Finite Group Databasebeta

← C2×#27144 #6447780

Subgroups — C2×#27144 #6447780

Order 512.

1886 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#27144C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2065C211yesmaximal, characteristic
6C2×#2058C211yesmaximal, characteristic, complemented
7C2×#2023C211yesmaximal, characteristic
8C2×#1949C211yesmaximal, characteristic, complemented
9C2×#1936C211yesmaximal, characteristic
10C2×#1926C211yesmaximal, characteristic
11C2×#1926C211yesmaximal, characteristic
12C2×#1925C211yesmaximal, characteristic
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1920C211yesmaximal, characteristic, complemented
15C2×#1917C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8—21no—
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2513—24no—
30#2513C2218yes—
31C2×#397—21no—
32C2×#397—21no—
33#2365—24no—
34#2365C2218yescomplemented
35C2×#391—21no—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#387—21no—
45C2×#387—21no—
46C2×#382C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2065C2218yes—
54#2058C2218yescomplemented
55#2023C2218yes—
56#1949C2218yescomplemented
57#1936C2218yes—
58#1926C2218yes—
59#1926C2218yes—
60#1925C2218yes—
61#1920C2218yescomplemented
62#1920C2218yescomplemented
63#1917C2218yescomplemented
64#1776C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2314yes—
87C2×C4×Q8—28no—
88C2×C4×Q8—48no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C42—21no—
109C22×C42—21no—
110C22×C42—21no—
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C2×#107—24no—
116C2×#107C2314yes—
117C2×#106—24no—
118C2×#106—24no—
119C2×#106—24no—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8—24no—
124C2×C4⋊C8C2314yes—
125C2×#102—24no—
126C2×#102—24no—
127C2×#102—24no—
128C2×#102—24no—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#97—24no—
134C2×#97C2314yes—
135#397—28no—
136#397—28no—
137#391—28no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390C2314yes—
142#390C2314yes—
143#390C2314yes—
144#389—22no—
145#389—22no—
146#389—22no—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#388—28no—
151#388—28no—
152#387—28no—
153#387—28no—
154#382—22no—
155#382C2314yes—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4D414yes—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95D411yescharacteristic
176C2×#95D411yescharacteristic
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×Q8—28no—
191C22×Q8—24no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4C2412yes—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2411yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8#4114yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C4⋊Q8—216no—
233C4⋊Q8—28no—
234C4⋊Q8—28no—
235C4×Q8—432no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4#4114yes—
267C2×C4⋊C4#4114yes—
268C2×C4⋊C4#4114yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—82no—
285C2×C22⋊C4—82no—
286C2×C22⋊C4—82no—
287C2×C22⋊C4—82no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—82no—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2412yes—
306C2×C42C2412yes—
307C2×C42C2412yes—
308#107—28no—
309#106—28no—
310#106—28no—
311#106—28no—
312C4⋊C8—28no—
313#102—28no—
314#102—28no—
315#102—28no—
316#102—28no—
317#97—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—22no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24C22×D411yescharacteristic
338C2×Q8—216no—
339C2×Q8—432no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—42no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—44no—
354C2×D4—44no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4AGL(1,ℤ/8)14yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C42−1+412yes—
413C22×C42−1+412yes—
414C22×C4C2511yescharacteristic
415C2×C8—28no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—44no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8C2×#4114yes—
431C4⋊C4—216no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—84no—
455C4⋊C4C2×#4114yes—
456C4⋊C4C2×#4114yes—
457C4⋊C4C2×#4114yes—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—48no—
474C42—48no—
475C42—48no—
476C42—84no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23#52512yes—
492C23#54712yes—
493C23#54812yes—
494C23C2×AGL(1,ℤ/8)12yes—
495C23#57512yes—
496C23#57712yes—
497C23#57712yes—
498C23C23×D411yescharacteristic
499Q8—432no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—84no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4#52512yes—
557C2×C4#52914yes—
558C2×C4#54114yes—
559C2×C4#54712yes—
560C2×C4#54812yes—
561C2×C4#55614yes—
562C2×C4#55614yes—
563C2×C4C2×AGL(1,ℤ/8)14yes—
564C2×C4C22×#4111yescharacteristic
565C2×C4C22×#4111yescharacteristic
566C2×C4C22×#4111yescharacteristic
567C2×C4C22×#4111yescharacteristic
568C2×C4C2×2−1+411yescharacteristic
569C2×C4C2×2−1+411yescharacteristic
570C8—24no—
571C8—42no—
572C8—42no—
573C8—42no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—168no—
582C22#246914yes—
583C22#311414yes—
584C22#312614yes—
585C22#335014yes—
586C22#335414yes—
587C22#337114yes—
588C22#337314yes—
589C22C2×#52511yescharacteristic
590C22C2×#54711yescharacteristic
591C22C2×#54811yescharacteristic
592C22C22×AGL(1,ℤ/8)11yescharacteristic
593C22C2×#57511yescharacteristic
594C22C2×#57711yescharacteristic
595C22C2×#57711yescharacteristic
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—42no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4#246914yes—
610C4C2×#52911yescharacteristic
611C4C2×#54111yescharacteristic
612C4C2×#55611yescharacteristic
613C4C2×#55611yescharacteristic
614C2—162no—
615C2#2714418yesdirect product factor
616C2C2×#246911yescharacteristic
617C2C2×#311411yescharacteristic
618C2C2×#312611yescharacteristic
619C2C2×#335011yescharacteristic
620C2C2×#335411yescharacteristic
621C2C2×#337311yescharacteristic
622C2C2×#337111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. C2×#2365C2semidirect
3. #27144C2direct (N × H)
4. C2×#2101C2semidirect
6. C2×#2058C2semidirect
8. C2×#1949C2semidirect
13. C2×#1920C2semidirect
14. C2×#1920C2semidirect
15. C2×#1917C2semidirect
16. C2×#1776C2semidirect
34. #2365C22semidirect
52. #2101C22semidirect
54. #2058C22semidirect
56. #1949C22semidirect
61. #1920C22semidirect
62. #1920C22semidirect
63. #1917C22semidirect
64. #1776C22semidirect
615. C2#27144direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#23652
#271442
C2×#21012
C2×#20652
C2×#20582
C2×#20232
C2×#19492
C2×#19362
C2×#19262
C2×#19252
C2×#19202
C2×#19172
C2×#17762