Finite Group Databasebeta

← C2×#28043 #6457481

Subgroups — C2×#28043 #6457481

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2C2×#2506C211yesmaximal, characteristic, complemented
3#28043C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#1943C211yesmaximal, characteristic, complemented
9C2×#1936C211yesmaximal, characteristic
10C2×#1936C211yesmaximal, characteristic
11C2×#1934C211yesmaximal, characteristic
12C2×#1925C211yesmaximal, characteristic
13C2×#1919C211yesmaximal, characteristic, complemented
14C2×#1836C211yesmaximal, characteristic, complemented
15C2×C4×#106C211yesmaximal, characteristic, complemented
16C2×#1771C211yesmaximal, characteristic, complemented
17C22×#125—21no—
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2513—24no—
31#2513C2218yes—
32#2506—24no—
33#2506C2218yescomplemented
34C2×#400—21no—
35C2×#398—21no—
36C2×#397—21no—
37C2×#397—21no—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#388—21no—
48C2×#387—21no—
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2094C2218yes—
55#2088C2218yescomplemented
56#2085C2218yescomplemented
57#2065C2218yes—
58#1943C2218yescomplemented
59#1936C2218yes—
60#1936C2218yes—
61#1934C2218yes—
62#1925C2218yes—
63#1919C2218yescomplemented
64#1836C2218yescomplemented
65C4×#106C2218yescomplemented
66#1771C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×#125—28no—
83C2×#125—48no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4D411yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C2×#107—24no—
112C2×#107C2314yes—
113C2×#106—24no—
114C2×#106—24no—
115C2×#106—24no—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8—24no—
120C2×C4⋊C8C2314yes—
121C2×#102—24no—
122C2×#102—24no—
123C2×#102—24no—
124C2×#102—24no—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#97—24no—
130C2×#97—24no—
131C2×#97C2314yes—
132C2×#97C2314yes—
133C2×C4×C8—24no—
134C2×C4×C8C2314yes—
135#400—28no—
136#398—28no—
137#397—28no—
138#397—28no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390C2314yes—
143#390C2314yes—
144#390C2314yes—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#388—28no—
156#388—28no—
157#387—28no—
158#384—22no—
159#384C2314yes—
160#382—22no—
161#382C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4D414yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95D411yescharacteristic
178C2×#95D411yescharacteristic
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4C2412yes—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2411yescharacteristic
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8#4114yes—
222C22×C8#4114yes—
223C22×C8#4114yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228#125—432no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4#4114yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—82no—
284C2×C22⋊C4—82no—
285C2×C22⋊C4—82no—
286C2×C22⋊C4—82no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42—82no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2412yes—
301C2×C42C2412yes—
302C2×C42C2412yes—
303#107—28no—
304#106—28no—
305#106—28no—
306#106—28no—
307C4⋊C8—28no—
308#102—28no—
309#102—28no—
310#102—28no—
311#102—28no—
312#97—28no—
313#97—28no—
314C4×C8—28no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95C2×D414yes—
325#95C2×D414yes—
326C24—22no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24C22×D411yescharacteristic
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—44no—
342C2×D4—42no—
343C2×D4—44no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—44no—
347C2×D4—44no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
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—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
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×C4AGL(1,ℤ/8)14yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C42−1+412yes—
406C22×C42−1+412yes—
407C22×C4C2511yescharacteristic
408C2×C8—28no—
409C2×C8—28no—
410C2×C8—28no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—44no—
421C2×C8C2×#4114yes—
422C2×C8C2×#4114yes—
423C2×C8C2×#4114yes—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—48no—
452C4⋊C4—84no—
453C4⋊C4C2×#4114yes—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C42—24no—
461C42—24no—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—48no—
470C42—84no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—84no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23#53212yes—
486C23#54712yes—
487C23#54812yes—
488C23C2×AGL(1,ℤ/8)12yes—
489C23#57512yes—
490C23#57712yes—
491C23#57712yes—
492C23C23×D411yescharacteristic
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—84no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—48no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
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—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4#52914yes—
550C2×C4#53212yes—
551C2×C4#54114yes—
552C2×C4#54712yes—
553C2×C4#54812yes—
554C2×C4#55214yes—
555C2×C4#55214yes—
556C2×C4C2×AGL(1,ℤ/8)14yes—
557C2×C4C22×#4111yescharacteristic
558C2×C4C22×#4111yescharacteristic
559C2×C4C22×#4111yescharacteristic
560C2×C4C22×#4111yescharacteristic
561C2×C4C2×2−1+411yescharacteristic
562C2×C4C2×2−1+411yescharacteristic
563C8—24no—
564C8—24no—
565C8—24no—
566C8—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—168no—
575C22#251914yes—
576C22#317114yes—
577C22#317414yes—
578C22#334514yes—
579C22#334814yes—
580C22#337114yes—
581C22#337314yes—
582C22C2×#53211yescharacteristic
583C22C2×#54711yescharacteristic
584C22C2×#54811yescharacteristic
585C22C22×AGL(1,ℤ/8)11yescharacteristic
586C22C2×#57511yescharacteristic
587C22C2×#57711yescharacteristic
588C22C2×#57711yescharacteristic
589C4—24no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—44no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4#251914yes—
603C4C2×#52911yescharacteristic
604C4C2×#54111yescharacteristic
605C4C2×#55211yescharacteristic
606C4C2×#55211yescharacteristic
607C2—162no—
608C2#2804318yesdirect product factor
609C2C2×#251911yescharacteristic
610C2C2×#317111yescharacteristic
611C2C2×#317411yescharacteristic
612C2C2×#334511yescharacteristic
613C2C2×#334811yescharacteristic
614C2C2×#337311yescharacteristic
615C2C2×#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×#2506C2semidirect
3. #28043C2direct (N × H)
5. C2×#2088C2semidirect
6. C2×#2085C2semidirect
8. C2×#1943C2semidirect
13. C2×#1919C2semidirect
14. C2×#1836C2semidirect
15. C2×C4×#106C2semidirect
16. C2×#1771C2semidirect
33. #2506C22semidirect
55. #2088C22semidirect
56. #2085C22semidirect
58. #1943C22semidirect
63. #1919C22semidirect
64. #1836C22semidirect
65. C4×#106C22semidirect
66. #1771C22semidirect
608. C2#28043direct (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×#25062
#280432
C2×#20942
C2×#20882
C2×#20852
C2×#20652
C2×#19432
C2×#19362
C2×#19342
C2×#19252
C2×#19192
C2×#18362
C2×C4×#1062
C2×#17712