Finite Group Databasebeta

← C2×#29054 #6460397

Subgroups — C2×#29054 #6460397

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic, complemented
2C2×#2426C211yesmaximal, characteristic
3#29054C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2042C211yesmaximal, characteristic, complemented
6C2×#2035C211yesmaximal, characteristic, complemented
7C2×#2033C211yesmaximal, characteristic
8C2×#1959C211yesmaximal, characteristic, complemented
9C2×#1943C211yesmaximal, characteristic, complemented
10C2×#1936C211yesmaximal, characteristic, complemented
11C2×#1926C211yesmaximal, characteristic, complemented
12C2×#1893C211yesmaximal, characteristic
13C2×#1892C211yesmaximal, characteristic, complemented
14C2×#1863C211yesmaximal, characteristic, complemented
15C2×#1763C211yesmaximal, characteristic, complemented
16C2×#1752C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#126—21no—
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2496—24no—
37#2496C2218yescomplemented
38#2426—24no—
39#2426C2218yes—
40C2×#400—21no—
41C2×#397—21no—
42C2×#396—21no—
43C2×#394—21no—
44C2×#390—21no—
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#387—21no—
51C2×#384C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#2100C2218yescomplemented
57#2042C2218yescomplemented
58#2035C2218yescomplemented
59#2033C2218yes—
60#1959C2218yescomplemented
61#1943C2218yescomplemented
62#1936C2218yescomplemented
63#1926C2218yescomplemented
64#1893C2218yes—
65#1892C2218yescomplemented
66#1863C2218yescomplemented
67#1763C2218yescomplemented
68#1752C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—42no—
83C23×C8—42no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2314yes—
89C2×#126—28no—
90C2×#126—48no—
91C2×#124—24no—
92C2×#124—22no—
93C2×#124—42no—
94C2×#124C2314yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—42no—
98C2×C22⋊Q8C2314yes—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121—42no—
102C2×#121C2314yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—24no—
105C2×C4×Q8—42no—
106C2×C4×Q8C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C2×#107—24no—
135C2×#107C2314yes—
136C2×#106—42no—
137C2×#106C2314yes—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103—24no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8—24no—
155C2×C22⋊C8C2314yes—
156C2×#97—42no—
157C2×#97—42no—
158C2×#97C2314yes—
159C2×#97C2314yes—
160#400—28no—
161#397—28no—
162#396—28no—
163#394—28no—
164#390—28no—
165#390—22no—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#388—28no—
174#387—28no—
175#384—22no—
176#384C2314yes—
177#380—22no—
178#380—22no—
179#380C2314yes—
180#380C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4D414yes—
184C4×C22⋊C4C2314yes—
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#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8#4114yes—
210C22×Q8C2412yes—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—82no—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
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—28no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8—416no—
258C22×C8#4114yes—
259C22×C8#4114yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264#126—432no—
265#124—28no—
266#124—44no—
267#124—48no—
268C22⋊Q8—28no—
269C22⋊Q8—44no—
270C22⋊Q8—48no—
271#121—28no—
272#121—44no—
273#121—48no—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8—48no—
277C2×C4⋊C4—21no—
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—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347#107—28no—
348#106—44no—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—44no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357C22⋊C8—28no—
358#97—44no—
359#97—44no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24AGL(1,ℤ/8)14yes—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×#4114yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—84no—
414C2×D4—816no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
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—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4C2511yescharacteristic
484C2×C8—28no—
485C2×C8—28no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—416no—
497C2×C8—44no—
498C2×C8—44no—
499C2×C8C2×#4114yes—
500C2×C8C2×#4114yes—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—84no—
525C4⋊C4C2×#4114yes—
526C22⋊C4—28no—
527C22⋊C4—44no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—48no—
543C42—48no—
544C23—28no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—28no—
549C23—28no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23#53212yes—
572C23#54612yes—
573C23#54612yes—
574C23C2×AGL(1,ℤ/8)14yes—
575C23C2×AGL(1,ℤ/8)12yes—
576C23#57512yes—
577C23#57712yes—
578C23#57812yes—
579C23C23×D411yescharacteristic
580Q8—24no—
581Q8—24no—
582Q8—42no—
583Q8—42no—
584Q8—48no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—816no—
590D4—84no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
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—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4#52914yes—
645C2×C4#53212yes—
646C2×C4#53814yes—
647C2×C4#54612yes—
648C2×C4#54612yes—
649C2×C4#55314yes—
650C2×C4#55314yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C8—24no—
658C8—24no—
659C8—42no—
660C8—42no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—42no—
665C22—42no—
666C22—416no—
667C22—42no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—168no—
677C22#255914yes—
678C22#317814yes—
679C22#318014yes—
680C22#330614yes—
681C22#330814yes—
682C22#331314yes—
683C22#331714yes—
684C22C2×#53211yescharacteristic
685C22C2×#54611yescharacteristic
686C22C2×#54611yescharacteristic
687C22C22×AGL(1,ℤ/8)11yescharacteristic
688C22C2×#57511yescharacteristic
689C22C2×#57711yescharacteristic
690C22C2×#57811yescharacteristic
691C4—42no—
692C4—44no—
693C4—42no—
694C4—44no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#255914yes—
704C4C2×#52911yescharacteristic
705C4C2×#53811yescharacteristic
706C4C2×#55311yescharacteristic
707C4C2×#55311yescharacteristic
708C2—44no—
709C2—162no—
710C2#2905418yesdirect product factor
711C2C2×#255911yescharacteristic
712C2C2×#317811yescharacteristic
713C2C2×#318011yescharacteristic
714C2C2×#330611yescharacteristic
715C2C2×#330811yescharacteristic
716C2C2×#331711yescharacteristic
717C2C2×#331311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
1. C2×#2496C2semidirect
3. #29054C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2042C2semidirect
6. C2×#2035C2semidirect
8. C2×#1959C2semidirect
9. C2×#1943C2semidirect
10. C2×#1936C2semidirect
11. C2×#1926C2semidirect
13. C2×#1892C2semidirect
14. C2×#1863C2semidirect
15. C2×#1763C2semidirect
16. C2×#1752C2semidirect
37. #2496C22semidirect
56. #2100C22semidirect
57. #2042C22semidirect
58. #2035C22semidirect
60. #1959C22semidirect
61. #1943C22semidirect
62. #1936C22semidirect
63. #1926C22semidirect
65. #1892C22semidirect
66. #1863C22semidirect
67. #1763C22semidirect
68. #1752C22semidirect
710. C2#29054direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#24262
#290542
C2×#21002
C2×#20422
C2×#20352
C2×#20332
C2×#19592
C2×#19432
C2×#19362
C2×#19262
C2×#18932
C2×#18922
C2×#18632
C2×#17632
C2×#17522