—
256C22×C8—22no—
257C22×C8#4114yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262#127—216no—
263#127—28no—
264#127C2×Q8116yes—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—432no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
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—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4#4114yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42—82no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×Q814yes—
332C2×C42#4114yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336#107—28no—
337#107—44no—
338#106—28no—
339#106—44no—
340C4⋊C8—28no—
341C4⋊C8—44no—
342C4⋊C8—44no—
343#102—28no—
344#102—28no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102—44no—
349#102—44no—
350#102—44no—
351#97—28no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—24no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24C22×D411yescharacteristic
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—48no—
398C2×D4—432no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—816no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4C22×Q814yes—
412C2×D4C22×Q814yes—
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—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—816no—
459C22×C4—816no—
460C22×C4C22⋊Q814yes—
461C22×C4C22⋊Q814yes—
462C22×C4AGL(1,ℤ/8)14yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×Q812yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4C2511yescharacteristic
477C2×C8—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—44no—
490C2×C8—44no—
491C2×C8—44no—
492C2×C8C2×#4114yes—
493C4⋊C4—24no—
494C4⋊C4—28no—
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—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—48no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C2×#4114yes—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—48no—
530C42—84no—
531C42C22⋊Q818yes—
532C42C22×Q814yes—
533C42C2×#4114yes—
534C23—24no—
535C23—24no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23C2×C22⋊Q812yes—
563C23#54612yes—
564C23#55012yes—
565C23C2×AGL(1,ℤ/8)12yes—
566C23#57512yes—
567C23#57612yes—
568C23#57712yes—
569C23C23×D411yescharacteristic
570D4—24no—
571D4—24no—
572D4—24no—
573D4—24no—
574D4—432no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—48no—
580D4—48no—
581D4—816no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—816no—
623C2×C4—816no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4C2×C22⋊Q814yes—
630C2×C4C2×C22⋊Q814yes—
631C2×C4C2×C22⋊Q812yes—
632C2×C4#52914yes—
633C2×C4#54014yes—
634C2×C4#54612yes—
635C2×C4#55012yes—
636C2×C4#55414yes—
637C2×C4#55414yes—
638C2×C4C2×AGL(1,ℤ/8)14yes—
639C2×C4C23×Q811yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C8—24no—
646C8—42no—
647C8—42no—
648C8—42no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—816no—
656C22—816no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22#245414yes—
666C22#307714yes—
667C22#308114yes—
668C22#330914yes—
669C22#331014yes—
670C22#339914yes—
671C22#340114yes—
672C22C22×C22⋊Q811yescharacteristic
673C22C2×#54611yescharacteristic
674C22C2×#55011yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C2×#57511yescharacteristic
677C22C2×#57611yescharacteristic
678C22C2×#57711yescharacteristic
679C4—24no—
680C4—24no—
681C4—24no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#245414yes—
692C4C2×#52911yescharacteristic
693C4C2×#54011yescharacteristic
694C4C2×#55411yescharacteristic
695C4C2×#55411yescharacteristic
696C2—84no—
697C2—84no—
698C2#2654118yesdirect product factor
699C2C2×#245411yescharacteristic
700C2C2×#307711yescharacteristic
701C2C2×#308111yescharacteristic
702C2C2×#331011yescharacteristic
703C2C2×#330911yescharacteristic
704C2C2×#339911yescharacteristic
705C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2570C2semidirect
3. #26541C2direct (N × H)
4. C2×#2093C2semidirect
5. C2×#2092C2semidirect
6. C2×#2067C2semidirect
7. C2×#2065C2semidirect
10. C2×#1949C2semidirect
11. C2×#1940C2semidirect
12. C2×#1864C2semidirect
13. C2×#1864C2semidirect
14. C2×#1804C2semidirect
15. C2×#1802C2semidirect
16. C2×#1776C2semidirect
38. #2570C22semidirect
58. #2093C22semidirect
59. #2092C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
64. #1949C22semidirect
65. #1940C22semidirect
66. #1864C22semidirect
67. #1864C22semidirect
68. #1804C22semidirect
69. #1802C22semidirect
70. #1776C22semidirect
698. C2#26541direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25702
C2×#23662
#265412
C2×#20932
C2×#20922
C2×#20672
C2×#20652
C2×#19822
C2×#19612
C2×#19492
C2×#19402
C2×#18642
C2×#18042
C2×#18022
C2×#17762