Finite Group Databasebeta

← C2×#26096 #6450654

Subgroups — C2×#26096 #6450654

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#26096C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic
5C2×#2092C211yesmaximal, characteristic
6C2×#2067C211yesmaximal, characteristic
7C2×#2067C211yesmaximal, characteristic
8C2×#1949C211yesmaximal, characteristic, complemented
9C2×#1949C211yesmaximal, characteristic, complemented
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1939C211yesmaximal, characteristic, complemented
12C2×#1919C211yesmaximal, characteristic, complemented
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1846C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×C4×D4C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2570—24no—
33#2570C2218yescomplemented
34C2×#400—21no—
35C2×#398—21no—
36#2357—24no—
37#2357C2218yes—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#387—21no—
45C2×#387—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#381—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4—21no—
56#2094C2218yes—
57#2092C2218yes—
58#2067C2218yes—
59#2067C2218yes—
60#1949C2218yescomplemented
61#1949C2218yescomplemented
62#1947C2218yescomplemented
63#1939C2218yescomplemented
64#1920C2218yescomplemented
65#1919C2218yescomplemented
66#1846C2218yescomplemented
67#1804C2218yes—
68C4×#102C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×C4×D4—22no—
84C2×C4×D4—24no—
85C2×C4×D4Q818yes—
86C2×C4×D4C2314yes—
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×C42—21no—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C2×#107—42no—
112C2×#107—42no—
113C2×#107—42no—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106—24no—
118C2×#106—24no—
119C2×#106—42no—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8—24no—
124C2×C4⋊C8—24no—
125C2×C4⋊C8—24no—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—42no—
132C2×#102—42no—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C4×C8—24no—
138C2×C4×C8C2314yes—
139#400—28no—
140#398—28no—
141#390—22no—
142#390—22no—
143#390C2314yes—
144#390C2314yes—
145#389—22no—
146#389—22no—
147#389C2314yes—
148#389C2314yes—
149#388—28no—
150#388—28no—
151#387—28no—
152#387—28no—
153#384—22no—
154#384C2314yes—
155#382—22no—
156#382—22no—
157#382—22no—
158#382C2314yes—
159#382C2314yes—
160#382C2314yes—
161#381—28no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4D414yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C22⋊C4—28no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4C2×Q814yes—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—82no—
211C23×C4—82no—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
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—22no—
228C22×C8—22no—
229C22×C8#4114yes—
230C22×C8#4114yes—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C4×D4—216no—
236C4×D4—28no—
237C4×D4C2×Q8116yes—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
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—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×Q814yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—82no—
287C2×C22⋊C4—82no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—82no—
296C2×C42—82no—
297C2×C42—82no—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×Q814yes—
302C2×C42#4114yes—
303C2×C42C2412yes—
304C2×C42C2412yes—
305C2×C42C2412yes—
306#107—44no—
307#107—44no—
308#107—44no—
309#106—28no—
310#106—28no—
311#106—44no—
312C4⋊C8—28no—
313C4⋊C8—28no—
314C4⋊C8—28no—
315#102—28no—
316#102—28no—
317#102—44no—
318#102—44no—
319C4×C8—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24—24no—
334C24—22no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24C22×D411yescharacteristic
342C2×D4—24no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4C22×Q814yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
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—216no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
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×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—816no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C22⋊Q814yes—
403C22×C4C22⋊Q814yes—
404C22×C4#13514yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q812yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+412yes—
418C22×C4C2511yescharacteristic
419C2×C8—28no—
420C2×C8—28no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8C2×#4114yes—
434C2×C8C2×#4114yes—
435C4⋊C4—24no—
436C4⋊C4—28no—
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—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4C22×Q814yes—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—24no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—84no—
474C42—84no—
475C42—84no—
476C42C22⋊Q818yes—
477C42C22×Q814yes—
478C42C2×#4114yes—
479C23—24no—
480C23—48no—
481C23—48no—
482C23—42no—
483C23—48no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23C2×C22⋊Q812yes—
494C23#54712yes—
495C23#55012yes—
496C23C2×#13512yes—
497C23#57512yes—
498C23#57612yes—
499C23#57712yes—
500C23C23×D411yescharacteristic
501D4—24no—
502D4—24no—
503D4—48no—
504D4—42no—
505D4—42no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
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—48no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
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—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—816no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4C2×C22⋊Q814yes—
557C2×C4C2×C22⋊Q814yes—
558C2×C4C2×C22⋊Q812yes—
559C2×C4#53314yes—
560C2×C4#54414yes—
561C2×C4#54712yes—
562C2×C4#55012yes—
563C2×C4#55214yes—
564C2×C4#55314yes—
565C2×C4C2×#13514yes—
566C2×C4C23×Q811yescharacteristic
567C2×C4C22×#4111yescharacteristic
568C2×C4C22×#4111yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C2×H(5,𝔽2)11yescharacteristic
571C2×C4C2×2−1+411yescharacteristic
572C8—24no—
573C8—24no—
574C8—42no—
575C8—42no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—816no—
584C22#244114yes—
585C22#307614yes—
586C22#308114yes—
587C22#333914yes—
588C22#335114yes—
589C22#339414yes—
590C22#340314yes—
591C22C22×C22⋊Q811yescharacteristic
592C22C2×#54711yescharacteristic
593C22C2×#55011yescharacteristic
594C22C22×#13511yescharacteristic
595C22C2×#57511yescharacteristic
596C22C2×#57611yescharacteristic
597C22C2×#57711yescharacteristic
598C4—24no—
599C4—24no—
600C4—24no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4#244114yes—
612C4C2×#53311yescharacteristic
613C4C2×#54411yescharacteristic
614C4C2×#55211yescharacteristic
615C4C2×#55311yescharacteristic
616C2—84no—
617C2#2609618yesdirect product factor
618C2C2×#244111yescharacteristic
619C2C2×#307611yescharacteristic
620C2C2×#308111yescharacteristic
621C2C2×#333911yescharacteristic
622C2C2×#335111yescharacteristic
623C2C2×#339411yescharacteristic
624C2C2×#340311yescharacteristic
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
1. C2×#2570C2semidirect
3. #26096C2direct (N × H)
8. C2×#1949C2semidirect
9. C2×#1949C2semidirect
10. C2×#1947C2semidirect
11. C2×#1939C2semidirect
12. C2×#1919C2semidirect
13. C2×#1920C2semidirect
14. C2×#1846C2semidirect
33. #2570C22semidirect
60. #1949C22semidirect
61. #1949C22semidirect
62. #1947C22semidirect
63. #1939C22semidirect
64. #1920C22semidirect
65. #1919C22semidirect
66. #1846C22semidirect
617. C2#26096direct (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×#23572
#260962
C2×#20942
C2×#20922
C2×#20672
C2×#19492
C2×#19472
C2×#19392
C2×#19192
C2×#19202
C2×#18462
C2×#18042
C2×C4×#1022